Faraday Discuss. Chem. SOC., 1983, 76, 37-52 Properties of Concentrated Polystyrene Latex Dispersions BY DERYCK J. CEBULA,~ JAMES W. GOODWIN, G. CHARLES JEFFREY, RONALD H. OTTEWILL, ANTHONY PARENTICH f AND RACHEL A. RICHARDSON School of Chemistry, University of Bristol, Cantock’s Close, Bristol BS8 1TS Received 24th May, 1983 In order to understand the properties of concentrated dispersions it is necessary to obtain the spatial and temporal locations of the particles in the system as a function of both the volume fraction and the electrolyte concentration and to correlate this with theoretical models. A useful experimental technique for examination of the static structure factor is provided by small-angle neutron scattering, using cold neutrons with a wavelength of ca. 10 A, since the scattering vectors then available are appropriate for systems containing small particles over a wide range of volume fractions.Polystyrene latices consisting of spherical particles of radius 157 A, with a narrow distribution of particle sizes, provide an excellent system for such studies. Structures are reported for these latices covering a range of volume fractions from 0.01 to 0.14 and electrolyte concentrations from very low values, ion- exchanged systems, up to 5 x mol dmP3 sodium chloride. The structure factors ob- tained are compared with various theoretical models for electrostatically interacting systems. An important factor in determining the properties of stable aqueous colloidal dispersions is the range over which the electrostatic field of one particle can be felt by another.For example, at very low electrolyte concentrations the range of particle-particle electrostatic interactions can be of the order of one micrometre, whereas in high salt concentrations it is only of the order of ten gngstroms or so. Although the form of the pair potential for the electrostatic interaction between colloidal particles has been examined in some detail,’ - 4 there are uncertainties as to how this can be used to predict the bulk properties of colloidal dispersions in which there is considerable interaction between the particle^.^ Consequently, detailed struc- tural information is required on this type of system to enable progress to be made in this field and this requires careful selection of both the system and the investigating technique.As the system we have chosen polystyrene latices since these can be produced as dispersions of spherical particles in which the spread of sizes is very narrow. Furthermore, the particles are rigid spheres which maintain a constant size and where the size can be well defined by a number of different techniques. In addition the basic surface charge density of the surface groups can be obtained by conduc- tometric titrations and the zeta-potential of the particles can be obtained in very dilute dispersions by electrophoresis. In earlier work7-12 it has been shown that the light-scattering technique could be used to obtain both structural and dynamic information on latices. However, its t Present address: Rutherford Appleton Laboratory, S.E.R.C., Chilton, Didcot, Oxfordshire OX1 1 $ Present address: The Western Australian Institute of Technology, South Bentley, 6 102 Australia.OQX.38 CONCENTRATED LATEX DISPERSIONS use is limited to dilute dispersions because multiple scattering becomes pronounced at concentrations in excess of ca. 0.01%. Moreover, the range of scattering vectors which can be obtained with light scattering restricts the range of reciprocal space which the technique can probe. A much wider range is available using cold neutrons since the wavelengths available (3-16 A) are much shorter than those available using visible light. In addition, the weak nature of the interaction of neutrons with matter (i.e. scattering by nuclei rather than electrons) allows deep penetration with little attenuation of the radiation.Consequently much more concentrated dispersions can be examined with a neutron beam than with light. In this communication we report a small-angle neutron-scattering examination of polystyrene latices containing particles of radius 157 A, over a wide range of volume fractions and electrolyte concentrations. EXPERIMENTAL MATERIALS The distilled water used throughout the investigation was doubly distilled from an all- Pyrex apparatus. The sodium chloride used was B.D.H. AnalaR material. The anion- and cation-exchange resins were supplied by Bio-rad Ltd in a specially pre- cleaned form suitable for analytical use. The resins were further cleaned by the method des- cribed by van den Hul and Vanderhoffs in order to remove any soluble polyelectrolyte species.Particular care was taken to ensure that the final wash water from the resins had low ultraviolet absorption at a wavelength of 280 nm. The resins were stored under double- distilled water which was changed at frequent intervals. The polystyrene latex used for the experiments was SLRR1, whose preparation has been previously described. The particle size of this latex was determined by transmission electron microscopy and also by time-average light scattering. The former technique gave a number- average particle radius of 156 %, with a standard deviation on the average value of 28 & and light scattering gave a weight-average radius of 161 5 A. The agreement between the number-average and weight-average results indicated that the latex was monodisperse and free of aggregated material.12 SURFACE CHARGE DENSITY OF THE POLYSTYRENE LATEX PARTICLES Following dialysis and ion-exchange treatment, samples of the latex were titrated with standard sodium hydroxide solutions.The titration was followed conductometrically. l 2 The surface charge density of the particle surface was found to be 4.2 pC cm-2. SMALL-ANGLE NEUTRON SCATTERING, SANS The SANS measurements were all carried out at the Institut Laue-Langevin (ILL), The intensity of scattering was measured over a range of scattering vectors (momentum Grenoble, France, using the small-angle neutron-scattering apparatus D 1 1. transfers) Q defined by Q = (47r/A) sin (8/2) M (2z/A)(rO/D) where 8 is the total scattering angle, A is the wavelength of the incident neutrons, ro IS the radial distance on the detector measured from the undeviated beam in the plane of the two- dimensional detector and D is the distance between the detector and the sample. An incident wavelength of 10 A was used in most of the measurements and all samples were examined at sampIe-to-detector distances of 2.66, 5.66 and 10.66 m. The smallest setting was utilised in order to obtain a good estimate of the incoherent background for the concen- trated samples by making measurements at high Q values.The procedure used will be de- scribed in detail elsewhere. l4D. J. CEBULA et al. 39 PREPARATION FOR SCATTERING MEASUREMENTS All the neutron-scattering measurements were carried out with the samples in optical- quality quartz cells having a path length of 1 mm.For some of the latex samples a small amount of mixed-bed ion-exchange resin was placed in the cell and then latex of known volume fraction was added. Other samples were prepared in sodium chloride solution. PTFE stoppers were inserted into the cells and then the top of the cell was covered with Parafilm to prevent evaporation. The samples containing ion-exchange resin were allowed to stand for several weeks before examination in order to ensure that exchange was as complete as pos- sible; it was found to be a slow process. The volume fraction of the latex was determined by taking a known weight of latex in a glass container and then drying to constant weight in an oven at 70 “C. THEORY SMALL-ANGLE NEUTRON SCATTERING The coherent scattered intensity at a scattering vector Q, i.e.I(Q), from a beam normally incident to the target is directly related to the target’s differential cross- section with respect to angle, and in the static approximation is given by where ( ) denotes the time-average value and Nt is the number of particles in the scattering volume; multiple scattering has been assumed to be absent. In eqn (2) rp and r4 are the position coordinates of the pth and qth scattering elements and ap and a, are the scattering amplitudes of the particles. For a monodisperse system, with non-interacting spherical particles, as shown by Guinier and Fournet,15 eqn (2) takes the form (3) where A is a constant containing instrumental factors, Np is the number of particles in the dispersion per cm3, Vp is the volume of each particle of radius R, pp is the coherent neutron-scattering length density of the particle, pm is the coherent neutron-scattering length density of the dispersion medium and P(Q) is the particle form factor, as given by I(Q> = A(pp - ~m)’ VP’ NP p(Q) P(Q) = [3(sin QR - QR cos QR)/(QR)3]2.(4) Vp = 4 nR3/3 ( 5 ) q~ = Np Vp. ( 6 ) For a system of monodisperse spheres Vp is given by and the volume fraction of the latex by In practice the latex is not completely monodisperse and this has to be taken into account in addition to the small-wavelength spread in the incident beam and the finite size of the detector elements. This procedure has been described previously. - * In concentrated dispersions, when particle-particle interactions are present, the scattering of the beam by the particles is no longer independent and the spatial correlations between the particles have to be allowed for by the introduction of a structure factor.The intensity of scattering at scattering vector Q is then given by40 CONCENTRATED LATEX DISPERSIONS where S(Q) is given by S ( Q ) = 1 + ~ 71NP j: [ g ( r ) - 11 sin (Qr) r dr Q where g(r) is a distribution function giving the radial distribution of the particles relative to a reference particle. In terms of volume fractions eqn ( 3 ) and (7) can be written in the form and where rp, is chosen as the volume fraction for the particles such that S(Q) = 1 at all the Q values investigated. For this system if the instrumental constants are deter- mined and I(Q) is expressed as an absolute intensity then P(Q) can be obtained directly.Alternatively, since I(Q) is a sensitive function of Q a fitting procedure can be used to obtain P(Q). In all cases qn, Vp and (pp - pm) are well known. Once P(Q) is known then in terms of absolute intensities S(Q) is given by eqn (7a). Alternatively, and conveniently for a system such as polystyrene latex, S(Q) can be written as It is important that the experiments in this case should be carried out at exactly the same conditions of sample container, beam intensity, beam aperture and wavelength and that the incoherent background corrections to the concentrated sample should be carefully determined.14 The results reported in this paper were obtained using this procedure. RESULTS SCATTERING RESULTS For a dilute dispersion of non-interacting monodisperse spherical particles the scattered intensity at a particular Q value is given by eqn (3) or (3a).Since the volume fraction of the latex was accurately known it was possible to obtain the radius of the latex using a fitting procedure. In order to do this it was assumed that the particle size distribution was a zeroth-order log-normal distribution. In ad- dition, the procedure of Chauvin l 7 was followed in order to allow for the small spread of wavelengths in the incident beam and the finite size of the detector ele- ments. Fig. l(a) shows the experimental points obtained with SLRRl at a latex volume fraction of 0.01 1 in a 1.3 x mol dm-3 sodium chloride solution. A good fit to the data was obtained using a particle radius of 157 f 14 A, a standard deviation on the mean of 9.0%.The values for the coherent neutron-scattering length densities of polystyrene and water were taken as pp = 1.41 x 1O1O cm2 and pm = -0.56 x 1Olo cm2, respectively.20 The value of R obtained by SANS, namely 157 A, was in good agreement with the values of 156 and 161 A obtained by electron microscopy and light scattering. In concentrated systems the effect of S(Q) becomes important and this has a substantial effect on the form of the I(Q) against Q data. This is illustrated in fig. l(b), where results are given for the latex at a volume fraction of 0.14 in molD. J. CEBULA et al. 200 41 0 - oo*oooo 0 0 I I -0bOoc 00 00 OO 4001 x " t 800 6ool 400 00 0 0 0 0 0 0 0 0 0.005 0.01 0.015 0.020 Q1A-l Fig.1. (a) Plot of I(Q) against Q for latex at cp = 0.011 in 1.3 x chloride solution. (b) Plot of I@) against Q for latex at 9 = 0.14 in mol dmP3 sodium mol dm-3 sodium chloride solution. dm-3 sodium chloride solution. A pronounced peak in the intensity is now apparent at Q = 0.0137 A - l . The two curves given in fig. l(a) and (6) represent examples of the basic data obtained for non-interacting systems, as represented by eqn (3a), and for interacting systems, as represented by eqn (7a). From results of this type using eqn (9) it was possible to obtain curves of S(Q) against Q. S(Q) AGAINST Q The results obtained under various conditions of volume fraction and electrolyte concentration are shown in fig. 2 and 3. Fig. 2(a) shows the results obtained with ion-exchanged samples at volume fractions of 0.04, 0.08 and 0.13.Fig. 2(b) gives the results obtained at an electrolyte concentration of mol dm-3 for volume frac- tions of 0.01, 0.04 and 0.13. Similar sets of results obtained at sodium chloride concentrations of mol dm-3 and 5 x In fig. 4 a comparison is given of curves of S(Q) against Q obtained at the same volume fraction, 0.04, at different salt concentrations. The decrease of the main peak in S(Q) is clearly discerned as well as an increase in the magnitude of S(Q) at low Q as the salt concentration is increased. rnol dm-3 are given in fig. 3. DISCUSSION The experimental curves of S(Q) against Q obtained indicate clearly that a structure is built up in dispersions of polystyrene latices which is dependent on both42 CONCENTRATED LATEX DISPERSIONS 2.0 1 .o G o G, 2.0 1 .o 0 d I I 0.01 Q1A-l 0.0 2 Fig.2. (a) Plot of S(Q) against Q for an ion-exchanged latex: @, cp = 0.04; A, cp = 0.08; 0, cp = 0.13. (b) Plot of S(Q) against Q for latex in lov4 mol dm-3 sodium chloride: @, cp = 0.01; A, cp = 0.04; 0, cp = 0.13. volume fraction and electrolyte concentration. This is indicated by the increase in magnitude of the peak in S(Q) and the decrease in S(Q) at low values of Q as the volume fraction is increased. The first peak also moves to higher Q values as the volume fraction is increased, the variation of peak position with Q being consistent with the formation of a regular array with the particles in lattice sites in the system. The effect of electrolyte is visible in the sequence of curves given in fig.2 and 3 and in the results presented in fig. 4 which show the effect of different electrolyte con- centrations at a constant volume fraction of 0.04. The softening of the peak with increase in electrolyte concentration and its movement to higher Q values (fig. 4) indicate that the repulsion between the particles becomes of shorter range and that a loosening of the structure occurs. The sharpening of the peaks with increasing volume fraction, which is commen- surate with an increase in the force of interaction between the particles, also suggests an increase in ordering with volume fraction, and consequently it is of interest to determine at which volume fraction of latex the array conforms to a regularD. J. CEBULA et al. 43 2.0 1 .o 6i K O 1.5c 0 l .o ~ * 5 0 0.01 QIA- 0.02 Fig. 3. (a) Plot of S(Q) against Q for latex in A , cp = 0.04; 0, mol dm-3 sodium chloride: e, cp = 0.01; mol dm-3 = 0.13. (b) Plot of S(Q) against Q for latex in 5 x sodium chloride: 0, cp = 0.01; 0 , cp = 0.04; A, cp = 0.08. 'crystalline' lattice. In many of the optical studies of iridescent latices2l, 22 it is es- sentially only the position of this first peak which can be observed, and hence the information which can be obtained from such studies is, of necessity, limited. How- ever, by the use of small-angle neutron scattering information can be obtained with a different range of particle sizes over a much wider range of scattering vectors and the data so obtained are not seriously affected by multiple scattering.The method of approach based on peak position alone is to consider that the primary peak in S(Q) constitutes the first Bragg peak.8i21,22 As shown previously8p22 for a face-centred cubic array the value of 8 at the peak position Omax is given by sin(O,,,/2) = J3 A(3cp/16~)'/~/2R (10) for the first-order reflection which corresponds to a reflection from a (1 11) plane. Hence Qmax is given by (1 1) Qmax = 47~ ~in(omax/2)/A-44 CONCENTRATED LATEX DISPERSIONS I I I 1 0 0 .OO 5 0.010 0.01 5 0.020 Q/A - ‘ Fig. 4. Plot of S(Q) against Q for latex at cp = 0.04: 0, IER; A, mol dm-3 NaC1; U, 5 x mol dm-3 NaCl. The reflection for an f.c.c. lattice in fact is close to that for the (1 10) reflection of a b.c.c. lattice and we will not discuss the distinctions here. From eqn (1 1) it is possible to calculate Qmax as a function of 9 and to compare these values with those obtained experimentally.This is shown in fig. 5(a). It can be seen from this plot that for the ion-exchanged samples all the measured values of Qmax lie on the calculated curve. As the salt concentration is increased, so there are deviations from the calculated curve at the lower volume fractions and correspon- dence with the calculated curve at the higher volume fractions. The point of intersec- tion can be taken as a possible transition point from an ‘ordered-liquid lattice’ to a 0 0.0 5 0.10 0 0.0 5 0.10 0.1 5 cp Fig. 5. (a) Plot of em,, against cp: 0, IER; A, NaCI; 0, 5 x mol dm-3 NaCI; 0 , loT3 mol dmF3 mol dm-3 NaC1. (b) Plot of log (salt concentration) against cp showing transition line.D.J. CEBULA et al. 45 ‘disordered liquid-like’ array. The transition volume fraction is plotted against electrolyte concentration in fig. 5(b). The words used to describe this effect have to be carefully chosen since the transition is a continuous one with the particles gradu- ally moving together until all the particles take up lattice sites. All the particles appear to be separated by the continuous phase and no nucleation effects have been observed as would be expected in the formation of an ordered solid phase. THE DISTRIBUTION FUNCTION g(r) The distribution function g(r) given in eqn (8) is related to the number con- g(r) = N(r)/Np (12) where N(r) is the number concentration of the particles at a distance Y from a reference particle.The quantity g(r) can be obtained directly by Fourier transform- ation of eqn (8), when we obtain centration of the system, Np, by This equation can be used in the form of a summation to obtain g(r) against r from the experimental curves of S(Q) against Q provided that certain precautions are taken. First, to extrapolate the experimental curve to small Q, and hence to Q = 0, which can be achieved if it is recalled that as Q --* 0 1 2 , 2 3 S(Q) = S(0) + b e 2 . . . . (14) Secondly, S(Q) at very small increments of Q must be taken from a smoothed curve. Thirdly, truncation of the summation is preferable at a point where S(Q) = 1. Curves of g(r) against r are shown in fig. 6 for volume fractions of 0.01, 0.04 and 0.13 in mol dmF3 sodium chloride solutions. In fig.6(a), the form of the g(r) against r curve appears to resemble closely that expected for a vapour-like system; the exclusion region is clearly visible and the first peak is rather small. With an increase in volume fraction to 0.04 the initial slope has increased, indicating a streng- thening of the repulsive force between the particles, and clear first and second peaks are visible. At q~ = 0.13 the initial gradient is very steep, suggesting a much harder interaction. The curve provides evidence for a particle arrangement in which there is considerable short-range order, as evidenced by the substantial peaks in g(r) in the range 400-1500 A, and long-range disorder, as suggested by the fact that g(r) is clearly tending to unity as r tends to values > 2000 A. The form of this curve suggests ‘liquid-like’ order of the particles in the system. THE POTENTIAL OF MEAN FORCE From eqn (12) it also follows that g(r) = W Y N P = exP[w)/kTl @(r) = W ) + W ) (15) (16) where CD is the potential of mean force between the particles such that where V(r) is the pair potential for interaction between the particles and $(r) is a perturbation potential which allows for the many-body interactions present in the s ys tem.46 CONCENTRATED LATEX DISPERSIONS 1 .o 0.5 0 1 .5 1 .o n Z 0 . 5 0 1.5 1 .o 0.5 0 I I I 500 1000 1500 2 000 rlA Fig. 6. Plot of g(r) against r for latex in mol dm-3 sodium chloride: (a) cp = 0.01, (b) cp = 0.04, (c) cp = 0.13. Fig. 7 shows the potential of mean force for a volume fraction of 0.13 at a sodium chloride concentration of lop4 mol dmP3.It is clear from this curve that the central particle is interacting with more than one layer of particles and that many particle interactions are involved. On the other hand, the results for a volume frac- tion of 0.01 appear to resemble very closely a simple pair potential. The steeply rising portion implies that very strong repulsion can occur at @/kT values of the order of 1 kT, and hence we conclude that it is the ‘tail’ of the potential-energy curve which strongly influences particle-particle interactions in colloidal dispersions.1 .O 0.5 0 -0.5 h s o 8 0.5 0 -0.5 -1 .o D. J. CEBULA et al. \ (4 I 1 I ( b ) 47 0 500 1000 1500 rlA Fig. 7. Plot of @/kT against r for latex in mol dm-3 sodium chloride: (a) cp = 0.01, (b) q = 0.13.COMPARISON WITH THEORETICAL MODELS In attempting to consider the theoretical aspects of particle-particle interaction one of the simplest models to use is the hard-sphere model, based on the idea of a hard-sphere potential. This has been utilised to explain the physics of simple liquids and also in recent work to explain the properties of microemulsions and colloidal d i s p e r ~ i o n s . ~ * ~ ~ The model, in colloidal terms, suggests that the particle has an effective radius & which is larger than its actual radius R and that when two particles are separated by a distance 2Rff the potential energy of repulsion becomes infinite. The particles then have an effective volume fraction given by q e K = 4zR& Np/3. (17) A straightforward solution in order to obtain S(Q), based on the original theory of Percus and Y e v i ~ k , ~ ~ has been given by Ashcroft and Lekner 26 in the form48 with CONCENTRATED LATEX DISPERSIONS where a, p and y are given by a = (1 + 2 (Ped2/(1 - p = -6 q e m (1 + 0.5 cpeff)2/(l - ( ~ e f f ) ~ Y = 0.5 (Peff (1 + 2 (Peff)2/(l - ( ~ e d ' .Fig. 8 shows the fits obtained to the experimental curves of S(Q) against Q by allowing the values of Reff to float until the best fit is obtained. It can be seen that at cp = 0.01 a reasonable fit is obtained using Reff = 435 A and at cp = 0.1 1 using Reff = 251 A. Although the fit is reasonable in terms of amplitude and position of the first peak, at low Q values the hard-sphere model gives values lower than those determined experimentally and at high Q values the modulation appears to be slight- ly out-of-phase with respect to the position of the second peak.The hard-sphere model, although qualitatively of interest, is formally incorrect for 2.0 1.5 1 .o 0.5 0 Ql.8- Fig. 8. Plot of S(Q) against Q showing fits using a hard-sphere model: 0, experimental points; (-) theoretical fit. (a) with = 435 A, (b) with Rerf = 251 A.D. J. CEBULA et al. 49 charged particles in that it ignores the essential features of the particle-particle interaction, i.e. the electrical double layer. However, as pointed out in an earlier paper,12 a link can be made via the theory of Barker and Henderson27-29 in order to take some account of the softness of the potential. In this approach the effective interaction radius of the particles can be written in the form R Rem = R + J: ( 1 - exp[ - V(r)/kT]) d(r/R) where V(r) is the pair potential for the interaction between two particles with their centres separated by a distance r.In the conditions we have used the repulsive interaction is dominated by electrostatic repulsion and hence as a first approach for the pair potential the form given by Verwey and Overbeek2 can be used, namely V(r) = 4 E ~ E , R2 $: exp(2~R) exp( - w ) / r (21) where Er the relative permittivity of the medium, E~ is the permittivity of free space, $s is the surface potential of the particles and K is the Debye-Huckel reciprocal double-layer thickness of the bulk electrolyte, which for a 1 : 1 electrolyte can be written in the form (22) where e is the fundamental charge on the electron, N is Avogadro's number and c is the concentration of the electrolyte in mol dm-3.From eqn (18) and (19) using ReK values of 435 and 251 A, i.e. the values ob- tained in fig. 7, and an electrolyte concentration of lop4 mol dmV3 the value of rl/s is found to change from 20 mV at cp = 0.01 to 1 1 mV at cp = 0.1 1. These appear to be low potentials compared with the [-potential of 58 & 10 mV obtained on a very dilute dispersion using laser electrophoresis. Low potentials would account for the fit obtained to the hard-sphere model 30* but it might be unwise from this evidence alone to conclude that the potential is changing with cp. This approach appears to be more appropriate for low volume fractions.12 An alternative method to obtain a theoretical estimate of the form of g(r) against r is to utilise eqn (21) in a Brownian-dynamics computation, an approach which has been used by several a ~ t h o r s .~ ~ - ~ ~ In the approach utilised by Ermak,36* 3 7 if hydro- dynamic effects are neglected, the charged particle equation of motion can be put in the form = 2e2 N c 103/&,~, kT where Ri ( t ) denotes a random displacement of the particle determined by a Gaus- sian probability function that has an average value of zero and a mean-square value given by (Ri ( A t ) 2 ) = 6 D kT (24) The term Fi ( t ) is the force component32 given by in which V(rij) is taken as the pair potential. values of r the diffusion coefficient D was calculated from In the computational procedure adopted to obtain values of g(r) at various50 CONCENTRATED LATEX DISPERSIONS where q is the viscosity of water and D(u) is the diffusion coefficient for the approach of one particle to another to a distance uR along the line of the centres.The quantity P(u) has been calculated by Honig et al.38 who found p -+ 1 as u + 00. In the computations values of Ri were generated using a random-number procedure 39 using a box containing 108 particles and taking a time step of lo-’ s. Two initial configurations, a face-centred cubic lattice and a random distribution, were used. g(r) against Y was calculated as a histogram of separations and plotted at various 1 .o 0.5 0 1.5 1.0 - 0.5 0 1.5 1 .o 0.5 0 0 3 P I I I ( c) 500 1000 1500 2000 rlA Fig. 9. Plot of g(r) against Y showing computations using Brownian dynamics: 0, comput- ations; (-) experimental data.(a) cp = 0.01, $s = 30 mV; (b) cp = 0.04, @s = 40 mV; (c) cp = 0.13, $s = 50 mV. mol dm-3 NaC1.D. J. CEBULA et al. 51 times until a constant result was obtained. The use of two starting configurations ensured that an equilibrium situation was reached. The data were then collected over several hundred steps to reduce the statistical noise. The results obtained using this procedure are shown in fig. 9 wherein they are compared with the experimentally determined values of g(r) against r obtained in mol dm-3 sodium chloride solution (fig. 7). The latter electrolyte concen- tration was used to calculate K and eqn (21) was used for the pair potential, @s, until the best fit was obtained. As can be seen from fig.9, this was 30 mV at q = 0.01,40 mV at cp = 0.04 and 50 mV at cp = 0.13. Again the value of 30 mV obtained at the lowest volume fraction is lower than the -58 mV obtained for the zeta-potential of the particles by laser electrophoresis using very dilute dispersions at the same electrolyte con~entration.~~ The apparent increase of potential suggested by this approach is interesting since, if correct, it would suggest the possibility of interaction occurring at constant charge. What is clear is that at a mol dmP3 sodium chloride concentration the effective interaction surface charge density of the par- ticles, corresponding to t,hs = 30 mV, is 0.21 pC cm-2, a value very substantially below the value of 4.2 pC cm-2 found for the actual surface charge density of the latex particles.It can therefore be concluded that it is the diffuse part of the electrical double layer which controls the interaction between the particles in concentrated dispersions. However, further wprk is required to determine whether the zeta- potential measured under dilute conditions is the appropriate potential to use for modelling the behaviour of concentrated dispersions. In this context it should be pointed out that there are a number of problems attached to the use of eqn (21) for strongly interacting systems. These include the assumption of constant surface potential, the assumption of constant electrolyte concentration with increasing volume fraction and the assumption that there is always present a reservoir of bulk electrolyte as a reference state for the calculation of K .It is clear that in concentrated systems these conditions are not fulfilled, par- ticularly in the absence of added electrolyte, and that the concentration of the counter-ions and its dependence on latex volume fraction should be taken into ac- count, as suggested by Beresford-Smith and Chan.41 Our preliminary calculations 42 indicate that their approach provides a clearer interpretation of the behaviour of ion-exchanged systems, and this work will be reported in due course. We thank the S.E.R.C. for support of this work and the Institut Laue-Langevin for neutron-beam facilities. One of us (R. H. 0.) thanks Dr. Derek Chan for a number of stimulating and useful discussions. €3. V. Derjaguin and L. 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