摘要:

Application of Modern Concepts in Liquid State Theory to Concentrated Particle Dispersions BY A. VRIJ, E. A. NIEUWENHUIS, H. M. FIJNAUT AND W. G. M. AGTEROF Van? Hoff Laboratory for Physical and Colloid Chemistry, University of Utrecht, Transitorium 3, Padualaan 8, Utrecht, The Netherlands Received 12th December, 1977 Theories on liquid structure properties are applied to concentrated colloidal dispersions in non- polar media. Light scattering experiments on microemulsions (W/O) with benzene and toluene as the continuous phases are characterized in terms of a hard sphere repulsion and a perturbative attraction. The origin of the attraction and the influence of polydispersity on particle size are dis- cussed. Light scattering of crosslinked PMMA-latex particles in benzene as a function of scattering angle showed structure formation at higher latex concentrations.The results were discussed in terms of the structure factor S(K), susceptibility x(K) and the radial distribution function g ( v ) of the particle centres. The latex behaves as an “ expanded ” liquid structure. This structure formation was also reflected in the diffusion coefficient measured by quasi-elastic light scattering. 1. INTRODUCTION Structure and interparticle interactions in colloidal dispersions may be treated in the same way as in simple liquids, at least as far as the equilibrium properties are concerned. This idea originated at the beginning of the century when Einstein1 derived Van? Hoff’s law for suspended particles and Perrin2 and Costantin3 meas- ured a second virial coefficient of a dilute dispersion which was discussed in the con- text of van der Waals’ gas law.Later, the theoretical basis for such a treatment was given by McMillan and Mayer4 and Kirkwood and Buff.s It turns out that the interaction potential between molecules (in vacuo) has to be replaced by a potential of mean force (with a free energy character) between all the dispersed particles. The solvent properties appear as parameters in the potential of mean force. Often the solvent may be treated as a homogeneous medium if the particle size is sufficiently large. Early investigators in the field were Onsager,6 Riley and Oster7 and Kirkwood and Mazur.’ However, accurate fluid models were not available at that time. At present more is known (mainly due to computer simulation studies) about (simple) liquids with spherically symmetric (one-atomic) molecules.The picture emerged that in dense liquids the spatial structure, represented by the radial distribu- tion function g(r), and also some thermodynamic properties, are mainly determined by the hard repulsions between the particles in close proximity. The effect of the weaker, less steep attractive or repulsive tails may be treated as perturbations. For recent reviews see e.g., ref. (9) and (10). This places the hard sphere (HS) fluid as a “ reference ” model in a central position. This result is completely in accordance with the original view of van der Waals. The techniques originally applied to low molecular weight liquids are now being incorporated more and more into the field of colloidal dispersions.In aqueous dis-1 02 LIQUID THEORY A N D PARTICLE DISPERSIONS persions stabilized by extended electrical double layers, structural properties 11-16 and (osmotic) pressures 17918 have been measured and interpreted. Recently much interest has been shown in the “ order-disorder ” transitions found in these ~ystems.l~-~* Fewer studies have been made of these phenomena in dispersions in non-aqueous In this paper we shall apply some of the ideas taken from the field of liquids to concentrated dispersions in non-aqueous 2. S US CEPTIBILI T Y An instructive way to describe a system with centrosymmetric particles in terms of structural and thermodynamic properties is to formulate how the mean particle density, which has a constant value po in a uniform system, is perturbed under the influence of a weak external field, 6u(r) < kT.This can be formulated as follows34 Here 6p(r) = p(r) - po; p = ( k o h l ; r = IrI ; h(r) = g(r) - 1 is the total correlation function, i.e., the relative mean number density deviation at a distance r from the centre of a given particle. The integration is performed over all space. This exact relation can, heuristically, be explained as follows. The first term on the r.h.s., which predominates at small po, stems from the familiar Boltzmann relation: 6p(r) = pod exp[-/3v(r)] = p,[-pGv(r)]. The second term reflects the influence of the surround- ings on the number density deviation in r. This term, in fact, is the product of a local density variation in r’, (p,[-pGu(r’)]) times the interaction between particles in r and r’, [poh(Ir - r‘l)].The influence of the total surroundings is found by integrating this product over all space. We apply eqn (2.1) to some simple cases. GRAVITATIONAL OR CENTRIFUGAL FIELD In a macroscopic field (e.g., gravity) the spatial variation [6v(r) = -mgz] has a Under this condition 6u much larger scale than Iz(r), (except near a critical point). can be placed outside the integral, giving, But, in a macroscopic field -6u is equal to the local variation in the chemical potential Sp of the particles. Thus, co (p0pap/ap)-l = (paP/ap)-l = 1 + 4np, I, r2h(r)dr (2.3) where P is the (osmotic) pressure. This is the well-known Ornstein-Zernike 35 relation which provides a route connecting statistical with thermodynamical properties. In experiments in gravita- tional or centrifugal fields, eqn (2.3) can be applied locally at every z.PERIODIC FIELDS Another simple case is a spatially sinusoidal field with wavenumber K = 27t/A, and amplitude 6vK, &(x) = 6vK cos Kx. (2.4)A . VRIJ, E . A . NIEUWENIIUIS, H . M . FIJNAUT, w. G . M . AGTEROF 103 Substitution of eqn (2.4) into (2.1) gives after performing an integration over solid angles, ~ P ( x ) = dpK cos KX = -pOPd~Kx(K) cos Kx, (2.5) (2.6) N co x(K) = 1 + p,h(K) = 1 + 4np0 r*h(r)[(Kr)-l sin(Kr)]dr. 0 N Note that h(K) is the 3-dimensional Fourier transform of h(r). The function x(K) is usually regarded 3 4 9 3 6 as a generalized susceptibility since it measures the (linear) response of the system to an impressed, weak, periodic potential of wave number K.In the limit of small K(1arge A) it describes how the density reacts to compression and dilation on a macroscopic scale. On comparing eqn (2.6) with (2.3) one finds, x(K = 0) = P-'ap/aP, (2.7) i.e., x is proportional to the (osmotic) compressibility of the system, as is expected. For K # 0, one expects x to contain maxima around K-values where the system yields most to the external field and to become unity when A = 2n/K, the field periodicity, becomes (much) smaller than the particle size. x becomes large for small K near a critical point and presumably also near a " solid-liquid " transition37 when A is comparable to a lattice spacing. Since the function x(K) may be determined from scattering experiments it is not only of conceptual value.Indeed, scattering (e.g., of light) at an angle 8, with respect to the direction of incidence, probes the spontaneous concentration fluctuations of a wave number K = (4n/.;l) sin (6j2). In a subsequent section we shall see that x(K) is simply the so-called structure factor S(K). Thus x(K) can be considered as a convenient system property which connects different experimental techniques in a unifying way. 3. PARTICLES WITH A HARD CORE The prototype of a harsh repulsion between particle cores is the hard-sphere (HS) potential given by, m , r < a 0, r 3 a ' @(r) = where a is the hard sphere diameter. Even for this simple pair potential no exact theory is available to calculate P and h(r). This difficult many-particle problem, which in the HS-case is purely geometric in character, could be attacked, however, with the help of computer simulation studies.This opened the way to test approximate theories in a direct manner. For HS, the scaled particle theory3* and the Percus- Yevick theory (compressibility version) give a fairly good r e ~ ~ I t , ~ ~ * ~ ~ * ~ ~ P(ap/aP>PY = (1 + 2d2(1 - 9F4, (3.2) 1 6 where rp = - npa3 is the volume fraction of hard spheres. The Carnahan-Starling41 equation, p(ap/ap)cs = (1 + 4p + 4q2 - 4v3 + p4)(1 - (3.3) is a semi-empirical extension of eqn (3.2) and fits the computer simulation results within computational accuracy. Both equations are exact up to and including the third virial coefficient. The values obtained by eqn (3.2) are higher (e.g., -8% at p = 0.4) than those of eqn (3.3).For more realistic systems one possible way to104 LIQUID THEORY AND PARTICLE DISPERSIONS attack the problem is to add to the HS-potential a repulsive or attractive part as a perturbation. For colloidal particles in non-polar media these interactions are not known quantitatively. Therefore, we shall resort to a semi-empirical approach. The simplest example of such an approach is adding a van der Waals-type attractive term, This eqn (3.4) gives much better results42 for gases than does van der Waals’ original equation which contained a repulsive term of the form (1 - 4 ~ ) - ~ . In the future one could think of more refined models like the square-well potential, when more experimental results become available. An important difference between atomic liquids and colloidal systems is that dispersed particles are seldom mono- disperse.Before considering too detailed models of interaction potentials one should have an indication of the importance of polydispersity. P W P = P(apPP>Cs - 2aY. (3 -4) 4. LIGHT SCATTERING EQUATIONS One of the techniques available to obtain structural and thermodynamic properties is light scattering. For a monodisperse system of spherically symmetric particles one may write for unpolarized light,’ with R(K) = (1 + cos2 8)YcM P(K) S(K) K = (4nn/iZo) sin (812). Here R(K) = Re is the reduced scattering intensity (Rayleigh ratio) of the dispersion over that of the solvent, 8 is the scattering angle, A4 the particle molar mass, P(K) = Po is the particle scattering factor, c = pM/N and N is Avogadro’s number.Further, Z = 27t2n2(dn/d~)2(?~oN)-1 (4.3) where n is the refractive index of the dispersion and Lo the wavelength of the light in vacuo. The structure factor is given by, S(K) = 1 + 4np r2h(r)[(Kr)-l sin(Kr)]dr. lorn Note that S(K), is identical to x(K), eqn (2.6). For small 6, K, eqn (4.1) reduces to the D e b ~ e ~ ~ equation, (1 + cos2 B)Xc - 1 aP --- R(K1 RT ac* (4.4) (4.5) This equation is to be used in conjunction with eqn (3.2)-(3.4). Recently, one of us was able to derive44 the influence of polydispersity on light scattering for HS in the Percus-Yevick approximation. Assuming that dn/dp is a constant and that the particle size distribution is given by a generalized exponential, the exact result45 turns out to be, (1 + cos2 e ) x c - - R(K) ( M ) , (’ (1 4- - P I 2’r [l + 12(---f-)L- 1 + 5 y 2 1 - + 9 ( A 2 ) (&)21--1.(4.6) 1 Here y2 = ((a2)N/(a)&) - 1 ; p = nZo:pi. iA . VRIJ, E . A . NIEUWENHUIS, H . M . FIJNAUT, W. G . M . AGTEROF 105 The subscripts w and N denote weight and number averages. Observe from eqn (3.2) that the polydispersity correction is simply the expression inside the brackets of eqn (4.6). As expected, R(K) increases in the case of polydispersity. This effect becomes more pronounced if y and p increase. Eqn (4.6) will also apply to any size distribution when y is small enough. 5. LIGHT SCATTERING OF A CONCENTRATED MICROEMULSION Microemulsions (W/O) of the S~hulman-type,~~ containing water, potassium oleate, hexanol and benzene or toluene were investigated. The microemulsions Bl and B2, with benzene as the continuous phase, contained numbers of water and oleate mole- cules in a ratio Nw/Ns = 23.5 and 49.5, respectively.For sample T1, with toluene as the continuous phase: Nw/Ns = 25.0. The monolayers on the surfaces of the aqueous droplets contain hexanol and oleate molecules in a ratio of 1.28 and 1.88 for B1 and B2, respectively and in a ratio of 1.12 for T1, as determined from titrations. Stock emulsions were diluted with the continuous phase (containing hexanol) and filtered (Millipore 0.45 pm). Light scattering intensities were measured in a Fica-50 photometer at 1, = 546 nm. Because scattering dissymmetry was absent [PQ = 1, S(K) = S(K = O)], only RgO [= R(K) at 8 = 90'1 is reported (T N 21 "C).The microemulsion concentration c is expressed as total mass of dispersed particles per unit volume, as calculated from titration plots. The measured refractive index increments, dn/dc were found to be constant over the whole concentration range. Diffusion coefficients, D, of the emulsion particles were determined by means of quasi-elastic light scattering. The instrument was designed and built in our labora- tory and contains a 5 W Spectra Physics Model 165 Ar+ ion laser, a Saicor SA 1-42 correlator and a Nova 1200 (Data General Corp.) computer. The time auto- correlation functions of the scattered light were analysed according to a single exponen- tial. The l/e value of the exponentials corresponds to the relaxation time T. Then D was obtained with the equation r1 = 2DK2 from a (linear) plot of r1 against K2, and extrapolated to c = 0 to obtain the " hydrodynamic " diameter uhy from the Einstein relation Ohy = kT/(3nyDo).More details are given e l ~ e w h e r e . ~ ~ . ~ ~ RESULTS trations .Xc/RgO gives aP/ac according to eqn (4.5), which can be written as follows, Measured values of RgO of Bl and T1 are plotted in fig. 1. For the lower concen- (RT)-laP/ac = M-yl + 2B,c + . . .) (5.1) where B2 is the second virial coefficient. Values of M and B2 are given in table 1. From M, the composition of the particles and the densities of the components one may calculate the mass and volume occupied by aqueous core, monolayer and total particle, assuming volume additivity. It was further assumed that the aqueous core contained water, -COOK from the oleate and -OH from the hexanol with an overall density of 1.05 g ~ m - ~ for B1 and T1 and 1.03 for B2.From these quantities the over- all particle density was calculated (0.97 g cmW3 in all cases) and the diameter of the aqueous core, CT,, (see table 1). At higher concentrations RgO goes through a maximum indicating that the decrease in S(K = 0) = x(K = 0) = kT ap/aP is larger than the increase in c itself. In search- ing for the type of interaction between the particles, a plot was made of aP/ap, as obtained from the experiments and eqn (4.5). The results are plotted in fig. 2 and106 LIQUID THEORY AND PARTICLE DISPERSIONS I 1 1 1 0 0.1 0.2 0.3 0.4 FIG. 1.-Reduced light scattering intensity at 8 = 90" of (W/O) microemulsions as a function of c / g ~ r n - ~ emulsion concentration; 1: emulsion B1 (0); 2: emulsion T1 (0); theory (-).TABLE 1 .-CHARACTERISTIC PROPERTIES OF MICROEMULSIONS (anlac) ~ 1 1 0 5 g /cm3 g-l mol-I q,Jnm a,,/nm aHs/nm y/cm3 g-l ~ J c r n ~ g - l a B1 -0.113 1.25 11 6.1 7.5 1.06 -0 5.4 B2 -0.129 5.25 15 10.5 11.8 0.99 -0 5.3 T1 -0.103 3.10 13 8.4 10.25 1.0g5 3.2 2.4 compared with a hard sphere repulsion as described by the Carnahan-Starling eqn (3.3). Since the exact relation between p and c is not known, three theoretical curves are drawn for three different ratios of q = v/c. As can be seen from the difference between theory and experiment, a pure hard sphere interaction must be rejected. From the experimental results at lower concentrations in fig. 2 we can find an indica- tion of a more realistic type of interaction.Since the slope of p W/ap for small c is nearly zero, the second virial coefficient B2 [see eqn (5.1)] is also about zero. This implies that the repulsive contribution to B2 is nearly compensated by an attraction. Therefore, our next approximation for the interactions is that of a hard sphere repulsion with a van der Waals type of attraction as described by eqn (3.4). In fig. 3 we have plotted p(iP/8p)cs - XcM/R9, against c. Here again we have to adjust the ratio q = p/c. The " best " overall linear fit was obtained with q = 1.06 cm3 g-l. From the value of the slope we found cc = 5.4. From q one calculates a hard sphere diameter: aHS = (6qMj7~N)~'~ = 7.5 nm (see further table 1). DISCUSSION In table 1 one observes that (ohy - a,,) is 4- 5 nm and that (aHs - oaq) is only 1.3 - 1.8 nm.Because the surface layers on the particles consist of a mixture of oleate and hexanol chains, this suggests that a h y is determined by the extremities of the particle formed by the oleate chains. Their double extended length is indeed in the range of 4 - 5 nm. The much smaller values of oHS suggest the possibility of large interpenetration of the surface layers, which is the basis for the interparticle attraction. The second virial coefficient B2 is about zero in benzene and positive in toluene,A . VRIJ, E . A . NIEUWENHUIS, H . M. FIJNAUT, w. G . M. AGTEROF 107 2 3 a CD 5 0 0.1 0.2 0.3 0.4 c l g C K 3 FIG. 2.-Derivative of the osmotic pressure P with respect to the number density p for microemulsion B1; HS theoretical curves are drawn for different values of q = p/c; 1 : q = 1.06; 2: q = 0.94; 3: q = 0.87 cm3 g-l, c = concentration of emulsion particles in g ~ r n - ~ , q = hard sphere volume but smaller than the HS value, which would be 4q N 4.4 cm3 8-l.Thus in benzene the HS repulsion is about compensated by attractions in the overlapping surface layers for low values of c. In toluene the attraction and also the overlap of surface layers is apparently smaller. We think that the attractions are a result of the exchange of solvent and chain segments. If solvent and chain segments have different energy densities, the (exchange) energy is lowered when the components ~egregate,~~ i.e., when chain segments of opposing particle surfaces are in contact.This exchange is diminished when chain and solvent segments become more similar. On these grounds one would indeed expect that the attraction in benzene is larger than in toluene, which contains an extra alkane group, and that it would become negligible in an alkane solvent. Preliminary results from microemulsions in cyclohexane support this view. In addition to an energetic contribution an entropic contribution should have been considered. This last contribution is expected to be repulsive, but has not been considered here. Calculations that macroscopic attractive van der Waals’ forces between the aqueous cores are too small to explain the zero-value of B2 in benzene. At high concentrations aP/+ is large, clearly because of the large contribution of the hard repulsions; the attractions apparently play a less important role.From fraction, p = (kT)-l.108 LIQUID THEORY AND PARTICLE DISPERSIONS 0 0.1 0.2 0.3 c f g C n i 3 FIG. 3.-Plots of /3(aPlap),, - S C M / R ~ ~ against c for emulsion B1; 1: q = 1.10; 2: q = 1.06; 3: q = 1.02 cm3 g-l. theoretical considerations it may be expected34 that at high concentrations the attractions are well described by a van der Waals type of eqn (3.4). In the dilute range, which can be described by a second virial coefficient, this is of course also the case, but not necessarily in the intermediate range. This implies that the values of cc are not necessarily equal in both extreme ranges. This makes the simple procedure we applied in fig. 3 semi-empirical. This is also apparent from the values obtained for tc, by a forced fit over all concentrations, which are not wholly consistent with the obtained B,.In the dilute range one must have B2 = (4 - tc)q. Although B2 and (4 - a)q are larger for T1 than for B1 and B2, the numerical values deviate. There are indications that the overall fit can be improved by the introduction of an attractive well.33 Hitherto, we have assumed that the particle dispersions were monodisperse, which seldom occurs in practice. To assess the influence of polydispersity on our analysis we corrected the hard sphere contribution to aP]ap with the term in brackets in eqn (4.6). This has the tendency to increase the values of q and a, eg., for Bl, tak- ing y = 0.15 (M,/M, = 1.2) one finds that cc increases by some tenths, q by about 7% and oHS by about 2%.A similar effect to that of polydispersity would be caused by a soft repulsion. It is not possible at the moment to discriminate between these possibilities. This effect might show up more clearly in good solvents. The overall fit of our analysis is shown in the drawn curves of fig. 1.A . VRIJ, E. A . NIEUWENHUIS, H . M. FIJNAUT, W. G. M . AGTEROF 109 6. LIGHT SCATTERING O F SWOLLEN LATEX PARTICLES As another example of concentration effects and the liquid structure formation of dispersed particles, we report some preliminary results on the light scattering of internally crosslinked latex particles dispersed in a non-polar solvent. A poly(-methylmethacrylate) (PMMA) latex was prepared by emulsion polymer- ization in water in the presence of 5% ethylene glycol dimethacrylate as a cross- linker.The latex was dialysed, deionized, dried and dispersed in benzene. From sedimentation experiments it was found that the amount of non-crosslinked PMMA was less than a few percent. The preparation is a modification of those of Kose and Hachi~u,~' Ono and Saeki4* and Fitch and T ~ a i , ~ ~ and will be described in more detail elsewhere. The latex was fairly homodisperse with a sphere diameter ow of 206 & 10 nm as determined by electron microscopy. In benzene the hydrodynamic diameter was found to be ohy = 335 nm, and the molar mass determined from the diffusion co- efficient and the sedimentation rate of the particles was found to be M = 3.8 x lo9 g mol-I. The dnldc, which is extremely small, could not be measured accurately.We found dn/dc N 0.0075 at 2, = 546 nm and 0.0010 at 2, = 436 nm both at 20 "C. RESULTS For c < 0.001 g cfn3, log (1 + cos2 R(K) against K2 [K = scattering wave- number as defined in eqn (4.2)] plots were linear over a large K-range. From the slope of such a so-called Guinier-plot the slope of log P(K) follows, which gives the radius of gyration rg and a particle diameter ore = (20 r:/3)&, which was found to be 330 nm. At higher concentrations characteristic maxima developed, as shown in fig. 4. Similarly, structured plots were found for the diffusion coefficient shown by the circles in fig. 5. 1 I I 1 I I I I I 0 1 2 3 4 5 6 7 8 X 2 / 10" ~ r n - ~ FIG. 4.-Reduced light scattering intensity, Re = R(K), against K2 for PMMA latex particles in benzene at several latex concentrations; 1 : c = 0.023; 2: c = 0.040; 3 : c = 0.058; 4: c = 0.077 g cm-3, with K = (4nn/A0) sin (8/2).110 LIQUID THEORY AND PARTICLE DISPERSIONS FIG.5.-Structure factor S(K) and reciprocal effective diffusion coefficient Dert (K;p), (O), against K, for PMMA latex particles in benzene at c = 0.077 g crnF3. DISCUSSION The numbers obtained for the diameters show that in benzene, which is a good solvent for PMMA, the particles are highly swollen: uhy/uw - 1.6. It can, therefore, be understood that the particles repel each other even at large separations, i.e., at quite low concentrations. The liquid-like structure formation is clearly shown in fig. 4 by the maxima in R(K) which are also found in X-ray scattering of simple liquids, like argon.5o A similar structure in R(K) was found recentlyl1~l4 for small latex particles dispersed in deionized water.According to eqn (4.1) it should be possible to extract from these data S(K), from which h(r) can be obtained by the following Fourier transform, h(r) = (2n2p)-l lom K2[S(K) - l][(Kr)-l sin(Kr)]dlY: (6.1) It turned out, however, that this could not be done in the usual way, i.e., by dividing R(K) by the particle scattering factor P(K) found at low c (which leads to awkward results). It appeared that a somewhat different P(K) had to be chosen with a smaller magnitude and a smaller angular dependence in the lower K-range. We surmise that with increasing concentration of the latex particles, their peripheric segment clouds are compressed.This results in two effects. First, the angular dependence will be less pronounced since the particles are optically smaller. Secondly, the average segment density distribution between the particles is more uniform, which results in a higher average refractive index of the background and thus in a lower scattering power of the particles. These two effects complicate the extraction of S(K) from the experiments. However, with the following procedure we can estimate S(K). In the zero limit of particle separation, one knows that g(r) = 0 (no particle over- lap), thus h(r = 0) = 1. By taking the limit for small r one finds from eqn (6.1), -2n2p = low K2[S(K)- l]dK.A . VRIJ, E. A. NIEUWENHUIS, H . M . FIJNAUT, W. G . M. AGTEROF 111 By estimation of P(K) at small K one calculates S(K) from R(K).This S(K) is inserted in the integral of eqn (6.2) and by performing the integration one can check the estimate since the identity (6.2) should hold. If not, new estimates of P(K) are used until eqn (6.2) is satisfactorily fulfilled. An example of S(K) thus obtained is given in fig. 5. The positions of the maxima in S(K), which are at slightly larger K than those of R(K), are given in table 2. TABLE 2.-sOME CHARACTERISTIC PROPERTIES OF PMMA-LATEX IN BENZENE c/g ~ r n - ~ Km/105 cm-1 d, = (3/2)+A,/nm c A ~ ~ / ~ O - ~ ~ g 0.023 0.040 0.059 0.078 - 1.26 1.45 1.58 - 610 530 490 - 5.0 4.8 4.9 According to our theoretical discussion in a previous paragraph, Am = 2n/Km is the wavelength where x(K) is largest and the system yields most to a perturbing field. It is interesting to compare A, with a mean distance of particle centres which is proportional to c-'I3.From table 2 one observes that Am3c is nearly constant, whereas c varies by a factor of 2. This suggests that the particles occupy '' minimum free energy positions " created by (repulsive) interactions with surrounding particles. There exists a kind of" expanded " structure, in which all particles are held at mean, equidistant positions, that fills the whole volume. Similar behaviour was found in concentrated protein solutions and in aqueous latex dispersions with expanded double layer^.^^^^^ This must be contrasted with HS-fluids or compressed argon, where A, is nearly constant. In " expanded " structures of the close-packing type, the mean interparticle centre dis- tanceis d, = (3/2)% A, for a closest cubic packing (f.c.c.).Obtained numbers for dc are shown in table 2. From the number of lattice points per unit cell, M follows from M = 1.30 NcAm3, which was found to be 3.8 5 0.2 x lo9 in accordance with the value obtained from sedimentation plus diffusion. In a further analysis we performed the integration in eqn (6.1) for K = 0.5 to 3.0 x lo5 cm-I to obtain h(r), (see fig. 6). A spurious peak at low r, which often appears in these transformsSo was omitted. The details of the procedure will be discussed elsewhere. The position of the first maximum in h(r) is at rmaX = 470 nm, which is indeed nearly the same as the mean interparticle distance which was d, 21 490 nm for this case.From the observation that d, is much larger than Ghy we conclude that the periphery of the particles contains loosely dangling chains that protude very far into the solvent in order to give a sufficient interaction (of the order of kTper particle) with surround- ing particles to keep all of them separated at the mean interparticle distance. The steeply rising part of g(r), where strong repulsions prohibit further overlap, however, is in a range of Y that is comparable to Ghy, which, presumably, measures the size of the more dense, crosslinked particle core. Finally one finds also that the effective diffusion coefficients1 is structured at higher concentrations, as shown in fig. 5. Without going into details, we remark that one apparently observes here how ap, relaxes in time when a perturbing field is turned off.For a macroscopic field (K = 0), one has from irreversible thermo- dynamics 51 An equivalent discussion can be given on the basis of " Bragg-spacings Deff = kTf-l(P)(l - dP0P aPlaP. (6.3)112 LIQUID THEORY AND PARTICLE DISPERSIONS If one accepts the " Ansatz " that at finite K, 6p should be replaced by --dvK, one obtains with eqn (2.5), In fig. 5 one, indeed, observes that the extrema in Dzfl coincide with those of x(K) = S(K). The friction factorfis presumably a smoother function of K. r J lo2 nm FIG. 6.-Total correlation function h(r) for PMMA latex particles in benzene at c = 0.077 g cm-'. Here h(r) = g(r) - 1, with g(r) the radial distribution function of particle centres separated by a distance r.Similar phenomena were observed and d i s c ~ s s e d ~ l * ~ ~ * ~ ~ for latex dispersions in extremely dilute electrolyte solutions. We think that in both cases the interaction must be characterized as soft, long range repulsions. It was found by others (see Introduction) and by ourselves that in both systems these interactions may ultimately lead to a " crystalline " phase. We thank Dr. W. van der Drift for advice and help with the sedimentation experi- ments and Mr. H. Mos for his help with the quasi-elastic light scattering experiments. We also thank Miss H. Miltenburg for typing the manuscript and Mr. W. den Hartog for drawing the illustrations. A. Einstein, Investigations on the Theory of the Brownian Movemerit (Dover Publications, New York, 1956), p. 1.J. Perrin, Compt. rend., 1914,158, 1168. R. Costantin, Compt. rend., 1914,158, 1171. W. G. McMillan and J. E,. Mayer, J. Chem. Phys., 1945, 13,276. J. G. Kirkwood and F. P. Buff, J. Chem. Phys., 1954, 19,774. L. Onsager, Ann. N. Y. Acad. Sci., 1949, 51, 638. ' D. P. Riley and G. Oster, Disc. Faraday Suc., 1951, 11, 107. J. G. Kirkwood and J. Mazur, J. Polymer Sci., 1952,9, 519. H. C. Andersen, D. Chandler and J. D. Weeks, Adv. Chem. Phys., 1976, 34, 105. J. A. Barker and D. Henderson, Rev. Mod. Phys., 1976,48, 587.A . V R I J , E. A . NIEUWENHUIS, H . M . FIJNAUT, w. G . M. AGTEROF 113 l1 J. C. Brown, P. N. Pusey, J. W. Goodwin and R. H. Ottewill, J. Phys. A: Math. Gen., 1975, 8, l2 W. van Megen and I. Snook, J. Colloid Interface Sci., 1975, 53, 172.l3 R. P. Keavey and P. Richmond, J.C.S. Faraday II, 1976,72,773. l4 D. Schaefer, J. Chem. Phys., 1977,66, 3980. l5 W. van Megen and I. Snook, J. Chem. Phys., 1977,66,813. l6 J. C. Brown, J. W. Goodwin, R. H. Ottewill and P. N. Pusey, Colloid and Interface Science, ed. M. Kerker (Academic Press, New York, 1976), vol. 4, p. 59. l7 R. H. Ottewill, Progr. Colloid Polymer Sci., 1976, 59, 14. A. Homola, I. Snook and W. van Megen, J. Colloid Interface Sci., 1977, 61, 493. l9 W. Luck, M. Klier and H. Wesslau, Ber. Bunsenges. phys. Chem., 1963,67, 75, 84. 2o P. A. Hiltner and I. M. Krieger, J. Phys. Chem., 1969, 73, 2386. 21 S. Hachisu, Y. Kobayashi and A. Kose, J. Colloid Interface Sci., 1973, 42,342. 22 R. Williams and R. S . Crandall, Phys. Letters, 1974, 48A, 225.23 D. W. Schaefer and B. J. Ackerson, Phys. Rev. Letters, 1975,35, 1448. 24 I. Snook and W. van Megen, J.C.S. Faraday ZI, 1976,72,216. 25 R. Williams, R. S. Crandall and P. J. Wojtowicz, Phys. Rev. Letters, 1976, 37, 348. 26 S. Marcelja, D. J. Mitchell and B. W. Ninham, Chem. Phys. Letters, 1976, 43, 353. 27 S. L. Brenner, J. Phys. Chem., 1976,80, 1473. 28 K. Takano and S . Hachisu, J. Chem. Phys., 1977,67,2604. 29 P. A. Hiltner, Y. S. Papir and I. M. Krieger, J. Phys. Chem., 1971,75, 1881. 30 A. Kose and S . Hachisu, J. Colloid Interface Sci., 1974,46,460. R. J. R. Cairns and R. H. Ottewill, J. Colloid Interface Sci., 1978, in press. 32 W. G. M. Agterof, J. A. J. van Zomeren and A. Vrij, Chem. Phys. Letters, 1976, 43, 363. 33 A. A. Calj6, W. G. M. Agterof and A. Vrij, Micellization, Solubilization and Microemulsions, 34 J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic Press, London, 1976), 35 L. S. Ornstein and F. Zernike, Proc. h a d . Sci. (Amsterdam), 1914, 17, 793. 36 L. Mistura and D. Sette, J. Chem. Phys., 1968, 49, 1422. 37 R. Lovett, J. Chem. Phys., 1977, 66, 1225. 38 H. Reiss, H. Frisch and J. L. Lebowitz, J. Chem. Phys., 1959, 31, 369. 39 E. Thiele, J. Chem. Phys., 1963, 39, 474. 40 M. S. Wertheim, Phys. Rev. Letters, 1963,10, 321. 41 N. F. Carnahan and K. E. Starling, J. Chem. Phys., 1969,51,635. 42 E. A. Guggenheim, Mol. Phys., 1965,9,43. 43 P. J. W. Debye, J. Phys. Chem., 1947, 51, 18. 44 A. Vrij, Chem. Phys. Letters, 1978, 53, 144. 45 A. Vrij, to be published, in J. Chem Phys. 46 J. H. Schulman and J. A. Friend, Kolloid Z., 1949, 115, 67. 47 J. H. Hildebrand and R. L. Scott, The solubility of Non-electrolytes (Dover, New York, 1964). 48 H. Ono and H. Saeki, Colloid Polymer Sci., 1975, 253, 744. 49 R. M. Fitch and C. H. Tsai, Polymer Colloids, ed. R. M. Fitch (Plenum Press, New York, 1971), 50 J. Karnicky, H. Hollis-Reamer and C. Pings, J. Chem. Phys., 1976, 64, 4592. 51 B. J. Berne and R. Pecora, Dynamic Light Scattering (John Wiley, New York, 1976), chap. 13. 664. ed. K. L. Mittal (Plenum Press, New York, 1977), vol. 2, p. 779. chap. 5, 6. p. 73, 103.

ISSN:0301-7249    

DOI:10.1039/DC9786500101

出版商:RSC    

年代:1978

数据来源: RSC

摘要:

GENERAL DISCUSSION Dr. R. Buscall (Bristol) said: I was interested to see in Mewis’ paper that the presence of floc structures lead to wide relaxation spectra which could be interpreted using a ladder network model. We have recently made creep and dynamic measure- ments on ordered latex dispersi0ns.l The viscoelasticity of these systems arises from interparticle repulsion rather than aggregation. The results could be accounted for using a very simple model with two relaxation times, or a spectrum with two closely- spaced lines. It appears, therefore, that the spectral shapes for disperse systems are characteristic of the state of dispersion. Prof. J. Mewis (Leuven) said : Depending on the degree of dispersion certain differ- ences in spectral shape can be expected. Stable, closely packed systems and those containing rigid aggregates do not seem to give a broad distribution of relaxation times.Loose, elastic flocs can eventually vibrate in various modes. This gives rise to a well defined spectrum. Chainlike structures can be considered an extreme case and most structural dispersions will give spectra in between those reported here and those obtained by Buscall. The previous argument only holds for a mechanical response due to small changes in interparticle distance. As such it is valid for the explanation of the high frequency part of the spectrum. At very small frequencies more complicated rearrangements can occur in partially flocculated systems, which are still very difficult to model in terms of structure. Hence the correlation between spectra and the state of dispersion should be made with care.Dr. M. van den Tempe1 (Vlaardingen) said: The value of the spring constant calcu- lated by Mewis from his relaxation spectra is several orders of magnitude lower than one would expect on the basis of van der Waals forces. It is difficult to imagine any other type of attractive force that holds the particles together in a chain, and I wonder whether this strange result could be due to the interpretation of the relaxation spectra by means of a kind of Rouse theory, which is probably not valid for these particle chains. Prof. J. Mewis (Leuven) said: The stability of carbon black dispersions, as those discussed here, is not well understood to date. From various measurements it is clear that loose aggregates and networks can be formed.They can be broken down again easily by shearing and can take a long time (i.e., weeks) to reconstitute a kind of equilibrium structure again. It is not certain that van der Waals forces alone govern this behaviour. With respect to the possible explanation of the spectrum, it is true that no final proof is available for the interpretation we suggest. It should be mentioned however that the Rouse spectrum is definitely a possibility, even for particle chains, even when the particles are not too close together. Rouse has derived his spectrum by replacing the molecular chains by equivalent particle chains. Awaiting further experimental evidence on the same systems the present data allow R. Buscall, J. W. Goodwin, M. Hawkins and R.H. Ottewill, in preparation.GENERAL DISCUSSION 115 one additional control, on internal consistency. In table 1 of our paper, model chain sizes have been calculated from the spectrum height at zmi, based on the Ruckenstein- Mewis analysis. The same parameter N , can be estimated from the spectrum width, neglecting the chain size distribution. The resulting values are about 50% higher than the values from table 1. Considering the various assumptions, the correspondence is still reasonable. Dr. W. D. Cooper (Edinburgh) said: Ramsay et al. found no significant variation of electrophoretic mobility with electrolyte concentration up to 10-1 mol d ~ l l - ~ , how can this lack of dependence on ionic strength be explained? What are the shapes of the electrophoretic mobility against pH curves and how might they be affected by the presence or absence of polymeric aluminium cations of the type the authors suggest could be formed at the surface of boehmite particles? Dr.J. D. F. Ramsay (Harwell) said: Micro electrophoresis measurements were made on aggregates of boehmite particles which were observed to grow in size as the electrolyte concentration was increased. The charge of these aggregates can be con- sidered to originate first by ion adsorption (viz. H+ and OH-) and secondly by ionisa- tion of polynuclear aluminium species adsorbed at the surface of the boehmite. In the former case the mobility would be expected to decrease with increases in electrolyte concentration, due to a contraction in the thickness of the double layer surrounding individual particles.In contrast, however, the ionisation of polynuclear species would probably not be suppressed significantly by added electrolyte and hence little change in mobility would occur as is shown in fig. 1. At a pH 2 7 precipitation of FIG. 1 .-Mobility plotted against pH for boehmite sol particles in different concentrations (mol drn-j) of potassium nitrate: - x-, -0-, lo-’; -- , lo-’. hydrous oxides from solutions of polynuclear aluminium cations would be expected, which would correspond with the marked decrease in mobility observed. It is also noteworthy that the electrophoretic mobility of particles of unpeptised boehmite (at pH 21 4; [KNOJ = 5 x pm s-l V-I m) than that of the sols measured under similar conditions. mol dm-3) was lower (u E 1.5 x Prof.T. W. Healy (Melbourne) said : The microelectrophoresis results reported by the authors do not appear to indicate any anomalous increase in mobility for sols peptised at pH values below, say, 4.1. Perhaps this and other effects has forced the authors to seek a non-electrostatic origin for the repulsion observed. However,116 GENERAL DISCUSSION an array of plate-shaped particles represents a difficult geometry for analysis in terms of traditional DLVO concepts ; specifically, there must be structural (statistical mechanical) control that allows the system of repelling plates to order or organise itself to minimize the total energy of the array. I would suggest, therefore, that the authors might reconsider the physics of such arrays before they need to invoke inter- facial structuring promoted by polynuclear transients.Again, the concentration of such species is strongly dependent on the anion present and peptization studies with F- counter-ions may prove useful in that they will minimize the concentration of polynuclears. Dr. J. D. F. Ramsay (HarweZZ) said: We would agree that any detailed calculations of interparticle potentials for a concentrated dispersion of plate-shaped particles would be considerably more difficult than those which have already been made for systems of monodispersed latex spheres using DLVO theory for examp1e.l However, the short range repulsion forces between the boehmite particles cannot be accounted for by double-layer repulsion, as is apparent in some of the properties of the dispersions which have been described, in particular the lyophilic nature of the gels which is featured by their redispersibility.Concentrated boehmite sols in common with other hydrous oxides can be pre- pared with certain acids (e.g., HN03, HCl, HC104, HC103, CH,COOH). With acids containing strongly complexing anions (e.g., F- , SO;-, 10;) similar peptisation can- not be achieved, presumably because the formation of polynuclear ions is inhibited. Prof. J. Th. G. Overbeek (Utrecht) said: (1) Since the interpretation of the pH as measured in a concentrated suspension is complicated, due to the so-called suspension effect, it would be useful to know the pH of the intermicellar liquid, prepared by equilibrium dialysis, or by (ultra) centri- fugation or by ultrafiltration.(2) How much of the HN03 has reacted with the boehmite in the preparation of the sols, how much has remained free? Have such measurements been made? Dr. J. D. F. Ramsay (HarweZZ) said: (1) Measurements of the pH of concentrated sols and their equilibrium dialysates showed significant, although not marked, differences. Thus for a typical sol (-10% w/w), where [HNO,]/[AlOOH] - 2 x the pH measured in the dispersion was 4.17 compared to that of 3.96 in the dialysate. At higher dispersion concentrations this difference increased only slightly e.g., ApH - 0.25 at 40% w/w. (2) During the peptisation of the boehmite the extent of reaction of HN03 will depend on the [HNO3]/[A100H] ratio and the specific surface area of the powder (190 mz g-' in this case).If the boehmite is peptised with the minimum of HN03 ([HN03]/[A100H] -2 x then dissolution is probably limited to the surface since the pH rises rapidly and reaches a constant value (pH -4) within about 1 h. On average this would correspond to approximately one proton per 1 nm2. Analysis of the ultra filtrate (pore size -2 nm) of such a sol showed that the proportion of dissolved aluminium was [Alrl']/[A1OOH] -2.8 x This would not include those polynuclear ions which were strongly bound to the boehmite surface or retained by the ultrafilter however. the reaction is probably P. A. Forsyth, S. Marcelja, D. J. Mitchell and B. W. Ninham, Adv. Colloid Interface Sci., 1978, 9, 37. At higher ratios of acid to boehmite [(4 - 6) xGENERAL DISCUSSION 117 mori extensive since the increase in pH is much slower and can take several days to reach a pH of z4.Dr. Th. F. Tadros (Jealott’s HilZ) said: One of the problems in dealing with con- centrated dispersions of oxides is that the pH drifts with time. How was this con- trolled? Moreoever, addition of salts will have an effect on the pH of the suspen- sion and if there is specific adsorption of ions, this will be accompanied by a shift of the i.e.p. Could the authors indicate how these variations were accounted for. Dr. J. D. F. Ramsay (Harwell) said: Slow increases in pH were observed during the peptisation of boehmite with HN03 as has been described (reply to Overbeek). However with the small quantities of acid employed here a stable value (pH w 4) was achieved after a few hours.Addition of certain salts (e.g., KN03, KClO,, KBr) only produced slight increases in the pH of the sols whereas with KIO,, and especially KF, more marked increases were noted, presumably due to the specific interaction of the anions with the boehmite surface. The concentrations of these electrolytes which were required to coagulate dilute sols ( 5 10% w/w) were sufficiently low however (see e.g., table 2 of our paper) to cause only a modest increase in pH viz. from pH w 4 to pH ~ 5 . Dr. H. N. Stein (Eindhoven) said: Daish et al. have adduced evidence for a stabiliz- ing effect of polymeric aluminium containing ions on AlOOH sols. In a recent in- vestigation on the coagulation of Ca3A12(OH)12 suspensions we found a dependence of the coagulation rate on electrolyte concentration which was interpreted as due to polyaluminates [the pH in our experiments was higher than that corresponding to the IEP of Al(OH),].At relatively low electrolyte concentrations ((0.1 mol drn-,), the coagulation rate showed a trend contrary to that predicted by theory for coagulation in either the primary or the secondary minimum: the stability ratio increases with in- creasing electrolyte concentration. The increase in stability ratio appears to be inde- pendent of the type of electrolyte present: NaNO,, NaOH, KN03 and Ca(OH), had at equal values of the Debye-Hiickel parameter IC, the same influence on the stability ratio although they effected quite different 5 potentials [thus, in NaOH or KOH solutions, ICl was larger than in NaNO, or KNO, solutions; in all these solutions ( was >O; in saturated Ca(OH), solutions, on the other hand, [ was > O ] .We ascribed, therefore, the increase in stability ratio with increasing IC to changes of a property connected with ionic strength in the solution rather than to changes in surface or Stern potential. Formation of polyaluminate ions was suspected. Alumin- ate ions were present in the solutions [from dissolution of the Ca,Al,(OH),,], the presence of polyaluminate in the solutions concerned, although not demonstrated conclusively, is likely because the concentrations are close to the region where rapid precipitation of amorphous Al(OH), is observed.2 In general, polyions can have a flocculating or a deflocculating action. What type of influence is envisaged as an explanation for the observations mentioned, depends on whether larger electrolyte concentrations are thought to stimulate polyion formation, or to promote degradation of polyions.In the former case, the large stability ratio in the Ca3A1(OH)12 suspen- sion at intermediate electrolyte concentrations (IC c 1 x lo9 m-l) can be ascribed to the presence of polyions, adsorbed on the Ca3A12(OH)12 particles and affording some steric stabilization; in the latter case, the polyions should bridge the distance between G. A. C. M. Spierings, Ph.D. Thesis (Eindhoven 1977); G . A. C. M. Spierings and H. N. Stein, Colloid Polymer Sci., 1979, in press. F. E. Jones and M. H. Roberts, Building Res., Curr. Pap. Ser. I , June 1963.118 GENERAL DISCUSSION the particles and thus be responsible for the relatively low stability of the Ca3A12(OH)12 suspension at low electrolyte concentrations.A similar action of polymeric Zn(OH), species in solution has been postulated by Healy and Jealett for explaining the coagula- tion behaviour of 2nO.l At present, it is not clear which of these alternatives should be chosen. The former alternative might explain the fact that coagulation in the suspensions concerned is remarkably slow when compared with the von Smoluc- kowski-Muller theory of fast flocculation ; the latter alternative however better explains the electrophoretic data, which indicate that the greater part of the surfaces concerned is not covered by the stabilizing ions. At any rate the assumption of poly- aluminate ions as being responsible for the remarkable stability of Ca3A12(OH)12 suspensions can account for the facts only if the degree of ionization of the aluminate ions is independent of their degree of polymerization; for in the other case NaOH or KOH should certainly act differently from NaNO, or KN03.Dr. Th. F. Tadros (Jealott’s Hill) said: In reply to Stein’s observation concerning the unusual stability of alumina on addition of Ca2 + salts, it is well known that bivalent cations can specifically adsorb on the surface of oxides leading to charge reversal and hence restabilisation. Has Stein done any experiments e.g., electrophoresis and/or titration to see whether this was the case? Dr. H. N. Stein (Eindhoven) said: By electrophoresis, [ potentials were measured.The observations indicated that changes in surface potential due to Ca2+ adsorption could not be held responsible for the effects observed. Dr. T. van Vliet (Wageningen) said: The authors suggest that the differences in the values of the cohesive energy e, per particle, obtained at different dispersion concentra- tions result from a change in the alignment of the particles to each other. What is the reason to expect such a change in alignment? Would it not be more appropriate to FIG 1 .-Aggregate model. assume that the dispersions are built up of aggregates consisting of particles; these aggregates being connected by rather small chains of particles (fig. 1). Then the number and the thickness of these particle chains would mainly determine the rheo- logical properties of the boehmite dispersions whereas the cohesive energy per particle is constant or nearly constant.The ratio between the number of particles in the T. W. Healy and V. R. Jealett, J. Colloid Interface Sci., 1967, 24,41. H. Muller, Kolloidchem, Beihefte, 1928, 27, 223.GENERAL DISCUSSION 119 chains and the total number is a function of the volume fraction p and so are the number of stress carrying particles and the ensuing mechanical properties. At low p there would be only a relatively small number of particles in the chains, most of them being in the aggregates. Dr. J. D. F. Ramsay (Harwell) said: A model similar to that suggested by van Vliet could explain viscoelastic behaviour in dilute dispersions of aggregates, in particular those composed of spherical particles as depicted in the figure of van Vliet’s question.10 c I ol P E \ .I s 5 + E w -0 0 0.5 relative pressure, pip, FIG. 1 .-Adsorption isotherm of nitrogen at 77 K for outgassed (423 K) boehmite gel. In the boehmite dispersions, waere the particles are platelets, the structure of the aggregates is difficult to define, although it is not independent of dispersion concentra- tion, as is implicit in this model. Thus at low concentration (p < the particle density in the aggregates, as indicated by light scattering, is low ( E particles m-3). However when the dispersions are concentrated these aggregates must undergo extensive contraction and association, until eventually a rigid gel is produced, con- taining 21023 particles ~ll-~. The surface and porous properties of these gels (as determined from adsorption isotherms of nitrogen on the outgassed solid, cf.fig. 1) are always very similar and can be ascribed to a compact structure (p w 0.5) in which the boehmite plates are uniformly arranged to give slit-shaped pores with a narrow size distribution. Dr. J. W. White (Grenoble) said: The quasi-elastic neutron scattering observed for water in boehmite sols closely resembles the data that we took a number of years ago for water in fumed silica (Anderson and White).l The interpretation of such data R. G. W. Anderson and J. W. White, Special Disc. Faraday SOC., 1970,1, 205.120 GENERAL DISCUSSION depends upon a good knowledge of the chemical composition of the system. The two extreme models, which can be distinguished from such a knowledge, are (I) Scattering from trapped molecules (where the range of gas-like motion or free liquid diffusion is hindered by walls), and (2) 2-site diffusion (where the water may be for part of its time considered to be anchored at a site, and for the other part of its time to be freely moving in a more or less bulk liquid state).The first model would be appropriate for molecules adsorbed in very small cavities, even of hydrophobic surfaces, and would produce a scattering law similar to the one that you have shown. The second law has been considered specifically by Richter, Kehr and Springer for the case of diffusion of hydrogen in niobium with interstitial traps. Again, a scattering law similar to the one that you have shown arises except that there is a high frequency component as well.It would be extremely interesting to see whether the Q dependence that you have measured could be used to distinguish between these models. Prof. A. Vrij (Utrecht) said: Now nearly 20 years ago, I proposed and used in my dissertation2. a contrast variation method to obtain the correct molecular weight of polyelectrolytes and other charged colloidal particles and the (negative) " adsorption '' of the supporting electrolyte. It was successfully applied to ionic micelle~.~*~ The procedure is to measure the light scattering of the particles in a series of electrolytes with identical counterions but different co-ions with varying scattering power, e.g., the sodium halides for a negatively charged polyelectrolyte or micelle. Around the charged particles an electrical double layer is present.The counterions are attracted and the co-ions are repelled from the particle surface. The overall effect is a negative adsorption of the electrolyte. Because the co-ions are repelled from the particle surface one may expect that spec$c interactions with the particle will be small or absent, so that the negative adsorption will be identical for the different electrolytes. By plotting the light scattering as a function of the contrast (=dn/dc) of the supporting electrolyte an extrapolation to zero contrast of the supporting electrolyte can be per- formed. The intercept of the plot is simply related to the molecular weight of the particle and from the slope the negative adsorption can be calculated.Dr. J . W. White (Grenoble) said: Thank you very much for bringing your experi- ment, done 20 years ago on contrast variation for the study of adsorption on ionic micelles, to my attention. I regret that I had not known it before but, of course, it certainly embodies the principles which have been used once again in the contrast variation with neutron scattering. The chief virtue of the neutron method is that one produces contrast by isotopic variation, thereby changing to a relatively small degree the chemistry of the solution. I am sure there are many things that this use of iso- topic contrast can learn by considering experiments done in X-ray and in light scatter- ing. Dr. S . P . Stoylov (SoJia> (communicated) : The application of neutron scattering to the investigation of the structure of aggregates reported by Ramsay et al.and D. Richter, K. W. Kehr and T. Springer, Proc. Conf. Neutron Scattering (Gatlinburg, Tennessee, U.S.A. June6-10, 1976), vol. 1, p. 568. A. Vrij, Dissertation (Utrecht, 1959). A. Vrij and J. Th. G. Overbeek, J . CoZZoid Sci., 1962,17, 570. H. F. Huisman, Proc. Kon. Ned. Akad. Wetenschap. Ser., 1964, B67, 367,376, 388,407. J. Th. G. Overbeek, A. Vrij and H. F. Huisman, in Proceedings of the Interdisc@Zinary Confer- ence on EZectromagnetic Scattering, ed., M. Kerker (Pergamon, London, 1963), p. 321.GENERAL DISCUSSION 121 Cebula et al. provides the very stimulating possibility for the deeper understanding of this problem. However I should like to point out that on the way from the simpler techniques like light scattering to these, most complicated techniques, like neutron scattering there exist a number of techniques of intermediate complexity.These are the electro-optic techniques. So for example the light scattering in an electric field follows the aggregates simultaneously both optically and hydrodynamically. The hydrodynamic data come from the study of the transient processes which gives both the rotational diffusion constants and their distributions. There exist some examples for a quite successful application of these techniques for studying the structure of aggregates of clays1 and aerosils. Dr. J. W. White (Grenoble) said : I am grateful to you for mentioning the technique of light scattering in an electric field. In fact, although neutron scattering can only be done at relatively few places in the world at present, it is a relatively simple technique and the experiments described in our paper require only a few minutes of measuring time ; modern computer interfacing to low angle scattering instruments allows the Guinier plots and other analyses to be made on line so that the experiments can be followed during their course.Dr. C. J. Wright (Harwell) said: I would like to clarify one particular aspect ofthe published neutron scattering measurements of diffusion coefficients which I think has not been fully appreciated in the preceding discussions. The observed scattering from a sample of H2Q is a sum of the scattering from its individual scattering centres. Consequently if the sample contains dissimilar mole- cules, and if only a single diffusion coefficient is extracted from the experimental data, then this will be a complex average of the diffusion coefficients of all the mole- cules present.Mathematically the averaging arises because a sum of lorentzians, each with a different half width, is treated as a single lorentzian. In published measurements on thin films of water held between solid plane surfaces the observation of a diffusion coefficient for that water which is significantly different from that of bulk water does not necessarily imply that the water is " structured " over the distance between the plates. The measurement is an average over water molecules close to the interface which may be significantly perturbed from bulk water, and water molecules at the centre of the film which may be indistinguishable from bulk water.Dr. J. W. White (Grenoble) said: As you correctly point out, the neutron scattering measurement of diffusion coefficients give an average of the diffusion coefficient for protons in different sites of the sample. I should like to make it clear that this is not simply an arithmetical average of the diffusion coefficients for the two sites but is an average which depends upon the momentum transfer of the experiment. At small values of the momentum transfer, the uncertainty of the momentum in the scattering event is obviously small, and therefore the coherence length for the neutron and the particle causing the scattering must necessarily be long. This leads to neutron measurements over distances of the order of the reciprocal of the momentum transfer (in A-1), which may be up to 20 A.Given the time-scale of such a measure- ment, the diffusing particle may have time to sample a number of different environ- ments and one arrives at a measurement of the effective diffusion coefficient, Deffective. At large momentum transfers the observation range is necessarily short. The J. Schweitzer and B. R. Jennings, J. CoEIoid Interface Sci., 1971,37,443.122 GENERAL DISCUSSION neutron then sees the superposition of the scattering laws from the physically different dynamics in the system. In the simplest possible case one sees the addition of the scattering from bound water molecules and the scattering from free water molecules away from the surface.Since these models are rather different from a physical point of view, it is essential to have good chemical characterisation of the surfaces. Prof. R. €5. Ottewill (Bristol) said: The lyotropic series of monovalent cations, Li + , Na+ , K+, Rb+ , Cs+ are frequently quoted in colloid science and the difference in their specificities invoked to explain various phenomena. The small angle neutron scattering experiments described in the paper give an extremely clear demonstration of the considerable difference in specificity of lithium, potassium and caesium with respect to montmorillonite. Not only does this tech- nique clearly distinguish the specificity, it also appears to offer a very sensitive means of obtaining an indication of floc morphology. Would the authors like to comment on the major advantages that the neutron scattering technique offers for the study of clay dispersions compared with other techniques ? Mr.D. J. Cebula (Oxford) said: The major advantages of using neutron small- angle scattering to study clay dispersions compared with the use of other techniques originate largely froni: (a) the expected shape and size of single clay platelets and of flocs of platelets in comparison to the wavelength of the neutron radiation used and the angles at which scattering may be observed, (6) the neutron scattering power of the clay versus that of the dispersion medium, be it H20 or D20 and (c) the convenience of studying long range order [spacings of 10 2: &A) 2: 5001 with the long neutron wavelengths currently available.(a) Single clay platelets, and even flocs of clay platelets with up to about ten com- ponent platelets stacked face-to-face, are highly anisotropic in shape. So the scatter- ing will be well separated into a low Q region and a higher Q region. The large di- mension of the platelet, R, is best studied by light scattering but the small dimension H, the thickness and multiples of the thickness falls into a Q region easily observable with neutrons of wavelength 4 < IJ% < 16. * r of D,O FIG.1.-Scattering densities of clay as a function of solvent concentration. The full lines for the clay assume no H/D exchange in the clay with the solvent. Dashed lines represent full isotopic exchange.GENERAL DISCUSSION 123 (b) The neutron scattering power, 6, of clay is different from that of water; see fig.1. Moreoever the scattering length profile, p(R), across the thickness of the clay platelet is constant, pm. Since for water the scattering length density, ps, depends on the ratio of D,O: H20, it is possible to vary the contrast, p , for the solvent against the clay. Such a property, realised only by neutrons, will be exploited in the investiga- tion of adsorbed species at the clay-water interface. Such absorption will affect p(R) and only in the H direction when observation in the correct Q range is made: then the absorbed layer thickness is accessible. (c) Order-disorder transitions and phase separations in colloids have been studied theoretica1ly.l Experimental work has been performed on systems with model model shapes, i.e., T.M.V.and latices but only few data exist on plate-like systems. The long-range separations predicted are easily resolvable using long-wavelength neutron diffraction. In fact the case of the sodium clay dispersion showed a definite peak in the small-angle region of -400 A, however, interpretation of this feature is only tentative at this stage. Mr. A. K. van Helden and Mr. E. A. Nieuwenhuis (Utrecht) said: Recently we have performed conventional light scattering studies, which were quite analogous to Cebula’s neutron scattering experiments. The light scattering method was used in studying monodisperse silica dispersions in apolar solvents. The silica particles were stabilized by CI8 aliphatic chains. We were able to characterize the dispersed particles optically by changing the com- position of a binary mixture of solvents.Starting from cyclohexane the refractive index of the medium was increased upon addition of t-decahydronaphthalene. Some characteristic scattering curves are shown in fig. 1. It is seen that these results are similar to Cebula’s Guinier plots. The contrast-matching point was found to be in the mixture with 0.77 mole-fraction cyclohexane. When the contrast was large a good linear dependence was obtained. At small angles a curvature is found, especially near 3 1.0 t, 4 Q? 0.3 0.1 0 0 5 1.0 1.5 K2x 10 5,A-2 FIG. 1 .-Guinier plots for the light scattering experiments from silica dispersions in various mixtures of cyclohexane and t-decahydronaphthalene at 1, = 4360 A. The mole fraction of cyclohexane is 0, 0.00 17, 0.46 A, 0.66 A, 0.91 0, 1.00.P. A. Forsyth, S. Macelja, D. J. Mitchell and B. W. Ninham, Adv. CoZloidInterface Sci., 1978, 9, 37-60.124 GENERAL DISCUSSION the contrast matching point, probably indicating the presence of some large particles due to cluster formation. The slope of the Guinier plot, determined by extrapolating from high scattering angles, depends on the contrast. The variation of RE2 is plotted against the reciprocal of the contrast (fig. 2). Extrapolation ff '9 x 105/ii2 of (contrast)-'+ 0 yields the radius of -2 -1 0 1 2 (contcastl-'x 10 -* FIG. 2.-Square of the radius of gyration is a linear function of the reciprocal of the contrast. 0, A, = 4360 A; @, l o = 5460 A. gyration of the equivalent homogeneous particle (R, = 520 A).This result agrees well with the photon correlation spectroscopy result a = 625 A. The slope in this plot is positive, indicating a higher optical density at the periphery of the particle and a lower optical density in the core. Clearly conventional light scattering too has been shown a powerful tool in characterizing dispersion particles optically. A drawback, compared with neutron scattering, is that the contrast is changed by varying the chemical composition of the medium, so the thermodynamic interactions are altered as well. In a way light scattering is a complementary technique however, because larger dimensions are accessible, owing to the different range of the wave vector. Dr. J. W. White (Grenoble) said: I am delighted to see the results of the work on contrast variation using refractive index variation for adsorbed monolayers on silica sols. This is very elegant work and in every way comparable to the things which are done with neutron scattering in our own paper.The important differences for light scattering and neutron scattering are that we are able to work down to much lower particle sizes in the neutron scattering experiments, and typically between radii of 10 and 1000 A, but more particularly because the variation of contrast around the scattering particle, e.g., by H20/D20 contrast, (but one could obviously use other combinations of deuterated/protonated solvents), produces the minimum disturbance to the chemical potential of the adsorbed molecules and of these molecules in the solution at equilibrium.We believe that this is a point of particular interest for the future when studies on adsorbed polymers will be made. Prof. R. H. Ottewill (Bristol) said: The determination of an adsorption isotherm for a surface active molecule adsorbed onto a solid substrate from solution is often deceptively simple. The problem is that although this tells us directly the number of molecules removed from the solution phase on to the solid substrate, it gives little direct information about the configuration of the molecule on the surface,GENERAL DISCUSSION 125 I should like to congratulate White and his colleagues on applying the neutron beam technique to the problem of determining the thickness of the adsorbed layer of a simple molecule on a solid particle and also obtaining directly the area occupied per adsorbed molecule.It is clear that this technique has considerable potential and as experience increases in its utilization for this type of experiment, substantial increases in accuracy will be achieved. In the field of adsorption from solution, however, we still badly need techniques for studying the configuration of adsorbed molecules, and their modes of movement relative to the surface. I wonder if White would comment on whether inelastic neutron scattering and neutron beam spectroscopy are likely to develop in the future to provide this information. Dr. J. W. White (Grenoble) said: What you say is quite true. In the paper as presented, we have only covered the aspects of neutron diffraction, and in particular of neutron low angle scattering from molecules adsorbed at the solid liquid interface.There is a very exciting further development possible associated with measuring the inelastic scattering of such adsorbed molecules. Such measurements are now being made for molecules adsorbed at the gas-solid interface and it has been possible, for example for adsorbed methane on carbon, to distinguish the phases present at different temperatures and coverages. Whereas neutron diffraction alone can only give the structure of the adsorbed phase, it cannot distinguish between, for example, an ad- sorbed two-dimensional liquid phase and an adsorbed amorphous solid phase. By measuring the inelastic scattering, and hence the motions of the adsorbed molecules, one can distinguish these.Because the cross-sections for low angle neutron scattering are so large, it seems to me to be quite a reasonable proposition to be able soon to observe the inelastic scattering from adsorbed materials, especially on such favourable substrates as mono- dispersed polystyrene sols or micro emulsions. Because we can use isotopic substitu- tion, it will be possible to observe the inelastic scattering from protonated molecules separately from the scattering from other arnphiphiles and materials in the adsorbed state. Some discussion of this has been given ear1ier.l Prof. A. Vrij (Utrecht) said: In your paper you say that the technique promises to give information on the swelling process when styrene is added to preformed latex. Could you explain this in more detail? Secondly could you comment on the influence of polydispersity in size and composition of the particles studied by this method? Mr. N.M. Harris (Grenoble) said: The study of latex swelling is being carried out in collaboration with Ottewill. There is some uncertainty as to the mechanism of this swelling process, and in particular whether this occurs homogeneously throughout the latex particle or whether there is an accumulation at the particle surface. By swelling a protonated polystyrene latex with deuterated styrene we can use neutron low angle scattering to determine very readily the distribution of the swollen material in the particles. Prof. A. Vrij (Utrecht) said: But why use styrene to swell the latex? Mr. N. M. Harris (Grenoble) said: We are also planning to look at swelling of latices using toluene at some stage. J.W. White, Proc. Roy. SOC. A , 1975, 345, 119.126 GENERAL DISCUSSION Prof. R. H. Ottewill (Bristol) said: With regard to the swelling of polystyrene latex particles raised by Harris, it should be emphasised that the internal structure of poly- styrene latex particles and the mechanism by which the particles nucleate and grow is still a problem of very considerable interest. Some of our recent studies have indicated that the mechanism of particle formation involves the coagulation of the initially formed particles, which then become colloidally stable units and subsequently grow by incorporation of monomer.' This would lead one to suspect that, unless the coagulated structure completely anneals out during swelling, the structure of the particle could be expected to be somewhat heterogeneous.Other structures have also been proposed for the structure of the particles, including the idea of " core-shell " morphology.2 These various proposed models should lead to density heterogeneities within the particle which would both affect the mechanism of swelling and also be revealed during the swelling process. Neutron scattering provides a direct method of examining the kinetics of swelling and also a method for determining the density profile of the particle. Swelling of polystyrene by the monomer, deuterostyrene, in D20 + H20 mixtures enables the contrast to be balanced and thus increases the sensitivity of the technique. The experiment can also, of course, be carried out by swelling poly- (deuterostyrene) with styrene.Other swelling materials, e.g., toluene, can be used, but deuterostyrene is the most directly related to the synthetic method of producing polymer latices. Mr. N. M. Harris (Grenoble) said: In reply to the question of Vrij on the effect of non-sphericity and polydispersity, neutron low angle scattering data can be most accurately analysed for monodispersed systems of spherical particles. Naturally, polydispersity of such systems introduces certain errors, but these are systematic and calculable for a given particle size distribution, and less than other experimental errors in, for example, the adsorbed layer studies for size variations of up to E 10% on the substrate diameter. Non-spherically symmetrical systems are more complicated in their analysis, but certain parameters, e.g., surface coverage per adsorbed molecule, can still be readily obtained for reasonably monodisperse systems.We are concentrating on measurements using spherically symmetrical systems as these are the most suited to neutron scattering measurements. For studies of adsorbed layer structure the use of spherical substrate particles, e.g. polystyrene latex, enables spherical symmetry to be obtained for a large range of systems. The micro- emulsion systems as presented by Overbeek are also evidently well suited to study by these techniques. Dr. J. W. White (Grenoble) said: The measurements on carbon sols and on clay platelets reported here are just the first illustration of the possibility of using low angle neutron scattering with contrast variation for less than ideal sols.It can be seen that polydispersity has a complicating effect on interpretation, but that, nevertheless, one can use such simple parameters as the extrapolated intensity at zero angle scattering to determine the density of the adsorbed layer. For modelling such particles the scattering length density can, of course, be ex- panded as in spherical harmonics. Provided that enough contrasts etc. are made, J. W. Goodwin, R. H. Ottewill, R. Pelton, G. Vianello and D. E. Yates, Brit. Polymer J., 1978, 10, in press. M. R. Grancio and D. J. Williams, J. Pulynrer Sci., 1970, 8, 2617.GENERAL DISCUSSION 127 enough coefficients can be found. This has been done for biological particles by S tuhrmann.l Another technique that one could exploit is that, for particles with different chemi- cal constitutions, it may be possible to specifically adsorb into these different areas materials of different scattering length density, for example deuterated methane or deuterium gas.One could then mark the areas where preferential adsorption oc- curred and use the contrast variation techniques to elaborate the structure. This technique would resemble what is currently being done for determining three dimensional structure of the protein subunits of ribosomes. In particular, it has been possible to deuterate selectively the proteins in the ribosome and reconstitute the organelle with only two proteins deuterated. The neutron scattering is then the scattering from the dumb-bell and the radius of gyration of the dumb-bell can be worked out very easily from the pattern.In a subsequent experiment, after re- constituting with one common deuterated protein and a deuterium marker in a further protein subunit, another radius of gyration can be built up. Hence, by triangulation, the relative position of the three is found. Successfully, one can build up by triangula- tion the whole ribosome s t r u c t ~ r e . ~ ~ ~ Prof. S. G. Whittington (Toronto) said: For long range forces the boundary condi- tions used in a Metropolis style Monte Carlo treatment can have an important effect. For instance, in the case of a two-component plasma, the estimated energy of the system changes dramatically when a minimum image approximation is replaced by an Ewald ~ummation.~ What boundary conditions were used in your calculation and to what extent were the results sensitive to the boundary conditions chosen? Dr.W. van Megen and Dr. I. Snook (Melbourne) said: For high electrolyte con- centrations (rca >> 1) the pair potential is of sufficiently short range so that the usual minimum image, with the spherical (Wood-Parker) truncation, is quite sufficient even with 32 particles. We have substantiated this assertion by occasionally comparing the results of computations using 32 particles with those using 108 particles. For low electrolyte concentrations (rca < 1) we have examined dispersions of very low volume fractions (< 1 %). Thus the long range potential is partly accounted for by the increased dimensions of the central box of particles.The calculations for these cases have been carried out with 108 particles and we find no substantial difference in g(r) when using the minimum image or an Ewald sum (or repeating the calculations using 256 particles) to include a greater part of the pair p~tential.~ This is, of course, consistent with findings in liquid state calculations where the structure of dense fluids is primarily determined by the hard repulsive part of the pair potential; the effective density of these dispersions is indeed high. Dr. E. Dickinson (Lee&) said: Snook has reported6 the successful application to assemblies of spherical colloidal particles of some methods of statistical mechanics developed originally to describe low molecular weight liquids.However, one funda- H. B. Stuhrmann, Acta Cryst., 1970, A26, 297. P. B. Moore and D. M. Engelman, Brookhaven Synip. Neutron Scattering in Biology, 1976, pv-12. W. Hoppe, R. Mayr, P. Stockel, S. Lorenz, V. A. Erdmann, H. G. Wittmann, H. L. Crespi, J. J. Katz and K. Ibel, Brookhaven Symp. Neutron Scattering in Biology, 1976, p. IV-38. J. P. Valleau and S. G. Whittington, in Statistical Mechanics: Equilibrium Techniques, ed. B. Berne (Plenum, 1977). W. van Megen and I. Snook, J. Chenz. Phys., 1977,66,8 13. W. van Megen and I. Snook, this Discussion.128 GENERAL DISCUSSION mental feature distinguishes a real colloidal dispersion from its molecular liquid analogue : the presence of a distribution of particle sizes. As part of our study into the effect of polydispersity on colloid stability, I should like to report some preliminary Monte Carlo results for assemblies of 125 particles having diameters randomly distri- buted with a gaussian distribution of half-width a,.We consider a pair potential u(rij) between spheres of radii a, and a,: u(rij) = uR(rij) + uA(rij)* The double-layer repulsion uR(rij) is assumed to be1 uR(ri,) = 2n(aiaj/aij)~yiyj In ( 1 + exp (-aijRij/d)), where aij = (a, + aj)/2, R,, = (rij - 2aij)/aij, E is the dielectric constant of the con- tinuous phase, 'y, and ' y j are surface potentials, and d is the double-layer thickness. The van der Waals attraction uA(rij) is taken as2 uA(ri,) = -(A/6)[2a,uj/(rfj - (ai - aj)2) + 2 a , ~ ~ / ( r ; ~ - 4 a 3 + In (r& - 4a:j) - In {r:j - (a, - aj)2)], where A is Hamaker's constant.With ai = a,, the potential reduces to that used by van Megen and Snook for monodisperse systems. The Monte Carlo method of Metropolis et aL3 was used with normal periodic boundary conditions. Pair potentials were truncated at rij = 6aij to reduce com- puter time. The runs recorded in table 1 were taken over (7-12) x lo5 configurations, allowing about lo6 configurations for equilibration. The pressure p at temperature T is calculated from i < j where V is the volume of the N particles, k is Boltzmann's constant and angular brackets represent the canonical average. We set at zero the probability of finding a particle with radius outside the limits (a} &- 3a,, and, following van Megen and Snook, consider a 1 : 1 electrolyte with 'y = 0.06 V, E = ~ O E , , A = 2.5 x J, T = 300 K, and (a) = 590 nm.At low volume fraction (q = 0.1 10) the (osmotic) pressure of a polydisperse system with a,/(a) = 0.25 is indistinguishable from that of the equivalent mono- disperse system within the estimated computational error (taken to be twice the stand- ard deviation). At 9 = 0.3 and electrolyte concentrations of 0.2 and 1.0 mol m-3, the two pressures differ by about 10%; at an electrolyte concentration of 0.16 mol M - ~ , a moderate degree of polydispersity leads to about a 20% change in pressure. It appears that if the volume fraction is low, polydispersity tends to reduce the pressure, but the converse is true if the pressure is high. This can be rationalized by assuming that at low pressures the dispersion is disorderd and liquid-like: it is known4 that mixing together hard spheres of different sizes reduces the non-combinatorial free energy, and hence the pressure, a principle which operates similarly with poly- disperse hard sphere^.^ At high pressures, however, where an ordered solid-like state is thermodynamically the more stable, the free energy and pressure will likely be in- H.R. Kruyt, Colloid Science (Elsevier, Amsterdam, 1952), vol. 1 . J. Mahanty and B. W. Ninham, Dispersion Forces (Academic Press, London, 1976), p. 16. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, J. Chew. Phys., 1953,21, 1087. J. S. Rowlinson, Disc. Faraday SOC., 1970, 49, 30. E. Dickinson, Chem. Phys. Letters, 1978, 57, 148.GENERAL DISCUSSION 129 creased by deviations from monodispersity.This is perhaps best illustrated in fig. 1 showing the radial distribution functions obtained in runs 7-9. The monodisperse system exhibits considerable solid-like order (notwithstanding the choice of N which tends to inhibit complete crystallisation into a close-packed hexagonal array). For I FIG. 1 .-Effect of polydispersity on the radial distribution function g(r/<a>) in a colloidal dispersion of volume fraction 0.3 and 1 :1 electrolyte concentration 0.16 mol m-3: - , a,/(a> = 0 ; ---, u&a> = 0.05; - * - * -, o,/(a) = 0.1. degrees of polydispersity of only 0.05 and 0.10, the peaks have become much broader and more liquid-like, and presumably there is a degree of polydispersity above which the concept of a solid phase becomes meaningless.On the basis of our preliminary results, I would suggest that even a small degree of polydispersity may significantly affect the position of the order-disorder transitions reported by Snook in his paper. TABLE 1 .-DEPENDENCE OF p V/NkT ON DEGREE OF POLYDISPERSITY aa/(a) AS A FUNCTION OF VOLUME FRACTION $!) AND ELECTROLYTE CONCENTRATION c run aa/(a> p C/molm-3 pV/NkT 1 2 3 4 5 6 7 8 9 0 0.25 0 0.25 0 0.25 0 0.05 0.10 0.1 10 0.110 0.300 0.300 0.300 0.300 0.300 0.300 0.300 1 .o 1 .o 1 .o 1 .o 0.2 0.2 0.16 0.16 0.16 1.50 f 0.04 1.44 f 0.04 4.43 4 0.02 3.89 f 0.28 12.5 f 0.3 13.3 =t 0.5 16.3 f 0.7 20.1 rt 1.1 20.0 4 0.9130 GENERAL DISCUSSION Dr. W. van Megen and Dr. I. Snook (Melbourne) said: It is certainly interesting to see a statistical mechanical treatment of polydisperse systems.With regard to the osmotic pressures, we also find a reduction in this quantity for heterodisperse systems at low to moderate 9 (unpublished results). Care must be taken, however, with the definition of a solid like phase and the criteria one uses to define it for polydisperse systems. With increasing polydispersity it is the crystallized state (that particular solid phase which gives rise to the iridescence in polystyrene dispersions) which becomes less perfect and ultimately loses meaning. The shape of the indiscriminate radial distribution function is then no longer a meaningful criterion for determining the phase of the system. A loose molecular analogy is a glass; this would have a radial distribution function qualitatively like a dense liquid, yet we are speaking of a solid not a crystal.A better criterion for determining the phase of a polydisperse system of particles is the r.m.s. displacement. In a MC simulation of the liquid phase the r.m.s. displacement is an increasing function of the number of configurations sampled, whereas for a solid it has an upper limit. Dr. H. M. Fijnaut (Utrecht) said: From Monte Carlo calculations the radial distri- bution function g(r) can be found for a given pair potential. From g(r) the structure factor S(K) can be obtained straightforwardly. In your paper one can see that there is no marked difference between a hard sphere and a Coulomb repulsion g(r). How is this difference manifested in S(K)? If S(K) is only experimentally available in a limited K range, is it then possible to distinguish properly between different types of pair interaction ? Dr.W. van Megen and Dr. I. Snook (Melbourne) said: We shall assume that you are referring to the similarity of the radial distribution functions displayed in fig. 3 of our paper. The hard sphere radial distribution function, go(r), normally has a much larger first peak [than the correct g(r)] with a discontinuous drop to zero at the effective hard sphere diameter. The hard sphere result shown in fig. 3, however, has the first order correction, gl(r), added to it; which brings this result into much closer agreement with the corresponding MC result. The effective hard sphere diameter and the consequent g(r) N go(r) + gl(r) depend not only on the precise form of the pair potential but also on the particle number density.Any evidence of structure in g(r) must clearly be replicated in S(K) and vice versa. The inversion of an experimental S(K), known only over a limited range of K, to obtain the pair potential is too inaccurate unless the data can be completed by some reliable interpolation scheme. Another (but laborious) approach is to use statistical mechanics, with a parameterized pair potential, to calculate S(K) and compare with available experimental data. Prof. T. W. Mealy (Melbourne) said: In a series of theoretical papers, Wadati et aI.1-4 attempted to describe the order-disorder results of Hachisu and co-w~rkers~-~ in M. Wadati, A. Kose and M.Toda, Kagaku (Science, Japan), 1972,42,646. M. Wadati and M. Toda, Oyo Butsuri (Appl. Phys. Japan), 1973,42,1160. M. Wadati, Kotai Butsuri (Solid State Phys., Japan), 1973, 8, 49. M. Wadati and M. Toda, Proc. van der Waals Centennial Con5 Statistical Mechanics (Am- sterdam, August 27-31, 1973), pp. 180-181. S. Hachisu, Y . Kobayashi and A. Kose, J. Colloid Interface Sci., 1973,42, 342. A. Kose and S . Hachisu, J. Colloid Interface Sci., 1974, 46,460. S. Hachisu and Y. Kobayashi, J. Colloid Interface Sci., 1974, 46,470. K. Takanov and S . Hachisu, J. Phys. SOC. Japan, 1977, 42, 1775; J. Chem. Phys., 1977, 67, 2604.GENERAL DISCUSSION 131 terms of a hard sphere first order phase transition with the interaction represented as V(R) = a R < (d + P / K ) R > (d + p/7c) and V(R) = 0 where d is the latex particle diameter, 7 c - l is the Debye length and p is a constant of order 1.At p = 1.3, for example, the familiar coexistence regime in the (volume fraction, salt concentration) diagram is well represented at high volume fraction-high salt conditions but, as expected, poorer agreement is obtained at low volume fraction- low electrolyte conditions. On behalf of Hachisu and his colleagues I wish to bring this earlier work to the attention of the authors. Dr. W. van Megen and Dr. I. Snook (Melbourne) said: We are aware of some of the work of this group and have also referred to your point [which was also expressed in ref. (l)] when dealing with a quantitative hard-sphere characterization of electro- statically established dispersions with 7ca $ 1 .2 These calculations used the Barker- Henderson criterion and yielded effective hard-sphere diameters given by P deff = do + - 7c where do is the particle diameter and the value for P was around 7 depending precisely on the electrolyte concentration.This result, which is independent of the volume fraction, produced the D-0 transition in good agreement with predictions of more complete theories over a range of particle diameters and electrolyte concentrations (for 7ca $ l).4 The present discussion paper takes these calculations a step further by comparing the corrected (to first order) hard-sphere radial distribution functions with the corre- sponding MC results for colloidal solutions with Ica 9 1 and ica < 1. For the latter system deff, determined by a variational technique,’ now also depends on the volume fraction q, Thus the simple results that the effective hard-sphere diameter is obtained by adding a number of Debye screening lengths to the particle diameter is not applic- able for systems with 7ca < 1.We wish to point out that the hard-sphere model is only a first approximation to the properties of the real system, and any property based on this model is clearly very sensitive to the choice of deff. Thus, if this model is to yield a reasonable first approxi- mation, say for the location of the D-0 transition or the osmotic pressure, an accurate and tested (by comparing with more complete or exact calculations for a given pair potential) criterion for deff is required. Any simple “ rule of thumb ” will, at best, have a very limited range of validity.In this and earlier work we have merely at- tempted to expose and unify already available quantitative methods for selecting the equivalent hard-sphere system and determine their applicability to colloidal solutions. Since there is still a great deal of uncertainty in characterizing colloidal solutions, even in terms of the parameters of DLVO potentials, it is extremely important that the theory used to calculate the properties of colloidal solutions is sound. Prof. J. Th. G. Overbeek (Utrecht) said : In concentrated suspensions, at low con- centrations of supporting electrolyte, all double layers may overlap and nowhere in the M. Wadati and M . Toda, J. Phys. SOC. Japan, 1972,32,1147. W. van Megen and I.Snook, Chem. Phys. Letters, 1975,35,399. J. A. Barker and D. Henderson, J. Chem. Phys., 1967,74,4714. See for example, I. Snook and W. van Megen, J. Colloid Interface Sci., 1976,57,47. D. Henderson and J. A. Barker, Phys. Rev. A , 1970,1,1266.132 GENERAL DISCUSSION suspension would the equilibrium concentration of electrolyte be present. Never- theless this equilibrium concentration can in principle be determined by equilibrium dialysis. It can also be calculated in good approximation using a cell model for the suspension, in which each particle is in the centre of a spherical cell, the volume of the cell being equal to the inverse of the particle concentration. Prof. S . Hachisu (Tsukuba) (communicated) : As was pointed out by Overbeek, the electrolyte concentration of a concentrated latex is a quantity definable only by means of " equilibrium dialysis " or Donnan equilibrium.This recognition leads to some important conclusions concerning the phase separating phenomenon in monodisperse latex. One of them relates to the correction of the phase diagram (the graphical presentation of phase separating condition, using the particle concentration q as ordinate and the logarithm of the electrolyte concentration C as abscissa). The fol- lowing is a brief explanation. Considering a latex system consisting of pure neutral latex * and added neutral salt,? and being in the state of dialysis equilibrium against KCl solution of concentra- tion Coo. Let the amount of added salt be M and the volume of the aqueous phase be V, then, the average concentration of the added neutral salt is given by C = M/V.The relation between C, C, and the particle concentration 9 is very complicated but can be written down as follows c = f (9, (7, Coo) where (7 is the number of electrolyte charge per one particle. Generally C is smaller than Coo and this situation is more pronounced when C, is low and qp is high. When the latex is not uniform but in the state of phase separation, the KCl concentration C, in the ordered phase may be larger than that c d in the disordered phase, because the particle concentration is smaller in the ordered phase than in the disordered. The expectation is that and CCO > c d > CO c d > c > co The measurement was done on these quantities by the following way (the details will be published elsewhere).A latex was prepared, which was in the state of phase separation in a dialysing tube immersed in a KC1 solution. After equilibrium attained (one month after prepara- tion), the tube was taken out of the solution, pinched at the separating position and the two phases were taken out separately. Then the C1-ion concentrations in three liquids, the dialysing solution, the ordered phase and the disordered phase, were determined. (Each Cl-concentration is equal to KCl concentration of each liquid.) The result is shown in table 1. As seen, the differences between Co, c d and C, are obvious and pronounced when C, is low. This fact makes it necessary to correct the phase diagram in our earlier work on phase separati0n.l There, it was postulated that Co = c d , and the diagram construc- * A latex extensively dialysed and then neutralised by KOH. 7 KCI is used here.S. Hachisu, J. Colloid Interface Sci., 1973,42,842.GENERAL DISCUSSION 133 TABLE 1 .-THREE CLASSES OF THE NEUTRAL SALT CONCENTRATION EXPERIMENTALLY DETERMINED. C, : concentration of dialysing solution. Co: concentration in the ordered phase. disordered phase. Cd: concentration in the disordered phase. pd: particle concentration (in volume fraction) of po : particle concentration in the ordered phase. dialysing solution disordered phase ordered phase c, = 9.2 x 10-5 ca = 6.3 x 10-5 co = 5.3 x 10-5 (pd = 0.066) (po = 0.078) ~~ * At this Coo-value, c d becomes equal to Co, but COO > cd(=cO = C). tion was made by locating the two p-values (po and p,) on one vertical line represent- ing average KC1 concentration C, as shown in fig.1 (a). But in view of the fact above described, the points representing the ordered and disordered phases must split apart from the vertical C-line [cf. (b) in fig. 11. This correction is not much in magnitude but works to reduce the discrepancy in the diagram. ‘ordered L Qbisorde rud t d - log c (b) - log c OD Id FIG. 1.-Three modes of presentation of the states of the phase separated latex. (a) Uncorrected (p, C) expression, (b) corrected (p, C) expression, (c) Coo-expression (C, > C). C: average con- The best method of diagram construction is to use the (p, C,) representation [cf. (c) in fig. 11, in which the points corresponding to the ordered and disordered phases lie on one vertical line representing C,.Considering that every theory on concentrated latex is necessarily based on C,, the diagram should be of (q, C,) type. Poor agree- ment with the experiment of our effective-radius-theory at low electrolyte concentra- tions would partly be due to the fact that the diagram was of the uncorrected (p, C ) type. centration of the neutral salt. COO: the neutral salt concentration in the dialysing solution. Dr. R. Buscall (Bristol) said: In their calculations of the various equilibrium proper- ties of latex colloids, van Megen and Snook have assumed pair-wise additivity of the134 GENERAL DISCUSSION particle interaction potential. Comparison of their results with experiment suggests that this is a fair approximation and consequently DLVO theory offers a good starting point for the development of theories of concentrated systems.We have recently investigated the linear viscoelastic properties of ordered latices and have compared data for the high-frequency (Hookean) shear moduli with theoretical estimates based on the assumption of pairwise additivity of the DLVO potential. Good agreement was obtained. For particles in an ordered array the theoretical modulus is related to the pair potential by where k is a constant of order one which depends upon the type of packing (0.5300 for a face-centred cubic array). Combining eqn (1) with a suitable expression for U(riJ) given Go to within the pair-wise additivity approximately, for example for 7ca < 3 we have (2) 476 E~ &,a2 t,v$ U(rij) = exp [ - K ( r i J - 2a)l ri j and Gi becomes, In general, the evaluation of this expression is complicated by the fact that K and yo are not usually known for concentrated systems.The latices we used were dialysed against large reservoirs of salt solution so that volume fraction and particle size could be varied at a constant, known, value of K . Comparison of experimental moduli, measured as a function of volume fraction (0.1 < tp < 0.41, with the appropriate expression for Gh showed that the data could be fitted exactly if yo was assigned a constant value. The very pronounced particle size dependence (in the range 25 < a < 100 nm) was also very well accounted for by the theory. In both cases physically realistic value for yo were obtained (50 to 90 mV dependent on particle size and salt concentration). At first sight the excellent agreement between theory and experiment suggests that pairwise additivity is a very good approximation indeed, even for quite concentrated system, however, the following points should be borne in mind.First, ordered latex dispersions are known to be polycrystalline in nature, whereas the presence of grain- boundaries was ignored in the calculations. Secondly, although the derived values for yo were deemed reasonable (on the grounds that they are of the same order as electrokinetic potentials measured for dilute systems) they may conceal deviations from the idealised model arising from mu1 tibody effects, polydispersity, crystal im- perfections and the like. Nevertheless, given these reservations, we concur with van Megen and Snook's view that the simple pair potentials provide a good basis for the understanding of concentrated dispersions.Dr. S. Levine (Manchester) said: I would expect that pair-wise additivity of the electric double layer forces becomes inaccurate in a concentrated colloidal dispersion when the thickness of the diffuse layer exceeds the mean separation between the centres of two adjacent particles, ix., if the thickness is greater than the distance of the first R. Buscall, J. W. Goodwin, M. Hawkins and R. H. Ottewill, in preparation. J. W. Goodwin and A. M. Khidher, Colloid Interface Science (Academic Press, N.Y., 1976), vol. IV, p. 529.GENERAL DISCUSSION 135 peak in the radial distribution function for two particles.This would be the case if Ica < 1, when the double layer particle interactions could be obtained by using a cell model. When the range of particle size and electrolyte concentration is such that rca 3 1, then the familiar Derjaguin approximate method is available for evaluating the interaction of two particles at small separations, as illustrated by eqn (2.4) of the paper under discussion. Pair-wise additivity should be a good approximation when the Derjaguin force formula applies. The diffuse layers of three particles will overlap in three separate regions and the interaction energy will be governed by these overlaps. Although the electrolyte concentrations quoted are low enough to validate the Poisson- Boltzmann equation, it should be noted that the Debye-Hiickel parameter IC which appears in the pair-potential formulae (2.4)-(2.5) may differ from that defined by the bulk electrolyte concentration. This difference is particularly relevant when the con- centration of potential-determining ions in the colloidal system becomes comparable with that of the supporting electrolyte.The simple formulae (2.4)-(2.5) must then be a1 tered. Dr. I. Snook and Dr. W. van Megen (Melbourne) (communicated) : We certainly agree with your comment that three and higher body non-additive potentials may, in some special cases, be important. In practice, however, we generally find that the qualitative features of the properties of dispersions are well reproduced by the use of pair-wise additive potentials ; these qualitative features are seemingly insensitive to the fine details of the interaction potentials. Furthermore, we also find that we usually obtain reasonable quantitative agreement with experiment for osmotic pressure and for radial distribution functions.2 One would expect these latter systems to be influenced by many body potentials as Ka < 1 and the double layers are strongly over- lapping.However, the range of particle concentration which is experimentally relevant for these systems is very low (q 5 0.1 %) thus making the overlap of more than a few double layers not very significant energetically. From this reasonable agreement between calculation and experiment we can con- clude that the simple D.L.V.O. pair potential is a reasonable effective pair potential in most cases.However, for some systems (particularly for rca < 1) there is difficulty in experimentally defining the surface charge and bulk electrolyte concentration so some calibration of these parameters is usually required. This calibration may, in fact, be masking the effect of the many body potentials. Thus it would be extremely valuable to have expressions for the potential energy of interaction of 3 or more double layers which could then be used to directly evaluate the effect of these non-additive potentials on observable properties. Finally, there are cases where the assumptions upon which the D.L.V.O. pair interaction potential is based are invalid (for example, when the separation of the particles is of the order of molecular dimensions) and we are currently examining the structure and overlap of double layers from a microscopic viewpoint in an attempt to shed some light on these problems.Prof. R. H. Ottewill (Bristol) said : With the small particles, % 50 nm in diameter, at the low volume fractions ( E which have been used in light scattering experi- m e n t ~ ~ of the type referred to in the paper by van Megen and Snook, it has been our experience that two particular difficulties are encountered. These are: (1) to deter- A. Homola, I. Snook and W. van Megen, J. Colloid Interface Sci., 1977,66,493. W. van Megen and I. Snook, J. Chem. Phys., 1977,66,8 13 ; and this Discussion. J. C. Brown, P. N. Pusey, J. W. Goodwin and R. H. Ottewill, J. Phys. A . Math. Gen., 1975, 8, 664.136 GENERAL DISCUSSION mine the magnitude of the surface potential at very low electrolyte concentrations, (2) to determine accurately the bulk electrolyte concentration and pH without dis- turbing the system.It must be remembered that to obtain ordered arrays with these small particles (x 50 nm diameter) the latex is contained in a stoppered silica cell for days, or even weeks, in the presence of a small quantity of mixed-bed ion-exchange resin. Once ordering occurs the angular scattering intensity envelopes can be obtained. The latter are very sensitive to addition of traces of electrolyte and in the more sensitive cases even removing the stopper of the cell and admitting COP can lead to the dis- appearance of order. We do not currently have a detailed phase diagram for the latex which we used for , .&existence, ' / / 0 / 4 ' order 0.0 1 0.02 volume fraction FIG. 1.-Phase behaviour for a polystyrene latex of diameter 104.5 nm showing the dependence of ordered region on volume fraction and sodium chloride concentration. light scattering experiments1 but Miss S. M. Lyon in Bristol has recently determined a diagram for a latex of diameter 104.5 nm. This diagram is shown in fig. 1. From this it can be seen that even at volume fractions of the order of the ordered phase only exists at electrolyte concentrations below mol dme3 and also that the co-existence region becomes very narrow. Herein lies the reason for the electrolyte sensitivity of the ordering process at low volume fractions with small particles. It is also worthy of note that our optical diffraction studies indicate that the struc- ture of the ordered phase is a body centred cubic arrangement of particles. Dr.J. W. Goodwin (Bristol) said: The data illustrated in fig. 1 have been calculated for a 1.2 pm particle diameter latex at an electrolyte concentration of mol dmV3. These calculations predict, and this is stated in the text, that at volume fractions of tp > 0.35 a long range order should occur. This does not agree with experiment where it was found that latices with particle diameters > 1 pm gave disordered struc- tures at high volume fractions and order arrays were only found for latices with particle diameters < 1 pm.2 Experimentally, we find a marked particle size dependence on the ordering process and this does not appear to be predicted by these calculations.J. C. Brown, P. N. Pusey, J. W. Goodwin and R. H. Ottewill, J. Phys. A. Math Gen, 1975,8,664. L. Barclay, A. Harrington and R. H. Ottewill, Kolloid Z., 1972,250,655.GENERAL DISCUSSION 137 Dr. I. Snook and W. van Megen (Melbourne) said: We, in fact, find a disorder- order transition in which an ordered phase is stable at high volume fraction for particle radii between 250 and 5950 A. However, one should define what one means by this statement. We usually calculate the free energy of each phase and find which phase has the lowest free energy. This tells us which phase is thermodynamically stable; however, it does not tell us how long it will take to produce a phase change. In particular, it does not tell us how long it would take to make a disordered array crystallise into an ordered one.Now the total pair potential energy is a balance be- tween a screened coulomb repulsion and a van der Waals attraction and for large particles at high electrolyte concentration this leads to a deep secondary minimum. As I believe you have found experimentally such a deep secondary minimum en- courages the formation of clusters and inhibits the formation of the ordered phase. Thus, even though this ordered phase is the thermodynamically stable phase at high volume fraction it may take an enormous time to form this phase from a disordered array (in other words, we have a long relaxation time). We believe it is very important to draw a clear distinction between thermodynamic stability and the kinetic problem of how long it takes to produce a new phase.Prof. R. €3. Ottewill (Bristol) said: This is a very interesting technique and should provide a very useful method for the examination of concentrated colloidal dis- persions. The method has some problems, however, and I would be grateful if Vrij would comment on the following questions : (1) Since the diffusion coefficient of a particle in quasi-elastic light scattering enters into the analysis as an exponential term, if there is more than one diffusion coefficient, for example, coefficients for particles interacting in a lattice, those diffusing through the lattice, and those in free Brownian motion, a family of exponentials would be obtained. How were the data analysed to give the Deff given in fig. 5 of your paper? (2) With rather concentrated dispersions multiple scattering can be a problem.How did you check that your results were not influenced by multiple scattering? (3) The Fourier transformation of S(K) against K to obtain the radial distribution function can be rather sensitive to the data obtained at low angles since it involves a cumulative integration. Could you mention the lowest angles at which measurements were made and whether you carried out the integration directly from the experimenta data or from smoothed curves ? Prof. A. Vrij (Utrecht) said: (1) Data were analysed according to a cumulant method. The value of Deff was found from the first cumulant. It appeared, however, that the relative difference be- tween the cumulant value and an overall exponential fit was only a few per cent.(2) We found the attenuation of the primary beam in our 2 cm diameter cylindrical cell (surrounded by a toluene bath) to be maximal 25%. The angular variation of the scattering was not very different in a cell with a diameter of 1 cm. This suggests that multiple scattering is not so important. The influence of multiple scattering certainly deserves more attention. (3) Measurements were made between 8 = 15" and 150". Integration was carried out from smoothed curves. Dr. J. D. F. Ramsay (Harwell) said: Is it possible that, in the light scattering studies of PMMA latex particles, the anomaly in P(K), which was found at low values of K using the data for dilute dispersions, could have resulted from polydispersity or the138 GENERAL DISCUSSION presence of a small fraction of aggregates? Is it likely that at higher concentrations such effects could complicate the derivations of S(K) and h(r)? Prof.A. Vrij (Utrecht) said: The experiments in the dilute region give in no sense occasion to suppose large polydispersity or the presence of aggregates (log Re against sin2 8/2 shows a straight line in the lower range of 8). Also from sedimentation and diffusion experiments in the dilute region no such indication was found. Dr. J. W. Goodwin and Prof. R. H. Ottewill (Bristol) said: The light scattering method described by the authors provides a direct method of obtaining the second differential of the energy of interaction, V, with respect to the distance of separation between interacting surfaces, h, i.e., d2V/d1z2 against h.It is restricted, however, to systems which are sufficiently optically transparent to enable light scattering measure- ments to be made without complications arising from multiple scattering. There are, however, also other methods which can be used to obtain basic informa- tion of this type on concentrated non-aqueous dispersions which are not optically transparent. For example, the measurement of the excess osmotic pressure of a dis- persion as a function of volume fraction, 9, can be made using a pressure cell of the type we have previously described.l These data then lead directly to dV/dh against h if the structural arrangement of the particles is known. In addition, measurements of the velocity of transmission of an audio frequency wave, pulse shearometer, can be 100 80 r*I E Z Y \ 60 2 2 n v) 40 20 0.59 0.58 0.57 0.56 voIurne fraction 5 4 v, 3 e 3 3 , E Q \ r Z 2 3, h) 1 FIG.1 .-Pressure and shear modulus as a function of the volume fraction of the cores for a polymethyl- methacrylate latex with a core diameter of 402 nm; -0-, pressure; -0-, shear modulus. R. J. R. Cairns, R. H. Ottewill, D. W. J. Osmond and I. Wagstaff, J. CoZZoid Interface Sci., 1976, 54, 45.GENERAL DISCUSSION 139 used to obtain the shear modulus1S2 as a function of p and thence essentially a plot of d2 V/dh2 against h. Both these techniques can be used on systems which have reached an " equilibrium " state and involve a minimal disturbance of the system. Work carried out in Bristol by Dr.L. Ant1 and Mr. R. Hill has systematically utilised both these techniques to examine a series of latices, of various particle sizes, prepared with polymethylmethacrylate cores and with poly-1Zhydroxy stearic acid chains chemically grafted to the core to give sterically stabilised particles in d0decane.l Fig. 1 shows the results of pressure and shear modulus obtained on a latex, with a core diameter of 402 nm, as a function of the volume fraction, p, of the cores. The onset of steric interactions can be seen from the (pressure, 9) curve to occur at z p = 0.565. The shear modulus, however, has detected the onset of steric interactions at an even lower volume fraction. Hence it appears that shear modulus determinations, de- pendent on d2V/dh2, are a very sensitive indicator of interactions in non-aqueous dis- persions. Dr.J. D. F. Ramsay (Harwell) said: Vrij has demonstrated how the radial distribu- tion functions of latex dispersions can be obtained by light scattering. A similar approach has recently been used3 to study silica sols of much smaller diameter (< 10 nm) by small angle neutron scattering (SANS). In such systems, where distances of interparticle separation are considerably smaller, interference effects occur at much larger momentum transfers, which although beyond the range of light scattering (K < 5 x lo5 crn-l) are readily achieved in SANS. This feature is demonstrated in fig. 1 which shows the dependence of normalised scattered neutron (A = 4.78 A) intensity, I@), on momentum transfer, Q(n.b. 1 Q/A-l = lo8 K/cm-I) for a silica sol (diameter ~8 nm) at several different concen- trations.As the concentration is increased the interference becomes more marked and the maxima move to higher Q, which indicates a reduction in the equilibrium separa- tion between particles. Structure factors, S( Q), obtained from these measurements, 0 I 5 10 15 momentum transfer, 102 Q A-' FIG. 1 .-Small angle neutron scattering curves for silica sols of different concentration (% w/w SiOJ : I, 5 ; 11, 15; 111, 23; IVY 35. N.B. Data have been normalised at high Q when S(Q) 1. J. W. Goodwin and R. W. Smith, Faraday Disc. Chem. SOC., 1974,57,126. J. W. Goodwin and A. M. Khidher, Colloid and Interface Sci., 1976, IV, 529. A. J. Leadbetter and J. D. F. Ramsay, to be published.140 GENERAL DISCUSSION were used to obtain g(r) as a function of interparticle separation r (cffig.2) by the following Fourier transformation : where N/ Y is the particle number density. - /bJ 'I I I I I 30 0 rlA 2oo 100 FIG. 2.Radial distribution functions of silica sols (R = 4 nm) of different concentration ('A w/w SOz): (a) 15, (b) 23, (c) 35, p is respectively 0.074,0.12 and 0.19. Dr. C. Taupin (Paris) said : I have four comments : (1) A small angle neutron scatter- ingstudy of microemulsion was performed by Dr. R. Ober and myself in the ILL. I would like to compare some of our results to those Vrij reports. The variable contrast method is a very powerful technique for providing information about the Composition of the interfacial film. We showed that this film is penetrated by the continuous phase and we determined the '' non penetrated " volume (corre- sponding to soap and excess alcohol).The corresponding radius is in excellent agree- ment with the hard-sphere radius determined by the light scattering experiments of Vrij. (2) Neutron scattering experiments give also information about the correlations between the droplets. We always found a very small second virial coefficient, which corresponds to an attraction. (3) I want to raise a point about the difficulty of working with such systems. Usually, the extension of the domain of existence of microemulsion in the pseudo- ternary (oil, water, surfactant) diagram is very wide and it is necessary to define the dilution procedure carefully ; i.e., varying the concentration without changing the structure of the elementary droplet.(4) The last point may be an answer to the first question. We measured the polydispersity of our systems by analytical ultracentrifugation technique. The dis- persity of the radius was within < 10%.GENERAL DISCUSSION 141 Prof. A. Vrij (Utrecht) (communicated) : Following Schulman we assumed all the soap molecules to be present in the surface layer of the droplets. In the preparation of our micro-emulsions we held the water to soap ratio constant and diluted the system by a mixture of oil and alcohol. The composition of this continuous phase is deter- mined by titration experiments along the straight borderline between the region of existence of the microemulsion and the coarse emulsion in the pseudoternary diagram (water/soap, alcohol, oil).In this way we are certain not to change the composition of the surface layer of the droplet during the dilution procedure. Dr. Th. F. Tadros (Jealott’s Hill) said: In the case of the water/benzene micro- emulsion, your light scattering results indicate a zero second virial coefficient. How would you explain the thermodynamic stability of such a microemulsion? Is this attributed to the mixing term as given by the Percus-Yevick equation? Prof. A. Vrij (Utrecht) (communicated) : Thermodynamic stability does not require the second virial coefficient to be positive. The equation for the (osmotic) pressure used by us is very similar to the classical van der Waals equation of state. The Percus-Yevick or rather the Carnahan-Starling, however, gives a much better repre- sentation of the repulsive interaction than van der Waals’ original equation (see sec- tion 3).According to eqn (3.4) one obtains = 1 + (8 - 261)~ + . . pG Thus the second virial coefficient is: (8 - 261). A system is stable when it is above the critical point. The critical point is found from the conditions: aPlap = a2P/ap2 = 0. Applying this to eqn (3.4) one finds tpc N 0.129 and ~CC, N 21.32. Thus at the critical point the second virial coefficient is strongly negative. For smaller a values the system is stable but still may have a negative second virial coefficient. Prof. E. Ruckenstein (Buflalo) said: I would like to stress that in contrast to regular emulsions or other colloidal dispersions, microemulsions can be absolutely stable from a thermodynamic point of view.The very small or even negative second virial coeffi- cient found experimentally for the osmotic pressure by several researchers1P2 (and in the paper presented at this meeting by Vrij et al.) can probably be related to their thermodynamic stability. A very small (or negative) second virial coefficient implies the existence of an attraction which compensates (or overcomes) the repulsion. The attraction cannot be caused by van der Waals interactions, because the latter are too sma1l.l It can be, however, the result of some of the effects which insure thermo- dynamic stability of microemulsions. Indeed, thermodynamic equilibrium fixes the state of the system and therefore, via the chemical potential of the solvent, the osmotic pressure. My comment is, therefore, an attempt to explain the origin of this thermo- dynamic stability.A microemulsion is composed of five components : oil, water, surfactant, cosurfac- tant and salt. Oil and water are immiscible. If a surfactant (such as an alkali metal soap) and a cosurfactant (such as an alkyl alcohol) are dissolved into the two immis- cible phases a spontaneous dispersion of globules below about 100 nm size of one phase in the other can occur. Let us assume that a large interface has been spontaneously W. G. M. Agterof, J. A. J. van Zomeren and A. Vrij, Chem. Phys. Letters, 1976,43,363. C. Taupin, J. P. Cotton and R. Ober, J. Appl. Cryst., in press.142 GENERAL DISCUSSION created between the two immiscible phases.The surfactant and cosurfactant are accumulated mostly at the interface. This accumulation has two important effects. On the one hand it decreases the interfacial tension of the water-oil interface to very low values. On the other hand, because of the large decrease in the concentrations of surfactant and cosurfactant in the two bulk phases, the molecules of surfactant and cosurfactant have a lower chemical potential, thus decreasing appreciably the free energy of the system. I call this phenomenon, the dilution effect. In addition to these effects, an ionic surfactant generates a charge on the interface and, therefore, double layers form spontaneously and interglobular repulsive forces develop. Spon- taneous dispersion occurs because the free energy change due to the dilution effect, which is negative, overcomes that due to the small, positive, surface tension.The cosurfactant has a double influence on stability (1) it contributes to a higher decrease of the surface tension than that achieved by the surfactant alone; (2) it generates an additional dilution effect. The pressures in the globules and in the continuous phase are different, while the volume of the system remains essentially constant during the formation of micro- emulsions. Therefore, the Helmholtz free energy dF of formation of' one cm3 of microemulsion is used to extract information about the thermodynamic stability.' This is defined as the difference between the free energy of a microemulsion having globules of a given radius and the free energy of the same system when the globules are very large.The quantity dF has to be negative for a microemulsion which is thermodynamically stable. The size of the globules is the radius for which dF has a minimum value. The free energy of the generated globule interface is equal to the product of the surface area 39/R per unit volume and the specific interfacial free energy. Here 9 is the volume fraction of the globules and R is their radius, which is, for the sake of simplicity, assumed to be uniform. The specific interfacial free energy is equal to y + ZI',puf, where y is the surface tension, Ti is the surface excess of species i and pi is the corresponding chemical potential in the state with globules of radius R (final state). Because of thermodynamic equilibrium, the chemical potentials are the same for the adsorbed molecules and for those in solution.The Helmholtz free energy change due to the formation of the interface and dilution from the initial State (corresponding to the large globules) to the final molar fractions (corresponding to globules of radius R) is given by Here ni is the number of molecules of species i per cm3 of microemulsion, p i is the electrochemical potential, p is the pressure (the subscripts 1 and 2 refer to the continu- ous and dispersed phases and pa is the pressure of the state with large globules). The superscript prime refers to the final state and the chemical potentials without superscript correspond to the state with very large globules (initial state). The present treatment implies that the continuous and dispersed phases can be considered bulk phases.39 In eqn (1) the term - y has a positive value, while the dilution term, Crzi(p; - &), R has a negative value. l E. Ruckenstein, Chern. Phys. Letters, 1978, 57, 517.GENERAL DISCUSSION 143 For ionic surfactants, the surface tension y depends on the surface potential y and surface charge density a. To obtain an equation for y we start with Gibbs adsorption equation which, for isothermal conditions, has the form dy = - ZI’idpi (24 Integrating at constant electrochemical potential for all species with the exception of surfactant and cosurfactant one obtains = 70 - Z’JI’idp;. (24 Here yo is the water-oil interfacial tension free of surfactant and cosurfactant. Be- cause of thermodynamic equilibrium the electrochemical potential at the interface is equal to that in the bulk.Such an equality provides a relation between Ti, bulk concentrations and surface potential. The surface charge a is proportional to the surface excess of surfactant and the surface potential can be expressed in terms of surface charge and of globule radius. In the end, the integral in eqn (2b) can be per- formed to obtain y as a function of the bulk concentrations of surfactant and co- surfactant and of the radius R. A more simple procedure consists, however, in decomposing the electrochemical potentials pi in their chemical (p’t) and electrical (ety) components at the interface and, hence, in rewriting y as y = yo - ZfJTidp’{ - Jady = y’ - lady. Expressing p‘; as a function of the surface excesses, the integrals JTidp’[ can be carried out.Although the specific free energy of the double layer (-Jody) was subtracted from y, y’ still includes electrical contributions. Indeed, Ti depends on the surface potential via the adsorp- tion isotherm which relates Ti to the bulk concentrations. Further it is convenient to decompose the specific free energy --Jody of the double c layer in the form - ady = (-fadl)a + [(I ody), - I’ody] = y1 + 72. The Ic subscript co indicates that the integration is performed assuming large distances be- tween globules (non-overlapping double layers). The quantity yl, which is negative, is the specific free energy of the non-overlapping double layers, while the quantity yz, which is positive, is the specific free energy due to the double layers overlap (repul- sion).Approximate expressions for both y1 and y2 have been proposedl in terms of the surface potential. The surface potential can be related, however, to the surface charge density, which, in turn, can be related to the specific adsorption of surfactant. In addition to AFd the free energy of formation AF contains the free energy change Me due to the entropy generated by the dispersion of the globules, which is a negative quantity. Expressions for Me have been established in ref. (1). The effect of van der Waals interactions is, in general, negligible compared to the other contributions. The quantities y‘, yl, y2, dFe and the chemical potentials pi depend upon the radius. This is due among others to the fact that the initial amounts of surfactant and co- surfactant are distributed between the bulk phases and interface and the distribution depends upon the area of the interface and hence upon the radius.In general, AF- + o(> for R ---+ 0 and AF + 0 for R -B- 00. In some circumstances there is a range of value of R for which AF < 0 and d F has a minimum for R = Re. This happens because the negative contribution of dFe and, in particular, of the dilu- tion effect overcomes the positive contribution due to the small surface tension ?(Re). The latter quantity has to be so small that y(R,) E 0 can be, in general, used to calculate Re. The present theory can explain the occurrence of stable microemulsions for both ’ E. Ruckenstein and J. C. Chi, J.C.S. Funzday 11, 1975,71, 1690.144 GENERAL DISCUSSION non-ionic and ionic surfactants. In the former case the stability is a consequence of the competition between the free energy contribution due to the interfacial tension y' and the dilution effect, while in the latter one has to account, in addition, for the terms containing y1 and y2. In conclusion, the thermodynamic stability of microemulsions is due to (1) the interfacial tension is decreased to very low values, and (2) the free energy change due to the dilution effect and to the entropy of dispersion of the globules, which is negative, overcomes that due to the small, positive, interfacial tension. Ref. (1) contains a more detailed presentation of the theory. Prof. J. Th. G. Overbeek (Utrecht) said: In describing the electrical contribution to the surface excess Gibbs energy, Ruckenstein has used the expression -jadly/, (a 2 surface charge density, 'yo = surface potential) whereas I have used +Jly,da. In order to avoid misunderstanding the following explanation may be useful. The surface excess Gibbs energy, Gu, is defined as G" = A(y + m p i ) where A is the surface area, y the surface tension, Ti the surface excess concentration of component i and pi its chemical potential. y is a measurable quantity, but it may be desirable to split it into an electrical and a non-electrical part, although these parts cannot be measured independently. If y is written as y = y' + jlyoda, Jyoda represents the work needed to charge the surface and the double layer. y' would be the surface tension if all concentrations and surface excesses would remain the same, but the molecules would not carry any charge. If, on the other hand y is written as y = y" - Sadly,, then --Sadly, does not only contain the electrical work, but also chemical work (negative) due to the preference of the charge carrying molecules for the surface. Both approaches, if carried through correctly, should lead to the same result for any measurable quantity. The choice between the two may be based on expediency or on personal preference. Prof. E. Ruckenstein (Bucfalo) said: In reply to Overbeek's remark let me explain why in my treatment of the stability of microemulsions I have chosen the form y = y" - fadw (4 for the surface tension y. the form I start with the Gibbs adsorption equation, which, for isothermal conditions, has dy = -2I'idPi (B) Here Ti is the surface excess and pi is the electrochemical potential. Ti for water and for oil are assumed zero. By integration at constant electrochemical potentials for all species with the exception of surfactant and cosurfactant, eqn (B) becomes y = yo - ~tJllidp;, (C) where yo is the surface tension of the oil-water interface and the second term accounts for the effect of adsorption of the charged surfactant and of the cosurfactant. The electrochemical potential can be further decomposed into its chemical (p'fl) and electrical (zety) parts at the interface, to lead to y = yo - C'jridP;' - Jadv, (D) E. Ruckenstein, Chern. Phys. Letters, 1978, 57, 517.GENERAL DISCUSSION 145 where a is the surface charge density due to the charge of the surfactant. Compared to eqn (C), in eqn (D) the specific free energy of the double layer, -[Sadly, which is a negative quantity, is extracted in a separate term. Assuming, for illustrative purposes, ideal gas laws for the interface, one has ,u’; = kTln rt + const and, hence, eqn (D), becomes where I-‘, and rCs are the surface excesses of surfactant and cosurfactant. In con- clusion, the surface tension y can be decomposed as in eqn (A). If instead of eqn (A) one uses y = yl + jvdc, (F) then - av yI = y“ includes in addition to y” the product between the surface charge density and the sur- face potential.

ISSN:0301-7249    

DOI:10.1039/DC9786500114

出版商:RSC    

年代:1978

数据来源: RSC

摘要:

On Coagulation in the Primary Minimum BY G. FRENS Philips Research Laboratories, Eindhoven, The Netherlands Received 1st December, 1977 The DLVO theory offers conceptual problems now that it appears to be established that double layer potentials are low at the critical coagulation concentration in colloids which follow the Schulze- Hardy rule, and that the predicted relation between the rate of slow coagulation and the particle size is not found experimentally. In this paper we explore the hypothesis that competition between aggregation and repeptization determines the net rate of slow coagulation, and that aggregates become progressively more irreversible after their formation. A theory of this kind could account for many experimental observations. It predicts that the double layer potential q~ at c, increases with the counterion valency and that there is some critical potential q)"s below which coagulation can begin.From the effects of the concentration and valency of counterions on the rate of coagulation it is evident that the stability of electrocratic colloids depends on the repulsion of electrical double layer~.l-~ Fuchs' theory4 of diffusion in a field of long range (double layer repulsion and van der Waals attraction) forces enables calculations to be made of the retardation of the rate of coagulation relative to the rate of rapid coagulation which was derived by von Smoluch~wski.~ One obtains a retardation of the coagulation by a factor W = 121 x - ~ exp( V,/RT)dx since a fraction of the Brownian collisions is prevented from occurring by the repulsive forces.V, is the potential energy of interaction at an interparticle distance x = H + 2a between two spherical particles of radius a. The DLVO theory2g3 computes V,, by addition of the contributions V, of the double layer repulsion and VA of the van der Waals forces. The largest contributions to the Fuchs-integral [eqn (l)] are from the range of x values where V, has a positive maximum. Relating changes in V,, and especially in its maximum value, to changes in the electrolyte con- centration c, Verwey and Overbeek found that a plot of log W against log c should consist of two linear parts, intersecting at the critical coagulation concentration c,. Identifying c, as the concentration where the positive maximum vanishes, they also gave a derivation which shows that (for large double layer potentials) c, x z6 is constant. This is the DLVO expression for the empirical Schulze-Hardy rule: the counterion valency has a very large effect on coagulation rates.These two results, the derivation of a Schulze-Hardy rule and the prediction plus experimental verifica- tion6 of the shape of (log W, log c) curves have been taken as evidence for the validity of the DLVO approach. As one of the first theories for the absolute rate of a reaction the DLVO theory was remarkably successful. However, some questions remain; and not only about the application of this In the absence of such a maximum, W = 1 approximately.G . FRENS 147 theory to practical colloids. It is difficult to reconcile the observation that colloids with electro-kinetic [-potentials as low as 25 mV obey the Schulze-Hardy rule with the assumption that the diffuse layer potential ya is high enough to allow the DLVO derivation of this rule (tanh zeya14kT z 1, i.e., z y ~ > 200 mV). Also, now that well defined monodisperse colloids are available, it has been found that d log W/d log c is hardly dependent on the particle size.7 This result is in conflict with the theory, which predicts that it should be proportional to a.In this paper we discuss an alternative possibility for the way in which double layer repulsion and van der Waals attraction affect the coagulation process via the potential energy diagram. We assume that there is coagulation in the primary minimum at short interparticle distances, but that the net rate of slow coagulation is determined by the probability of spontaneous (Brownian) repeptization of the aggregates.The idea that a fraction of the particles may break away from an aggregate is not altogether new,*-15 but has been explored before in formal analyses of the kinetic implications of aggregate reversibility, applicable to coagulation in the primary as well as in the secondary minimum of the interaction curve. For small (colloidal) particles the primary minimum is the only relevant potential energy mini- mum. In this paper we attempt to discuss some of the physical concepts involved and experimental conditions required for the reversibility of aggregates consisting of two particles in this primary minimum. In von Smoluchowski's calculation of the rate of rapid coagulation, and also in the DLVO theory, it is assumed that every collision results in the formation of a permanent aggregate.These theories count the number of collisions per unit time. Long range forces may prevent a fraction of the collisions [eqn (l)] and thus retard the coagulation. At short interparticle distances they can also account for irreversi- bility: VR varies as exp(--H) and VA as H-l so that V, becomes strongly negative as H--+ 0. Taken together these arguments explain why slow coagulation often proceeds at a " retarded Srnoluchowski " ate,^^*'^ in which a constant fraction of the collisions remains ineffective. The probability that a collision between two particles is prevented by an energy barrier V, remains independent of the number of primary particles and of the number and size of the aggregates.With reversible aggregates one would expect an aggregation-repeptization equilibrium to be established.1° The coagulation process would slow down (limited coagulation), as has been observed with polymer stabilized suspensions and in secondary coagulation. Indeed, there are a few observations of limited coagulation with small particles and highly irreversible ~ o l l o i d s , l ~ - ~ ~ which seem puzzling. If it is assumed that the rates of slow coagu- lation are retarded because of spontaneous repeptization of aggregates, as in this paper, then it has to be explained why in so many other experiments one observes " retarded Smoluchowski " rather than " limited " coagulation kinetics.Relating these observations to the rates at which double layers are equilibrated after a Brownian collision we obtain an alternative explanation for irreversibility. Some years ago in studies of the repeptization of electrocratic colloids after co- agulation in the primary minimum,20-22 it was found that the experimental observa- tions can be explained with the traditional type of potential energy diagrams ( VR + V' as a function of H ) if it is assumed that (a) the charge CT rather than the potential y of the double layer remains constant during a Brownian collision ; (b) that there is, therefore, a distance of closest approach H = 26 in an aggregate. 6 is the distance between a particle surface and the corresponding Outer Helmholtz Plane;148 COAGULATION I N THE PRIMARY MINIMUM (c) the potential 96 in the O.H.P., which varies with the electrolyte concentration c, must be used to compute the potential energy diagrams.Ageing of the aggregates was given as an explanation for the progressive irreversibility of flocs which could be repeptized shortly after their formation. In fig. 1 it is shown that these assumptions have a considerable effect on the shape of the potential energy diagrams. The pronounced maxima which govern the DLVO theory have disappeared, since the introduction of a (small) distance of closest approach has obscured the sharp increase of the attraction for H --t 0. Another feature becomes manifest when the “ critical ” parameter 22 is introduced.20 2, = I I I I I I 5 10 15 20 FIG. 1.-Potential energy diagrams.I: 2, = 3, c = 15 mmol dm”, 1-1 electrolyte. 11: Zto = 2; c = 60 mmol drnw3, 1-1 electrolyte. 111: Zoo = 1, c = 150 mmol dmj, 1-1 electrolyte. a = 500 A; A = 5 x 10-l~ erg; 6 = 2 A. zeVa/kTfor H = co . It is a measure of the diffuse layer potential 96 of a single particle, which can be identified tentatively with the electro-kinetic [-potential. Zm is the value of 2, for which VR + V’ = 0 at H = 26, i.c., in the primary minimum. A condition for coagulation in the primary minimum is that Zm < Zm; coagula will repeptize if 2, > 2& . From these studies the picture emerges of a primary minimum, which may be deep (compared with kT), shallow or nonexistent depending on the experimental conditions. The probability of two particles remaining aggregated for some length of time after a collision depends on the depth of the primary minimum.By Fuchs’ theory the rate of repeptization is related to the potential energy diagram. The depthG . FRENS 149 of the primary minimum itself is related to the electrolyte concentration, and c, is the concentration where the probability of repeptization of aggregates becomes negligible. The maximum of Vx in the DLVO theory and the depth of the minimum in the theories of secondary coagulation are rather insensitive to double layer potentials. They follow from variations in the range of the repulsion relative to that of the van der Waals attraction. The depth of the primary minimum depends on 96 and especially on the changes in 2, relative to Z. Thus the coagulation rate is related to the ionic strength via the effects of the electrolytes on the structure and the capacity of the electrical double layer23 rather than via its mere thickness.POTENTIAL ENERGY DIAGRAMS Fig. 2 shows the repulsive energies V g and Vg for constant ts and pa, respectively, for spherical double layers at a number of values 2, (G = z2/aVR). At H = 26 it is found that G is approximately proportional to Z:, more so for constant 9 6 than for constant 0. This implies that the potential energy at H = 26 is proportional both to a and to 9;. For a given 2, it is independent of K. Since the attraction energy VA (=aA/12Hfor small H/a) is also fixed for a given A and 6 this means that changes in the depth of the primary minimum, resulting in changes in coagulation rate, can only be brought about by changes in 96.The potential 96 depends on the kind and the concentration of the electrolyte. The dependence of pa on c is discussed in the Stern theory of the double layer24 and in more sophisticated treatments based upon it.23 We shall, in this paper, use a simple formula from this theory, originated by Fr~mkin,’~ and write for p6 3 0, This approximate formula is rather effective in the description of experimental results. It can account not only for electrokinetic data, but also for electrode kinetics and the effects on it of varied concentrations of inert e l e c t r ~ l y t e . ~ ~ * ~ ~ We have introduced Z, which implies that there exists a value of q 6 and, therefore, some electrolyte concentration c,, for which V, = - VA at H = 26.For c > c, the potential is lowered, and the depth of the primary minimum is the difference in the values of Vg at c and at c,. In a discussion of one colloid we may define c, as the unit of concentration. Then it is seen that the constant in eqn (2) becomes the value 9: at c = c,, for this colloid. The depth of the primary minimum is then found to be assuming G = KZ: . It would be interesting if a concentration c, and a corresponding potential pz could be determined as a threshold for the coagulation of an electrocratic sol. Below this concentration there would be no coagulation in a primary minimum, although there might remain secondary coagulation and Ostwald ripening to coarsen the suspension. These threshold values would directly be related to the strength (A) of the van der Waals attraction.Aggregates age ; their double layer potentials revert to the (Nernst) equilibrium value during the period after their formation. The repulsion energy diminishes accordingly, from Vg to Vg at the same Z,, which is a decrease of the order of 10 kT in a typical colloid. If two particles remain trapped for some time in (even aI50 50 40 30 b 0 X v) Q, c > Q li 20 10 COAGULATION IN THE PRIMARY MINIMUM FIG. 2.-G = z2/aVR as a function of K(H - 26) for different values of Z m . Full line, constant Q. Broken line, constant q6.G. FRENS 151 shallow) primary minimum, they will sink irreversibly into a rather deep potential well. Such a mechanism is not effective in, e.g., secondary coagulation. There the double layers are virtually unaffected by the interaction and screen the surface charge.But aggregates in a primary minimum must eventually be rendered irreversible by the relaxation to equilibrium of the strongly interacting double layer systems. COAGULATION KINETICS Once it is established that there exists a probability of repeptization for newly formed aggregates, the treatment of coagulation kinetics becomes analogous with that for secondary coagulation. In a suspension of Brownian particles the number of collisions with one particle per unit time is: J," = 8nD&, (4) where D is the diffusion coefficient of the particles, no their concentration and R w 2a is the distance x at which an encounter is counted as a collision. Without interaction these collisions remain without consequence and doublet dissociation (J:) occurs at the same rate as doublet formation. Interaction forces hinder the collisions and also the breaking up of doublets. According to eqn (1) the analogous expression for the repeptization of aggregates is where Vl is the potential energy at a point x measured relative to Vmin, the depth of the primary minimum.V1 = V' - Vmln, and the integration is in both cases over the whole " reaction coordinate " which contains all interparticle distances. The net number of collisions which result in aggregation at the beginning of the coagulation is a fraction of J:, given by The two Fuchs' integrals in this expression are equal, and approximately equal to W = 1 if the potential energy diagram has a primary minimum and no pronounced positive maximum.Thus 1 J, = J: w [l - exp (Vmin/kT)] and the effective retardation factor becomes analogous with expressions for secondary coagulation 1 2 9 1 3 which have been derived making the, too simple, assumption that there is a Boltzmann distribution of aggregate energies, and that aggregates with energies above -Vmin will break up. More rigorous treatments of coagulation kinetics with reversible aggregates follow the Fuchs a p p r o a ~ h . ~ * ~ ~ * ~ ~ Eqn (9), in combination with an expression for Vmin, likeI52 COAGULATION IN THE PRIMARY MINIMUM eqn (3), should permit construction of (log W, log c) diagrams. Expanding the exponential it is easily seen that log W = [some constant] - log (ln clc,). The constant contains a, which makes the value of the critical coagulation concentration (corresponding to Vmi, = kT) dependent on the particle size, though dlog W/dlog c is independent of a.The descending branch of the (log W, log c) curve is not really straight, but it resembles a straight line in the practical range of slow coagulation. Near the intersection at c, (Weff = 1) the curve becomes rounded, [cf., ref. (12)]. However, reversible aggregation can not be the whole story in slow coagulation kinetics. So far we have used a model which predicts limited coagulation and the establishment of a coagulation-repeptization equilibrium. Limited coagulation of electrocratic colloids has only been observed 16*18~19 at extremely low initial rates of aggregation. The more common processes (10 > Weff > 1) show retarded Smolu- chowski k i n e t i ~ s .~ * ~ ~ ~ ~ ~ This implies that near the critical coagulation concentration there may be a constant fraction of the newly formed aggregates which break up, but that the remaining older flocs stick irreversibly together. It has been suggestedz7 that there is a progressive irreversibility as the aggregates increase in size because primary particles would almost never succeed in breaking more than one contact at a time. This may be an explanation as to why limited coagulation processes do continue after some time and has been given as such; it cannot account for retarded Smoluchowski kinetics during the initial stages of coagulation. The decrease in the potential energy of an aggregate from Vg to Vg makes it irreversible during a period after its formation. Two particles will cling together, even if there was only a shallow primary minimum during the collision, unless their aggregate is broken up by their independent Brownian motion before it is too late.Thus the fraction of the doublets which will eventually break up becomes dependent on the relative rates of two processes. The rate of spontaneous repeptization depends on the momentary value of Vmin and continues to decrease, until Vmin has reached the value at which repeptization stops. The changes in Vmin are caused by the adjustment of the double layer potential to the Nernst equilibrium value; the rate of this adjustment is determined by the exchange current density at the interface. The rates of repeptization and of ageing are, therefore, independent of the number and the size of aggregates.Their ratio remains constant during the coagulation process ; this results, again, in retarded Smoluchowski kinetics. A constant fraction of the fresh aggregates can dissolve, but the remaining, older, aggregates no longer participate in spontaneous repep tization. Double layer relaxation will be most effective in making aggregates irreversible when they stay intact for a comparatively long time. Therefore, it is indeed expected that limited coagulation is observed in very slow coagulations, whereas retarded Smoluchowski rates occur with rather small values of Weff and rather deep primary minima. This effect makes the descending branch of experimental (log W, log c) curves steeper than indicated by eqn (9); the change in slope near c, also becomes more abrupt.The rate of adaptation for double layers on different materials may differ consider- ably. This must affect the shapes of (log W, log c) curves and also the values of c,. For one material, a shallow primary minimum will give sufficient stability to most aggregates to make them irreversible before they can break. In another sol the particles would need to be kept together for a long time, in a primary minimum of several kT, to make them stick. In this model the critical coagulation concentration is not only dependent on A and p6, but also on the electrode kinetics at the surface of the particle.G. FRENS 153 CRITICAL COAGULATION CONCENTRATIONS For a given colloid the depth of the minimum at c, should not depend on the kind of electrolyte present. This, rather than the condition of a vanishing maximum becomes the criterion for c, and the basis for a discussion of the Schulze-Hardy rule in terms of the Gouy-Stern double l a ~ e r .~ ? ~ ~ In fig. 2 it is seen that if 9 6 = 25 mV at c, for monovalent counterions (i.e., 2, = 1) it must become 35 mV with divalent and 40 mV with trivalent ions to give the same value of Vg, i.e., of Vmjn. In fig. 3 it is shown how the Stern model permits an estimate of the electrolyte concentrations at which these conditions will obtain for one equilibrium value of the total double layer potential po. A Schulze-Hardy type variation of c, with z is obtained, not very dependent on the potential q0 and at low potentials 9 6 of the diffuse layer.The model can, of course, be refined considerably by including for example, ion size, specific adsorption and discreteness of charge in the double layer model. But it is to be expected that a more complete theory will have many of the same features, since experimental, e.g., electrokinetic, diffuse layer potentials have a "Schulze-Hardy-like behaviour " . 2 9 9 3 0 Eqn (3) was based on the even cruder approximation that G is proportional to 2%. This would give the simple formula (l/z)ln c/co = constant for the Schulze- Hardy rule, which is an expression for a constant " critical potential " in a given sol and for the ancient " logarithmic " version31 of the Schulze-Hardy rule. Something > tm € 9- - 70 i 60 50 40 30 20 10 FIG. 3.-The Schulze-Hardy rule.Stern model without specific adsorption3 and different values of p0 (full line qo = 200 mV, broken line qo = 300 mV).154 COAGULATION I N THE PRIMARY MINIMUM of this kind had to be expected: a theory which involves Vmi, is a theory of qs at c,, and has, therefore, a link with the old adsorption theories' of colloid stability. However, the approximation in eqn (3) is not very good; the more complete theory using numerical computation of Vg values gives better results. It predicts that the diffuse layer potential at c, should increase with z. Vmi, is constant at c, for a given sol, and Vmin is also proportional to the radius a of particles in an aggregate. This explains the observation of fractionated coagula- tion. Experiments on fractionated coagulation have been carried out as follows : sets of monodisperse colloids were prepared, all of one material and of the same (electrolyte) composition, but with different particle sizes.In these sets c, can be determined as a function of a and it has been found that c, decreases when a increases. Extrapolation of c, to a = co gives a value for the coagulation threshold c, since for a large a even the shallowest primary minimum in the potential energy diagram is enough to make an aggregate irreversible. We have repeated such an experiment and found that in sets of citrate containing Au-sols, which presumably had equal H = 26 values, c, was about 15 mmol dme3 of a 1-1 electrolyte. In experiments of this kind one might identify a threshold concentration c, for different colloids.Comparable experiment^,^^^^^^ and also those in which larger particles are removed from a polydisperse colloid by fractionated coagulation, indicate that c, decreases as a increases. This decrease implies that the feature of the potential energy diagram which is proportional to a and which constitutes the criterion that c, has been reached was a minimum rather than a vanishing maximum in all these experiments. In some experiments with large particles, secondary coagula- tion cannot be ruled out completely as an explanation. But this seems unlikely in the experiments with colloidal gold where a was of the order of 100 A. CONCLUSION In this paper we have discussed in a qualitative way what the implications would be if it is assumed that the rate of slow coagulation of electrocratic colloids is deter- mined by competition between aggregation and spontaneous repeptization from the primary minimum.Such a hypothesis, in combination with the idea that the relaxa- tion of the interacting double layers causes progressive irreversibility in aggregates, seems to give a reasonable account of (log W, log c) diagrams, the Schulze-Hardy rule, limited coagulation, (retarded) Smoluchowski kinetics and fractionated coagulation, under the experimental conditions where these phenomena are observed. In such a theory the potential 96 across the diffuse part of the double layer becomes, once more, the key factor in colloid stability. It would predict that a critical value 9: can be found which constitutes the threshold value for coagulation. At the critical coagula- tion concentration (c,), q~ should increase with the valency z of the counterions.These predictions can be verified experimentally, especially since it seems possible to do experiments with stable colloids with fine particles under conditions where coarser sols of the same material coagulate. H. Freundlich, Kapillarchenzie (1909). B. V. Derjaguin and L. Landau, Actn Physicochini, 1941, 14, 63. E. J. W. Verwey and J. Th. G. Overbeek, Theory of the Stability of Lyophobic CoIIoids (Elsevier, Amsterdam, 1948). N. Fuchs, 2. Phys., 1934,89,736. M. von Smoluchowski, Phys. Z., 1916, 17, 557. H. Reerink and J. Th. G. Overbeek, Disc. Faraday Soc., 1954, 18, 74. ' R. H. Ottewill and J. N. Shaw, Disc. Faraday Soc., 1966, 42, 154.G . FRENS 155 S . R. Logan, Trans, Faraday Soc., 1967,63, 1712. R. H. Ottewill and T. Walker, J.C.S. Faraday Z, 1974, 70, 917. lo Th. Gillespie, J. Colloid Sci., 1960, 15, 313. l1 G. R. Wiese and T. W. Healy, Trans. Faraday SOC., 1970,66,490. l2 R. Hogg and K. C. Yang, J. Colloid Interface Sci., 1976, 56, 573. l3 P. Bagchi, Adu. Chem. Series, 1975, 145. l4 G. A. Martynov and V. M. Muller, Doklady Akad. Nauk S.S.S. R., 1972,207, 370, 1 161. l5 E. Ruckenstein and D. C. Prieve, AZCHE J., 1976, 22, 276. l6 A. Westgren, Ark. Kern. Mineral. Geol., 1918,7, 6. l7 H. R. Kruyt and A. E. van Arkel, Kolloid Z., 1923, 32, 39. l8 B. V. Derjaguin and N. M. Kudryavtseva, Research on Surface Forces (Moscow, 1961), p. 183. l9 B. V. Enustun and J. Turkevich, J . Amer. Chem. Soc., 1963, 85, 3317. 2o G. Frens, Thesis (Utrecht, 1968). 21 G. Frens and J. Th. G. Overbeek, J. Colloid Interface Sci., 1971,36, 286. 22 G. Frens and J. Th. G. Overbeek, J. Colloid Interface Sci., 1972,31, 376. 23 M. J. Sparnaay, The Electrical Double Layer (Pergamon, Oxford, 1972). 24 0. Stern, 2. Elektrochem., 1924,30, 508. 25 A. Frumkin, 2. phys. Chern., l933,164Ay 121. 26 P. Delahay, Double Layer and Electrode Kinetics (Interscience, Wiley, New York, 1965). 27 B. V. Derjaguin, Pure Appl. Chem., 1976,48, 387. 28 S. Levine and G. M. Bell, J. Colloid Sci., 1962, 17, 838. 29 S. A. Troelstra, Thesis (Utrecht, 1941). 30 J. Th. G. Overbeek, Colloid and Surface Chemistry ZZZ (M.I.T., Cambridge, Mass., 1973). 31 B. Tezak, 2. phys. Chem. A , 1942,191,270. 32 P. Tuorila, Kolloidchem. Beih., 1926,22, 279. 33 G. Frens, Kolloid Z., 1972, 250, 736. 34 A. M. Joseph Petit, F. Dumont and A. Watillon, J. Colloid Interface Sci., 1973, 43, 649.

ISSN:0301-7249    

DOI:10.1039/DC9786500146

出版商:RSC    

年代:1978

数据来源: RSC

摘要:

Coagulation of Amphoteric Latex Colloids : Reversibility and Specific Ion Effects BY THOMAS w. HEALY, ANDREW HOMOLA AND ROBERT 0. JAMES Colloid and Surface Chemistry Group, Department of Physical Chemistry, University of Melbourne, Parkville, Victoria 3052, Australia AND ROBERT J. HUNTER Department of Physical Chemistry, University of Sydney, Sydney, N.S.W., Australia Received 5th December, 1977 Coagulation studies of amphoteric latex sols of various i.e.p. values in LiN03, KNO3 and CsNOl solutions are reported for a wide range of salt concentrations and pH. Three different techniques for the study of coagulation phenomena all indicate reversibility, in that sols coagulated by pH or salt are able to be redispersed. K+ and Li+ counter ions are able to stabilize these sols in the high salt concentration region.With NO; and Cs+, the expected narrow zone of coagulation at low salt (i.e.p. coagulation) expands into the usual broad coagulation zone at high salt, where coagulation is observed at all pH values. The stabilizing effect of Li+ and K+ counter ions is attributed to a hydration barrier at the interface. The coagulation of lyophobic colloids is, in many respects, reasonably well under- stood in terms of the DLVO theory.lP2 However, experimentalists continue to probe two important features of coagulation which are more difficult to understand in terms of the classical theory; viz., (i) reversible coagulation, or the repeptization of previously coagulated sols, and (ii) specific ion effects (Hoffmeister Series) in coagulation.The present study is aimed at obtaining experimental results on both these areas of coagulation with a view to subsequent analysis in terms of classical DLVO theory and more recent " interaction under regulation " description^.^^^ The sols used are the amphoteric latex colloids of Homola and J a m e ~ , ~ which have all the usual advantages of conventional soap-free latex colloids but with the added advantage that they have a particular iso-electric point (i.e.p.). Again, since the i.e.p. is due to both -COO- and -NH$ groups in the latex surface, controlled variation in the carboxyl/amine ratio at polymerization leads to sols with different i.e.p. values, that are identical in every other respect. The amphoteric latex colloids have higher * Currently at CSIRO Division of Textile Physics, Sydney, Australia.T .W. HEALY, A . HOMOLA, R . 0. JAMES AND R . J . HUNTER 157 maximum surface charges than conventional carboxylate or sulphonate latex sols and, in this respect, they model more closely biocolloids and the important oxide colloids on which many ion sequence studies have been performed [see, e.g., ref. (6)]. FinaIly, since H+/OH- are potential determining ions (p.d.i.), it is possible to design experi- ments where surface potential (yo), surface charge (oo) and ionic strength (Debye K) are independently controlled, i.e., fix pH, vary ionic strength (i.e.? K and oo vary at ly, constant) fix ionic strength, vary pH (i.e., lyo and a, vary at fixed K ) . This flexibility allows consideration of " iso-electric point coagulation and " double layer compression " coag~lation.'-~ In a further publication lo we shall consider the theoretical analysis of the present data in terms of coagulation at constant a,, constant ly, or under regulation.EXPERIMENTAL The preparation and characterization of the amphoteric latex sols is described else- w h e ~ e . ~ ~ ~ Electron microscopy indicated a particle diameter of 1850 f 10 A. In the present study, sols with various i.e.p. values have been used with most of the work on sols of i.e.p. values at pH 5.2, 6.6 and 7.2. These i.e.p. values are those determined by micro- electrophoresis at low salt concentrations where there is a unique pH of zero electrophoretic mobility independent of 1 : 1 electrolyte concentration. As presented in detail el~ewhere,~*~ these i.e.p.values determined by electrophoresis and coagulation are identical within experi- mental error and coincide with the potentiometric point-of-zero-charge (P.z.c.) values ob- tained at low salt concentration.ll All microelectrophoresis results were obtained with the standard Rank Microelectro- phoresis apparatus. Triply-distilled water and recrystallized AR grade salt solutions were used throughout the study. All experiments were conducted at 25 f 0.5 "C. Coagulation effects were studied by three separate techniques, identified as follows : Method A . An optical density-time technique described in detail elsewhere.' This is similar in principle to the adder-mixer technique of O t t e ~ i l l , ' ~ * ~ ~ in which one measures, using a recording spectrophotometer, the time rate of change of optical density at fixed wavelength .MethodB. A 24 h residual turbidity technique similar to that used in classical coagulation studies. 2.5 cm of 0.02% latex and 2.5 cm3 of a salt solution were mixed in 10 cm3 capacity tubes and allowed to stand for 24 h. The latex suspension and the salt solution were adjusted to the required pH prior to mixing and care was taken to exclude COz for the period of the experiment. Visual observation against a black background, or arbitrary optical density measurements, permitted the identification of the presence or absence of coagulation. Method C. A coagulation-titration technique. The apparatus consists of a standard thermostatted titration cell with a pH electrode, automatic burette and nitrogen input-output ports. Light from a 2.5 mW red laser passes through the suspension in the cell and a photo- diode detector inserted into the suspension is located at right angles to the beam in order to detect light at 90" to the incident beam.By careful preparation of the glass tube, into the end of which is sealed the photodiode, it is possible to locate the detector surface within 1-2 mm of the laser light beam. The method is continuous in the sense that a direct output of scattered light intensity as a function of pH can be obtained on an X-Yrecorder during the course of a conventional potentiometric titration of the latex suspension. The procedure is to begin a titration at some 2 - 2+ pH units below or above the iso-electric point of the latex.Acid or base is then titrated in via a micro-burette over a period of approximately 20 - 30 min to a corresponding pH value 2 - 23 pH units below or above the i.e.p. The typical output is shown in fig. 1 ; for low salt concentrations the i.e.p. is identified and confirmed by electrophoresis, as the intersection of the up-scale and down-scale titration runs. The intersection pH is quite clearly the i.e.p. for the lower concentrations of KNOB.158 COAGULATION OF AhlPHOTERIC LATEX COLLOIDS Indeed, it is more useful to define a traditional critical coagulation concentration (c.c.c.) and a critical stabilization concentration (c.s.c.) value [see fig. l(c)] for H+ as the coagulating ion. In the low salt cases, the intersection pH is half-way between the C.C.C.and C.S.C. pH values and is a reliable estimate of the pH of the i.e.p. At higher salt concentrations where K+, H+ and NO;, OH- are competitive and/or complementary coagulating ions, the identifica- FIG. 1 .-Coagulation-titration curves for an amphoteric latex dispersion in mol dm-3 KN03 (i.e.p. by microelectrophoresis pH 6.6). The titration was started at pH 5.15 (curve a), sonicated for 1-2 s at pH 9.90 and then back-titrated (curve ,!I) to pH 4.15. In curve c the smooth lines drawn through the chart lines are superimposed to give a point of intersection at pH 6.6 and C.C.C. and C.S.C. values as shown. tion of the intersection pH with the i.e.p. is not possible; the sol is dispersed at the start of the down-scale titration and a C.S.C. value is obtained unequivocably.In contrast, the sol is coagulated at the start of the up-scale titration for, say, KN03 greater than 0.3 mol dm-3 and is slow to redisperse during the titration up-scale. For CsN03 as the coagulating ion, intersection pH values cannot be determined for high salt concentrations since the sol is coagulated at all pH values. RESULTS The coagulation behaviour of an amphoteric latex colloid as a function of pH and KN03 concentration is shown in fig. 2 (optical density-time, Method A), fig. 3 (24 h turbidity, Method B) and table 1 (coagulation-titration, Method C). The general pattern of coagulation is reproduced but with understandably minor differences between the three sets of measurements; the methods measure three different but related aspects of extent and/or rate of coagulation.The general bending of the coagulation domain or rapid coagulation regime to lower pH values is perhaps indicative of specific adsorption of the anion. While this may occur, the major feature of all the KN03 coagulation data is that stability is observed on the high pH side of the coagulation domain at very high salt concentrations. The coagulation zone one would expect is one in which the narrow coagulation domain at low salt (i.e., iso-electric point coagulation) would broaden on either side of the i.e.p., such that coagulation would be observed at all pH values at high salt. If specific adsorption occurs, it may promote stability to somewhat higher salt concen- trations on the low pH side compared with the high pH side. Such is not observed with KN03 as the 1 : 1 electrolyte; inordinate stability appears in the high pH, nega- tive, cation counter ion region and is confirmed by all three methods.The effect of varying the type of cation (counter ion) on this high pH, high saltT . w. HEALY, A . HOMOLA, K. 0. JAMES AND K. J . HUNTER 159 PH FIG. 2.-Coagulation rate studies using the optical density-time method (Method A) for an ampho- teric latex of i.e.p. pH 6.6 (by electrophoresis) in KNOJ solutions. 9, W = 1 corresponds to rapid coagulation. FIG. 3.-Coagulation domain diagram for an amphoteric latex of i.e.p. pH 7.2 (by microelectrophore- sis) in KNOJ solutions. Coagulation (a) or dispersion (0) is assessed after 24 h (Method B).160 COAGULATION OF AMPHOTERlC LATEX COLLOIDS region is shown in fig.4 and table 2. With Cs+ as the counter ion, the expected symmetrical shape of the coagulation domain is observed; thus at low salt, coagula- tion is observed only at pH values close to the i.e.p., whilst at high salt, coagulation is observed at all pH values. In contrast, moving from K+ to Li+ causes the domain TABLE 1 .-ISO-ELECTRIC POINT AND COAGULATION-STABILIZATION pH VALUES DETERMINED BY THE COAGULATION-TITRATION TECHNIQUE (METHOD c). THE pH VALUES LISTED ARE DEFINED IN FIG. l(C). THE ELECTROLYTE IS KNo3 AND THE AMPHOTERIC LATEX HAS AN I.E.P. BY MICRO- ELECTROPHORESIS OF pH 6.5 - 6.7. KN03 conc. /mol drn” C.S.C. pH intersection pH C.C.C. pH 10-4 10-3 10-1 3 x 10-l 6 x 10-1 9 x 10-1 6.42 6.40 6.35 5.6 5.3 - - 6.6 6.6 6.6 6.1 5.8 5.4 * 5.0 * 6.79 6.80 6.90 6.4 6.1 5.8 5.6 * Estimated value only because of the lack of stability at start of up-scale titration.to skew even further to low pH with again, stability at high pH and high salt. With K+, and to a greater extent with Li+, these sols are stable at pH values one or two units and more above the i.e.p., even at 1 mol dm-3 salt. Coagulation-titration runs with LiN03, KNO, and CsNO, were conducted on latices with i.e.p. values (by electrophoresis) of 5.2, 6.6 and 7.2. Normalization of FIG. 4.-Coagulation domain diagram for an amphoteric latex of i.e.p. pH 7.2 (by microelectro- phoresis) in CsN03 solutions. Coagulation (a) or dispersion (0) is assessed after 24 h (Method B).T . w. HEALY, A . HOMOLA, R. 0. JAMES AND R. J . HUNTER 161 TABLE 2.-COAGULATION-TITRATION DATA FOR AN AMPHOTERIC LATEX (I.E.P.BY MICRO- DEFINED IN FIG. 1(C). ELECTROPHORESIS OF pH 7.2). THE INTERSECTION AND C.C.C. AND C.S.C. pH VALUES ARE conc. intersection electrolyte /mol dm-3 C.S.C. pH PH C.C.C. pH LiN03 10-3 KN03 10-3 10-1 1 1 o-2 5 x 1 0-1 1 ON03 10-3 10-1 1 6.8 - - 6.8 6.7 6.0 6.1 6.62 6.5 6.4 - - 7.1 6.7 * 4.8 * 7.2 7.15 7.1 6.7 5.2 * 7.2 7.3 7.2 --t 7.4 6.9 5.0 7.5 7.6 7.7 7.15 5.3 7.62 7.85 7.80 I * Estimated values only because of instability at start of up-scale titration. 7 Coagulated at all pH values. the data for each salt to the i.e.p. (Le., ApH) resulted in superimposition of all C.C.C. or C.S.C. or intersection values. DISCUSSION Of the several interesting aspects of the present results, it is appropriate to focus attention on two of the more striking features, viz., the ease with which coagulated ,-ls are redispersed by changes in pH (yo) and/or ionic strength ( K ) and secondly, the uLexpected stability of the sols at high ionic strength with K+ and Li+ as counter ions, but not with Cs+.The specific cation effects are difficult to understand in quantitative terms. While it is possible that they reflect specific effects of each ion on the interfacial water struc- ture, it is not clear how changes in colloid stability are thereby induced. What is required is an understanding of the added stability to electrolyte coagulation with Li+ and, to a lesser extent, K+ counter ions, and the almost theoretical or ideal behaviour with Cs+ as the counter ion. In terms of water structure effects, the presence of Lif as a counter ion may mean enhancement of the interfacial structure such that a steric hydration barrier prevents close particle-particle aggregation into a primary well.A simpler, more plausible interpretation of the phenomenon, is that the hydrated cation counter ions would need to be partially or completely dehydrated to allow close particle-particle aggrega- tion; thus the van der Waals force is insufficient to allow dehydration of Li+ and K+ but sufficient to allow dehydration of Cs+ . Again, it also follows that nitrate counter ions do not produce a large hydration barrier to coagulation. While steric hydration barriers to coagulation or simply '' hydration stabilization " are an intuitively satisfy- ing concept, it is far from understood and must now be modelled theoretically.In their consideration of the reversibility to coagulation of AgI sols to both iso- electric and IC coagulation, Frens and Overbeek14 conclude that the re-establishment of the e.d.1. is able to provide the necessary electrostatic repulsion to restabilize AgI and oxide sols. The present high charge amphoteric sols are clearly of this same162 COAGULATION OF AMPHOTERIC LATEX COLLOIDS kind. In contrast, it is commonly observed that conventional low-charged sulphonate or carboxylate sols cannot be redispersed. It is important to stress that there must be some mechanism that prevents the development of an extremely deep coagulation well at short distance. This may be provided by a hydrated immobile surface layer, a charge adjustment or a regulation mechani~m.~*~ The 1-5 pC cm-2 maximum charge of low charge conventional latex sols must be insufficient to produce restabil- ization.Qualitative experiments suggest that maximum surface charges in the range 10 - 15 pC cnr2 are necessary before restabilization is observed in the poly- styrene + water + polystyrene system with cations such as Li+ and K+ as counter ions. It is possible to examine certain aspects of the present data in terms of classical DLVO theory and to obtain a quantitative understanding of some of the observed phenomena. For example, consider (i) the width of the pH range of coagulation for low salt, and (ii) the C.C.C. value, due to NO- counter ions at high salt and pH values well below the i.e.p. In both cases, cal~ulation~~ shows that coagulation is observed when the primary maximum gets below about 15 - 10 kT and the yo values correspon- ing to coagulation calculated from the low salt region are &20 mV.The C.C.C. value (at pH < pHiae..,) predicted from V,,, = 0 simplified DLVO theory16 is 0.3 - 0.4 mol dmF3 compared with approximately 0.4 - 0.5 mol dm-3 observed experimentally. Furthermore, the general pattern of repeptization with change in pH or salt cannot in general be attributed to coagulation into a secondary minimum. Specifically, secondary minimum coagulation would, following Wiese and Healy,17 be apparent at IApHI > 2 (i.e., 2 pH units above or below the i.e.p.) and at 1 : 1 electrolyte con- centrations of 2 to 4 x low2 mol dm-3. The present latices are all stable in this region.That repeptization is observed at high and low salt over a wide span of ApH values further precludes secondary minima effects as a general explanation. We conclude, as does Overbeek,18 that the present and earlier l9 observations of repeptiza- tion are real and cannot be seen as minor exceptions. Frens and Overbeek have s ~ g g e s t e d l ~ * ~ ~ that a solvation barrier is present at the particle surface to prevent close penetration into a primary well. This concept is useful in the present case, provided that the solvation is present to an equal extent for NO- and Cs+ counter ion systems, but present to such an extent for K+ and Lif counter ion systems that it renders these later systems stable. The authors acknowledge support from the Australian Research Grants Commit- tee and from the University of Melbourne for provision of an Emergency Research Grant.B. V. Derjaguin and L. D. Landau, Actu Physicochem. U.S.S.R., 1941,14,633. E. J. W. Verwey and J. Th. G. Overbeek, Theory of the Stability of Lyophobic Colloids (Elsevier, New York, 1948). D. Chan, J. W. Perram, L. R. White and T. W. Healy, J.C.S. Furaday I, 1975, 71, 1046. D. Chan, T. W. Healy and L. R. White, J.C.S. Furuduy I, 1976,72, 2844. A. Homola and R. 0. James, J. Colloid Interface Sci., 1977,59, 123. ti F. Dumont and A. Watillon, Disc. Furuduy SOC., 1971,52, 352. ' R. 0. James, A. Homola and T. W. Healy, J.C.S. Furuday I, 1977,73, 1436. E. MatijeviC, in Twenty Years of Colloid and Surface Chemistry-The Kendull Award Address, ed. K. J . Mysels (A.C.S. Washington, D.C., 1973), p. 283. R. 0. James, G. R. Wiese and T. W. Healy, J. Colloid Interface Sci., 1977, 59, 381. lo L. R. White and T. W. Healy, to be published. l1 J. Lyklema, Disc. Furaday SOC., 1971, 52, 317. l3 R. H. Ottewill and J. N. Shaw, Disc. Furaday SOC., 1966,42, 154. l4 G. Frens and J. Th. G . Overbeek, J. Colloid Interface Sci., 1971 , 36, 286. R. H. Ottewill, J. Colloid Interfuce Sci., 1977, 58, 357.T . tv. HEALY, A . HOMOLA, R . O. JAMES AND R. J . HUNTER 163 G. R. Wiese, P1i.D. Thesis (University of Melbourne, Australia, 1973). l6 H. Reerink and J. Th. G. Overbeek, Disc. Faraday Sac., 1954, 18, 74. l7 G. R. Wiese and T. W. Healy, Trans. Faraday SOC., 1970, 66, 490. l9 R. Benitez and F. MacRitchie, J. Colloid Interface Sci., 1972, 40, 310. J. Th. G. Overbeek, J. Colloid Interface Sci., 1977, 58,408.

ISSN:0301-7249    

DOI:10.1039/DC9786500156

出版商:RSC    

年代:1978

数据来源: RSC

摘要:

On Enzymatic Clotting Processes. Part 3.-Flocculation Rate Constants of Paracasein and Fibrin BY T. A. J. PAYENS Netherlands Institute for Dairy Research, Y.Q. Box 20, Ede, The Netherlands Received 24th November, 1977 FIocculation rate constants of micellar paracasein and fibrin have been determined from the lag phase in the enzymatic clotting. Without exception these rate constants turn out to be very small. Different arguments are put forward to demonstrate that the retardation in the clotting of paracasein is chiefly of a steric nature. The activation energy of the clotting process becomes negative beyond 310 K, which is ascribed to the growing influence of the London-van der Waals attraction as the micelles become discharged by the enzyme. The rate constants of fibrin measured from the lag phase have been compared with those estimated from early light scattering results of Steiner and Laki.In both cases rate constants are found to decrease with increasing ionic strength. This is consistent with the proposed mechanism of fibrin polymerization (Ferry; Doolittle), in which rodlike particles form staggered overlaps which are held together by electrostatic forces. In the preceding papers of this series,lB2 the kinetics of enzyme-triggered coagula- tion reactions, such as the clotting of blood3 and milk4 have been analysed. The rate of change in the concentrations of particles of different degrees of aggregation was calculated by accounting for the enzymatic production of the " monomer '' in von Smoluchowski's theory for the rate of coagulation of unstable colloid^.^ Once the particle concentrations are known, it is relatively simple to calculate the rate of change of the weight-average particle-weight, Mw, of the clotting enzymatic product and that of the whole solute. The result is:2 avlMl = 1 - MO(1 -.f>(~V/ks)+'(f(t/4 - (1 -f)(tiz)3/3>/C,. (1) In this equation M, is the molecular weight of the substrate,fthe ratio of the molecular weight of the peptide split off by the enzyme and M,, V the rate of production of the clotting species, ks its flocculation rate constant, t the reaction time and C, the solute concentration in g ~ m - ~ .The parameter z appearing in eqn (1) is the so-called enzym- atic clotting time, defined as z = (ksV/2)-*. (2) It is evident from eqn (1) that a dramatic change in the average particle weight is only to be expected for reaction times t >, z.To a fair approximation we thus have for the clotting time, t,, of blood or milk: t, = O(z), ( 3 ) its actual value depending on the magnitude offand the technique chosen to monitor the clotting. For reaction times smaller than t, the average particle weight may even pass through a shallow minimum as a result of the dominant effect of proteolysis during the lag phase. This behaviour is exemplified in fig. 1.T. A . J . PAYENS 165 120 90 *" az \ 60 30 0 0.30 0.60 0.90 1.20 t1.r FIG. 1 .-Model calculation of the time-dependence of the weight-average molecular weight of micellar paracasein during clotting. Model parameters [cf. eqn (l)]: M,, = 5 x lo8; f = 0.04; C , = 0.03 g ~ m - ~ ; k, = lo5 cm3 mol-l s-l; V = 10-l2 mol ~ r n - ~ s-l.Note the shallow minimum in the plot before the clotting gets underway. It is clear from eqn (1) and (2) that the clotting time of blood (i.e., of fibrin) and of milk (i.e., of paracasein) is determined by the flocculation rate as well as by the enzymatic velocity. The present paper deals in particular with the determination of the flocculation rate constants of paracasein micelles and fibrin. As such are defined the products of the limited proteolysis of casein and fibrinogen by chymosin (E.C. 3.4.23.4) and thrombin (E.C. 3.4.1.21.5) respectively. The structure of the (para-)casein micelle is fairly well understood nowadays.6 Its average radius is m50 nm and it consists of several hundreds or even thousands of casein subunits, 12% of which is ~c-casein.Normally Ic-casein protects the micelle against flocculation, but its limited proteolysis by the enzyme triggers the clotting. It is thus clear that the average micelle contains a large number of potential sites of attack for the enzyme. From hydrodynamic and electron microscopic studies, fibrinogen emerges as a rodlike particle with an axial ratio of x 18 and m 50 nm long. The molecular weight is 340 000 20 000. Chemically it consists of three different peptide-chains which are held together by disulphide bridges and which are doubled by gene-duplication. From these a maximum of four minor peptides is split off by thrombin to set the clotting going.166 FLOCCULATION RATE CONSTANTS OF PARACASEIN AND FIBRIN - - - - - - - - - - - EXPERIMENTAL AND RESULTS Suspensions of casein micelles were prepared from low-heat skim milk powder, diluted with 0.01 mol dm-3 CaC12 to prevent the disintegration of the mi~elles.~ Clotting was brought about by appropriately diluted rennets (CSKF, Leeuwarden, Netherlands).The clotting of fibrin was studied with fibrinogen Kabi (AB Kabi, Stockholm, Sweden). The clotting was monitored in 0.5 or 1 cin cuvettes in the Cary 14 spectrophoto- meter at 500 nm. Rapid mixing (mixing times ~2 s) of substrate and enzyme was achieved as previously described.2 Clotting times were estimated either by extrapolation of the steep part of the absorbance curve to zero absorbance increase or by establishing the time of the incipient rise in the turbidity.The first method has been criticized in the preceding papers;lS2 the latter can be understood as follows. At the moment of the incipient rise in the turbidity &fw/M,, = 1 and therefore by eqn (1): If we accept that the average casein rnicelle of molecular weight 5 x 10' contains 12% of ic-casein,6 from which all the potential macropeptide is split off during the lag period, then we have f= 0.04 and t, z 0.35 z. With fibrin f= 0.0235 ' and therefore t, s 0.27 z. As can easily be verified froin the particle concentrations computed previously,' at t < 0.35 z the system virtually contains only singlet and doublet particles. Thus the flocculation rate constants found from t, correspond to kll, the rate constant for doublet formation, which facilitates its theoretical interpretationg All computations have been carried out on the Hewlett-Packard 9830A calculator, extended with the 9862A plotter.Blood-clotting thrombin was from Merck (article no. 12374). (tclr> = (3f/(1 - f )I*+ (4) 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 c 0.8 0.7 1 0.6 - 0.5 - 0.4 - 0.3 - 0.2 - 0.1 - 0.0 - I 1 FIG. 2.-Typical absorbance measurements of clotting paracasein micelles (A) and fibrin (B). Meas- urements with the Cary 14 spectrophotometer at 500 nm. Enzyme addition indicated by arrow. Experimental conditions: (0) skim milk diluted 16 times with 0.01 mol dm-3 CaC12; 35 "C; Y 21 2 x 10-lo mol ~ m - ~ s-l. Note the shallow minimum in the absorbance in line with the model calcula- tion shown in fig. 1 ; (6) 0.5% fibrinogen; I = 0.32; 31.7 "C; Y z= 7.2 x mol s-'.T.A . J . PAYENS 167 Fig. 2 shows typical absorbance plots of clotting paracasein and fibrin. The influence of the temperature and the ionic strnegth on the enzymatic clotting time of paracasein is shown in fig. 3 and 4. The least-squares regression line through the experimental points in fig. 3 is : In z = 108.36 - 6.6624 x 104/T + 1.0696 x 107/T2. (5) A similar influence of the ionic strength as with casein is found with fibrin as can be seen from the data collected in fig. 5. DISCUSSION From eqn (1) and the results presented above it is clear that the enzymatic clotting time, z = (k,V/2)-+, is the only kinetic parameter that can be extracted readily from the absorbance measurements. Assay of the enzyme from the length of the lag periodl*lO*lf is, therefore, only possible provided k, is known beforehand or its con- stancy can be guaranteed.It is true that in principle Y could be obtained directly from the initial slope of the absorbance plot, because for t < z, eqn (1) reduces to, but application of this relation seems hardly feasible under the present experimental conditions. * Independent knowledge of V is indispensable, however, if k, is to be derived from the measured clotting times. Fortunately, rough estimates of Y are available from the rate of proteolysis of natural or synthetic substrates.13-15 These have been used below in the computation of the flocculation rate constants from z. Systematic errors in k, may further arise from the fact that the theory underlying eqn ( 1 ) has been set up for constant 1V.l If considerable exhaustion of the substrate occurred during the lag phase, this would also aEect the k,-values given below.A rough estimate, based on the coniplete Michaelis-Menten expression for V shows, however, that it is improbable that the rate constants would change by as much as an order of magnitude as a result of beginning exhaustion. Thus the general con- clusions arrived at further on, will remain valid. FLOCCULATION RATE OF PARACASEIN MICELLES Flocculation rate constants of micellar paracasein at I = 0.03 and different temperatures have been collected in table 1. Compared with the value of 6 x 10l2 cm3 mol-l s-l expected for diffusion-controlled flocc~lation,~*~~ these rate constants appear very low. Various arguments can be put forward to demonstrate that this is not to be explained primarily by the occurrence of some sort of energy barrier, in particular not by electrostatic repulsion.(1) Addition of salt (cf. fig. 4) leads to an increase in the clotting time? in contrast to what is expected for the case of electrostatic repulsion. (2) The 5-potential of the micelle increases from about - 13 to -6 mV as a conse- quence of the splitting off of a highly acidic peptide by the e n ~ y m e . ~ * ~ * ~ ~ We have recently estimated interaction potentials of two micelles using such (-potentials and choosing, rather arbitrarily, the relatively small Hamaker coefficient of 1 0-21 J * It is worth while mentioning that Scott Blair and Oosthuizen12 have indeed observed an initial slope in agreement with eqn (6) in delicate rheological experiments on the clotting of milk.t It can be shown that the salt effect in this case is primarily to reduce V, not to affect k, (T. A. J. Payens and P. Both, to be published).168 FLOCCULATION RATE CONSTANTS OF PARACASEIN AND FIBRIN FIG. 3.-Influence of temperature on the clotting time of paracasein micelles. Experimental con- ditions: skim milk diluted 20 times with 0.01 mol dm” CaC12; Y z 2 x mol cmA3 s-l. I FIG. 4.--Influence of the ionic strength on the clotting time of paracasein micelles. Experimental conditions: skim milk diluted 16 times with 0.01 mol dm-3 CaC12; 35 “C; V 21 2 x mol cmA3 s-l. because of the large porosity of the micelles.2s18 Some of these computations have been gathered in fig.6 and table 2. It is clear that on the basis of these calculations the retardation factor, W, appears too small to account for the low rate constants of table 1. On the contrary, one expects the clotting to become accelerated by the pre- dominancy of the London-van der Waals forces as soon as the [-potential exceeds, say, -10 mV. We will come back to this point later on. (3) Beyond 310 K the activation energies of the clotting process become negative, which suggests that the clotting rate is limited first of all by steric factors. The argu- ment is as follows.T . A . J . PAYENS 169 --_ 3 I-- 0 0.15 0.30 0.45 I FIG. 5.-Influence of ionic strength on the clotting time of fibrin. Experimental details: 0.5% fibrinogen; 25.5 "C; V N 7.2 x lo-'' mol cm-j s-l.TABLE FLOCCULATION RATE CONSTANTS OF MICELLAR PARACASEIN AT I = 0.03 AND DIFFERENT TEMPERATURES, COMPUTED FROM THE ENZYMATIC CLOTTING TIMES WITH kcat = 84 s-l, [REF. (13); EQN (7)]. 10" x k,/cm3 temperaturePC d S mol-' s-I 17.5 327 0.5 21.8 225 1.1 25 185 1.7 30 149 2.6 34.5 126 3.6 38 114 4.4 41.3 103 5.3 TABLE 2.-sURFACB CHARGE AND POTENTIAL AND RETARDATION FACTOR, w, COMPUTED FOR TYPICAL CASEIN MICELLES BY THE DLVO-THEORY.~ charge/& cm-2 potential/mV log w 1 - 15 3.01 0.83 - 12.5 0.18 0.67 - 10 0.01 0.50 -7.5 -2.81 0.33 - 5 -3.33 From the definition of z and with v = kcat (E), (7) (8) we have where k,,, is the catalytic constant, (E) the enzyme concentration and E , the activation energy of the proteolysis proper. The latter has been measured by Nitschmann and Bohren l9 as 41.8 kJ from the temperature-coefficient of peptide-liberation.It has previously been postulated2 that the low flocculation rate constants of table 1 d In z/d(l/T) = --(d In k,/d(l/T) - E,,/R]/~,170 FLOCCULATION RATE CONSTANTS OF PARACASEIN AND FIBRIN 20 10 0 CI * 5 -10 -20 -3 0 -4 0 FIG. 6.-Potential energy curves for a pair of casein miceIles of constant charge at different stages of proteolysis. The plots are for c-potentials of 15, 12.5, 10, 7.5 and 5 mV, respectively. Other para- meters: particle radius 50 nm; thickness of the electrical double layer 1 nm; Hamaker coefficient J. should be explained by the occurrence of a number of ineffective collisions and that the “ hot ” sites on the micelle surface which do lead to sticking are to be identified with the sites of attack by the enzyme.In a classical paper20 Collins and Kimball have elaborated upon the intricacies encountered with von Smoluchowski’s rate theory in the case that only a fraction of the particle collisions is effective. In particular, they pointed out that in that case the rate constant ‘‘ is of the nature of a specific reaction rate ”. Let us therefore write quite generally16 k, = v exp{AS*/R}exp{-AH*/RT), (9) where 1’ is a frequency factor of the order of 1013 s-l and AS* and AH* account for the activation entropy (the “ steric factor ”) and enthalpy respectively. It needs no further comment that in the present case AH* z E,, (10) and exp(AS*/R) cc y, (1 1) where I, is the activation energy of the clotting process and p the probability of successful collisions. If p is constant we thus find from eqn (8), (9) and (10) E, = 2R d In z/d(l/T) - E,.The &,-values computed with this equation from the temperature-dependence of zT . A . J . PAYENS 171 have been collected in the 3rd column of table 3. It is seen that the activation energy becomes negative at about 310 K. This is in qualitative agreement with the DLVO theory, which predicts that beyond, say, - 10 mV the van der Waals attraction should become dominant (cf. table 2). TABLE ~.-L~CTIVATION ENERGIES, Es, OF THE CLOTTING OF PARACASEIN MICELLES COMPUTED WITH EQN (12) AND (15) AND E, = 41.8 kJ mol-l [REF. (19)] 2R[d In z/d (1,491 &,/kJ mol-'; e,/kJ mol-l; t emperature/K /kJ mol-1 eqn (12) eqn (15) 286 234 303 313 135.3 101.6 65.8 28.5 93.5 59.8 24.0 - 13.3 9.9 -23.8 - 59.6 -96.9 The above computation does not reckon, however, with a possible temperature- dependence of p , though it is to be expected that activation of the enzyme will lead to an increased number, y1, of hot sites on the micellar surface.The magnitude of this effect is difficult to assess, because we do not know to what extent p1 can increase. An upper limit can be calculated as follows. If i z could grow without limit we obviously have for the probability of successful collisions where n cc V. dependence of p : Combining eqn (8), (9), (1 1) and (14) then yields for E,: P = n2, (13) With the help of eqn (7) we, therefore, find for the upper limit of the temgerature- d In p/d( 1/T) = - ~ E J R . E, = 2R d In T/d(l/T) - 3 ~ ~ .(14) (1 5) The &,-values computed in this way have been collected in the 4th column of table 3. They appear much more negative than those estimated from the DLVO theory. A possible explanation is that d In p/d(l/T) is considerably smaller than predicted by eqn (14), because n has reached its maximum. It should also be remembered that we do not know the type of bond which is formed at the hot sites,2 nor the way it contributes to k,. In view of such uncertainties an interpretation of the figures of table 3 is rather hazardous at present. Anyhow it is clear that the activation energy becomes negative at the higher temperatures. Having thus established the probable occurrence of ineffective collisions in the clotting of paracasein, one may wonder whether the rotational Brownian movement does not speed up the clotting rate to the diffusion-controlled value.It can indeed be shown by an argument due to PollardZ1 (see also Lyklema)22 that this is unlikely within the small range of attraction predicted by the potential energy curves of fig. 6. With the Einstein relations for translational and rotational diffusion, one easily finds for the angle of rotation, 8, traversed during the time that the particles are within each other's sphere of attraction where a is the particle radius and d the distance over which the attraction prevails. 8 = 0.866 d/a, (16)172 FLOCCULATION RATE CONSTANTS OF PARACASEIN AND FIBRIN Since a is of the order of 50 nm6 and d N" 1 nm at the highest, it is clear that rotational diffusion cannot restore the effect of unfavourable orientations. FLOCCULATION RATE OF FIBRIN Ideally, flocculation rate constants should be measured from the rate of change in the weight-average molecular weight itself. Absorbance measurements suffer from multiple scattering effects.2 Direct light scattering measurements are difficult to carry out, however, because with the particles studied here the angular dependence of the scattered light should be measured instantaneously.It is interesting that Steiner and Laki23 as early as 1951 have carried out light scattering studies on the clotting of fibrin by thrombin, which may be used to check the present theory and to estimate the flocculation rate of fibrin. FIG. 7.-Light scattering measurements of Steiner and Laki [ref.(23)] on the clotting of fibrin re- plotted according to eqn (17). Experimental details: protein conc. 1.33 g d ~ n - ~ ; pH 6.35; I = 0.48. To this end eqn (1) is rewritten as: (iVW - Mo)/Mot = -2M0(1 -f)fV/co + M,(l -f>2V2kst2/3~o, (17) which predicts that the graph of (mW - Mo)/Mot against t 2 should be linear with a slope proportional to k, and an intercept proportional to Y. In fig. 7 we have replotted the original data of Steiner and Laki at I = 0.48 according to this equation. The best fitting line is y = -5.62 x + 1.40 x 10-lox, (18) (r2 = 0.9884; q = 7). mol cmW3 s-l and from the slope k, = 7.8 x lo4 cm3 mol-1 s-l. The value of Y corresponds to a catalytic constant of rn 18 s-l, which compares not unfavourably with the reported15 value of 24 s-l.In the same way their data at I = 0.24 yields k, rn 7.4 x lo5 cm3 mol-' s-l. These figures should be compared with our own results obtained from the measurement of the clotting time [cf. fig. 2(b); table 41. The following conclusions then seem pertinent. (1) The rate constants emerging from both sets of data are again very low, but they differ by less than one order of magnitude. The reason for this failing overlap Using their other experimental data, we find from the intercept Y = 4.6 xT. A . J . PAYENS I73 TABLE 4. -FLOCCULATION RATE CONSTANTS OF THE THROMBIN-INDUCED CLOTTING OF FIBRIN AT DIFFERENT IONIC STRENGTHS COMPUTED WITH kcat = 24 s-'.15 ionic strength k,/~rn-~ mo1-1 s-l exptl. technique 0.24 0.48 0.16 0.32 0.48 7.4 x 105 7.8 x 1 0 4 4.3 x 107 absorbance 2.8 x lo6 absorbance 5.9 x 105 absorbance light scattering [ref.(23)] light scattering [ref. (23)] is not understood, although it has been known for a long time that different fibrin preparations may show widely different clottabilities. (2) Rate constants increase with decreasing ionic strength, which is in line with the idea3*24 that the aggregation of fibrin proceeds via the formation of electrostatic bonds between negatively and positively charged patches on the fibrin surface. Generally speaking then, the present results are consistent with the proposed mechanism of fibrin p o l y m e r i ~ a t i o n , ~ * ~ ~ ~ ~ ~ in which trinodular or cylindrical rods form staggered overlaps that are held together by electrostatic forces. At present there exists no rate theory that could account for such highly oriented flocculation of elongated particles. Finally we should mention that the low flocculation rates discussed here are by no means restricted to micellar paracasein and fibrin.We have observed such small rate constants also with the clotting of rc-casein (to be published) and in the cryo- precipitation 26 of milk euglobulins. Such observations seem to reflect the mosaic structure of the protein surface and the fact that the clotting proceeds via the bonding of specific amino acid side-chains. The author is indebted to Prof. Hans Lyklema, Agricultural University, Wagen- ingen, for fruitful discussions. T. A. J. Payens, A. K. Wiersma and J. Brinkhuis, Biophjs. Chem., 1977,6,253. T. A. J. Payens, Biophys. Chem., 1977, 6,263.R. F. Doolittle, Adv. Protein Chem., 1973, 27, 1. N. J. Berridge, Adv. Enzymol., 1954, 15,423. J. Th. G. Overbeek, in Colloid Science, ed. H. R. Kruyt (Elsevier, Amsterdam, 1952), vol. 1, chap. 7, p. 278. D. G. Schmidt and T. A. J. Payens, in Surface and Colloid Science, ed. E. MatijeviC (Wiley, New York, 1976), vol. 9, p. 165. P. F. Dyachenko, Investigation of Milk Proteins (Proc. AlIunion Dairy Res. Inst., Moscow, 1959), vol. 19. P. A. McKee, in Proteolysis and Physiological Regulation, ed. D. W. Ribbons and K. Brew (Miami Winter Symp., 1976) vol. 1 1, p. 240. J. W. Th. Lichtenbelt, C. Pathmamanoharan and P. H. Wiersema, J. Colloid Interface Sci., 1974, 49, 281. lo B. Foltmann, in Methods in Enzymology, XIX, ed. G. Perlmann and L. Lorand (Academic Press, New York, 1970), p. 421. l1 D. J. Baughman, in Methods in Enzymology, XIX, ed. G. Perlmann and L. Lorand (Academic Press, New York, 1970), p. 145. G. W. Scott Blair and J. C. Oosthuizen, J. Dairy Res., 1961, 28, 165. l3 J. Garnier, Biochim. Biophys. Acta, 1963, 66, 366. l4 S. Visser, P. J. van Rooyen, C. Schattenkerk and K. E. T. Kerling, Biochim. Biophys. Acta, l5 S . Magnusson, in The Enzymes, ed. P. D. Boyer (Academic Press, New York, 1971), vol. 111, 1976,438,265. p. 298.174 FLOCCULATION RATE CONSTANTS OF PARACASEIN AND FIBRIN l6 R. S . Hansen, in Physical Chemistry: Enriching Topics from Colloid and Surface Science, ed. l7 M. L. Green and G. Crutchfield, J . Dairy Res., 1971, 38, 151. l8 V. A. Parsegian, in Physical Cheniistry : Enriching Topics from Colloid and Surface Science, l9 H. Nitschmann and H. U. Bohren, Helv. Chim. Ada, 1955, 38, 1953. 'O F. C. Collins and G. E. Kimball, J. Colloid Sci., 1949, 4, 425. 21 R. B. Setlow and E. C. Pollard, Molecular Biophysics (Addison-Wesley, Reading, Mass., 1962), 22 J. Lyklema, Pure Appl. Chem., 1976, 48, 499. 23 R. F. Steiner and K. Laki, Arch. Biochem. Biophys., 1951, 34, 24. 24 M. W. Mosesson and J. S . Finlayson, in Progress irz Hemostasis and Thrombosis, ed. T. H. 25 N. Nemoto, F. H. M. Nestler, J. L. Schrag and J. D. Ferry, Biopolymers, 1977, 16, 1957. 26 T. A. J. Payens and P. Both, Immunochem., 1970,7, 869. H. van Olphen and K. J. Mysels (Theorex, La Jolla, Calif., 1975), chap. 13. ed. H. van Olphen and K. J. Mysels (Theorex, La JoIla, Calif., 1975), chap. 4. p. 501. Spaat (Grune and Stratten, New York, 1976), vol. 111, p. 61.

ISSN:0301-7249    

DOI:10.1039/DC9786500164

出版商:RSC    

年代:1978

数据来源: RSC

摘要:

GENERAL DISCUSSION Prof. J. Lyklema (Wageningen) said: An important factor in the picture developed by Frens appears to be the rate of change of the Stern-layer composition, as compared to the rate of approach. If the relaxation time z, of the Stern-layer is short as coni- pared to the time scale z, of an encounter between two particles, we have the constant potential case and particles may stick irreversibly. However, if z, > T,, interaction is at constant charge and the particles may bounce (provided the repulsive force is high enough). A more quantitative theory of the ideas, put forward by Frens would contain z, and 2, or for that matter, two rate constants, K, and K,. Such a theory would require some picture on the nature of the Stern-layer relaxation process and I would ask Frens if he has some suggestion in this respect.In connection with the choice of a Stern-layer adsorption isotherm equation and the effect of the counterion valeney, it may be recalled that on negative silver iodide the increase in capacitance with z+ is evidence for increasing specific adsorption in this order.’ Moreoever, also stability data are readily explained by assuming such an increase.2 All of this strongly suggests a decrease of tyd with increasing z+ which is at variance with Frens’ prediction. If the theory is further elaborated these facts deserve consideration. Dr. G. Frens (Eindhouen) said: I agree with Lyklema’s suggestion that Stern layer relaxation should be included in the complete description of coagulation in the prim- ary minimum.In my answer to Stoylov I have indicated how I think that this could be done. In Lyklema’s second question there seems to be some misunderstanding about what was predicted about pa and z. Specific adsorption increases when z increases. This implies that at comparable concentrations of electrolyte there is less charge in the diffuse layer, and a lower q a , if z is larger. In my paper it is said that cps at c, is higher for higher z. Of course there is the Schulze-Hardy rule to indicate that c, varies strongly with z. As it should, so that pa can have the values which make Vmin at c, independent of the counterion valency z. Dr. S. P. Stoylov (Sofia) said: Frens3 in his paper presented at this General Discussion suggests a very interesting hypothesis for the explanation of the stronger slowing down of the rate of slow coagulation for the bigger particle, based on competi- tion between aggregation and repeptization.Here I should like to suggest another hypothesis which scarcely could be regarded as less realistic. Overbeek4 in his remarkable article on the recent developments in the understand- ing of colloid stability has ruled out the relaxation time of the double layer as a possible rate effect, assuming that it is governed only by double layer thickness and giving for 5 mmol dm-3 aqueous electrolyte solution a value of s. However the same author in collaboration with Errera and Sack was the first to observe by the electric birefring- ence method relaxation times of the order of s for colloid solutions of to J. Lyklema and J.Th. G. Overbeek, J. Colloid Sci., 1961,16, 595. J. Lyklema, Croat. Chem. Acta, 1970, 42, 151. G. Frens, this Discussion. J. Th. G. Overbeek, J. Colloid Interfuce Sci., 1977,58,408. ’ J. Errera, J. Th. G. Overbeek and H. Sack, J. Chirxphys., 1935,32,681.176 GENERAL DISCUSSION particles of similar dimensions (but not spherical) and similar electrolyte concentra- tions which were attributed to double layer relaxation. The following years mainly as a result of dielectric and electro-optic studies on colloid systems, comparatively large relaxation times for a variety of colloid particles in colloid solutions were found. In the sixties mainly for the dielectric studies, theories of deformation of double electric layer or of the relaxation of the so called interfacial electric polarizability were de- veloped, allid6 giving a2 Gelax - - 201 where a2 is the square of the particle's radius (or longest dimension for anisodiametric particles) and D, is the ionic diffusion coefficient.This relaxation is experimentally well verified both for spherical and non spherical particles, in the latter case both by dielectric and electro-optic measurements. As an example for the experimental values, one can take the value for trelax obtained through dielectric measurements by Schwan et aL7 for polystyrene spheres of diameter 1, 17 pm, electrolyte concentration of the order of 1 mmol dm-3 KCl, trelax = 3. 5 x Respectively for a a lo3 A, Thus while the time of Brownian collision tBrown = 3nqa/rc2kT w 10-5-10-7 s for a = 1000 A and l/lc FS 10-100 A (where q is viscosity, K the reciprocal double layer thickness, k and T respectively Boltzman constant and absolute temperature) with increase of particle's dimensions rises linearly, the relaxation time of the double layer grows with the square of the dimension.From here it seems not difficult to imagine why bigger particles will have lower rate of slow coagulation than that calculated without taking into account the effect communicated above, especially when one has in mind that for the range for a experimentally investigated,* trelax 3 tBrown. In 1970 Oosawa9 wrote, " The fluctuation of counterion distribution along the polyion can be an origin of the attractive force between two polyions like the van der Waals force. The large polarizability results in strong attractive force. The fact that the fluctuation is composed by many modes with different relaxation times suggests that the force may depend on the velocity of approach of two polyions.The problem will be analysed elsewhere . . . ". s. trelax w 10-5-10-6 s. In conclusion I want to say that this explanation is not entirely new. Dr. G. Frens (Eindhoven) said: Stoylov and Lyklemapropose to include certain types of double layer relaxation in the theory of electrocratic colloid stability. The different mechanisms for double layer relaxation are each characterized by a relaxation time 2. Let us consider three examples of such mechanisms, with relaxation times zl, z2, 2, respectively. One is the Debye-Hiickel relaxation, mentioned by Overbeek, in which there is a rearrangement of free (diffuse layer) ions in a changing potential field.It is found, both theoretically and experimentally, that in aqueous electrolytes z1 = lO-"/c when c is given in mol dm-3. A second mechanism is Stern layer relaxation ( 2 3 in G. Schwarz, J. Phys. Chem., 1962,66,2636. M. Mandel, MoZ. Phys., 1961,4,489. J. M. Schurr, J. Phys. Chem., 1964,68,2407. S . S . Dukhin, Electrical Conductivity and Electrokinetic Properties of Disperse Systems (Naukova dumka, Kiev, 1975). F. Oosawa, PoZyekctroZytex (Marcel Dekker, N.Y., 1971). S . Takashima, in Digest of Dielectric Literature, 1977, in press. H. P. Schwan, G. Schwarz, J. Maczuk and Pauly, J. Phys. Chent., 1962,66,2626. G . Frens, this Discussion. F. Oosawa, Biopolymers, 1970,9, 677.GENERAL DISCUSSION 177 which the concentration of specifically adsorbed ions changes.A third mechanism with z3 M lov4 s is mentioned by Stoylov. Presumably these observations relate to some rearrangement of charges (polarization) in the particles when a field is applied. Reinerl has introduced the Deborah number D = zr/zo, where T~ is the time of observation and z, is the relaxation time of the observed phenomenon. In problems of colloid stability zo is the duration of a Brownian encounter during which the aggre- gating particles experience the interaction forces. z, are the different double layer relaxation times. A relaxation process for which D < 1 can keep up with the changes in the geometry of the double layer which are caused by the Brownian motion of the particles.The Debye-Hiickel relaxation is such a process, the diffusion of ions being faster than that of colloidal particles. In the computation of potential energy dia- grams a process with D < 1 should be treated in a thermodynamic way, e.g., via a charging process. For D 9 1 there is no relaxation during the collision. It is a boundary condition in the computation of the potential energy that the quantity, e.g., the surface charge, remains constant. The slow relaxation processes of this type constitute the ageing which will eventually render all aggregates irreversible. Things become complicated for D M 1, and one expects rather peculiar size effects since zo depends on the particles size. In most cases the z3 phenomena fall in the D > 1 category, and my personal feeling is that z2 > zo too.Breaking specific bonds and discharging surfaces involves binding energies and activation energies which are, by definition, much larger than kT, so that the rates of such processes can be orders of magnitude lower than those in diffu- sion controlled processes, even if the diffusion involves, not too large, colloidal particles. However, to gain insight in these matters we shall need the experimental results of investigations on the structure and the relaxation of the double layer such as they are carried out by Dr. Hoiiig in our laboratory. Prof. A. Watillon and Dr. F. Dumont (Brussels) said: The author stresses that the limited coagulation of electrocratic colloids near the critical coagulation concentra- tion is interpreted by the fact that “ a constant fraction of the newly formed aggregates breaks up, but that the remaining older flocs stick irreversibly together ”.In the case of Fe,O, hydrosols,2 we observed that in rapid coagulation conditions (at electro- lyte concentrations greater than the c.c.c.) there does exist a true coagulation-re- peptization equilibrium which strongly depends on the initial number of primary particles ; how can the author’s considerations explain this fact? Dr. G. Frens (Eindhoven) said: Quite the contrary: in the paper it is said that the older flocs become irreversible in retarded Smoluchowski kinetics, whereas limited coagulation follows from the establishment of a coagulation-repeptization equilibrum which implies the permanent reversibility of the aggregates.The quoted experiments with colloidal Fe,O, fit nicely into this pattern. In your paper it is demonstrated that aggregates of hematite particles remain reversible for considerable times. These reversible aggregates depend in size on the sol concentration, i.e., on the number of collisions. It seems that this represents the situation where the number of bonds between particles which is broken per unit time equals the number of bonds formed by collisions. Even with completely reversible aggregates there exists a critical coagu- lation concentration in the sense that the rate of coagulation will become rather independent of the salt concentration when Vmin becomes > 1 kT. Above this con- M. Reiner, Phys. Today, 1964,17, 62. F. Dumont, Dang van Tan and A.Watillon, J. Colloid Interface Sci., 1976,55,678.178 GENERAL DISCUSSION centration, and with reversible aggregates it is to be expected that the size of aggregates is reduced by diluting the unstable suspension. Prof. A. Watillon and Dr. A. M. Joseph-Petit, (Brussels) said: From his general considerations concerning the mechanism of the coagulation, the author deduced that, for a given material, the Clim (critical coagulation concentration) should decrease as the particle radius a increases; he found experimental support for this conclusion in the experimental data we obtained by studying Se hydroso1s.l Let us summarize these results: the qirn of different Se hydrosols were measured at pH 12 in the presence of NaC104 as a function of the particle size, from a = 47.5 to 137 nm.To ensure the reliability of the results, hydrosols coming from different batches and purified by using various techniques were used for a given a value. The main experimental results are : for a values < nm, Ciim increases with the particle size, passes then through a maximum and then decreases when a increases. The descending branch of the (CIim, a) curve was attributed to a coagulation in a developing secondary minimum. The ascending branch was more difficult to explain but, at any rate, may not be attributed to a rela- tive increase of the van der Waals attraction forces at low a values : the Se hydrosols were prepared by heterogeneous nucleation on gold nuclei (a = 2.5 nm) thus for the studied range of particle radius (47.6- 137 nm) the relative volume fraction of the gold in the selenium particle varies from furthermore, the Hamaker constant of Se in water amounts to -6.4 x erg, not significantly different from the Hamaker constant of gold, mentioned in Derjaquin's paper at this Discussion.In conclusion, it seems that these results are far from supporting the proposed mechan- ism of coagulation. to -2 x Dr. G. Frens (Eindhouen) said: The experimental results for the larger Se particles can indeed be explained in terms of coagulation in the secondary minimum. But to obtain such a minimum Joseph-Petit and her coworkers had to assume that there is a distance of closest approach (24 in their paper) between the particles. Without this assumption, i.e., with 24 = 0, the Hamaker constant A = 6 x erg*, the double layer potential of 30 mV and the electrolyte concentration of 0.2 mol dm-3 there would be no maximum and no secondary minimum in the interaction curve, but only a deep primary minimum.Fitting the theoretical curves for coagula- tion in the primary and in the secondary minimum around their experimental data they obtained an estimated value of 5 kTfor the maximum in the potential energy dia- gram at the critical coagulation concentration, and 24 = 10 for the distance of closest approach. It is just as easy to make the same experimental data consistent with coagulation in the primary minimum. For A = 6 x erg* and 2; = 1.2 (i.e., 96 = 30 mV) one obtains 26 = 10 A [ref. (22) of my paper], indicating that at 26 = 24 the primary minimum is shallow indeed! For the large Se particles 2, needs only be slightly smaller than 2; to reach the condition (Vmin = 1 kT) for the critical coagulation concentration according to the present theory.In this context it is interesting to note that Joseph-Petit et al. measured an increase in [ at c, when a increases. This could be indicative of Vmin being constant at c, and independent of the particle size, whereas Vmin at a given concentration is of course proportional with the particle radius. In this way one might even obtain a more precise and experimentally sound alternative for eqn (3) of my paper. It should not go unnoticed that a maximum of 5 kTin the interaction curve at c, * 1 erg = lO-'J. A. M. Joseph-Petit, F. Dumont and A. Watillon, J. Colloid Interface Sci., 1973, 43, 649.GENERAL DISCUSSION 179 would imply that the fastest coagulation rate for Se sols of this type is < 1 % of the Smoluchowski rate.This seems rather unlikely in view of other data concerning rates of rapid coagulations and especially since there is no reason in the theory as given by Joseph-Petit et al. why this maximum should not decrease upon the addition of electro- lyte. A slight difference in the value of Z%(A or 26) in the sols with small partides seems just as reasonable as an explanation of the observations. In that case rapid coagulation would proceed at the normal rate. Prof. A. Watillon and Dr. A. M. Joseph-Petit (Brussels) said: The author predicts a significant increase of the potential for coagulations (w!") with the valence of the counterion.(a) Kotera et a2.l studying the 5lim of PSL hydrosols in the presence of K+, Ba2+ and La3 + , measured values around 25 mV independent of the valence of the counterion in the three cases. (b) Ono et aL2 measured a Clim around 10 mV in the presence of NaC, K+ and Ba2+ on pure polystyrene and on a series of styrene-acrylonitrile copolymer latices, also independent of the valence of the counterion. Some experimental results in the literature do not confirm this view: Dr. G. Frens (Eindhouen) said: I doubt if such work with polymer latices can be used in discussions on electrocratic sols without a further study of the double layer properties of these colloids. As it is the quoted papers contain some rather unusual data. In one of them it is reported that the double layer potentials of the latex particles increase upon the addition of inert electrolytes.In the other it is observed that the difference in coagulation concentrations for ions of different valency is only a factor three or four. This is in striking contrast with the observations on normal electrocratic sols for which the Schulze-Hardy rule describes characteristic behaviour. Dr. A. E. Smith (Port Sunlight) said: One must surely have reservations about the use of eqn (2). The tern? before In C will be less than kT/ze by a factor itself a function of inner layer capacity and electrolyte concentration. How sensitive are the pre- dictions to this? In agreement with Frens' conjectures we have observed that particles aggregated in weak attraction conditions are readily separated by collision with other particles, not themselves aggregating.However, it should be pointed out that for so called rapid flocculation, brought about by excess electrolyte, the deviations from retarded Smoluchowski rates at finite particle concentration are in the direction of increasing rather than decreasing the net flocculation r a k 3 Dr. G. Frens (Eindhoven) said: Indeed one can have reservations about eqn (2) of my paper, or about any of the refinements of the Stern theory [ref. (23)-(26), (28)-(3O)l. But the general result is that q ~ a decreases as the concentration of inert electrolyte increases. In my paper this variation of Za relative to Ta is identified as the prime reason for changes in the rate of coagulation. The precise formulation of the relation between pa and c, as in eqn (2) affects the shape of the (log W, log c) dia- gram.This shape will also depend on other effects, such as the double layer relaxa- tion in the aggregates which is introduced later in the paper. A more detailed analysis of experimental ( W, c) relations, in combination with kinetic data (limited, or retarded Smoluchowski behaviour ?) seems necessary before any conclusion can be drawn. A. Kotera, F. Furusawa and K. Kudo, Kolloid-Z., 1970,240, 837. H. Ono, F. Sato, E. Jidai and K. Shibayama, Colloid and Polymer Sci., 1975,253,538. W. Hatton, P. McFadyen and A. L. Smith, Trans. Faraday SOC., 1974,70,655.180 GENERAL DISCUSSION Other predictions in the paper, such as the difference in tp3 at c, for counterions with different valencies, are based on the theory of the diffuse layer.These are not affected by the choice of eqn (2) as a description of the ( 9 8 , c) relationship. You mention that the " bimolecular " rate constants of rapid coagulation are found to be dependent on the particle concentration. They increase from about half the Smoluchowski value at low concentration (i.e., from the expected value if hydro- dynamic interaction is taken into account) to the Smoluchowski value in concentrated suspensions. This indicates, at least, that one should have a second look at the con- cept of hydrodynamic interaction and its applicability in concentrated suspensions. But, more along the lines of my paper, one might also consider this is an indication that something in the aggregates codetermines the rate of coagulation.This might be their size [ref. (27)], which depends on the particle concentration when there is an association-dissociation equilibrium between aggregates and primary particles (cf. my answer to the question by Watillon and Dumont). Dr. EX. N. Stein (Eindhoven) said: A good impression of the importance of re- dispersion in coagulation phenomena might be obtained by investigating coagulation in the presence of macroscopic shear. In the first stags of the coagulation, a shear will increase the coagulation rate, by increasing the number of collisions between the partic1es.l A shear will, however, increase the repeptisation rate as well; and if repeptisation is important then the coagulation rate will in the final stages of the coagulation be decreased by a shear.Thus, if the coagulation is followed by measuring the turbidity, then this quantity is expected to show behaviour such as in fig. 1. turbidity I without shear time FIG. 1.-Expected course of turbidity as a function of time in a coagulating sol with and without shear. Such experiments are expected to be especially easily observed in dispersions of Are there any observations in this respect? rather coarse particles (diameter, say 0.5- 1 ,urn). Dr. G. Frens (Eindhoven) said: In principle I agree with Stzin that coagulation experiments in shearing conditions could give an impression about the reversibility of aggregates. Tuorila et al. have shown that stirring effects are negligible in the coagulation of small particles, and mill- J.Th. G. Overbeek, in Colloid Science I, ed. H. R. Kruyt (Elsevier, Amsterdam, 1952), p. 289. However, I think that some cautioning is called fur.GENERAL DISCUSSION 181 ing experiments indicate that extreme shear is needed to affect aggregates of submicron size. So the experiments which Stein proposes should be done with fairly large particles, but in coarse suspensions it could be that coagulation in the secondary rather than in the primary minimum is observed, and shows reversible aggregates. Prof. E. Ruckenstein (Bufalo) (communicated) : Frens assumes that the rate of slow coagulation is determined by the competition between aggregation and repeptization at a kind of secondary minimum and uses eqn (8) to describe this reversible process.Concerning this treatment I observe that reversibility is introduced in eqn (8) on intuitive grounds. I have formulated recently the problem of reversible adsorption or coagulation of Brownian particles in a different manner : (a) A short range repulsive potential of the Born type was introduced into the expression of the interaction potential in addition to the London and double layer interactions. This assures the existence of a primary minimum, the depths of which depends upon the values of the parameters. For some parameter values, no primary minimum practically exists.2 (b) The process is assumed to occur in two steps. First the particles move from the bulk of the fluid to the secondary minimum and then a fraction of the particles that arrive at the secondary minimum moves further to the primary minimum.The main difference between our treatment and that of Fuchs consists in the splitting of the assumption of quasi-steady state diffusion in a force field into two separate steps. The first step is a quasi-steady state diffusion in a force field between the bulk of the fluid and the secondary minimum, while the second step is a quasi-steady state diffusion in a force field between the secondary and the primary minimum. Accumulation occurs at the two minima and expressions for both rates of accumulation have been derived. Of course, there are conditions under which accumulation occurs at one or at both of them. Computations are now being carried out to compare this theory with the experimental results of Ottewill and Shaw3 discussed by Frens. Dr.G. Frens (Eindhoven) (communicated): I wonder where Ruckenstein got the impression that my paper deals with coagulation in the secondary minimum. It is explicitly said, both in the title and in the text of the paper, that coagulation in the primary minimum will be discussed. The reversibility of aggregates in the primary minimum is introduced in the paper, not on intuitive grounds, but on the basis of the experimental evidence concerning the repeptization of coagula which I have given as ref. (20)-(22). In these same papers it is also shown that the introduction of a Born type repulsion potential, as proposed by Verwey & Overbeek and extensively computed by K r ~ p p , ~ is not sufficient to account for the experimental facts, repeptization in particular.The problem with computing potential energy diagrams is not to vary the para- meters and the boundary conditions, but to establish which parameters and conditions are physically realistic. This was the objective of our repeptization experiments. It resulted in the model for coagulation and repeptization which has been used in the present paper on slow coagulation. Essential features of the model are: constant CT in the diffuse layer for the duration of a Brownian collision; a distance 26 to accom- modate the necessary counterions; a variable Z,, dependent on the electrolyte con- centration, and a constant of the material Z, which discriminates between experi- mental conditions for coagulation and for repeptization in a given colloid; and E.Ruckenstein, J. Colloid Interface Sci., in press. E. Ruckenstein and D. Prieve, A.I.Ch.E.J., 1976,22,276. R. H. Ottewill and J. N. Shaw, Disc. Faraduy SOC., 1966,42,154. H. Krupp, Dechema Monogr., 1960,38,115.182 GENERAL DISCUSSION finally, the slow equilibration of the double layers which have been disturbed by the changes in geometry, i.e., capacity, during a Brownian collision. Having convinced ourselves that this is a realistic model for the description of coagulation and repeptization we have used it in the present paper to explore the consequences of the reversibility of aggregates in the primary minimum, which is inherent in this model, for the description of slow coagulation. I do not claim that it can account for all experimental data.But it certainly puts some general observa- tions, such as the Schulze Hardy rule, in a different perspective and it leads to predic- tions which can be verified or controverted by experiments. Dr. R. M. Cornell, Dr. J. W. Goodwin and Prof. R. H. Ottewill (Bristol) said: Frens mentions in his paper the idea that particles may break away from an aggregate after flocculation has occurred. This is indeed the case as we have been able to verify recently by direct experimental observation. Using a microtube apparatus based upon Id) Fro. 1.-Tracings taken from a cinematographic recording of the flocculation behaviour of a 2 .um diameter polystyrene latex at various time intervals after making the latex 8 x mol dm-3 with sodium chloride. ( a ) 1.25 s, (b) 6.87 s, (c) 15.20 s, ( d ) 30.73 s.the design of Vadas, Goldsmith and Mason,' it has been possible to make cinemato- graphic observations of the flocculation of polystyrene latex particles with diameters of the order of 2 pm. A sequence of particle arrangements traced from the projected image of a film taken at various time intervals after the addition of 8 x mol d ~ n - ~ sodium chloride is shown in fig. 1. In this type of experiment two phenomena were observed. Firstly, the break-up and reformation of aggregates and secondly the mobility of the particles within the aggregates. In addition to using the apparatus to study the Aoc morphology we have also examined the kinetics of flocculation of latex particles. Results are given in fig. 2 for observations of the percentage of particles remaining as singlets at various times after making the dispersions 3 x mol dm-3 with respect to sodium chloride.In both cases the concentration of singlets reached a constant value after a certain time interval, indicating that a steady state had been reached, in which the single particles were in equilibrium with doublets and higher multiplets. and Dr. W. D. Cooper (Edinburgh) (communicated): Can the authors give any indica- tion of the possible distribution of charged groups on the surface of the latex particles E. B. Vadas, H. L. Goldsmith and S. G. Mason, J . Colloid Interface Sci., 1973,43,630.GENERAL DISCUSSION 183 5 0 100 150 200 250 300 time/ min FIG. 2.-% particles as singlets at various time intervals after adding salt to the system: -0-, 3 x mol dmW3 sodium chloride; -@-, lod2 mol dm-3 sodium chloride.or of the effect of LiN03, KNO, or CsNO, on electrophoretic mobility? The surface charge density of amphoteric latex sols is unlikely to be uniform in the manner re- quired by simple Gouy-Chapman Theory but will fluctuate in magnitude to an extent governed by the number and distribution of -COO- and -NH3 + groups in the latex surface. Furthermore, if discrete areas consisting of charged groups of predominantly one type occur, a patchwork with respect to charge will be created on the particle surface. The measured electrophoretic mobility will then reflect some resultant of the " surface charge heterogeneity '' and would be a poor indicator of the charge density over a small region of surface if the heterogeneity were large." Surface charge heterogeneity " will affect coagulation behaviour perhaps causing a reduction in stability since at small particle separations the interaction could become similar to that found in heterocoagulation between oppositely charged particles. Under these circumstances not only would the zeta potential be expected to be a poor guide to the possible stability of the system but the measured stability would be greatly affected by the extent of the heterogeneity. The extent of the heterogeneity depends initially on the ratio of --COO- to -NNH3+ groups in the surface but would be modified by specific adsorption of ions from the dispersion medium. Thus differences in specific adsorption of Li+, K+ and Css ions at high pH for example would alter both the resultant electrophoretic mobility and the " surface charge heterogeneity ".Pre- sumably reduction of the latter would enhance stability. Have the authors zeta potential data for the latex samples referred to in their fig. 3 and 4 in the region of similar pH and electrolyte concentration but where dispersions in KNO, were stable but in CsNO, were unstable? Prof. T. W. Healy (Melbourne) said: Our electrophoresis studies of the amphoteric latex colloids are restricted to supporting electrolyte concentrations below 10-1 mol dm-, salt, which is below the region where " normal " and " abnormal " coagula- tion is observed with CsNO, and KN03 respectively. Our earlier work [see fig. 1 of ref. (l)] indicated that simple constant potential DLVO theory could account for a single latex in 1 : 1 electrolyte up to 10-1 mol d ~ n - ~ .In this sense, the coagulation, R. 0. James, A. Homola and T. W. Healey, J.C.S. Faradny I, 1977,73,1436.184 GENERAL DISCUSSION as predicted from the measured c-potentials, was " classical " and by inference, sug- gests a uniform surface charge. A more convincing proof of the uniform distribution of charges on the surface comes from some recent work by Harding and Healy (to be published) on the uptake of Cd" on latices of various isoelectric points @Hie,). The latex, irrespective of its isoelectric point, will bind Cd" (as) only at pH values just above the pHlep. If the surface were heterogeneous, one might expect binding below the pHiep. This was not observed. Again, such results, while not conclusive, provide further evidence that the amphoteric latex surface has a random array of ionizable groups.Dr. J. W. Goodwin (Bristol) said: The high surface charge of the amphoteric latex colloids, better termed " zwitterionic " latex colloids, and the total reversibility to coagulation suggest that the surface is polyelectrolyte in character and that the charge resides in depth. Could the authors comment on this proposition. Prof. T. W. Mealy (Melbourne) said : It is difficult to decide on the appropriate word to describe the surface of the present latices. Whatever the word, they are charac- terized by the fact that they possess an isoelectric point @Hie,) or a point-of-zero- charge (pH,,,) as determined by electrokinetic or titration respectively." Amphoteric " and " zwitter-ionic " both imply that the surfaces have an iso- electric point but neither describes precisely the chemical structure of the surface. Thus the separately bound carboxyl and amine groups are not bound through a peptide link and in that sense are not zwitterionic. Again, they are obviously not the same groups, as is required by the word amphoteric! The charge is high relative to conventional latex colloids but the maximum charge attained has always been less than a close-packed monolayer of chargeable groups; attempts to produce super-charged amphoteric latex have not raised the maximum charge above that of a monolayer, i.e. s &40 pC cm-2. In that there is obviously very little hydrophobic character (i.e., areas of bare polystyrene), then the description '' polyelectrolyte surface " is entirely appropriate.Dr. Th. F. Tadros (Jealotts HiZZ) said: The high charge densities encountered with these amphoteric latices can be either due to the presence of polyelectrolyte chains adsorbed or anchored to the particle surface (as discussed by Goodwin) or it could be due to the presence of pores in the polystyrene latex. Either or both could explain the specific ion effects. The second point regards the reversibility of the (ao, yo) curves. Have the authors checked these points? Prof. T. W. Healy (Melbourne) said: The surface charge densities of the amphoteric latices are high in relation to conventinal latices but not higher than one would expect for a " close-packed monolayer " of charge.In general terms, if 5 x 1014 ionizable groups per cm2 represent a maximum attainable surface charge, then for the case of an amphoteric latex of pHiep of 7 (approximately equal numbers of amine and carboxylate groups) the limits of charge are +40 or -40 pC cm-2. This is the maxi- mum charge that we have been able to attain. On the question of porosity, we must first ask the question-" Porous to what species?" Thus porosity to protons might be achieved but unless the pores admit counter-ions the internal structure cannot contribute towards the titratable surface charge. The (go, pH) isotherms are indeed reversible with KN03 as the supporting electrolyte. To date we have not determined the (go, pH) isothernis in other electrolytes. For these and other reasons, our current belief is that the amphoteric latex colloids are impermeable to simple counter-ionsGENERAL DISCUSSION 185 and can be thought of, as a first approximation, as possessing " smooth " or " hard " impermeable surfaces of high charge.Spectroscopic and other studies are in pro- gress to test this hypothesis. Dr. F. Dumont and Prof, A. Watillon (Brussels) said: The paper of Healy et al. suggests two comments. The first is concerned with the ionic adsorption sequence, the second with the high experimental stability of their PSL hydrosol. (1) The ionic adsorption sequence deduced from the experimental data is Cs+ > K+ > Li+. The authors said that it is difficult to explain the sequence by water structure effects. First, it is necessary to summarize the main concepts of Gurney1 concerning the ion-ion interactions. The ions may be classified in two groups : those around which the water molecules are more ordered than in the bulk phase and those around which the water molecules are less ordered.The first ones are the structure promoting ions (SP), they are characterized by a positive B coefficient of viscosity describing the state of hydration of the ion in the bulk phase [Jones-Dole equation],' among these, Li+ (B = +0.15), Na+ (B = +0.086), 10,- (B = +0.14). The second ones are structure breaker (SB), their B coefficients are negative: K+ (-0.007), Cs+ (-0.045), NO: (-0.046), ClO; (-0.056), I- (-0.068). The ion interactions in solution are classically described by the Debye-Huckel theory in which only the electrical interionic forces are accounted for.The new concept of Gurney consisted in adding to the electrical interaction energy a term depending on the relative effect of the interacting ions on the solvent structure: when a strongly hydrated (SP) cation interacts with a strongly hydrated (SP) anion, they share a part of their hydration shell giving rise to an extra attraction energy; when a weakly hydrated cation (SB) interacts with a weakly hydrated anion, they have more affinity for each other than for the sur- rounding water, it results also in a supplementary interionic attraction; finally, when a strongly (weakly) hydrated cation interacts with a weakly (strongly) hydrated anion, none of the previous situations being encountered, no extra interionic attraction will appear.The best confirmation of this theory is given by the activity coefficients of the alkali halides. The same concept may be generalized to the ion-surface interactions, if one con- siders the colloidal particle or the surface as a ma~ro-ion.~~~ It can be stated that an interface will strongly adsorb an ion when its action on the water structure is the same as the corresponding action of the ion: a strongly hydrated ion (SP) will be strongly adsorbed on a strongly hydrated surface (SP), a weakly hydrated ion (SB) will be adsorbed on a weakly hydrated surface (SB); in the opposite case, the ion will be re- jected from the surface. The adsorption sequence observed on Hg5, Ag16 surfaces which are both hydrophobic, thus SB, is Cs+ > K+ > Na+ > Li+; the sequence observed on Fe2037 and TiOZ4 surfaces which are hydrated (SB) is Li+ > Na+ > K+ > Cs".The surface of the PSL is hydrophobic and may be considered as SB, R. W. Gurney, Ionic Processes in Solution (Dover, N.Y., 1953). E. R. Nightingale, in Chemical Physics of Iottic Solutions, ed. Conway and Barradas (John Wiley and Son, N.Y., 1966), p. 87. L. Gierst, L. Vandenberghen, E. Nicolas and A. Fraboni, J. Electrochem. SOC., 1966, 113, 1025. Y. G. BCrub6 and P. L. De Bruyn, J. CoZloid Sci., 1968,28,92. D. C. Grahame, Chein. Rev., 1947, 41,441. H. R. Kruyt and M. A. M. Klompe, Koll. Beihefte, 1942,54,484. F. Dumont and A. Watiilon, Disc. Furaday SOC., 1971,52, 352; F. Dumont, Dang Van Tan and A. Watillon, J. Colloid Interface Sci., 1976, 55, 678.186 GENERAL DISCUSSION then it will show the Cs+ > K+ > Li+ sequence which was experimentally observed in this paper.The Li+ ion being held far from the surface, there is no reason to have a contribulion of the hydration shell of this ion to an eventual hydration layer of the surface. TABLE 1 .-HYDROSOL COAGULATION VALUES Cli,/mrnol dm-3 10-14 41 60 10156 21060 2 x 10-14 1040 2540 5265 5 x 1 0 - 1 4 166 406 842 10-13 42 42 211 Furthermore, the structure breaking properties of NO, and Cs+ are almost identi- cal as shown by their B coefficient (B,,+ = -0.045, BNos = -0.046), thus according to the proposed model, the coagulating power of these ions should be very similar: this is confirmed by the experimental results of this paper (Ciz N 700 mmol dm-3, CYf; 2: 600 mmol Finally some features may also be anticipated: the coagulating power of C10;; (B = -0.056) or I- (B = -0.068) should be higher than that of NO; (B = -0.046) (the Clim should be lower) whereas the coagulating power of 10; (B = +0.14) should be very similar to that of Li+ (B = +O.15). It should be very interesting to perform these experiments. (2) The coagulation values (Clim) of an hydrosol for different lyym potential at the coagulation and Hamaker constants A are given in table 1. The Clim were calcu- lated using some classical approximations in the DLVO theory. It essentially shows how low A values may give rise to very high Clim. Now, when an ion is not adsorbed in the Stern layer, y/ym remains high and a large amount of electrolyte is needed to coagulate the hydrosol, whereas, when the ion is adsorbed in the Stern layer, yiim is much lower and the coagulation can take place at lower electrolyte concentrations. So, with an adsorbed ion, t&'" will be lower than with an unadsorbed ion and the Clim will also be lower.This was experimentally observed with Fe203 hydro sol^.^ In this paper, the Clim of Cs+ far from the i.e.p. amounts to -600 mmol dm-3; assuming that the Hamaker constant of the PSL could be as low as 2 x erg,l lykim should be a little bit less than 20 mV (see table 1). In the presence of K+ and Li+ which are less adsorbed, &'" will be higher and the Clim will drastically increase. For instance, if &'" amounts to 30 mV (a very realistic value) the Clim will be higher than 5 MIL! The high stability of the PSL hydrosol is due to the low value of its Hamaker constant and is, in fact, not so surprising.Prof. T. W. Healy and Dr. R. J. Hunter (Australia) said: Dumont and Watillon raise several interesting points in their contribution. Under comment (1) they outline the earlier analysis of structure making/ breaking ions and interfaces, and their mutual interaction. Indeed, they seek a more detailed picture of the phenomenon we have observed, uiz., with K+ and Li+ in contrast to Cs+ we (and they) observe a stability of our amphoteric high charge latex at high salt. This would be an expression of the inability of Li+ to penetrate the latex surface water structure. We have expressed this concept as a hydration repulsion during particle/particle interaction.A. Watillon and A. M. Joseph-Petit, Disc. Furnday Soc,, 1966,42, 143,GENERAL DISCUSSION 187 The approximate calculations given in item (2) lead Dumont and Watillon to ascribe the high stability to the low Hamaker constant. We would prefer to suggest that the low Hamaker constant sensitizes latex coagulation to the extent that subtle effects, ascribed by us to hydration repulsion, are observed. With a material of higher Hamaker constant, it may not be possible to observe effects such as we have described. Referring to table 1, we should like to point out that we, as yet, have no way of know- ing whether or not the &'" values are realistic or not. Dr. P. C. Scholten (Eindhoven) said: Although unable to give a better explanation for the specific ion effect upon the stability at high salt concentrations, I don't think the size of the hydrated ion could be responsible.A good indicator for the size of an ion is its mobility. At 25 "C and infinite dilution the mobilities for Li+, K+, Cs+ and NO, are 3.4, 6.6, 7.1 and 6.4 x If size were the determining parameter, one would expect K+ and Cs+ to behave similarly, instead of K+ and Li + as is observed. Moreover, as the size of NO, is close to that of K+, this would predict a symmetrical stability against pH diagram for KN03. m2V-l s-l, respectively. Prof. T. W. Heaiy and Dr. R. J. Hunter (Australia) said: As Scholten points out, size alone cannot account for the stability sequence or, indeed, the fact of stability in the high salt rigime. Our concept of " stability related to hydration " relates to the effect on the chemical potential of water itself in the interparticle region due to the presence of counter ions.If the balancing charge that must be accommodated forces an unattainable change in the chemical potential of water in the interparticle region, then a repulsion results to regulate the chemical potential of water to an attainable value. Prof. J. Lyklema ( Wageningen) said : In connection with attempts to account for the added stability in K+ and Li+ but not in Cs+ solutions, I vote for hairiness of the articles, influenced by the solvent quality close to the surface. Several electrokinetic results obtained with latices in our laboratory are compatible with a picture of hairs, imparting some steric stabilization, that can be '' salted out " by electrolytes. That there is a difference in salting-out power, i.e., in solvent quality between the three alkali ions is perhaps not surprising if it is realized that the activity coefficients of the relevant electrolytes differ considerably.Specifically, the surface layer of a carboxy- late latex may be compared with a concentrated solution of, say, alkali acetates. For instance, in 1 mol dm-3 solutionsf, (LiAc) = 0.690 is much lower thanf,(CsAc) = 0.802; in 2 mol dm-3 solutions, these figures are 0.734 and 0.952, respectively.' Prof. T. W. Healy and Dr. R. J. Hunter (Australia) said: The mean ionic activity coefficient sequence cited by Lyklema is also consistent with our hydration repulsion suggestion between essentially smooth surfaces.Thus Li+ exerts a greater change in the nature (" structure ") of the interparticle water than does Cs+ . We interpret this as giving rise to an unrealistic change in the chemical potential of water in this region which can only be relieved by separation, i.e., stability. Hairiness of our spherical particles has not been detected by physical measurements we have tried to date; such work continues. Dr. B. Vincent (Bristol) said: The authors have studied a number of variables systematically, but there is one other variable of the system that might be worth H. S. Harned and €3. B. Owen, The Physical Chemistry of Electrolytic Solutions (Reinhold, N.Y., 3rd edn, 1964), p. 732.188 GENERAL DISCUSSION investigating, namely the particle number concentration.Several years ago Long, Osmond and I published a paper dealing with the reversible, equilibrium flocculation that is associated with relatively shallow energy minima in interparticle interactions. We demonstrated the existence of a critical particle number concentration, only above which is flocculation observed. We also showed that this critical particle number concentration can be related theoretically to the depth of the energy minimum. It may be, therefore, that if the authors were to vary the particle number concentration of their latex particles in the pH/Kf ion concentration region where the anomalous stability region occurs, they would find such a critical particle number concentration associated with reversible flocculation. If that were to be so then they could estimate the depth of the energy minimum involved, and perhaps decide whether the repulsive forces involved were of a relatively short range nature (i.e., associated with a relatively thin hydrated ion layer in the Stern plane) or of a longer range nature (i.e., associated with a polyelectrolyte type ‘‘ fuzzy ” layer at the surface).Prof. T. W. Healy (MeZbourne) said: The suggestion made by Vincent to examine the effect of particle number concentration is a useful one, which we shall pursue. - 2 -1 0 + I +2 APH FIG. 1 .--Coagulation results using the titration technique method C (see text of paper) for amphoteric The present studies using 1850 A diameter particles were conducted at ~ 0 . 0 1 % solids (by weight) (methods B and C), and of the order of 0.005% solids (by weight) for method A.Over this range, there was no effect of particle number concentration. The results shown in fig. 2 and 3 of our paper using Methods A and B can be compared with the results obtained by Method C (shown in fig. 1 of this Discussion Remark). The coagulation domains using all three techniques correspond very clearly to one another. latex colloids in KN03 electrolyte. ApH refers to (pH-pH,,,). J. A. Long, D. W. J. Osmond and B. Vincent, J. Colloid Interface Sci., 1973,42,545.GENERAL DISCUSSION 189 Dr. J. W. Goodwin (Bristol) said: We have also prepared amphoteric latices in Bristol (although zwitterionic would be a more correct description of this type of latex). However, a different synthetic route was chosen to the preparative conditions used for the latex described in your work, i.e., the addition of a cationic co-monomer to styrene and an anionic free radical initiator could result in the formation of bound poly- electrolyte and hence a swellable surface layer.Our route consisted of the addition of a mixture of anionic and cationic free radical initiators using styrene as the only monomer. In this way, only the end-groups to the polymer chains could be charged. With these latices we did not find that the coagulation was reversible or the very high surface changes (20-40 pUC cm-2) that you reported. As a result, I would suggest that both the ease of redispersion and the very high surface charge are evidence of a poly- electrolyte surface. It would be valuable to know how long the latices were held at a pH or electrolyte concentration corresponding to instability in the redispersion experiments.As we used a technique similar to your method B, ours were held in an unstable condition for up to 24 h before attempts were made to redisperse them. Prof. C. A. Smolders (Erzschede) said: I would like to point at the analogy of your coagulation diagrams for amphoteric latices and the stability diagrams for some blood proteins which we found in our work (see this Discussion, contribution by van der Scheer, et aZ.) Especially fibrinogen shows some remarkable effects in stability as a function of pH and salt concentration, such as a restabilization at high salt concentra- tion at the i.e.p. Dr. J. N. Israelachvili (Canberra) said: Since it is of some relevance to the work by Frens and by Healy et al.I should like to mention that our results on the adhesion of mica surfaces in a primary minimum show that the position and depth of a primary minimum depends both on changes in the long-range forces (e.g., double-layer forces) as well as on changes in some very short range (<1 nm) repulsive forces. For ex- ample, we found that in 1 : 1 electrolytes the adhesion energies (measured as described in my Discussion paper) were generally less in mol dmV3. This is readily accountable by noting that the double-layer interaction is at the same potential at these two concentrations. However, in mol dm-3 the adhesion ener- gies differed greatly in NaCl and KCl solutions, as well as at pH 5.5 and pH 7.0, even though the long-range (>2 nm) double-layer forces, van der Waals forces and " hy- dration " forces were unchanged.Further, weaker primary minima occurred at separations up to 8 A further out from stronger primary minima. These results appear to implicate a cation and pM specific " solvation barrier " whose range can vary by a few Angstrom beyond each surface, sufficient to alter a strong primary minimum into a weak or non-existent one. mol dm-3 than in Dr. J. Visser (Vlaardingen) said: I have two specific questions about applying the DLVO-theory 10 the clotting mechanism of milk. (1) How far does the known broad size distribution of the casein micellar system affect your results? For example, the van der Waals interaction you calculated is based on an average size of 100 nm, but we found for the fraction that sediments at 100 000 g had an average size of 250 nm, which would increase the van der Waals attraction by a factor of 2.5 together with a Hamaker constant that might be higher than the quoted value of J, so that the van der Waals attraction could be about seven times higher.In this respect it would be useful to J, let us say 3.5 x190 GENERAL DISCUSSION determine the Hamaker constant of your system by one of the well-known techniques in colloid chemistry independently, and to use those data for your calculation. (2) Secondly, theie is the question of the so-called free or serum k--casein dissoci- ated from the micelle into the solution. This free K-casein may be acted upon by rennin in the first instance, forming a positively charged species that could be respons- ible for the observed flocculation of the system, e.g., by a bridging mechanism.Have you considered this in the interpretation of your measurements ? In this connection I would also ask you about the need for calcium ions for the rennin action, and how far hydration of the micellar system plays a role in respect of the observed temperature effect. Dr. P. Walstra ( Wageningen) said: (1) For paracasein micelles you show both from calculation and experimentally a deep minimum in h?JM0, roughly 0.5. It is easily demonstrated, however, that the minimum can never become lower than (1 - f ) 2 , i.e., 0.92 in the present case. For fibrinogen no minimum is observed. (2) It appears to me that the DLVO-theory is not applicable to paracasein micelles.You calculate a repulsion maximum at about 0.5 nm distance, but the surface un- evenness of the particles is already of the order of 10 nm, and moreover at such small distances hydration forces and steric repulsion must already play a part. (3) In assuming p to be proportional to n2 [eqn (13)], you implicitly assume that two colliding particles must both have a “ hot site ” at the place of contact for coagula- tion. Though this is a reasonable assumption, other assumptions may also be reason- able, for instance that only one particle needs a “ hot site ”, or that an active site at one particle needs the splitting of more than one peptide bond. (4) Anyhow, it appears that n (i.e., the number of reactive sites on a single particle) increases with time [see also your paper, ref.(2)]. But this implies that eqn (1) is no longer valid and your calculation of activation energies would be wrong. I would agree on the existence of a “ steric factor’’ and on the observation that the strong temperature dependence of the flocculation reaction is caused by a change in activa- tion energy with temperature, but the actual values may be different. Moreover, the activation entropy (the “ steric ” factor) may change with temperature. Dr. D. F. Darling (Bedford) said: The logarithmic plot of clotting time against the reciprocal of the absolute temperature is depicted as a smooth curve which is indicative of a gradually changing activation energy as a function of temperature. Not all of the experiment points coincide with the line drawn, particularly at the highest temperature.We have measured the clotting line of milk over a range of temperatures and rennet concentrations and have observed that two linear relationships exist between log ( t ) and l/Twith a transition teniperature between 30 and 35 “C (see fig. 1). This has also been observed in more detail by Tuszynskil in his study of the kinetics of the enzymic and flocculation processes of rennet action on milk. If the process of flocculation is considered as a result of two consecutive reactions, enzymic followed by aggregation then the plot of log t against 1/T can be interpreted by simple reaction kinetic theory. At high temperatures where flocculation is very rapid the rate determining step is the enzymic process whilst at lower temperatures both reactions are important. The activation energy for flocculation in this tempera- ture range is a function of the sum of the individual activation energies but the logarithmic plot is still a straight line.It is therefore possible to determine the activa- tion energy of the enzymic phase and the aggregation phase from the (log t , 1/T) plot W. B. Tuszynski, Journal Dairy Res., 1971, 38, 115.191 31.0 32.0 33.0 360 35.0 I O ~ K / T ’ FIG. 1-Effect of temperature on the rennet clotting time of milk. 0, 0.5% rennet; 0 , 0.2% rennet. and there is no need to postulate a continuing change in activation energy (as impli- cated in fig. 3 of the paper) but simply a change in the rate determining step. I would question the need for so-called “ hot spots ” on a micelle before gelation can occur.The structure of a rennet gel on formation is very similar to that of a yoghourt-type gel where the structural building element is the casein micelle. In the development of a yoghourt gel the structure is formed by isoelectric precipitation of the casein in a quiescent state and the concept of specific hot spots becomes almost mean- ingless. Since very similar gels can be produced by two completely different processes the idea of hot spots does not necessarily contribute to an understanding of the gelation mechanism. Gelation only occurs in a quiescent state where particle-particle collisions are a result of Brownian motion. If the forces responsible for the particle stability are gradually removed then a point will be reached where the energy barrier for floccula- tion no longer exists and then aggregation takes place.If the particles are sufficiently concentrated such that collisions between particles occurs more rapidly than the re- orientation of particles within a floc then gelation is likely to occur. A precipitate will be formed when reorientation of flocs into a minimum energy configuration occurs at a rate comparable to that of the particle-particle collision frequency. Precipitation occurs for example in an agitated system where there is an added driving force for the reorientation of flocculated material. Prof. C. Smolders (Enschede) said : Could the enormous gap between flocculation rate constants as found from your experiments (a 105-106 cm3 mo1-l s-l) compared to simple diffusion controlled flocculation ( ~ 5 x 10’’ cm3 mol-’ s-l) be partially explained by a chance factor for the encounter of relatively small hot sites on the two molecules (e.g., area of a fibrinogen niolecule E 1500 nm2; active site on its surface z 1.5 nm’)?192 GENERAL DISCUSSION Dr.T. A. J. Payrens (NIZO, Ede) said: In reply to Visser, Walstra and Darling, the main conclusion of my paper is, indeed, that the relative stability of paracasein micelles cannot be explained by the DLVO theory, but is due, first of all, to a steric factor. The problem of interfering surface roughness I have discussed in ref. (2). The in- fluence of polydispersity on the DLVO pattern is, admittedly, not very well known. However, because long-range electrostatic repulsion and London-van der Waals attraction appear to play only a minor role in the stability our ignorance about this point is considered to be of no importance.In the computation of the potential energy curve the value of the Hamaker coefficient has been adjusted so as to yield a stable, native micelle and to account for its extreme sponginess. Normally, renneted micelles clot very slowly, and the rate of clotting can be en- hanced by adding calcium ions. Renneted rc-casein, however, clots also in the ab- sence of such ions. Calcium therefore appears not to interfere with the clotting mechanism itself, but only to decrease the overall stability level of the micelle through interaction with the other calcium-sensitive (asl - and p-) casein components.With regard to the remarks of Visser and Darling about the influence of the tem- perature on the rate of clotting, I would reply: (1) It is well known, indeed, that colloidal calcium phosphate shows an increased tendency to aggregate with increasing temperature,l but so does the protein constituent of the micelle on account of its hydrophobic association [cf. ref. (6)]. At present, it is not clear in how far each of these effects contributes to the observed temperature coefficient of the clotting. (2) The experimental data (cf. fig. 3 of my paper) suggest that the course of the clotting time with temperature is smooth and that there is no reason to distinguish between two different linear portions. Also no theoretical argument can be provided for the occurrence of two different activation mechanisms.Arising from Ball’s informal queries, negative activation energies could be ac- counted for in various ways. First, the computations presented in table 2 of my paper show that beyond - 10 mV the London-van der Waals attraction becomes dominant. Second, it is known [cf. ref. (6) of my paper] that the residue of para-rc-casein, which is left after the action of chymosin, carries a net positive charge. It is therefore conceiv- able that such residues could form an electrostatic bond with the remaining negative charges on the micelles surface. Which of these explanations is the right one, remains to be investigated. To complete my answer to Walstra I would add: (1) It is a simple matter to demcnstrate that the minimum molecular weight arrived at during the lag phase is proportional to the ratio V/k,. In model calcula- tions, such as presented in fig. 1 of my paper, care should therefore be taken to prevent exhaustion of the substrate, because the theory has been set up for constant V [cf. ref. (1) of my paper]. (2) I have computed the activation energies of the clotting process proper for two extreme cases of collision efficiency. In the first of these B accepted that, in principle, the whole micellar surface is available for enzymatic attack. Clearly we then have for the probability of successful collisions p -n2, YE being the number of hot sites on the surface. On the other hand, if the surface is saturated with respect to the enzymatic product, t’nenp is a constant. Intermediate cases may arise, if the R. M. Parry, in Fimdaineritals of Dairy CJwmistry, ed. B. H. Webb, A. H. Johnson and J, A. Alford (Avi. Westport, 1974), p. 608.GENERAL DISCUSSION 193 enzyme can also penetrate the micelle or the clotting is brought about through electro- static bond formation between positively and negatively charged surface sites. (3) With regard to your reniark about the constancy of ks, your conclusion that for polyfunctional substrates this cannot apply, is perfectly right. Some evidence for this obtained from the functional relationship between log (enzyme concentration) and log (clotting time) [cf. ref. (1) of my paper]. In practice, however, it is found that k, does not vary much during the lag phase [cf. fig. 7 of my paper]. I would draw Smolders’ attention to the data of my table 3 which show that retardation of the clotting is normally brought about by both a steric factor and an activation energy, which might be positive or negative. At 310 K, however, the activation energy vanishes and there we may hope to interpret the measured clotting rate in ternis of coupled translational and orientational diffusion. This is a complex problem that has not been solved in a general way as yet. Partial solutions can be obtained froin ref. (2)-(4) of this contribution. The applicability of these theories to the problem at hand is presently being investigated. B. Ribadeau Dumas and J. Garnier, J. Dairy Res., 1970,37,269. Solc and Stockmayer, J. Chem. Phys., 1971,54,2981; Int. J. Chem. Kinetics, 1973,5,733. Chou Kuo-chen and Jiang Shou-ping, Scientia Sinica, 1974, 17,664. Johnsson and Wennerstrom, Biuphys. Chem., 1978,7,285.

ISSN:0301-7249    

DOI:10.1039/DC9786500175

出版商:RSC    

年代:1978

数据来源: RSC

摘要:

Macromolecular Surface Phases and the Stability of Colloidal Dispersions BY ALEXANDER SILBERBERG Polymer Department, Weizmann Institute of Science, Rehovot, Israel Received 7th December, 1977 The effect of adsorbing a flexible polymer onto the surface of a system of colloidal particles in dispersion is considered from the point of view of polymer solution theory. The analogy between the free energy of mixing of polymer and solvent and colloid and solvent is stressed and the appropri- ate interaction parameters are defined. It is shown that the stabilizing effect of polymer adsorption (“ steric ” stabilization) on the free energy of mixing of the entire system is mainly due to a reduction (by adsorption) of the polymer concentration in the bulk solution. “ Steric ” stabilization is only rarely the thermodynamically preferred effect.In many instances where it is effective it is due to the persistence of a metastable condition. 1. 1NTRODUCTION Colloidally disperse systems are characterized by the large particle size of the solute component. Brownian motion is thus often inadequate to maintain homo- geneity, i.e., a uniform dispersion, when density differences are too large. Dispersions will separate, “ settle ” or ‘‘ cream ”, upon standing. In such concentrated phases energetically favourable interactions between particles will be of prime significance in inducing precipitate formation. Shielding the surfaces from each other, either by charging them up or by coating them with a diffuse film of adsorbed, flexible polymeric molecules, reduces the attraction energy per contact and stabilizes the system.Both these methods are designed to prevent close approach by the particle surfaces by using either the long range nature of the electrostatic field, or the interspersion of a “ gel ” layer of adsorbed polymer segments. The term ‘‘ steric ” stabilization has in particular been applied to the latter approach, a term not only vividly descriptive of what is supposed to occur, but also one which commits our thinking to this way of explanation only. If the polymer were indeed able to operate “ sterically ’’ it would be hard to under- stand why layers of the same polymer stabilized some systems and not others, why the polymer works only at certain temperatures and under certain environmental conditions and why the same polymer instead of inducing stabilization can in the long term produce flocculation.To understand what is going on it is necessary to relate steric excIusion to a thermo- dynamic argument. It will then appear that some “ sterically ” stabilized systems are absolutely stable in the thermodynamic sense and that others are only metastable. Some of these metastable systems may remain disperse, even for very long times, due to the very low rate at which surface-held macromolecules desorb or change configuration. In these cases ‘‘ steric ” effects do indeed provide protection even though thermodynamically these barriers should, and in the long run very often do, break down.A . SILBERBERG 195 In what is to follow the question of absolute stability conferred by adsorbed macromolecular layers is to be discussed.2. THERMODYNAMIC CONSIDERATIONS (i) FREE ENERGY OF MIXING OF COMPONENTS OF UNEQUAL SIZE The free energy of mixing (due to entropy considerations alone) of Nl molecules of volume V, and N2 molecules of volume Vz is approximated by AF = kT[N,ln ql 4- N21n q2] (1) where Pl = ~ l V I / V v == NJ, + NZV2 9 2 = N2WV P2 = VJV,. Expression (1) is the so-called Flory-Huggins equation and is generally assumed to apply to the mixing of linear polymer molecules with a low molecular weight solvent on a lattice only. In actual fact, as has been pointed out,l it applies to the mixing of any set of unequally sized components provided only that the shape of the particles is the same before and after mixing and that mixing occurs without volume change.Energy changes due to mixing have to be added to eqn (1); in order to do this the nature of the contact between the two components before and after mixing has to be properly assessed. For linear polymers at relatively high concentrations it is assumed that the interpenetration of the polymer chains is sufficiently high that a homogeneous mixture of polymer segments (segments of size equal to that of the solvent) results. Assuming in particular that, despite energy differences, this mixture remains random one obtains AF= kT"oln(1 - 93p) + ~plnvp, + ~ 0 9 P X P O l (2) where " o " is used to indicate the solvent molecule of volume V, and " p " the poly- mer molecule of volume V,, = PV,. In the above, qp is the volume fraction of polymer 9 p = PNPl(N0 + PN,) (3) and is the interaction parameter relating the energy level E,, of a solvent/polymer segment contact to the energy levels E,, and coo of a polymer segment/polymer segment and a solvent/solvent contact, respectively.It is assumed that each solvent molecule and each polymer segment make 2 contacts with their environment so that the mixture is characterized by Z(No + PN,)/2 contacts altogether. The contacts between the segments of the polymer chain which constitute the links tying the chains together are ignored in this accounting. Eqn (2) corresponds to the following changes in chemical potential It further predicts that the mixture will remain absolutely stable provided xp, I (1/2)(1 + P-1/2)2 (7)I96 MACROMOLECULAR SURFACE PHASES and that if xpo exceeds criterion (7) the system can form two phases, one dilute and one concentrated, with the critical concentration (at which phase separation first occurs) given by (pp = (1 + P1I2)-l.When P is very large the dilute phase is thus very dilute indeed. For very dilute solutions, however, eqn (2) is not a good representation of the mixing conditions. Most macromolecules are then isolated with a relatively high segment concentration in the coil space and zero concentration outside. An approach due to Flory is more appropriate.2 Coil volume changes are permitted and the free energy is optimized. The coil adopts that volume, by imbibing a sufficient quantity of solvent, where the sum of the free energy of mixing of the coil space and the free energy of coil expansion is minimized.This also corresponds to the point where the chemical potential of the solvent inside matches the chemical potential of the solvent outside (i.e., essentially pure solvent). Instead of eqn (5) we thus have Apo=O; pp*O (8) (9) and instead of eqn (6) AP, = It.T[lnpp + (1 - P) + P x p o l ; v p + 0. In very dilute suspension condition (8) will apply generally, i.e., also with respect to other structures involving polymeric diffuse phases capable of volume adjustment, such as binary, or higher clusters, or gels, or adsorbed surface layers in contact with a pure solvent phase. (ii) ADSORPTION OF FLEXIBLE MACROMOLECULES TO SURFACES As can be seen from eqn (9) even in dilute solution the chemical potential of the polymer is dominated by an essentially constant energy term when its size P is suffi- ciently large.Hence the removal of a polymer molecule from solution and its transfer to an interface, if contact per segment is in favour of the adsorbed state, even if only slightly, is a strongly preferred event due to the large number P of such segments per molecule. Creation of an adsorbed, highly concentrated polymeric layer is thus not at all difficult even in contact with very dilute solutions. The interaction with the surface produces a major distortion in mean macro- molecular shape to allow a high fraction, if not all segments, to make effective contact with the surface. Unless, however, the energy effect is very high the macromolecule will try to avoid a total two-dimensional collapse.Instead a diffuse, but concentrated, polymer surface adjoint phase is formed. The equation which governs the equili- brium of this surface phase has been derived and can, with sufficient precision, for present purposes, be written as1 ln[pB/(l - p ) ] - lapp = P[xs - 0.7 $- l/Ps + In(1 - 6) - ln(1 - qp) where 19 is the fraction of the surface which is covered by polymer segments, vB is the volume fraction (average) of polymer segments in the diffuse layer phase over the surface, p is the fraction of segments of an adsorbed polymer which contact the surface, P, is the mean length of the trains of adsorbed segments on the surface and + (0 + VJ3 - 2~,1XPOl (10) x s = (zJ/2)[(2&, - &a - a o ) - (2&p - Ep, - &aa)l/kT = ( ~ U / Z ) [ X ~ O - xmpl (1 1) where 2, is the number of contacts per surface site, cu0 is the energy level of the solvated surface, is the energy level of the surface in contact with a polymerA.SILBERBERG 197 segment, coo is the energy level of a direct surface to surface site contact and where, in analogy to eqn (4), xuo = (Z/2)[2&uO - - &,,]/kT; xop = (Z/2)[2&UP - ~ o u - C ~ J ~ T - (12) As eqn (11) shows the reference state for adsorption is the solvated surface, i.e., the state of the surface in contact with the pure solvent phase.l Balance between the two sides of eqn (10) requires that the expression in the square brackets be of order 1/P, i.e., be negligibly small as compared with 1. By way of approximation we can thus deduce that --dl - 6) = ks - 0.7 + I f P , ) + x,o(e 1- VB) (13) which is independent of both qp and P (if P is large enough and provided qp is small but not too small).Adsorption will only then occur if the parameter xs is large enough to make the r.h.s. of the equation positive and thus match -ln(l - 0). It is also seen that an increase in xpo [rendering the solvent a poorer solvent for the polymer by bringing the system closer to the criterion for phase separation, eqn (7)] also favours adsorption, as is indeed found experimentally. (iii) DISSOLUTION OF THE COLLOID PHASE If we ignore the problem associated with density differences and the gravitational field, the free energy of mixing of a dispersion of Nu colloid particles each of volume V, in No solvent molecules each of volume V, is thus again expressed by an equation of the form (2).Some thought must, however, be given to the question of the energy of mixing term. Let us suppose that each particle can bind M, molecules of the size of a solvent molecule and that each such interaction involves 2, contacts [see eqn (1 l)]. Hence if the colloid phase is precipitated, it forms NUM,ZD/2 contacts between its parts. The solvent phase without colloid has N 3 / 2 contacts so that the system as a whole involves (N0Mu&/2 + N0Z/2) contacts where 2 has been defined in conjunction with eqn (4). Assuming as before that contacts are randomly distributed, we can write At:= kT[N0ln(1 - qg) + Nulnqu + N,quxa] (14) is the volume fraction of colloid and where xa = ~~ozo/P17z)~uo. (16) (1 7) (18) (19) Hence [compare eqn (5)] Alto = kTCln(1 - Po) 3- Vo(1 - 1IPo) + xovzbl Apu = k m w 3 + (1 - @)(1 - Pu) 3- PaXo(1 - qcr))”].x u < (1/2)(1 + P 3 ) 2 . xu 2 x%m/4PCY3 and [see eqn (6)] This system will be stable provided Since Pu - Vo, Mu - V:I3 and Z / Z - 1/4, xu eqn (16) can be approximated by198 MACROMOLECULAR SURFACE PHASES and will be small unless x a o is particularly large. This [see eqn (12)] can easily occur when, as with many dispersions, cUu < EGO, ix., when there is a very strong tendency for the bare colloid particles to stick to one another rather than to solvent. (iV) DISTRIBUTION OF THE COLLOID PHASE IN THE PRESENCE OF A N ADSORBING POLYMER We shall now assume that in addition to the No colloid particles and No solvent molecules, the system contains Np polymer molecules which almost all become adherent to the colloid surface [i.e., xs (eqn 13) is sufficiently large] so that a small number NL of polymer molecules remain in solution where they produce a change in the chemical potential of the polymer phase [see eqn (9)] now of Here App = kT[lnqb + (1 - P) -1- PxpO].9; PNd/(Nd -1- PN;) where Ni is the number of solvent molecules which are not associated with the diffuse polymer segment layer over the colloid particle. The solvent chemical potential for reasons explained in connection with eqn (8) matches that of the pure solvent. Hence the correction to be applied to the free energy of mixing insofar as solvent and polymer are concerned amounts to N*App - N o b o = NpkTllnyl;, -t (1 - P) + Pxpol + NOkT[(1IPf7)% + (1/2 - xu)&] (22) where eqn (20) was used for App and eqn (17) for Apo.Since there is equilibrium between the adsorbed polymer phase and the bulk solution, the same App and Apo apply to all polymer molecules and to all solvent molecules. This includes, therefore, the energy change which results from coating the colloid surface with the polymer and from building up the diffuse surface phase. The fact that 9; is smaller than would be the concentration in the bulk without adsorption makes App more negative. Hence the effect of adsorption is to lower the free energy of the system to below where it would have been without adsorption. Adding eqn (22) to (14), expanding In( 1 - qu) and simplifying gives AF = RT{N,[ln& - (P - 1) + Pxpo] + Nf7[lnpu - (Pa - 1)(1 - 90) -t- PaXu(1 ~ n ) ~ ] ) .(23) It will be noticed from eqn (23) that the free energy of mixing indeed represents the sum (NpApp + N,Apu + NoApo) when App and Ap0 are given by eqn (20) and (18), respectively and Ap0 = 0. If v polymer molecules on the average are adsorbed per colloid particle N i = Np - NaV. Moreover, if a fraction P of the polymer molecules is adsorbed, so that andIt is then possible to rewrite eqn (23) in the form AF= kT(Np[lnpp - (P - 1) + PxpO] (26) The effect of adsorption on the free energy of the system is incorporated into the last term. All the other terms for a given mixture are constant. This last term is negative and increases in absolute magnitude as /3 -.B- 1 and as v increases. -1- N,[lnrp, - (Pa - 1)(1 - qa) i- Poxa(l -- rpd2 -I- (v/P)ln(l - PN>.3. DISCUSSION We have shown that for a given system the influence of the polymer on the free energy of mixing is embodied in the term (v/p)ln(l - p) and thus depends upon the two parameters v and p. In terms of the parameters already discussed we can write v = M,0fpP p = N , ~ ~ / N , (27) V‘ == 17 (1 - 90) /3’ == N ~ I ~ ’ / N ~ (28) if the entire surface of the colloid is available for binding or if only the fraction (1 - qg) of the sites, which are exposed to the solution on the average, are available for binding. In the former case the energy of mixing term in eqn (14) must also be modified and should be NoPoxn instead of Noqaxa. This is the case of total “ steric ” hindrance since the particle surface remains covered by polymer when the particles are in contact (i.e., through their polymer layers) and we must add kT [-No(Poxa + NrP,XG-l = kTN,[Pnx,q,] (total coverage) to eqn (23) to derive an expression giving the free energy of mixing of colloid, polymer and solvent where almost all the polymer is adsorbed and the colloid surface is now totally covered with polymer segments both in trains and in loops and tails.There are now no “ GO ” contacts. These have all been replaced by “ a p ” contacts. The ‘‘ surface ” of the colloid is now the surface of the adsorbed polymer layer. Each colloid particle is of size V, +- vVp and has some (vPZ - 41M,Z,) interaction sites. If we ignore the presence of unadsorbed polymer we can look upon the present case as that of a mixture of solvent and colloid particles whose size is PA =Po + vP instead of P, and whose interaction with the solvent is governed by x:, = [(VP - &ZCr/Z)/(Po + VP)IX,, instead of xu.becomes Hence the criterion for no phase separation, instead of eqn (19), or xpo .= (112) [1 4- ~ a / ~ o ( ~ / p ) ] [1 - z7/z(e/P)l-1 [1 + Pr?I2. (28) (no phase separation in total coverage) This criterion is similar to criterion (7) governing the stability of the polymer solution and as pointed out previously3 the present case of an “ irreversible ” cover of polymer molecules, produces a special extra large kind of polymer “ p ” which takes the place of “ G ” and governs the “ no phase separation ” criterion.200 MACROMOLECULAR SURFACE PHASES If polymer adsorption is reversible, desorption of polymers may occur when two particles are in contact.In that case eqn (26) represents the free energy but with V‘ substituted for v and p’ for p. Hence the difference in free energy of mixing between the case of total coverage and the case of partial coverage is given by ~ ~ ~ P a [ ~ O X o - (V/P%) + PP0/(1 --P)IW- (29) (partial + coverage) Therefore if (Z,X,o/Z> < [(v/~&3%1ln[l + PPCT/(l - P)1 (30) the case of total coverage is favoured. Clearly the larger xD0 the closer /? must approach 1 for the above to hold, and adsorption of polymer is particularly effective if the added polymer is almost totally adherent. The real issue in colloid stabilization is to prevent increase in the size of the colloidal particles, by growth at the expense of each other, at constant total colloidal mass.This, in terms of the parameters introduced here, implies growth of P, at constant P,N,. Such growth will occur spontaneously if (adbs/aPo)N,pO < 0. If the opposite is true, on the other hand, the particle size will reduce to the minimum consistent with the internal structure of the basic colloidal unit. Hence a stable dispersion requires: ( a A v a m N , P , 2 0. (31) ( ~ o x a o l a (v/M&v -P> + (3/M& ( V P O ) (32) Substituting eqn (26) into (31) then leads to (total coverage) in the case of total coverage. Comparison of condition (32) with (30), whichmust hold in this case, shows, however, that condition (30) is the sharper criterion and that if condition(30) holds condition (32) will also be fulfilled. If Pp0/(1 -p) 4 1 we can expand the logarithm in eqn (30) and find (Z*X~OlZ) (v/M*)/(l -PI - (v/M,Ipyl,/2(1 +02 BPol(1-P) < 1.(33) (tot a1 coverage) On the other hand, in the case of partial coverage AF is given by eqn (26) modified by the use of v’ instead of v and p’ instead of /?. Substituting this into eqn (31) then leads to ( ~ o x o o / ~ > (V/Ma>/[l -B(1 -%)I 1- t3/M0(1 -vo)lln(l/u70> (34) (partial coverage) and it is easily seen that for given v and P this is the weaker restriction on xoo. In terms of particle size, condition (34) can be rearranged to give p, {c3 ln(l/~O)l/[(zO~Ooiz) - (V/MJI(~ -%)13/2 (35) (partial coverage) noting that M, = P2I3. Criteria (34) and (35), with v=O, are the criteria for the stability of colloidal suspensions without added polymer. When polymer is added and (v/Ma), i.e.the number of polymer particles per surface site, increases, the critical value of P, increases and the adverse effects of large xoo on stability are overcome. When eventually criterion (33) is reversed and we are in the case governed by criterion (30), stability isA . SILBERBERG 201 absolute and there will be no tendency for the colloidal particles to grow. Note, however, that will in most cases of polymer adsorption be a small number of order P - l . The quantity OIp, the number of polymer segments per site, is usually found to range between 2 and 10. Only in cases of exceptionally strong terminal attachment, where pP N 1, will v/MO reach its theoretical limit of 1. Hence condition (33) is almost always the more likely situation and the conferment of absolute stability by polymer adsorption is orily possible in the exceptional case.Note that in the case of total coverage, discussed here, it is assumed that even after the adsorption of the polymer the colloidal particles " contact " each other only through polymer and solvent. In the case of partial coverage, on the other hand, the polymer film is totally missing over the contact area. Practically, however, the contact will most often be between these layers. It is easily seen that energetically this case lies in between the extremes discussed here. Allowance can be made for this effect and the result is a factor, less than unity, which multiplies xdo in relations (30) and (32) and thus eases them both. Where dispersions are prepared, say, by vigorous stirring, the contact time between colloid particles is minimal and coverage by the polymers will tend to reach com- pletion whether this is the thermodynamic equilibrium state or not.When stirring is stopped, contact time is increased and density differences will tend to enhance the colloid concentration in either the top or the bottom layer, thereby increasing the probability of binary encounters. In the case of polymers, however, the rate of desorption is much smaller than the rate of adsorption, so that it is not unlikely that no removal of polymer occurs for the period for which two coated particles are normally in contact. While apriori eqn (30) determines whether there is a tendency for particle growth or not, the system may remain " sterically " stabilized for considerable periods. Tf some desorption does occur during contact, the next stage may well be bridge formation by the transference of a train of adsorbed segments from one surface to the other. Such a bridge enhances the lifetime of the binary cluster. If this favours the approach to a lower free energy level, the next step will then be a gradual diffusion of polymer from out of the gap between the particles, and the creation of " 0-0 " contacts. While steric stabilization by polymer adsorption is thus not necessarily protective for thermodynamic reasons, it often functions and functions adequately, in a meta- stable situation. As long as the colloid particles remain effectively covered criterion (28) is operative. Note, however, that long range van der Waals attractions have been neglected in all these considerations. These indeed become less and less effective if an irreversibly adsorbed polymer coat prevents a very close approach. On the other hand, in the case of reversible equilibrium adsorption considered here, we feel entitled to ignore the long range effect against interaction between surface sites in their primary minimum. V/Md = 8/pP A. Silberberg, J. Chem. Phys., 1977,48,2835. T. L. Hill, Introduction f a Statistical Thermodynamics (Addison Wesley, Reading, Mass., 1960), p. 422. A. Silberberg, Progr. Colloid Polymer Sci., 1976, 59, 33. F. Th. Hesselink, A. Vrij and J. Th. G. Overbeek, J. Phys. Chem., 1971,75, 2094.

ISSN:0301-7249    

DOI:10.1039/DC9786500194

出版商:RSC    

年代:1978

数据来源: RSC

摘要:

Modes of Polymer Adsorption with Excluded Volume on Parallel Colloidal Plates and their Interaction BY SAMUEL LEVINE, MICHAEL M. THOMLINSON* AND KENNETH ROBINSON? Department of Mathematics, University of Manchester, Manchester M13 9PL Received 22nd December, 1977 By using a lattice model in which excluded volume and nearest-neighbour interactions are taken into account in the self-consistent mean-field approximation, a study is made of polymer adsorption on two parallel flat plates, which represent interacting colloidal particles. The segment density profile and free energy of polymers adsorbed on the two pIates are derived from a recurrence relation for the distribution of the last segment of a polymer chain. Various modes of adsorption of the polymers on the plates (tails, loops, bridges) are conveniently treated by formulating the recurrence relations in terms of matrices similar to those of Rubin and DiMarzio.In its linear form the recur- rence relation is equivalent to a one-dimensional self-consistent field equation of Dolan and Edwards. The interaction free energy between the adsorbed polymers is calculated in a range of plate separations for athermal and theta mixtures and far two adsorption energies of segments on the plates. 1. INTRODUCTION A theory of stabilization of colloidal dispersions by adsorbed non-ionic polymers requires a statistical mechanical treatment of polymer solutions which accounts for both exclusion volume effects and polymer-solvent interactions. Making use of a lattice model and introducing the above two effects as a self-consistent mean field approximation,1-6 a recurrence relation is formulated to determine the segment distribution of a polymer molecule which is situated between two parallel colloidal plates.In applying the mean field approximation to a lattice model, each lattice site is assigned a weighting factor for occupation by a polymer segment which depends on the local mean polymer concentration. This factor is determined by adapting a method due to Whittington7 for describing the configurations of a polymer in terms of a self-avoiding random walk on a lattice. Whittington’s theory for a single polymer chain is extended to include segment-solvent interactions and the effect of other polymer molecules located between the two plates. A matrix method similar to that of Rubin and DiMarzio8S9 is used to solve the recurrence equations for adsorbed polymer molecules situated on a cubic lattice which fills the volume between the two plates.Each polymer chain has a functional group at one end or both ends, which is attached to a plate to form a tail, loop or bridge and the other segments in the polymer may be assigned an adsorption energy next to a plate. The mean distribution of polymer segments and also the free energy of interaction between the polymers adsorbed as tails, loops and bridges on the two plates are obtained in a matrix form which can be evaluated numerically. The nearest-neighbour polymer-solvent inter- actions are expressed in terms of the energy of mixing parameter 01) as in the Flory- Huggins theory.l* By relating this energy o f mixing to their excluded volume para- * Present address : Doncaster Metropolitan Institute of Higher Education, Doncaster DN1 3EX.t Present address : Atlas Computer Laboratory, Chilton, Didcot, Oxfordshire.S . LEVINE, M . M. THOMLINSON AND K . ROBINSON 203 meter (u), the recurrence relation is found to be closely equivalent to the self- consistent-field diffusion equation of Edwards and Dolan and Edwards.I1 Just as in the work of these authors, the free energy expression used here avoids the artificial separation into entropic and mixing terms, characteristic of some previous r e ~ u 1 t s . l ~ ~ ~ ~ When applied to a single polymer chain, the self-consistent field method of Edwards1 has been criticized by Des Cloizeaux14 on the grounds that the fluctuations in the polymer segments are too large.In the case where a number of polymer chains are confined between two parallel plates and the density of polymers is in the “ semi- dilute ” region, where polymer chains can overlap, then it is claimed15 that the mean field theory of Edwards will still apply when the exclusion volume varies as (polymer concentration)l14. 2. SEGMENT DENSITY DISTRIBUTION I N THE MEAN FIELD APPROXIMATION We imagine the volume between the two parallel plates to be filled with a cubic lattice of sites, each of which is occupied either by a segment of a polymer molecule or by a solvent molecule. The cubic lattice is divided into A plane layers parallel to the plates, each forming a square lattice of sites.Denoting a particular layer by 5, where 5 takes the integral values 1 to A, we assign a weighting factor g(<) for occupa- tion by a polymer segment of any site in the layer. The end layers next to the two plates are designated by t = 1 and 5 = A. The form assumed for the function g(Q) will be discussed in section 3. Let ns be the total number of segments in a single poly- mer molecule. We attach a subscript n to g(t), which therefore becomes gn(5), to denote that the nth segment of a polymer molecule is situated in layer <, where n takes the values 1 to rz,. Consider a particular configuration of a polymer molecule in which segment 1 lies in layer l’, segment 2 in layer 2’, and so on, so that all the integers 1 ’, 2’, . . ., ni are restricted to the range 1 to A .In the mean field approxima- tion, the probability of this configuration is assumed to be proportional to the product g1(lf)gA2’) * + gn,(n3- (2.1) It follows that the probability of the configuration is where the symbol 2 denotes the summation over all configurations consistent with any constraints on the polymer. If we now sum over all configurations of the first n - 1 segments and also of the last n, - n segments of the polymer chain then we arrive at the probabilityp(<,n) that segment n is situated in layer 5, namely v n s In the sum g n - 1 of the numerator, n’ - 1 = 5 - 1, 5 or < + 1 and in the sum gn,-,, n’ + 1 = 5 - 1 , t or t + 1. Rubin and DiMarzio and Rubin used a matrix formulation on a lattice model to calculate the segment distribution of a single adsorbed polymer on one plate or204 POLYMER ADSORPTION WITH EXCLUDED VOLUME between two plates, in the random walk approximation.Here we shall use a similar approach for the case where excluded-volume effects and segment-solvent interactions are taken into account in the mean-field approximation. Consider now the portion of the polymer chain consisting of the first n + 1 segments. Let the first segment be located in layer 1 and the (n + 1)th segment in layer c. We introduce the sum of the weights of all configurations of these (n + 1) segments where n' = 5 - 1, 5 or < + 1. In the mean field approximation the unnormalised weighting factor q(&n + 1) for n + 1 segments is related to q(<,n) for n segments by where 5 = 2, . . ., A - 1.For the two layers t = 1 and 5 = A adjacent to a plate, we have These equations differ from those of Rubin and DiMarzio in two respects, Using the random-walk approximation they introduce weighting factors g( 1) = exp(6,) and g(A) = exp(6A) for the two end layers next to a plate, and put g(<) = 1 for the other layers < = 2, . . ., A - 1. Here 6, = el/kT, OA = EA/kT where cl and &A are the adsorption energies of a segment at the two plates, k is Boltzmann's constant and T is the absolute temperature. Secondly, they consider the whole of the polymer chain so that n + 1 = n,. Our derivation of an expression for the weighting factor g(Q depends on the imaginary process of adding the last polymer molecule, one segment at a time, to the cubic lattice in the presence of all the other complete polymer molecules, already introduced into the system.This means that in formulating the relations (2.5)-(2.7), for the n + 1 segments of the last polymer chain, the rest of this polymer, consisting of n, - (n + 1) segments, is assumed to be absent. Such an interpretation of (2.5)-(2.7) implies that g(<) depends not only on the location c of the layer but also on n. However for a sufficiently high concentration of polymer molecules adsorbed on the plates, we may neglect the dependence of g(5) on n. It should also be observed that the relations (2.5)-(2.7) imply multiple occupancy of a site by polymer segments. The correction for deviation from the random-walk model is entirely attributed to the weighting factor g(Q. In eqn (2.4), segment 1 is anchored to a given site on layer 1 whereas the site in layer c on which segment n + 1 is situated is not specified.The weighting factors cannot be regarded as probabilities because part of the polymer chain has been ignored in the relations (2.5)-(2.7). We need to introduce recurrence relations similar to (2.5)-(2.7) for the part of the polymer chain, consisting of n, - (n + 1) segments, which have been ignored up to now. It is convenient to reverse the ordering of the segments n + 1 to n, and to regard n, as the first segment of a chain of n, - n segments. We visualise the two parts of the whole polymer chain to share in common the (n + 1)th segment located in layer t. Formulation of the recurrence relations (2.5)-(2.7) in matrix notation, which we now describe, makes unnecessary writing down explicitly the recurrence relations for the second part of the poly- mer.S.LEVINE, M. M. THOMLlNSON AND K. ROBINSON 205 Introducing the matrix H and column vector q(n), given by n we can write eqn (2.5)-(2.7) as q(n + 1) = m4. (2.9) For a polymer chain of k segments, where the first segment is situated in layer 1, anchored to a given site on a plate, k - 1 repeated multiplication by the matrix H yields the relation (2.10) where (Hk-Ig)t is the sum of weights of all the configurations which start at the given site in layer 1 and end in the layer Q. For example, if k = 3 (2.11) On making use of the relation a general proof of eqn (2.10) with the interpretation of (Hk-'g)t stated above is reached by mathematical induction.Multiple occupancy of sites by segments, as illustrated by eqn (2.1 I), is assumed in the result (2.10), which was given by Rubin for his particular form of g(<). Now consider the expression (2.3) for the probability that segment n is situated in layer 5. Here (2.13) is the sum of the weights of all configurations of the first n - 1 segments of the polymer, given that the first segment is anchored to a definite site in layer 1 and the nth segment is on some site in layer 5. This means that we need only multiply the sum eqn (2.13) by g,(<) to obtain (H"-lg)C. Let us now reverse the numbering of the206 POLYMER ADSORPTION WITH EXCLUDED VOLUME segments n + 1 to n, and regard the last segment of the polymer n, as the first segment of a chain of n, - n segments.Then we can write 2 kn+l(n' + 1) - - gn,(nl)I gn(0 = (Hns-"g'k (2.14) Lns-,* I where g' is a column vector whose form will depend on whether the whole polymer chain, anchored in layer 1, has the configuration of a tail, a loop or a bridge. For these three categories, g' is given respectively by (2.15) [g, I g in eqn (2.10)]. Here the subscripts L and R refer to the left-hand and right-hand plates to which layers 1 and A are adjacent respectively. We have considered above tails (TL) or loops (LL) anchored to the left-hand plate, but they can equally be attached to the right-hand plate, when we use the labels (TR) and (LR). We distinguish five cases for which cqn (2.4) becoines where the last probability refers to a bridge and (Hns-'gL)c and (H"'-'&)t; are the sums of all the configurations of a complete polymer molecule which start at a site in layer 1 and A respectively and end in layer c.The appropriate vector g in eqn (2.16) is determined by the positions of the two ends of the polymer chain. Apart from additive constants, the Helmholtz free energies of a polymer molecule for the five cases are given by FLL = ki!k(H"s-lg~)l, FLR = -kkTh(Hns-'gR)A (2.17) It can be shown16 that our expressions (2.16) for p(Q, n) are equivalent to those of Rubin and DiMarzio .*s F B = -kTln(H"s - lg,)A. 3. THE WEIGHTING FACTOR g ( 0 We apply a method used by Whittington' to describe a polymer chain configura- tion in terms of a self-avoiding random walk on a cubic lattice. This approach will be extended to take into account segment-solvent interactions and the effect of other polymer molecules in a concentrated polymer solution.Consider the imaginary process of adding the last polymer, one segment at a time, to the cubic lattice. Each segment, brought from the pure polymer, replaces a solvent molecule which is removedS . LEVINE, h l . M . THOMLINSON A N D K . ROBINSON 207 to pure solvent. Suppose that the first TZ segments of the last polymer have already been introduced and that the (n + l)th segment is being added. Let @(r, n + 1) be the probability that insertion of the ( p z + 1)th segment occurs at a given site P of the cubic latice. The nth segment must be located at one of the six neighbouring sites Y + h, where h denotes one of the six vectors (&h, O,O), (O,&h,O), (0,0,&-1z). Let dE(r + h, n) be the change in the nearest-neighbour interaction energy between polymer segments and solvent molecules which results from placing segment n + 1 at Y, when it is known that segment n is at r + A.Then the generalisation of Whitting- ton’s recurrence relation is b(r, 12 + 1) -- A z @ ( r + h, n)[l -p*(rlr + h, n)] x exp[--dE(r + ti, i?)/kT]. (3.1) Here the sum 2 is taken over the six neighbouring sites to location r and A is a normal- isation constant determined by the condition that the (n + 1)th segment must occupy some site, i.e., I1 h C’(r, n 4- 1) = 1. (3 -2) Y Also p*(r[r + h, n) is the conditional probability that the site r is occupied by some other segment, either from one of the complete polymer chains or from the incomplete chain of n segments, given that segment n occupies site r + h.Let cPP, E, and cop be the interaction energies of the nearest-neighbour pairs segment-segment, solvent- solvent and segment-solvent, respectively. We introducep*:(r + h’lr + h, n; r, n + 1) as the conditional probability that some other polymer segment is situated at r + h’, given that segment n is at Y + h and segment n + 1 at r ; the corresponding probability when segment n is at r + h and a solvent molecule at r is p*(r + h’[r + Ih, n; r, 0). Then it can be shown16 that, apart from an additive constant which is absorbed into the constant A, AE(r + h, n) = (cPp - ~ , , ) z p * ( r + h’lr + h, n; r, n + 1) where the sum 2 is taken over the five nearest lattice neighbours of r, excluding the site r + h.The approximations made so far are the use of a cubic lattice and the restriction to nearest-neighbour interactions, but some ‘‘ closure ” procedure seems necessary to make further progress. A simple approximation is to assume that the occupation of r + h’ by a polymer segment is independent of occupation of r + h and r by polymer segments. Bearing in mind that on the cubic lattice r + h and r + h’ are not nearest neighbours, here the more serious assumption is the independent occupa- tion of sites Y + h and r. If which is the familiar interaction energy of mixing, and p*(r 4- h’) denotes the prob- ability that location r + h’ is occupied by some polymer segment, regardless of nearby occupations, then eqn (3.3) becomes A’ A€ = cop -- 3(&00 + cpp), (3.4) AE(r $- h, n) = --2AcZp8(r -+ hr).(3.5) ti Summing over the five terms in eqn (3.5) and ignoring the difference between AE(r + h, n) and AE(r), we replace eqn (3.5) by ARE@, n) == -lOAc-p*(r). (3 - 6 )208 POLYMER ADSORPTION WITH EXCLUDED VOLUME Making the similar approximation of equatingp*(rlr + it, n) top*@), eqn (3.1) reduces to fi(r,n + 1) = A[(1 - ~“(r))exp(llp*(v)>)~~(~ + A, 12) ( 3 3 A = lOA&JkT, (3.8) h where On averaging over all lattice sites in a plane parallel to the two plates, the set of eqn (3.7) takes the same form as (2.5) for 5 = 2, . . ., A - 1, with where p*(<) is the probability that a site in layer < is occupied by some polymer segment. If the adsorbed polymer molecules all have the same chain length and the same type of adsorption (distribution of tails, loops, trains and bridges) then (3.10) where n,, is the number of molecules per unit area situated between the two plates.For different types of adsorption the sum in eqn (3.10), will be divided into the various categories. When c = 1 or A, then eqn (3.9) is replaced by g(t) = (1 - P*(mexPwP*(o - Ew/kTl (3.1 1) where E, is the adsorption energy of a polymer segment on a plate. g(1) and g(A) in eqn (2.6) and (2.7) are given by eqn (3.1 1). The constant A in eqn (3.7) cancels ou On linearising the exponential factor in eqn (3.7) and expanding in a Taylor series, in P ( 5 , 4 * eqn (3.7) is approximated by h2 -- - -py(r, n) - (1 - A)[p*(r) - c p ( r , n)p*(r)@(r, n). (3.12) an r Edwards’ diffusion equation for a chain of n segments, in which the nth segment at r “ sees ” only the part of the complete polymer chain of length L = nh less than n,h, is obtained from eqn (3.11) on putting 1 - A = v/h3, where 21 is Edwards’ exclu- sion volume and assuming all other (complete) polymer chains to be absent.For an athermal mixture A = 0 and Y = h3. A theta solvent (random flight chain) is described in our theory by the value A = 1 which corresponds to a x value of 0.6 for the coordination number 6. The value x = 0.5 in the Flory-Huggins theory’O is obtained if eqn (3.8) is replaced by A = 12A&/kT, for which the following argument can be made. The assumption of independent occupation of adjacent sites by polymer segments made in arriving at eqn (3.5) means that definite knowledge of a polymer segment in one of the 6 neighbours of a given site has been lost, suggesting that the sum over 5 sites of p*(r + h’) in eqn (3.5) should be replaced by a sum over 6 sites.This changes the factor 10 to 12 in the expression (3.8) for A , and similarly the ratio 4/5 to 5/6 in eqn (3.11). The calculations in the next section are performed with these changes. The averaging process employed here has eliminated the original dependence of p*(Q on n, which facilitated the calculation of AE(r, n). 4. NUMERICAL RESULTS AND DISCUSSION The numerical method of calculating the mean segment density p * ( a on layer < will be described for the case of equal numbers n,/2 of polymer molecules per unit area, attached to each plate by an end segment as a tail. The total numbers ofS .LEVlNE, M . M . THOMLINSON AND K. ROBINSON 209 polymer segments and sites per unit area between the plates are nPn, and A/h2, respec- tively. As an initial guess, we choose an even distribution of polymer segments, so that p*(<) is a constant, equal to n,n,h2/A, the average number of segments per site. The eqri (3.9) and (3.11) yield g(<) for < = 1, . . ., A and hence the probabilities pTL(<, n) and pTR(<, n) are calculated from eqn (2.16) for each < and n. A new value of p*(c) is now obtained by using eqn (3.10), which reads and the iteration process is repeated until two successive values of p*(<) differ by < 10” for all 5. In fig. 1 - 3, we have plotted the segment density p * ( l ) at three plate separations d 48h, 24h and 12h for equal numbers of tails and loops on the two plates and for I \ FIG.1.-Plot of segment density p*(r) (fraction of sites occupied by polymer segments in a given layer) against e (distance from one plate in units of segment length h) for equal numbers of tails (polymer molecules) on the two plates and three separations between the plates d = (A + 1)h = 48 h, 24h and 12h. Full curves (I-VI) described an athermal mixture (A = 0) and broken curves (VII-XII) a theta mixture (A = 1). Number of segments in each tail n, = 432 and number of tails per site between the two plates n,h2 = 0.002. Number of lattice layers A = 47 for (I, IV, VII, X), A = 23 for (11, V, VIII, XI) and A = 11 for (111, VI, IX, XII). Adsorption energy E, = 0 for (1-111) and (VII-IX) and E, = -kT/4 for (IV-VI) and (X-XII).Vertical arrows mark median plane. For A = 47 only curves from left-hand plate to median plane shown. For A = 23 and A = 11, broken curves drawn only from median plane to right-hand plate.210 POLYMER ADSORPTION WITH EXCLUDED VOLUME FIG. 2.-Plot of p*(c) against t for equal numbers of loops (polymer molecules) on the two plates. Same chain length and concentration (number of loops) per site as in fig. 1. Roman numerals have the same meanings as in fig. 1. bridges between the two plates. In all cases, the chains have the same number of segments (n, = 432) and the coiicentration of tails, loops or bridges per site between the two plates is the same (nph2 = 0.002). Two values of A have been chosen, A = 1 (a theta mixture) and A = 0 (an athermal mixture) and two values of the adsorption energy of a segment on a plate, E, = 0 and E, = -0.25 kT.Further details are given in the legends. Dolan and Edwards l1 introduced x,, = h(n,/3)*, the component in any direction of the root-mean-square of the end-to-end separation of a random flight chain consisting of n, segments, each of length h and also a dimensionless excluded volume parameter u = 2.'n32,/2xO. This is related to our A by u = 0 for a theta mixture and u = 15.6 for n, = 432, nph2 = 0.002 and A = 0. The values u = 5 and u = 10 used by Dolan and Edwards are, therefore, intermediate between those considered here. They treat the adsorption of tails only and the dependence on adsorption energy E , is not considered. Figs.1-3 demonstrate that the density profilep*(<) is very sensitive to changes in E,, a result which is consistent with the findings of other author^.^*^^-^^ It is also observed that for the same E,, chain length and polymer concentration between the plates, the differences in p* (5)S . LEVINE, M. M. THOMLINSON AND K . ROBINSON 21 1 O.'* i FIG. 3.-Plot of ~ " ( 5 ) against 6 for bridges (polymer molecules) joining the two plates. Same chain length and concentration (number of bridges) per site as in fig. 1. Roman numerals have the same meanings as in fig. 1. Curve VII ( A = 1, A = 47, E, = - kT/4) very similar to curve I ( A = 0, A = 47, E, = 0) and therefore not shown. between tails, loops and bridges are not pronounced at the smaller plate separations. Indeed, a comparison of theta with athermal mixtures shows a more marked change.As the solvent becomes poorer (Le., as A is changed from 0 to 1) the polymer density near a wall increases. This is to be expected, since the poorer the solvent, the greater the tendency of the polymer to coil up. Note that we use the terms tail, loop and bridge to describe the permanent anchoring of one or both ends of a polymer by some functional group. The average number of segments in " trains " is represented by the probabilities p*(l) and p*(d), andp*(<) allows for a statistical distribution of The free energy expression corresponding to eqn (4.2) which is calculated here is " stlb-loops ". the difference. LW = 4%h2[wrL + %Id - (FTL + &R)dol (4.3) where do = 48h and similar energy differences are chosen for loops and bridges.Figs. 4 - 5 show APas a function of plate separation for the various density profiles in figs. 1 - 3 with the athermal and theta mixtures, respectively. Under similar conditions (same E , and d), AFis much larger in the athermal case, i.e., the repulsion between the plates due to polymer adsorption is much greater. Also AF is reduced as the polymer attachment to a plate changes from a tail to loop to bridge and as the adsorption212 0.4 i - POLYMER ADSORPTION WITH EXCLUDED VOLUME 3 0 d i h 35 0.04 0.03 5 =r U N D 0.02 -7 5 w 0.0 I - 4 10 d t rs 20 0 3 0 FIG. 4.-Plot of free energy change per lattice site from do = 48h to d as a function of separation d for an athermal mixture ( A = 0). Letters T, L, B denote tails, Ioops and bridges respectively and subscripts 0 and 1 adsorption energies 8, = 0 and e, = - kT/4, respectively.Distribution and con- centration of tails, loops and bridges same as in figs. 1-3. Left-hand vertical scale refers to curves with unprimed letters and right-hand scale to curves with primed letters. At T = 25 "C and h = 0.6 nm, multiply scales by 0.01 14 to express AF in J m-2. At h = 0.6 nm, area per polymer molecde between plates = 180 nm2. On B;, AF < 0 for d/h > 37. energy E , becomes more negative. attachment show a minimum AFm in AF when E, = -kT/4 (fig. 5). A& = -4.3 x Hamaker constant A = Indeed, in the theta solvent all three types of For a tail, J m-2 at d = 14h. For a segment length h = 0.6 nm and a J, this is comparable with the van der Waals energy for infinitely thick plates, AFv = - A/(12nd2)= -3.8 x J m--2.With the-bop, A m F R _ _ *n-6 f -3 . 7 4 ~ 7 , 1 i - , i A P( F t . . qn-6 T nr, = - 0 . i x i u r J m-- at a = 1111, to pe comparea witn fir, = 0.1 x 1 u - w J m-2. With the bridges, in all cases IAFmI exceeds IAFvI by at least one order of magnitude. The weighting factor g(<) is based on the assumption of independent probabilities of occupation by polymer segments of two adjacent lattice sites (r and r + h). By using scaling (renormalisation group) theory, Daoud et U Z . , ~ ~ have argued that with a good solvent (e.g., an athermal solution) in the semi-dilute region of polymer concentration, segments on adjacent sites repel one another (i.e., apart from the exclusion effect).Consequently they propose that, given a lattice site (r) to be occupied by a polymer segment, the conditional probability that a neighbouring site (r + 11) is occupied by another segment is proportional to p5l4 where p is the segment density, whereas we have assumed proportionality to p. Identifying p*(c) with p, this means p*(c) should be replaced by [ ~ * ( c ) ] ~ / ~ in eqn (3.9) and (3.1 1). Using the new form of g(c) we have calculatedp*(c) and AF for the athermal case A = 0. The alterations0.15 0.10 0.05 h c 2 0 Q r 4 c C a -0.05 S . LEVINE, M. M. THOMLINSON AND K . ROBINSON 21 3 0-0005 5 a- W h) *O t 5 Y --00005 '-0001 FIG. 5.-Plots of free energy change per lattice site as a function of d similar to those in fig. 4 for a theta mixture ( A = 1).Left-hand vertical scale refers to curves with unprimed letters, the inner right-hand vertical scale to curves T,, L;, Bi, and B; and the outer right-hand scale to curves Ti and f i. to the plots of p*(<) in figs. 1-3 are only moderate but the values of AF shown in fig. 4 exceed the new values of AF by a factor of 2 or more. However, we suggest that the particular power of p *(c) which describes the conditional probability in question will depend on the energy of mixing ALE. For A& > 0 a repulsion between adjacent seg- ments is reasonable but it can be argued that A& = 0 implies little interaction between such segments. In the case of a theta solvent Ae > 0 but since g(s) is less dependent on p * ( l ) , the choice between p*(c) and [p*(l)l5i4 is less significant. It would appear that away from the theta condition the form for g(<) will strongly affect the interaction free energy AF. We are indebted to the S.R.C. and Unilever Ltd for post-doctoraI research assistantships to M. M. T. and K. R., respectively. S . F. Edwards, Proc. Phys. Soc., 1965, 85, 613. H. Reiss, J. Chern. Phys., 1967, 47, 186. H. Yamakawa, J. Chem. Phys., 1968,48,3845. P. G. de Gennes, Rep. Prog. Phys., 1969, 32, 187. K. F. Freed, in Advances in Chemical Physics, ed. I. Prigogine and S. A. Rice, XXII, 1972. E. Helfand, Macromolecules, 1976,9, 307. ' S . G. Whittington, J. Phys. A. (Gen. Phys.), 1970, 3, 28. * R. J. Rubin, J . Chem. Phys., 1965, 43, 2392. E. A. DiMarzio and R. J, Rubin, J. Chem. Phys., 1971, 55,4318.214 POLYMER ADSORPTION WITH EXCLUDED VOLUME lo P. J. Flory, PrinclTples of Polymer Chemistry (Cornell University Press, Ithaca, New York, l1 A. K. Dolan and S. F. Edwards, Proc. Roy. SOC. A, 1975,343,427. l2 D. J. Meier, J. Phys. Chem., 1967, 71, 1861. l3 F. Th. Hesselink, A. Vrij and J. Th. G. Overbeek, J. Phys. Chem., 1971, 7§, 2094. l4 J. des Cloizeaux, J. Physique, 1970, 31, 715. l5 M. Daoud, J. P. Cotton, B. Farnoux, G. Jannink, G. Sarma, H. Benoit, R. Duplessix, C. Pic0 l6 S. Levine, M. M. Thomlinson and K. Robinson, to be published. l7 A. Silberberg, J. Phys. Chem., 1962, 66, 1972. l8 E. A. Di Marzio and F. L. McCrackin, J. Chem. Phys., 1965, 213, 539. l9 D. Chan, D. J. Mitchell, B. W. Ninham and L. R. White, J.C.S. Fizraday 11, 1975, 71, 235. 1953). and P. G. de Gennes, Macromolecules, 1975, 8, 804.

ISSN:0301-7249    

DOI:10.1039/DC9786500202

出版商:RSC    

年代:1978

数据来源: RSC

摘要:

GENERAL DISCUSSION Prof. D. H. Everett (Bristol) (communicated) : Steric stabilisation may be discussed thermodynamically in the following way, which differs somewhat from that pre- sented by Silberberg. In considering the effects of adsorption on interparticle forces, Ash, Everett and Radke assumed that, during an interparticle encounter, equilibrium was maintained between the adsorptive (2) in solution and in the adsorbed state. This led to the following equation relating the force,f, between parallel plates a distance h apart in the solution, to that,f:, in pure solvent (1): where I-'$') is the relative adsorption of component 2 with respect to 1 and p2 is the chemical potential of 2 in the bulk solution. In the case of " steric stabilisation " by adsorbed polymers, the time scale of a particle-particle collision is such that adsorption equilibrium is unlikely to be estab- lished, although the distribution of polymer segments within the adsorbed layer may adjust itself to the equilibrium form corresponding to the current particle separation.But in this case the chemical potential of the polymer (or of the individual segments) will be unrelated to that of polymer in the bulk solution. The collision will thus occur at constant adsorbed amount of polymer. A similar situation will arise if the polymer is chemically anchored to the polymer surface and the bulk phase may or may not contain dissolved polymer. Eqn (1) was derived by considering the difference between the Gibbs-p V-energy (9) of the system and that of a reference system (9") in which no adsorption occurs: Eqn (1) is therefore inapplicable to both these cases.d(9 - go) = -(S - So) dT + ,u: dnz + pz dnz -+- 2cr dA - Afdh (2) where dny and dnz are the surface excess amounts and t~ is the interfacial tension between the liquid phase and the solid surface. This was combined with the Gibbs- Duheni equation for the bulk phase (including the condition that py = py, pz = p;), integrated at constant intensive variables, and redifferentiated to yield an analogue of the Gibbs adsorption isotherm. Cross-differentiation of this led to of which eqn (1) is the integrated form. However, if the polymer adsorption remains constant and if the chemical potential of adsorbed polymer is unrelated to that in solution, this procedure cannot be followed beyond eqn (2).One can nevertheless cross-differentiate eqn (2) to obtain where ,& is the chemical potential of the adsorbed polymer. If nz is measured in terms of monomer units, pz may be identified with the chemical potential of a monomer S. G. Ash, D. H. Everett and C. J. Radke, J.C.S. Faraday 11, 1973,69, 1256.216 GENERAL DISCUSSION segment. When the polymer is in its equilibrium conformation, the segment chemical potential will be uniform throughout the polymer. Integration of eqn (4) then gives Introducing the surface concentration (a real excess) we obtain r2 = n;/2A, (const. T, h). of segments (5') Eqn (6) has a general form similar to eqn (1) except that the variables pz and r2 have been interchanged. Thus (a/f/ah)r, is the change in segment potential caused by change in plate separation at constant adsorption.The equation will apply both to the case in which adsorption equilibruim fails to be maintained during a collision, and to the case of chemically anchored polymer. The difference between the potential energy of interaction in the presence and absence of polymer at a separation h is ob- tained by integrating eqn (6) from 00 to h : 1: r, = o These arguments suggest, therefore, that theoretical calculations of steric stabilisa- tion by polymers should aim to evaluate the dependence of the segment chemical potential on ni and h: this will be determined by the adsorption field of the solid, the nature of the polymer and by polymer-solvent interactions. One wouId thus hope to incorporate all these effects into the theory at the outset, and the aesthetically unsatis- factory procedure of introducing successive effects additively at the end of the calcula- tion is avoided.In particular, eqn (7) emphasises that the potential energy difference has to be integrated with respect to adsorption. In many theories it is tacitly assumed that the potential energy difference is a linezr function of ut; so that the integral becomes [&(h) - p;(w)]T2: this will not be generally true. To illustrate the application of these equations, we consider a very simple model which, although it is unlikely to yield realistic potentials, nevertheless directs attention to some interesting fundamental problems. Consider two parallel plates a distance h apart, each carrying an adsorbed layer of polymer of thickness L, approximately given by a[rE2]*, where r is the number of seg- ments in the polymer, I the segment length and O! the expansion factor.Three distance regimes are considered : (i) for h < 2L: if the two polymer layers do not interact with one another, then ( However, van der Waals forces between the polymer layers, and between each polymer layer and the opposite plate, will cause pc to depend on h, even though the layers do not overlap. One will expect dispersion force interac- tion to lead to (aj&/lah),, < 0 and hence an attraction between the plates, in addition to that between the material of the plates themselves; (ii) for h < 2L: the two polymer layers make contact and, depending on the theory adopted will either compress one another or interpenetrate. The precise mechanism is not important for the moment: it is assumed that the segment density distribution readjusts itself so that the segment chemical potential is constant between the plates.vP(T2, h) - up@, h) = - ( f - . f o ) dh = 2 /21, bm) - P%41 d G . (7) ah), = 0 and f = fo.GENERAL DISCUSSION 21 7 In a very crude approximation we may assume constant segment density so that the average volume fraction of polymer segments between the plates is where v2 is the segment molar volume and r2 the segment adsorption. We can now follow Flory in writing the segment chemical potential as1 where q17 is the volume fraction of solvent (assumed for simplicity to have the same molar volume as a segment). Inserting eqn (8) into (9) gives Applying eqn (7) in the range 2L --w h gives the Contribution to the potential for separations between 2L and h, as ([u(r2, h) - v(0, h)] - [v(r, 2L) - v(0,2L)])lRT = where I’f now refers to the whole polymer adsorption.In the case in which no inter- action occurs between adsorbed layers for h > 2L, eqn (11) becomes All three terms are repulsive, but the potential is not linear in rf. The first term is the so-called osmotic term since the contribution to the force from this term is 2r2RT/h = RTc2, where cz is the mean polymer concentration between the plates. A similar term occurs in the equation derived by Dolan and Edwards: but inexplicably with the opposite sign, apparently giving an attractive contribution. Since L depends on the expansion factor a which itself depends on x, the whole potential depends on x, and not just the second and third term.The second term is called an osmotic term by Gerber and Moore: but it is more appropriately called a second osmotic virial coefficient term: it contributes a repulsive term when 31 < + (7), r - 1 and an attractive term when x > +(r+). When x = +( q), which when r 1 means under &conditions, the second term vanishes. A closely similar term occurs in the Dolan and Edwards theory2 [see Gerber and Moore,3 eqn (9)]. When the parameters of the two treatments are compared, the Dolan and Edwards result is identical with ours except for a factor of 2. In the theories of Meier and of P. J. Flory, Principles ufPolymer Chemistry (Cornell, U.P., Ithaca, 1953), p.513, eqn (34). A. K. Dolan and S. F. Edwards, Proc. Roy. Soc. A,, 1974,337, 509; 1975,343,427. P. R. Gerber and M. A. Moore, Macromolecules, 1977,10,476. D. J. Meier, J. Phys. Chem., 1967,71, 1861.21 8 C EN EK A 1, D 1 S C U SS I 0 N Hesselink, Vrij and Overbeek,l a term with a similar dependence on x and T2 occurs; but with a different distance dependence. At first sight, the last term in 1/h2 is similar to that of Dolan and Edwards; but their term is linear with Q, while in eqn (12) the potential difference depends on the cube of Tg. Clearly they arise from quite different sources ; (iii) for h < L : here the polymer chains on one surface, which for example under 0-conditions have mixed with those on the other surface without any mutual free energy change, come in contact with the opposing surface.This situation which cannot be handled within the simple Flory-Huggins assumptions, but requires (as it is getting) much more detailed study. Aspects of the change from regime (ii) to (iii), influenced by the dependence of M , and hence L, on x are discussed by Everett and Stageman.2 Prof. S. G. Whittington (Toronto) said: 1 should like to point out that scaling argu- ments due to Daoud and de Gennes3 also lead to a term in / z - ~ for the semi-dilute regime. If the dimension of the n-mer can be written as R, - n’ then we expect v to be 3/5 in the dilute good solvent regime and 1/2 in the semi-dilute regime. Assuming that dF(lz) - (R,/hX) and using the extensivity of the free energy gives dF(h) - h-litr. There is also evidence from series analysis4 that, for the dilute good solvent regime, there is a term in h-’.Prof. A. Silberberg (Rehouot) (communicated) : In discussions of steric stabiliza- tion, tremendous emphasis is placed on the forces between colloidal particles, almost to the total neglect of the main question involved: do these forces or do they not suffice to bring about phase separation or coagulation by aggregation. I find this very strange since a related colloidal problem, the solubility of polymer molecules, has been dis- cussed, extremely successfully, without such considerations ever having been brought into it. It is well known, moreoever, that almost all polyiiiers in solution are ener- getically attractive for each other and that it is entropic considerations alone which maintain solubility.Hence the concern with other colloidal suspensions as well should be in the overall picture. It is important to discuss either the overall free energy and the overall stability of the system to variation of its configurational parameter or if we do discuss chemical potentials (and other forces) then all the chemical potentials must be considered and referred to the proper reference background. There is almost always a finite concentration of polymer unadsorbed in solution and solvent chemical potential too has a fixed value. I thus agree with Everett’s main conclusion. I also agree with him the remainder of the way if it is our concern solely to measure the force between two approaching coated surfaces. In colloid stability, however, the colloid is a component of the sys- tem and the free energy must be treated as such.Eqn (2) of his comment is thus in- complete and the last two terms represent only a part of the colloid chemical potential. While Fisher’ and later Evans and Napper6 correctly emphasized the fact that adsorption of polymer converts colloid particles into special polymers of the same kind of mixing character as the polymer used in the coating, they failed to emphasize that even before adsorption the colloidal suspension could be treated in the same way F. Th. Hesselink, A. Vrij and J. Th. G. Overbeek, J. Phys. Chern., 1971,75,2094. D. H. Everett and J. F. Stageman, this Discussion, p. 230. M. Daoud and P. G. de Gennes, J. Physique, 1977,38,85. A. J. Guttrnann and S.G. Whittington, J. Phys. A, 1978,11, L107. E. W. Fischer, KoIfoid Z., 1958, 160, 120. R. Evans and D. M. Napper, Kolloid Z., 1973,251, 329.GENERAL DISCUSSION 219 as a polymer in solution. Moreoever, they did not treat the presence of unadsorbed polymer as part of the system. The Flory-Krigbaum theory used in these analyses is a very elegant approach to the dilute solution problem, but most colloid suspen- sions are not dilute. The same 0-point, however, is predicted so that treatments which deal with the colloid surface to colloid surface approach give an essentially correct result in this respect. Dr. I. D. Robb (Port Sunlight) said: An important contribution to the limit of the amount of polymer adsorbed (plateau level) at the solid/liquid interface in a good sol- vent comes from the osmotic repulsion between unadsorbed segments.Thus in good solvents and at saturation adsorption (where one normally looks for steric stability), for a polymer chain attached to one particle to be able to penetrate the unadsorbed segments of a polymer layer on a second particle, and adsorb onto that second particle, would require a high energy of adsorption between polymer segments and the solid surface. A low p value suggested by the final equation would correspond to weak energies of adsorption, this favouring steric stability. The final equation also suggests that stability should improve if P is small, i.e., as the molecular weight decreases, whereas general experience indicates the reverse, Prof. A. Silberberg (Rehovot) said : Protective action to achieve colloid stability is mainly directed at overcoming two problems (i) when the colloid is dispersed in a non-solvent or a poor solvent, (ii) when the colloid particles are so large that long range van der Waals attractions play an important part in producing their coagulation.Insofar as (i) is concerned, " protection " by a polymer coat apriori presents no par- ticular advantage. It is as effective as any other surface active material except that polymer adsorption occurs from solutions of much lower concentration. There exists, moreover, greater flexibility (in principle) in designing a material (a copolymer) which adheres well enough and is soluble enough so that its presence on the colloid surface will prevent phase separation and primary minimum coagulation. What my remark about@ aims to show is that it is advantageous to have the number of segments per polymer chain in actual contact with the surface as small as possible.Ideally, there- fore, pP = 1. Note however that the actual quantity involved is 8/pP = T/P, where r is the total number of polymer segments (adsorbed, in loops and in tails) per unit site, i.e., the number of equivalent monolayers. It is T/P, which should be as large as possible. While T tends to go up with molecular weight, so does p [in eqn (26)]. Hence, if protective action indeed uniformly improves as P increases to very large values, then this would be contrary to the present model. Insofar as (ii) above is concerned, I basically neglected the problem of long range van der Waals attractions.For these a displacement of the radius of contact of the colloid particles to larger values, because of steric repulsion, might be most important but the indications seem to be that effect (ii) is generally unimportant.' Poorly soluble polymers actually give the thickest and densest films (and lowest p-values), but their osmotic repulsion is also poor. Actually a thick layer of terminally covalently anchored, highly soluble chains, which do not adsorb to the surface, is best. The film will be thick, since the polymer chains are forced to adopt a " tail " distribution and osmotic repulsion is high. Dr. P. C. Scholten (Eindhoven) said: In paragraph (iii) Silberberg derived the ther- modynamic conditions for aggregative stability of a non-stabilized suspension.In this derivation the energy term was based on the assumption that in the aggregation process all particle-solvent contacts disappear. In practice, however, only a very R. Evans and D. M. Napper, KuZZuid Z., 1973,251,329.220 GENERAL DISCUSSION small fraction of the surface area is lost in aggregation. Consequently, only ex- tremely high surface energies can be responsible for aggregation; in other cases the van der Waals force has to be taken into account. The entropy term was taken from the Flory equation (2), which was derived for flexible molecules. For rigid particles, a mixing term based on the particle number fraction would be more appropriate. Prof. A. Silberberg (Rehovot) said: Scholten is, I am afraid, mistaken in both his remarks : (1) The energy term N,q,,x,, in eqn (14) assumes that solvent is expelled only over the area over which two particles come into contact.Only over this fraction of the surface is there colloid/colloid contact. The remainder of the particle surface remains coated with solvent and nothing of the particle surface is lost. (2) The Flory-Huggins equation was indeed derived for flexible chains but the result is gener- ally true for components of unequal size. The minimum assumptions involved are discussed in ref. (1) of my contribution. Dr. P. C. Scholten (Eindhoven) (communicated) : If I interpreted Silberberg’s paper correctly, the term Noq70xg in eqn (14) is the product of: (a) N,M,,Z,,, the total number of contacts with their surroundings which all particles together can make; (b) No/ (PUN,, + No), the fraction of these in which the partner is a solvent molecule [NoZ/ (N,,M,,Z,, + NoZ) would have been more appropriate]; (c) (24,,, - E,, - .~~,,)/2kT, the energy involved in replacing such a contact by particle-particle and solvent- solvent contacts. Could the author indicate in which one of these the factor occurs that determines the fraction of the particle surface involved in aggregation? Does this factor also affect, in some different way, the entropy term? I presume that the derivation in ref.(1) to which Silberberg refers is the one in the appendix, leading to eqn (A5). As the author stated, this was based only on first estimates. It seems to me that its universal apFlicability stems from the rather crude model involved.In the placement of the components on a grid, e.g., the first element of a particle (polymer molecule) was put down on an unoccupied site. The remaining elements of that particle were then placed in a contiguous volume V,,, regardless of whether at that place such a volume (with the appropriate shape) was still available. This may be correct in a dilute suspension, but it seems unrealistic for concentrated ones and for the pure particle (polymer) reference state. Prof. A. Silberberg (Rehovot) said: As Scholten correctly points out the estimate for the free energy of mixing is somewhat crude but not actually for the reasons he particularly selects. A packing assumption is indeed made. It is assumed that the same homogeneous contact exists between the particles (or polymer molecules) in the reference state and in the mixture. This presents no conceptual difficulty when the particles are compact spheres but may seem to be a problem for flexible, linear polymer chains as Scholten points out.It should be remembered, however, that the macro- molecules already on the lattice and the one to be placed are all undergoing conforma- tional change. Hence when wishing to place a selected macromolecule with its centre of gravity at a particular place, it will always be possible to wait for that moment in time when the arrangement of the environment of the selected point and the conforma- tion of the macromolecule to be placed are mutually compatible. In other words, even if at first the placement of a particular macromolecule at a particular location is sterically forbidden, it will become possible after a certain, average interval of time has elapsed.Hence, the only difference when working with macromolecules instead of spheres is an extension of the time scale (a reduction of the rate) at which the system can be populated from the reference state. For equilibrium this has no impact what-GENERAL DISCUSSION 221 soever. The configurational factor is thus unimportant (if it remains the same) and only the volume factor (the a priori placement probability factor) remains. In fact all particles which can be viewed or made up of substructures, held together and confined into an essentially spherical domain (which remains the same in the presence of the solvent) will pack homogeneously and conform to the Flory-Huggins equation.Excluded are particles which tend to favour ordered regions, say, rod-like particles which give rise to side to side aggregation, or particles having strong binding sites for each other. The energy of mixing term Nop,x, to which Scholten refers is conceived for essen- tially spherical particles mixing with solvent molecules. If there are at most 2 particles which simultaneously can near neighbour a central one to the total exclusion of solvent, there will be Zp, near neighbours on the average when the particle volume fraction is 9,. These make (Z,M,/Z)Zp, contacts of the e,, type per particle, leaving M,Z, (1 - 9,) contacts on the average to be of the E,, type. Hence there are +N,M,Z,( 1 - 9,) contacts in the mixture altogether which are c,,.Each pair of E, contacts is made from one E,, and one E,, contact. Hence the total energy of mixing is Noq7,x, remembering [see eqn (16)] that xu = (~,Za/P,Z~X,o xcro = (2/2)(2&,0 - Euu - &oo) where [see eqn (12)l. Two models are discussed in the paper: phase separation at constant particle size and particle growth, i.e., direct growth of Po (and A&). Both are helped if xoo in- creases, i.e., if the energy level &,, becomes more negative. In both instances the number of ,, contacts increases. Mr. J. M. H. M. Scheutjens and Dr. G. J. Fleer ( Wageningen) said : One of the main points of the theory of Levine et al. is that the probability of a chain in a given confor- mation is proportional to the multiple product of the weighting factors g(<) for all the individual segments of the chain.Each of these segment weighting factors is solely determined by the number c of the layer in which the segment is found. Through a modification of a well-known matrix procedure the probability p(& n) that the nth segment of a chain is in layer is expressed in the weighting factors g(<) for all c. After a summation of p(C, n) over all n, segments the segment volume fraction in layer c, p*(<), is obtained, again as a function of all weighting factors g(C). In order to evaluate g ( 0 , Levine et al. have adapted a method of Whittington. Their expression for g(5) [eqn (3.9) and (3.1 l)] can be written as { 1 - p * ( 0 ] exp {--u(O/kT}, where 1 - p*(<) is the solvent volume fraction in layer < and u(c) is the exchange energy for a segment and a solvent molecule in (.For all layers except the two in contact with the surfaces, the authors set -u(<) = 2xp*(c), where x(=A/2) is the Flory-Huggins poly- mer-solvent interaction parameter. For the two surface layers the adsorption energy E , has also to be taken into account. By solving numerically the resulting implicit equations in p*(<) for all 5, Levine et al. derive the segment concentration profile and the free energy of the polymer between two plates for the case that at least one of the ends is fixed on one of the surfaces. We have developed a theory for the adsorption of polymers along the same lines, leading to essentially the same results. However, our derivation of the equations is different and it is, in our opinion, an improvement of the theory presented by Levine et al.Moreover, we have elaborated the theory further, so that we can find not only222 GENERAL DISCUSSION the concentration profile and the free energy of terminally adsorbed polymers between two plates, but also the train, loop and tail./- size distribution for homopolymers. Calculations have been made for several types of lattices, for polymers between two plates as well as for polymer adsorbed on a single plate. The latter situation requires a sophisticated computing routine since the number of layers that has to be taken into account (and thus the size of the matrix) is much greater than it is in the case of two plates at relatively small separation. Full details of our theory will be published shortly. Here we point out the main differences with Levine’s theory and present a few typical results.The starting point of our theory is the derivation of the partition function Q{nc> for a mixture of polymer molecules in specified conformations (n, molecules in conforma- tion c, nd in conformation d, etc.) and solvent molecules in a lattice adjoining an ad- sorbing surface. In the derivation of Q{nJ we use the Bragg-Williams approximation of random mixing within each layer parallel to the surface. This approximation roughly corresponds to the mean field approximation used by Levine et al. By differ- entiation of In Q with respect to the number of chains in a given conformation we find the equilibrium properties of the system.The numbers of chains in each conforma- tion turn out to be a function of the weighting factors g(c) for an isolated segment. The expression for g ( 0 resembles that given by Levine et al., although it is not identical. We derive from the partition function that the polymer-solvent interaction term -u(<) = 2xp*(5) has to be replaced by a weighted average over the layers < - 1,< and + 1, in the following way: where A, is the fraction of nearest neighbours in the same layer, and L1 = (1 - &)/2 the fraction of neighbours in an adjoining layer. The physical background of taking this weighted average is, of course, that a segment in layer < interacts not only with seg- ments and solvent molecules in the same layer, but also with those in the two adjoining layers.In the further elaboration, our procedure is identical to that of Levine et al., including the use of the matrix formalism. However, we do not restrict ourselves to terminally adsorbing polymers and a cubic lattice, but make also calculations for hoinoyolymers and for other lattice types. Fortunately, only the quantitative detail is altered if another lattice type is chosen. Since the numbers of chains in each conformation are known we can obtain also the train, loop and tail size distri- bution. For a polymer between two plates our results are nearly identical to those given by Levine et al., and will not be shown here. Instead, we present a few results for the adsorption on a single plate. The data of fig. 1 apply to a siniple cubic lattice (Ao = 2/3, Al = 1/6; this is the same lattice as used by Levine et al.) and an adsorption energy parameter xs = 1 (xs = -q,,/kT - &x).The number of segments per chain (kz,) is indicated by the symbol Y, the segment volume fractionp*(<) by y ( Q ; the bulk volunie fraction (i.e., { - a) is denoted by v*. The adsorbed amount P i s expres- sed as the number of segments of adsorbed chains per surface site, the occupancy in the surface layer (0) is equal to ~ ( 1 ) . Results are given for a theta-solvent (x = 0.5) and for an athermal solvent (x = 0). Fig. l(a) gives adsorption isotherms with a logarithmic scale for the bulk volunie -1 We use the term “ tail ” for one of the ends of an adsorbed molecule, of which no single segment is on the surface. This concept is more generally used than Levine’s definition which considers a tail as the whole of a terminally adsorbed chain, even if other segments of the chain are also adsorbed.GENERAL DTSCUSSION 223 FIG.1 .-Summary of adsorption results for polymer adsorbed on a single plate. The results apply to a simple cubic lattice (A, r= 2/3) and an adsorption energy parameter xs = 1. Curves are given for athermal solvents, = 0, (broken lines) and for theta-solvents, x = 0.5, (solid lines). (a) Adsorption isotherms. Note that the equilibrium bulk volume fraction q* is plotted on a logarithmic scale. (b) Adsorbed amount as a function of chain length Y, at constant q*, (c) Surface coverage (= volume fraction in the first layer) as a function of Y. (d) Fraction of segments present in tails as a function of chain length.224 GENERAL DISCUSSION fraction p*.For low molecular weights Langmuir-type curves are obtained (this is shown more clearly if T is plotted against a linear scale for q*); for high r the adsorp- tion isotherms are of the high affinity character usually found in practice. As ex- pected the adsorption is higher for poorer solvents. The trends of fig. l a are similar to those of earlier theories,lB2 but a quantitative comparison is not easy. In fig. l(b), the adsorbed amount is shown as a function of molecular weight at constant p* . In theta-solvents r is proportional to log r, at least for higher molecular weights. Although calculations are not yet possible for r > 1000 (due to the high computer storage capacity required), there are no indications that T would level off at high r, which is predicted by Silbsrberg’s2 theory.This is probably due to the fact that the occurrence of tails (see below) is not properly accounted for in this theory. For good solvents we find a levelling-off for T at high r, in accordance with earlier theories.lP2 Fig. l(c) shows the dependence of the surface coverage 0 on molecular weight. At high chain length, 8 is independent of bulk volume fraction and of r. In that region, rstill increases with r, but the additional segments are accommodated in layers further from the surface. In theta-solvents the surface coverage is higher than in good sol- vents since in poor solvents the segments attract each other. The most interesting part of our results is given in fig.l(d), showing the fraction of segments in tails as a function of chain length. Nearly all of the earlier theories ignore the presence of tails, since end effects are expected to be negligible for infinitely high molecular weight. Hesselink3 has pointed out that for not too long chains tails may be important. Fig. l(d) shows that in the range usually encountered in practice (r = 100 to lOOO), tails indeed play a significant role, especially when the bulk volume fraction is not too low. This effect is bound to be of relevance for the application of polymers in stabilization or destabilization of colloidal systems, since a few tails in which up to 20% of the segments is accommodated protrude very far into the solution and strongly affect the free energy of interaction between colloidal particles.Starting from monomers (r = l), the fraction in tails increases until the onset of loop formation (at r N 4). Then the fraction in tails increases less steeply with increasing r or (at low p*) it even decreases. When the surface coverage 8 increases and approaches saturation the adsorbed amount still goes up and a higher fraction of segments is found in loops and tails. At high r increasing numbers of loops are formed. As the number of tails per chain can never exceed 2, eventually the fraction in tails decreases again with increasing r. However, this decrease is slow, indicating that the number of segments per tail increases nearly proportionally with r. This effect, together with the outcome (not shown in the figures) that in the region r = 100 to 1000 the average num- ber of tails per adsorbed molecule is only about 1, implies that a considerable fraction of the segments is found in tails relatively far from the surface.As stated above, this certainly has consequences for the stability of colloidal systems in the presence of polymers. Our method can also easily be extended to copolymers and heterogeneous surfaces. In forthcoming publications we shall deal with the structure of adsorbed polymer layers in more detail. The detailed shape of the curves of fig. l(d) is quite complicated. Dr. S . Levine (Manchester) (communicated) : It is gratifying to learn that by a dif- ferent method Scheutjens and Fleer (S.F.) have developed a theory for the adsorption C. A. J. Hoeve, J.Chem. Phys., 1966,44,1505; J. Polymer Sci., 1970, C-30,361. F. Th. Hesselink, J. Colloid Interface Sci., 1975, 50, 606. ’ A. Silberberg, J. Chem. Phys., 1968, 48, 2835.GENERAL DISCUSSION 225 of polymers similar to ours. I would comment as follows on the polymer-solvent interaction term --u(<) used by S.F., which for a cubic lattice reads expanding in a Taylor series. Thus S.F.s -u(t) differs only slightly from our formula 2xp*(5) unless p*(C) changes very rapidly between successive lattice layers. If one considers the form of p*(G n our fig. 1 to 3, then clearly the two formulae for -u(<) are virtually identical wherever p*(C) is nearly linear in 5. If we choose an example where p*(<) is definitely not linear, say curve I11 in fig. 1 and fit p*(<) empirically to a parabola, then we calculate that the second term in the above expansion for u(t) amounts to only 0.7% of the leading term.This difference becomes trivial when one examines the assumptions at the basis of the theory. Our original objective was to improve upon Edward's classical self-consistent field theory of the excluded volume effect. In particular, we sought an alternative method of relating his excluded volume parameter directly to the interactions between polymer segments and solvent molecules. Consequently, our results in section 3, particularly eqn (3.3) and (3.7) and the diffusion eqn (3.12), were obtained before section 2 was developed. The mean field approximation used in our work involves replacing eqn (3.3) by (3.6) where eqn (3.3) is exact for a cubic lattice model if only nearest-neighbour interactions are considered.In passing from eqn (3.3) to (3.6), we assume independent occupation by polymer segments of two neighbouring lattice sites which means that short-range correlation between segment occupations have been neglected, and this is the basic assumption in the mean field or Bragg-Williams approximation. I fail to understand how this correlation can be corrected for in a direction normal to the plates but ignored in a plane parallel to the plates. The mean field approximation implies that the factor g(<) depends on < only and there seems to be an inconsistency if g(<) explicitly depends on densities p*(< - 1) and p*(T + 1) in the two neighbouring layers as well as onp*(<).One way forward in polymer theory is to correct for the short-range correlations neglected in the mean-field or Bragg- Williams approximation and we have referred briefly to such an attempt based on scaling theory at the end of our paper. We have only considered the case where a polymer is anchored because it appears that in practice steric stabilisation usually occurs under such circumstances. The case of no anchoring is obtained by replacing gL and gR by 81. in eqn (2.16) and (2.17). Prof. A. Silberberg (Rehouot) said: Scheutjens and Fleer find a significant portion of the segments of the adsorbed macromolecules to be in tails when the bulk concentra- tion and the chain length is high. They correctly point out that earlier theories (in- cluding the paper of mine which they quote) did not take this into account.What they are finding, however, is the effect which I discussed in a later contribution, to Faraday Discussion 59 (p. 203, 1975). There I pointed out that the presence of segments near the surface, exerts a pull causing the adherent macromolecules to be less well adsorbed on the average, i.e., push more of the segments into the tail or tails, or, as I pointed out in this paper, shift the adherent macromolecules from out of configurations where the centre of gravity is close to the surface to configurations where it is further away. It is gratifying that the calculations of Scheutjens and Fleer con- firm this prediction. On the other hand, the authors should be careful before attribut- ing too much significance to the effect of tails in the molecular weight dependencies which they observe.Even when tails are ignored a strong molecular weight depend- ence results and only disappears at r-values some 100 to 1000 times larger than those226 GENgRAL DISCUSSION the authors have been able to consider. Molecular weight dependence should dis- appear at these high r-values in contact with very dilute solutions, a " prediction " which accords well with experimental observation. I also think that the authors overemphasize the importance of their expression for ~(5). If we make a Taylor expansion ofp* around one will find an expression differ- ing only from Levine's only by terms involving the second derivative of p*. Hence, the difference between the authors and Levine et al.will be important only where p* varies strongly with c, say between layers 1 and 2, where, however, Levine et al. introduced a specific term. Prof. G. H. Findenegg (Bochum) said: In connection with the paper by Levine et al. I should like to mention some experimental results which indicate that (i) the adsorption of linear chain molecules on flat surfaces is dominated by cooperative lateral interactions between neighbouring chains ; (ii) such cooperative effects do no exist if the chain molecules contain side groups. We studied the adsorption of long-chain n-alkanes (up to n-CS2HS6) from various solvents onto graphitized carbon (which is known to have a very uniform surface). The surface excess isotherms of these systems exhibit a positive initial curvature and a pronounced point of inflection before approaching a plateau corresponding to a close- packed monolayer of molecules disposed with their long axes parallel to the graphite surface.l The weak initial slope of the isotherms indicates that there is no strong tendency for replacement of solvent (e.g., heptane) at the surface by single long-chain n-alkane molecules.The steep increase of the isotherms at higher concentrations indicates an enhancement of adsorption by the presence of some long-chain molecules on the surface. Presumably there is a strong tendency for the paraffin chains to align parallel to each other at a flat surface giving rise to strong lateral interactions (as in the solid n-paraffins). The S-shaped isotherms of these systems can be accounted for by a simple lattice model in which the adsorbed polymer molecules are restricted to a single monolayer parallel to the solid surface2 and by choosing a high value of the inter- change energy parameter corresponding to A [eqn (3.8)] in the paper by Levine et al.It seems that multilayer models of polymer adsorption based on the mean-field approxi- mation do not account for this behavio~r.~ Would the authors comment on this question ? The shape of adsorption from solution isotherms is changed drastically by incorpor- ating side groups into the chain molecules. Thus, for example, the adsorption of squalane (2,6, 10, 15, 19,23-hexamethyltetracosane) from heptane is very much weaker than the adsorption of n-docosane and exhibits normal Langmuir-type iso- therms in dilute solutions. Clearly the methyl side groups prevent a close packing of squalane molecules on the surface.Obviously the side groups play an important role for the adsorption of polymers on substrates with uniform surfaces. Mr. J. M. H. M. Scheutjens and Dr. G. J. Fleer (Wageningen) (communicated): It is true that multilayer models of polymer adsorption cannot explain S-shaped adsorption isotherms, at least if the polymer-solvent interaction parameter x is taken to be independent of the segment volume fraction. Howeve, there are several experi- mental indications that x is a function of the polymer concentration. If x is low at low concentration and high at higher segment densities, one could imagine that with H. E. Kern, A. Piechocki, U. Brauer and G.H. Findenegg, Colloid Polymer Sci., in press. S . G. Ash, D. H. Everett and G. H. Findenegg, Trans. Faraday SOC., 1968,64,2639. S. G. Ash, D. H. Everett and G. H. Findenegg, Trans. Faradoy SOC., 1968,64,2645; 1970,66, 708.GENERAL DISCUSSION 227 increasing coverage of the surface the segments prefer other segments to solvent mole- cules in their environment. In combination with the adsorption energy, this could lead to a high segment density close to the surface, while the effect of x for the second layer is small due to the low segment concentration. Since in such a case the chains may assume a nearly flat conformation on the surface, a monolayer model might des- cribe the adsorption behaviour reasonably well. It would be interesting to make multilayer model calculations with a concentration dependent X-parameter.We agree that the absence of S-shaped isotherms for branched molecules probably has to do with the effect that these molecules can approach each other less closely than linear chains. This might be explained in terms of close-packing, but possibly also in a lower concentration dependence (or even a decrease with increasing concentra- tion) of the X-parameter. Dr. M. La1 and Mr. A. T. Clark (Port Sunlight) (communicated): Recent Monte Carlo calculations for a single 101 segment, excluded volume, chain, on a tetrahedral lattice, interacting with two parallel flat plates at a separation of 10 lattice layers, and with adsorption energies E , = 0, -kT/4, under athermal conditions produced the following results, table 1 and fig.1. The plot of total segment density, fig. 1 (composed of the loop, tail and bridge contributions) have their maximum values at the median plane for both adsorption energies, whereas in the calculations of Levine et al., there is a minimum for E, = It is possible that this is due to the different adsorption energies which would be -kT/4. 0.1 5 - c? v 0.10- 0.05 - I \ I I 2 3 4 5 6 7 a 9 1 0 1 1 FIG. I.-Plot of normalised segment density (f> against layer number h. Broken curve LIE, = 0, full curve As, = -kT/4.228 GENERAL DISCUSSION TABLE 1 E, = 0 E, = -kT/4 fraction of segments on the plates (trains) fraction of segments in loops fraction of segments in tails fraction of segments in bridges fraction of segments in unadsorbed chains mean number of trains mean number of loops mean number of tails mean number of bridges mean train size mean loop size mean tail size mean bridge size 0.05 0.10 0.57 0.11 0.17 1.85 0.6 1.5 0.3 2.5 17 38 37 0.10 0.22 0.44 0.21 0.03 3.4 1.7 1.5 0.65 2.8 13 27 32 required to produce a uniform segment density in the different lattice systems.(As reflected in the different critical adsorption energies for adsorption at a single surface.) This value of I --dew I being less than kT/4 for the cubic lattice, but greater than kT/4 for the tetrahedral lattice. Would Levine comment on this ? Prof. S. G. Whittington (Toronto) said : La1 and Clark compare their results with those of Levine et al. and find that, for a particular adsorption energy, the shapes of the distribution functions are different.One contributing factor is, as they suggest, that the lattices used are different. Calculations on the self-avoiding walk model for polymer adsorption at a single surface clearly show this effect,l and one would expect it to persist in the two surface problem. A second difference between the two sets of calculations is that La1 and Clark treat a single polymer molecule while Levine et al. consider a large number of interacting polymers, presumably at a concentration in the semi-dilute regime. These inter-polymer effects would be expected to have an im- portant part to play in determining the form of the distribution function at a particular value of e/kT. Dr. S. Levine (Manchester) said: Whittington has given a number of reasons why for a particular adsorption energy, the shapes of the distributions obtained by La1 and Clark from Monte-Carlo calculations differ from those obtained by us.1 should point out that at the smaller plate separations the total segment density in our work also has a maximum at the median plane. The minimum at the median plane occurs at the larger separation. La1 and Clark do not mention whether the ends of their polymer are anchored to the plates. This anchoring would tend to produce a higher concentra- tion of polymer segments near the plates and hence a smaller density at the median plane. It seems likely that this is the cause for the minimum. Prof. J. Lyklema ( Wageningen) said : In connection with the question of reversible or irreversible polymer adsorption it may be interesting to repeat a point, made in an earlier Faraday Discussion by Verwey2 that, phenomenologically speaking, steric See, e.g., G.M. Torrie, K. M. Middlemiss, S. H. P. Bly and S . G. Whittington, J. Chen?. Phys., 19 76,65,1867. ' E. J. W. Verwey, Disc. Faraday Soc., 1966, 42, 314.GENERAL DISCUSSION 229 interaction at polymer adsorption equilibrium is comparable with double layer inter- action at constant potential, whereas steric interaction without any polymer desorp- tion is on line with double layer interaction at constant charge. The quintessence is, that in the first case the adsorbed amounts are determined by the chemical potentials pi of the adsorbing species and the interaction of the second particle, whereas in the second case the relationship with pi is lost, the adsorbate relaxing too slowly to adjust itself during the encounter of the particles. Prof. A. Silberberg (Rehovot) said: The comparison is very interesting but an im- portant factor in determining the free energy change in the approach of two colloidal surfaces (whether protected by a double layer or by a polymer coat) is the effect on local solvent chemical potential. While overall free energy changes per pair of interacting colloid particles will be confined to kT (in either case) the changes in charge or segment redistribution actually adopted will be such as to keep solvent chemical pot en ti a1 balanced . Prof. A. Silberberg (Rehovot) said: The case treated by Levine et al. is of some con- siderable interest, but in the question of stabilization by polymer layers the assump- tion that the polymer is firmly anchored to the surface limits the discussion to the case of permanently, i.e., " irreversibly " attached polymers. Of interest in this case is the mean distance of closest approach in a medium consisting of solvent alone. One has to consider at what point of separation of the two opposing faces the solvent chemi- cal potential in the gap equals its overall value in the colloidal system. One may expect that fluctuations around this mean distance of closest approach are confined to free energy changes per particle pair which do not exceed kT. Only such an analysis has meaning in terms of defining the range of steric stabilization and the effectiveness of the steric barrier against coagulation tendencies. In the more general case of reversible adsorption the equilibrium approach of the two faces is (or can be) associated with polymer desorption, on the one hand, or bridge formation on the other, always in equilibrium with a solution of polymer of finite concentration. Treatment of this question, in the detail attempted in this paper, seems to be still outstanding.

ISSN:0301-7249    

DOI:10.1039/DC9786500215

出版商:RSC    

年代:1978

数据来源: RSC

摘要:

Preparation and Stability of Novel Polymer Colloids in a Range of Simple Liquids B Y DOUGLAS H. EVERETT AND JOHN F. STAGEMAN Department of Physical Chemistry, University of Bristol, Cantock’s Close, Bristol BS8 1TS Received 3rd January, 1978 This paper describes the preparation and properties of sterically stabilised polymer latex disper- sions in the lower liquid alkanes down to ethane, and in liquid xenon. They exhibit both upper and lower flocculation temperatures which for small latex particles and thick adsorbed layers correlate closely with the bulk phase properties of the stabilising polymer in the same liquid. The thermal range of stability of dispersions of larger particles with a thinner adsorbed layer is considerably reduced. These results are interpreted in terms of the temperature dependence of the dispersion force attraction and the change in adsorbed layer thickness as the @-temperature of the polymer is approached; the effect of particle size is accounted for by the dependence of attractive potential on particle size.Most work in colloidal dispersions has been concerned either with aqueous systems, or dispersions in relatively complex organic liquids. In the absence of a detailed molecular theory of the behaviour of such liquids, few quantitative compari- sons have been possible between experiment and theory. Recent progress in the theory of fluids has, however, led to a rather complete understanding of a wide range of properties of the atomic liquids formed by the inert gases.l This situation presents a challenge to colloid scientists to close the gap between experiment and theory by extending their studies to dispersions in simpler media, with the ultimate aim of working with inert gas liquids.This paper describes progress towards this objective. * In attempting to prepare new types of colloidal systems, conditions should be chosen to minimise attractive forces between particles and to create an appropriate repulsive barrier to flocculation. Polymer latices were chosen as the disperse phase, rather than inorganic oxides or carbon, to try to match as closely as possible the very low Hamaker constants of the experimental fluids to be studied. In media of low dielectric constant, charge stabilisation caused by ionisation of surface groups is un- likely to be effective, so that dipolar repulsion and steric stabilisation between adsorbed layers were considered as possible stabilising mechanisms.Preliminary experiments carried out by Dr. R. Bown indicated that the former approach was unpromising. The work reported in this paper was thus concerned with sterically-stabilised polymer latices of a kind which were already known to be stable in n-alkanes from hexane upwards.2 We, therefore, worked from hexane downwards. That working down the alkanes to the inert gases might be a successful strategy is suggested by a compari- son between some of the physical properties of the inert gases and the lower alkanes. In particular, the polarisability and critical temperatures of methane and ethane are similar to those of krypton and xenon, respectively.* We have just discovered that over 70 years ago Svedberg prepared colloidal dispersions of sodium and potassium in liquid methane and observed their flocculation (T, Svedberg, Nova Actn Reg. SOC. Sci. Upsaliensis, 1907, Ser. 4, vol. 2, p. 1).D. H . EVERETT AND J . F. STAGEMAN 23 1 Steric stabilisation by adsorbed polymers depends qualitatively on striking a balance between sufficiently strong adsorption of the polymer, which is favoured by low polymer solubility, and adequate extension of the loops and tails from the surface, which is favoured by high solubility. One method of achieving this balance is to use an ABA block copolymer of which the B component is strongly adsorbed while the A tails are soluble. One of the most flexible polymer chains is that of poly- (dimethylsiloxane) (PDMS), which is known to be very soluble in the lower alkane^,^ while poly(styrene) (PS) is virtually insoluble in hexane.These were therefore chosen as the A and B components respectively. The decisive factor in determining colloidal stability in reversible systems of the kind under consideration, which probably exhibit a secondary minimum in the inter- particle potential energy curve, is the depth of this minimum relative to the thermal energy. In most of the present work we have studied the interplay of factors influen- cing the depth of the minimum by varying the temperature and the nature of the dispersion medium. In addition, we have also studied the effect of particle size and molecular weight of the adsorbed polymer.However, in simple fluids another possibility presents itself. Since the attractive potential between two particles depends on the dielectric properties of the dispersion medium, which in turn are directly dependent on its density, the application of pressures greater than the vapour pressure of the liquid can affect both this potential and the solubility of the polymer chains. Preliminary experiments indicate that stable dispersions can be made in simple liquids up to and beyond the critical point. Thus, because of the high com- pressibility of fluids in and above the critical region, the possibility arises of studying the effect on dispersion stability of varying the density of the fluid phase at constant temperature and hence at constant kinetic energy. EXPERIMENTAL Stable poly(methylmethacry1ate) (PMMA) and poly(acrylonitrile) (PAN) latices were prepared in hexane by a dispersion polymerisation technique using as stabilisers PDMSIPSI- PDMS block copolymers. These stabilisers were synthesised separately by sequential anionic polymerisation of the monomers in 50% THF, 50% toluene as solvent.The reaction was initiated by dilithium naphthalene. The silicone monomer was hexamethylcyclotrisiloxane, which is known" to ring open under these conditions without the formation of appreciable amounts of homopolymer. The molecular weights and composition of two of the most efficient block copolymer stabilisers synthesised in this manner are shown in table 1. The dispersion polymerisations were free radical initiated and carried out in refluxing hexane.The larger latices were made by seeding and growing with slow addition of mono- mer. Smaller latices were obtained by a one-shot process in the presence of excess stabiliser and, as a consequence, they tended to be more polydisperse. All latices were washed free of excess stabiliser by repeated centrifugation and redispersion in fresh solvent. Details of TABLE 1 .-ABA BLOCK COPOLYMER DISPERSION STABILISERS. is the average number of polystyrene (B) segments per molecule based on BW. AT is one half the average number of dimethylsiloxane segments per molecule based on m.,, i.e., the average composition of segment A. code wt % [Si(Me)20] m PS DMS 3 1.9 x 104 2 x 104 42.5 PS DMS 6 - 3 x 104 49.0 110 60 145 100232 COLLOIDS IN SIMPLE LIQUIDS some of the latices used in this work are given in table 2.The core radii were estimated from transmission electron micrographs, at least two thousand particles being measured. The smaller latices were more difficult to measure because of the indistinct " fuzzy " layers, presumably of stabiliser (but not removed by further centrifugation and redispersion) which surrounded the more electron-dense core. Hydrodynamic radii were measured by photon correlation spectroscopy (PCS)5 using very dilute samples in hexane at 30 "C. This tech- nique was also used to investigate the change in layer thickness with temperature and also the thermally induced onset of flocculation of samples dispersed in propane. TABLE 2.-sOME LATICES PREPARED BY DISPERSION POLYMERISATION IN HEXANE.The letters in the latex code represent the core monomer whilst the k s t figure indicates which of the stabilisers shown in table 1 was used in the particular preparation; the second figure is the preoaration number. zero latex conc. flocculation latex average core average hydrodynamic temp. in propanePC code radius/nm radiuslnm upper lower MM31 111 f 14 123 f 6 56 3: 1 -82 & 4 AN61 85 f 8 110 f 5 64 Zt 1 -113 f 4 AN31 50 f 8 70 5 68 rt 1 -84 f 4 AN63 40 r t 6 65 f 5 76 f 1 -115 f 4 All the latices shown in table 2 have been stable in hexane for some two years. Latex MM31 could be freeze-dried to a powder which redispersed spontaneously in any of the lower alkanes from propane upwards.6 PCS confirmed that redispersion to individual particles had taken place.Freeze-drying of the smaller particle-size preparations gave materials which would not readily redisperse, and so for these latices concentrated stock dispersions were produced in pentane by centrifugation and redispersion. Thus experiments with latices, other than MM31, refer to systems containing small amounts of pentane used as the carrier medium. The thermal range of stability of MM31 and AN63 in a range of lower alkanes and, in the latter case in liquid xenon, was investigated. Known weights of solid in heavy wall glass capillary ampoules were frozen and evacuated, and a quantity of previously degassed medium distilled in and the ampoules sealed. Flocculation was observed visually at a cooling or heating rate of about 0.5 "C min-l. Despite the subjective nature of this observation, individual flocculation points at higher temperatures were reproducible to =k 1 "C; whilst at low temperatures the uncertainty was -+ 3 "C.Flocculation at low temperatures was always reversible; at high temperatures, however, the process was reversible only if samples were not dlowed to remain at or above the flocculation temperature for more than about 30 min. Cloud-point curves for various molecular weights of PDMS in ethane, propane and xenon were determined in the same manner as the flocculation boundaries. The polymer samples (Aldrich Chemicals) were used without further purification or fractionation. The manufacturers' calibrations of these polymers are given in table 3. Individual cloud points could be reproduced to well within k0.5 "C for adequately mixed and equilibrated samples.TABLE 3.-MANCJFACTURERS' CALIBRATION DETAILS OF PDMS SAMPLES USED IN THIS STUDY. code 01 7.74 x 104 3.08 x 104 2.5 02 1.66 x 105 4.72 x 104 3.5 03 6.09 x 105 1.5 x 105 4.1D. H . EVERETT AND J . F. STAGEMAN 233 The ethane, propane, butane and xenon were all 99.9% pure, C.P. grade obtained from Matheson Chemicals. RESULTS As reported previously,6 dispersions of MM31 exhibit both an upper (UFT) and a lower (LFT) flocculation temperature when dispersed in hexane, pentane, butane and propane. These all lie in the normal liquid ranges of these alkanes and depend slightly on the polymer latex concmtration. Linear extrapolation gives the UFT and LFT of an infinitely dilute latex. These are shown as a function of carbon number of the liquid in fig.1. Also shown are the critical temperatures of the liquids and 1 2 3 t 5 6 no. of carbon atoms FIG. l.-o, UFT and LFT values of latex MM31 extrapolated to zero concentration as a function of the number of carbon atoms in the alkane chain; @, dispersion medium critical temperatures; x , &temperatures for PDMS in the saturated liquids, extrapolated from our own and previous phase separation data?.'. estimates of the O-temperatures of poly(dimethylsi1oxane) in the liquids made on the basis of Patterson's cloud-point data,3s7 extrapolated to infinite molecular weight, and corrected to the saturated vapour pressure of the liquid. The thermal range of stability of the dispersions decreases with decreasing molecular weight of the liquid, and although extrapolation of the curves would indicate a small range of stability in ethane, it was not possible to obtain stable dispersions of MM31 in this medium.The upper flocculation boundary lies 10 - 40" below the critical temperatures and rather less below the O-temperatures in the various solvents. Decreasing the size of the latex particle and increasing the molecular weight of the stabilising layer has a profound effect on the stability characteristics. The data for MM3 1 and AN63 in propane are compared in fig. 2. The combination of smaller overall diameter and thicker stabilising layer extends the range of stability at both the upper and lower boundaries. That this is not caused by the presence of some pentane introduced with latex AN63 is shown by the fact that the extrapolated values234 / l l , ' l , , , , COLLOIDS I N SIMPLE LIQUIDS -,20 , II 1 I 1 1 0 0.5 1 .o 1.5 FIG.2.-UFT and LFT values of latices: 0, MM31; @, AN63, as a function of concentration in saturated liquid propane. of the UFT at zero particle concentration (and hence zero pentane contamination) are different. The change in the slope of these graphs from a small negative to small positive slope may, however, be caused by the presence of pentane. In fig. 3 the stability characteristics of AN63 in propane are compared with the W t 'fa wt '/a 0 4 8 12 16 0 0.28 0.56 10 20 pen tans ii I k t 8 2 wt % ID. H. EVERETT AND J . F. STAGEMAN 23 5 phase behaviour of a sample of poly(dimethylsi1oxane) of m.w. 77 400 in propane.The polymer shows a lower consolute temperature of 79°C at a concentration of about 6 wt %. Patterson’s estimates of the LCST values of two higher molecular weight polymers are indicated by the arrows marked (1) and (2). In this case, the UFT extrapolated to zero particle concentration lies quite close to the LCST of the highest m.w. polymer, and about 3” below the LCST of polymer measured in this work. The search for an upper consolute temperature for the polymer solutions revealed that the lower boundary of phase stability was a line corresponding to the separation of solid polymer, which was approximately horizontal (& 5” in the range of concentra- tions studied). The LFT for AN63 lies a few degrees above this temperature. 5 e a c d- s (11-3 phase J . ) I , 1 1 1 1 , 1 1 1 wt 8.2 0.3 2 4 pentane wt % 2 0 -2 -1: I I I - -0- - - a -110 -120 -130 (0) polymer ( b ) latex FIG.4.-(a) phase diagrams of PDMS, (i) Mw = 7.7 x lo4, (ii) MW = 6.1 x 10’; (b) UFT and LFT values of latex AN63 in saturated liquid ethane. The arrow (1) in (a) indicates the LCST of polymer ATw = 6.3 x lo5 extrapolated to our experimental conditions from Patterson’s re~ults.~ In (b) the abscissa scale indicates both particle concentration and weight percent of pentane in the liquid medium. The wider range of stability of AN63 dispersions was also reflected in the fact that stable dispersions could be prepared in liquid ethane. Fig. 4 shows both the latex stability diagram and the polymer solution phase diagram for two samples of polymer of m.w. 7.7 x lo4 and 6.1 x lo’.Again the arrow indicates Patterson’s estimate of the LCST of PDMS, 3n.w. = 6.3 x lo’ in ethane, which lies close to the infinite dilution UFT, but is lower than the LCST for a similar polymer studied in this work. In this case also the polymer solution precipitates polymer before an UCST is reached, and the lower limit of latex stability lies a few degrees above this temperature. The dependence of the UFT on latex particle concentration (and hence on the presence of pentane in the medium) for this system was some five greater times than for dispersions in propane (note different scales on abscissae of fig. 3 and 4); further experiments are needed to separate the effects of particle concentration and pentane concentration. Ethane and xenon have several properties (e.g., polarisability) of about the same magnitude, so it is not surprising that stable dispersions of AN63 in xenon were236 COLLOIDS I N SIMPLE LIQUIDS readily prepared.The data are shown in fig. 5 which also shows the phase diagram for three polymer samples (m.w. 7.7 x lo4, 1.1 x lo5 and 6.1 x lo5) in liquid xenon. These are believed to be the first examples of polymer solutions and the first stable colloidal dispersions in liquid inert gases. The zero particle concentration (and zero pentane concentration) UFT lies only 1.5" below the LCST of the highest m.w. polymer solution. In contrast to the alkane systems the phase diagram of the polymer solution is cut off by the freezing point of xenon, rather than by the separation of the (01 polymer ( b ) latex FIG.5.-(u) phase diagrams of PDMS (i) = 7.7 x lo4, (ii) 1.7 x lo5, (iii) 6.1 x lo5; (b) UFT and LFT values of latex AN63 in saturated liquid xenon. The lower boundaries of polymer solution and dispersion stability occur close to the melting point of xenon. In (b) the abscissa scale indicates both particle concentration and weight percent of pentane in the liquid medium. solid polymer. Similarly the dispersion solidifies without flocculation at or just below the freezing point of xenon, presumably dependent on the small quantities of pentane present. In a preliminary experiment to investigate behaviour at higher temperatures a sample of MM31 latex in propane was introduced into a specially designed pressure cell and studied at pressures up to 150 atm.Under these conditions the UFT was raised from 76 "C to above the critical temperature of propane. Further work at high pressures is in hand. Another observation which may be significant is that Latex AN63 was stable in the pure liquid polymer PDMS of molecular weight about 20 000. DISCUSSION The main features of our experimental results are (i) the existence of both an UFT and LFT in all the systems studied except for those in xenon where the liquid froze before LFT was reached; (ii) the close correlation between the UFT of dispersions of the smaller latex with the LCST of solutions of PDMS of high molecular weight in the same solvent; (iii) the much smaller range of stability of dispersions of the larger latex bothD . H. EVERETT AND J . F .STAGEMAN 237 with respect to temperature and to the range of dispersion media in which it is stable; (iv) the correlation of the LFT of the smaller latex with the temperature at which solid polymer is precipitated from solution in the same medium. Flocculation temperatures close to the supposed &temperature of the stabilising polymer in the medium concerned have been observed in a considerable number of more complex systems by Napper.8 No studies have been previously reported on the effect of particle size on the colloid stability diagram, and only one other system has been reported which has both an UFT and a LFT.9 FIG. 6.-Potential energy, Y(L), diagrams to describe the total interaction between two sterically stabilised particles as a function of separation distance, L.The repulsive potential is taken to be an infinitely steep barrier which prevents very close approach. (a) Stable dispersion; (b) flocculated dispersion, the instability being caused by (i) decrease in range of repulsive potential, (ii) collapse of polymer layer with increase in concentration and consequent increase in depth of the potential energy minimum; (c) flocculated dispersion-the instability being caused by an increase in attractive potential and the consequent increases in depth of the total energy minimum. For cases (6) and (c) the location of the original potential (case a) is shown by the dashed lines. The flocculation of these dispersions is probably a secondary minimum pheno- menon, and we assume for the purposes of a preliminary discussion that the steric repulsive barrier is steep so that the mutual potential energy of a pair of particles can be represented schematically by fig.6(a). Increase in the depth of the minimum, sufficient to cause flocculation, may be envisaged as arising either from a decrease in the range of the steric repulsion [fig. 6(b)(i)] or from an increase in the attractive potential [fig. 6(c)]. We believe that our measurements provide evidence for both these effects, although they do not operate independently: reduction of the thickness of the steric barrier must involve a change in the local density of polymer within the stabilising layer and we show below that this also increases the attractive potential [fig. 6(b)(ii) 1. According to current theories of polymer solutions the LCST and UCST of a solution of very high (in principle infinite) molecular weight identifies the &tempera- ture at which, because of an exact balance between the excluded volume effect and the net segment-segment attraction, a polymer chain of N links adopts an ideal random walk configuration with a root mean square radius of gyration ((R2)*ideal) propor- tional to N".In the one-phase region, the chains are more fully extended while beyond the 6-temperature they collapse and the favourable segment-segment inter- action leads to phase separation. The action of the repulsive barrier can now be interpreted from two alternative, but essentially equivalent, points of view. Since at the LCST of a polymer solution a polymer-rich phase separates out, so at the UFT the surface-attached polymer chains238 COLLOIDS I N SIMPLE LIQUIDS also “phase separate” from the dispersion medium and collapse on the surface.Alternatively, below the UFT the excluded volume effects dominate so that in the inter-penetration of adsorbed polymer chains attached to approaching surfaces the entropy effect is predominant and a repulsive barrier (possibly a relatively “ soft ” one) is established at a distance of approximately 4 x (R2)+’. At the &temperature the polymer chains can inter-penetrate freely, the entropy and energy terms just cancelling. Theoretically, no effect is felt until the surfaces are at a distance at which the con- figurations of polymer molecules attached to one surface are influenced by the presence of the other surface.This interaction, with and without excluded volume effects, has been discussed by Dolan and Edwards lo and by Middlemiss, Torrie and Whitting- ton? The range of this interaction will be approximately 2(R2>3idesl which is less than one half of that away from the @-temperature. Again this reduction in distance at which the steric barrier becomes effective may be sufficient to allow the attractive potential to form a sufficiently deep secondary minimum to cause flocculation. That the UFT is correlated with the LCST of polymer extrapolated to infinite molecular weight, rather than that of polymer of molecular weight more nearly equal to that of the stabilising chains, indicates, as suggested by Osmond et a2.,l2 that because of the severe reduction in translational entropy caused by terminal attach- ment, an adsorbed polymer chain behaves as though it were part of an infinite chain.The LFT is seen as a consequence of the effect of decreasing temperature leading to dominance of adsorption forces, and segment-segment attraction, and again to collapse of the polymer chain on the surface: this process is then analogous to the precipitation of solid polymer from the bulk solution. However, although the correlation between dispersion stability and bulk polymer solution properties is observed with Latex AN63, the latex of larger particle size, and with lower molecular weight stabiliser (MM31) has a much narrower stability range. One possible explanation is to be sought in the view that the UFT is more correctly identified with the temperature at which the second virial coefficient of the solvent activity vanishes (i.e., it is analogous to the Boyle temperature of a non-ideal gas).Arguments have been adduced13 to show that this situation is reached before the phase separation temperature, and that it depends on polymer molecular weight. The lower UFT of MM31 dispersions would then correspond to the lower m.w. of the stabilising chains. The limited evidence so far available (table 2) suggests, however, that the UFT extrapolated to zero particle concentration correlates more closely with the particle size than with the m.w. of the stabilising polymer. Consequently we believe that the difference in behaviour between AN63 and MM31 is to be sought mainly in the dependence of the attractive potential on particle size.To investigate this we have used the Lifshitz14 method to calculate the van der Waals attraction energies between particles carrying adsorbed layers ; this method is preferable to that due to HamakeP since it enables a fuller account to be taken of the effect of temperature, which in the Hamaker method comes in only through the dependence of the Hamaker constants on the density of the media involved. The available dielectric data, given in table 4, were used with theoretical relaxation func- tions in a manner similar to that used by previous authors.16 The fluids used have particularly simple dielectric functions which are, however, very temperature sensitive, especially in the neighbourhood of the critical temperature where the density drops rapidly.To illustrate the effect of the thickness of the adsorbed layer, fig. 7 shows the attractive interaction between two semi-infinite slabs of PAN, slabs of PAN with layers of varying thicknesses of PDMS, and two slabs of PDMS, separated by liquidD. H . EVERETT AND .I. F . SI'AGEMAN 239 TABLE 4.-DIELECTRIC PROPERTIES OF POLYMERS AND DISPERSION MEDIA dielectric relaxations : magnitude : characteristic static frequency/rad s-l material and dielectric microwave (temperature/K) constant and below infrared ultraviolet PMMA (293) PMMA (333) PAN (293) PAN (333) PDMS (293) PDMS (333) propane (293) propane (333) ethane (273) xenon (263) 3.9 1.32:63 4.4 1.9 :7.54 x lo2 4.15 1.12:4.4 x lo2 6.0 2.97t6.3 x 10' 2.74 0.18:8.12 x 10" 2.56 0.18:8.12 x 1OI1 1.67 none 1.56 none 1.53 none 1.58 none 0.36:3.3 x 1014 0.28:3.3 x 1014 0.74:4.1 x 1014 0.74:4.1 x lOI4 0.59:2.2 x 1014 0.42:2.2 x 1014 none none none none 1 22: 1.4 x 10I6 1.22:1.4 x loi6 1.29:1.5 x 10l6 1.29:1.5 x 10I6 0.96:1.55 x 10l6 0.96:1.55 x 10I6 0.67:1.46 x 10l6 0.56:1.46 x 10l6 0.53: 1.5 x 10I6 0.58:1.59 x 10I6 propane at 20 and 60 "C.A131 defined by where &(L,T) is the interaction energy per unit area of two semi-infinite slabs of (1 + adsorbed layer) across a thickness L of material 3 at temperature T. The Ham- aker theory itself would lead to A131 independent of L in the non-retarded region (up to about 10 nm). At small separations slabs with adsorbed layers interact as though The results are expressed in terms of the Hamaker function A131(L9T) == 12nL2S131(L,T) 1 /nm FIG.7.-The Hamaker function, as a function of temperature and distance (L) between two similar semi-infinite slabs, through saturated liquid propane. Solid and dashed Iines are for materials at 60 and 20 "C respectively. The composition of the slabs is: (i) and (vi) pure PAN; (ii) PAN with 5 nm layers of PDMS; (iii) PAN with 12 nm layers of PDMS; (iv) PAN with 25 nm layers of PDMS; (v) and (vii) pure PDMS.240 COLLOIDS I N SIMPLE LIQUIDS they consisted entirely of the material of the surface layer, while at large separations the interaction of the cores of the slabs break through to an extent which increases as the layer thickness is diminished. If the surface layer is supposed to consist not of pure polymer, but polymer diluted with solvent, then the core interaction makes itself felt at shorter distances. There is also a substantial increase in interaction energy between 20 and 60 "C.The interaction between spheres surrounded by adsorbed layers cannot be cal- culated exactly, so we have used the approximations employed by Parsegian and The change in configuration of the adsorbed polymer was simulated by considering the surface layer to be a uniform mixture of polymer segments and solvent molecules, and the effect on the attractive energy of changing the polymer volume frac- tion in the layer was examined. The dielectric properties of the diluted polymer layer were calculated from those of the pure polymer and pure solvent using a simple Debye formula. l8 To illustrate the conditions envisaged in fig.6, calculated potential energy curves for (a) simulated MM31 and (b) simulated AN63 are shown in fig. 8. For the larger 1 /nm L /nm 1 /nm FIG. 8.-Calculated attractive potential energics, V(L), betwccn two spherical colloidal particles, each with uniform diluted polymer layers, as a function of separation distance, L. Curves (a) are for PMMA cores of radius 110 nm, which correspond to MM31; (i) 20 "C with 12 nm thick layers of volume fraction 0.1 and 0.3; (ii) the same as (i) but at 60 "C; (iii) 60 "C with 6 nm thick layers of volume fraction 0.211 and 0.622, simulating the effect of layer collapse at constant amount of polymer. Curves (b) are for PAN cores of radius 45 nm corresponding to AN63: (i) 20 "C with 25 nm thick layers of volume fraction 0.1 and 0.3; (ii) the same as (i) but at 60 "C; (iii) 76 "C with 12.5 nm thick layers of volume fraction 0.254 and 0.762, simulating the effect of layer collapse at constant amount of polymer.particle size MM31, with layers of thickness 12 nm and volume fractions of polymer of 0.1 and 0.3 increase in temperature from 20 to 60 "C (approximately the observed UFT) leads, for the higher volume fraction, to a minimum of depth greater than 2kT; collapse of the layer to 6 nm thickness with corresponding increase in polymer volume fraction gives well depths of -2.5 and >6kT for the two volume fractions. For AN63, however, with smaller particle size and thicker surface layer (25 nm), increase in temperature from 20 to 60 "C causes a relatively small increase in well depth; how-D. H .EVERETT AND J . F . STAGEMAN 241 ever, increase in temperature to 76 "C (the observed UFT), accompanied by collapse of the surface layer to 12.5 nm, leads again to well depths -1 and >5kT. Thus, although these calculations are only approximate, they do illustrate in a semi-quantita- tive manner the effects which we believe to be operative in the present systems. More detailed calculations, and a more realistic consideration of the form and range of the repulsive barrier as a function of temperature approaching @-conditions are clearly needed before a convincing quantitative theory can be developed. It is also necessary to extend the experimental work to examine a wider range of parameters. In par- ticular it is proposed to investigate the effect of applied pressure to offset the decrease in density with increasing temperature which is largely responsible for flocculation : so far we have been able to maintain the stability of MM31 in propane up to about 100 "C (3 K above T,) by increasing the pressure to 100-150 atm.Since the net intersegment interaction between the polymer chains also depends on the liquid density, it will also be necessary for comparison to carry out phase equilibrium studies of polymers in supercritical fluids. We are indebted to the S.R.C. for support of this work. We also thank Mr. D. W. J. Osmond of ICI Paints Division and Dr. B. Vincent for stimulating discussion and helpful advice and Dr. D. A. Young for collaboration in the PCS measurements, which were carried out, by kind permission of Prof. G. Allen, in the Department of Chemical Engineering, Imperial College, London. See for instance, J. A. Barker, R. 0. Watts, J. K. Lee, T. P. Schafer and Y. T. Lee, J. Chem. Phys., 1974, 61, 3081. Dispersion Polymerisation in Organic Media, ed. K. J. Barrett (Wiley Interscience, New York, 1975). L. Zeman, J. Biros, G. Delmas and D. Patterson, J. Phys. Chem., 1972,76, 1206. 3. G. Zilliox, J. E. L. Roovers and S. Bywater, Macromolecules, 1975,8, 573. P . N. Pusey and J. M. Vaughan, in Dielectric and Related Molecular Processes, ed. M. Davies (Specialist Periodical Reports, The Chemical Society, London, 1975), vol. 2, p. 48. D. H. Everett and J. F. Stageman, CoZloid Polymer Sci., 1977,255,293. D. H. Napper, J. Colloid Interface Sci., 1977, 58, 390. R. Evans, Ph.D. Thesis (University of Sydney, 1976), referred to in ref. (8). A. K. Dolan and S. F. Edwards, Proc. Roy. SOC. A , 1974,337, 509; 1975,343,427. l1 K. M. Middlemiss, G. M. Torrie and S. G. Whittington, J. Chem. Phys., 1977, 66, 3227. l2 D. W. J. Osmond, B. Vincent and F. A. Waite, Colloid Polymer Sci., 1975,253, 676. l3 C. Domb, Polymer, 1974,15,259. l4 I. E. Dzyaloshinskii, E. M. Lifshitz and L. P. Pitaevskii, Ado. Phys., 1959,10, 165. l5 H. C. Hamaker, Physica, 1937, 4, 1058. l6 See for instance review by V. A. Parsegian, in Enriching Topics from Colloid and Surface l7 J. E. Kiefer, V. A. Parsegian and G. H. Weiss, J. Colloid Interface Sci., 1975, 51, 543. ' D. Patterson, G. Delmas and T. Somocynsky, Polymer, 1967,8, 503. Science, ed. H. van Olphen and K. J. Mysels (Theorex, La Jolla, California, 1975), p. 27. C. P. Smyth, Dielectric Behaviozir and Structure (McGraw Hill, New York, 1955), p. 18 et seq.

ISSN:0301-7249    

DOI:10.1039/DC9786500230

出版商:RSC    

年代:1978

数据来源: RSC