摘要:

Faraday Discuss. Chem. SOC., 1987, 83, 259-270 Determination of the Fractal Dimension using Turbidimetric Techniques Application to Aggregating Protein Systems David S . Horne Hannah Research Institute, Ayr KA6 5HL Scattering techniques are a powerful way of studying fractal structures. Depending on the characteristic length scale to be observed and the nature of the aggregate, it is possible to use the scattering of light, X-rays or neutrons, the fractal exhibiting power-law scaling of scattered intensity with wavevec- tor, q, when 1/R << q << l/a. Normally for long-length scales (small q ) angular dependence of laser light scattering is employed. It is shown here that the value of the fractal dimension of an aggregate is also accessible through measurement of the wavelength dependence of the turbidity of the solution, provided large enough aggregates are created.The remainder of the paper deals with specific applications of this technique to the study of aggregates of casein proteins both as individual proteins where aggregation is induced by the addition of calcium ions and as casein micelles where destabilization is brought about following enzyme (rennet) action or the addition of ethanol. Light scattering and the study of Brownian motion of colloidal particles have been intimately linked since the first observations of the phenomenon by Robert Brown almost two centuries ago. Since then light-scattering methods have been used to determine molecular weights and sizes of molecules and particles in solution. Their principle advantages are rapidity, simplicity and their provision of a non-invasive in situ measure- ment which avoids criticism regarding sample preparation artefacts. Instead of measur- ing scattered light intensity at finite angles, generally using specially designed apparatus, many workers have used the complementary technique of monitoring the non-scattered or transmitted light using conventional spectrophotometers.Thus Oster' followed the kinetics of aggregation in several colloidal systems, showing that in the initial stages the optical density or turbidity was directly proportional to time, whereas its rate of change with time was proportional to the square of the aggregating component concentra- tion. In the later stages of coagulation, however, when R 2 A/10, Rayleigh scattering theory and these simple relationships were no longer applicable. It is in such ranges of size that the use of Rayleigh-Gans-Debye theory' enables the determination of size and molecular weight through the angular variation of the scattered light.The com- plementary turbidimetric technique, developed by Doty and Steir~er,~ obtains the same size information by measuring the variation of solution turbidity with wavelength. Recently the results of computer simulation studies using different aggregation models have stimulated interest in defining the structure of colloidal aggregates in terms of the concept of the fractal dimen~ion.~ Qualitatively fractal objects show dilation symmetry, meaning that the essential geometric features are invariant to scale change. From a scattering viewpoint all fractals show a power-law dependence of the scattered intensity, I, on the momentum transfer or wavevector, q, i.e.I qPx when 1/R << q << l/a, a being the monomer radius and R a characteristic size of the aggregate. The wavevector has the normal definition q = (4rrnlA) sin 8/2 with n the refractive index, A the wavelength of the light and 8 the scattering angle. 259260 Structure of Aggregated Casein Systems For mass fractals the exponent, x, is equal to 0, the fractal dimension, which relates the size R of the object to its mass N through the definition Normally for long length scales, angular dependence of laser light scattering is employed to measure the fractal dimension. Since wavevector is also a function of wavelength, it is one objective of this paper to demonstrate that the fractal dimension of an aggregate is similarly accessible through measurement of the wavelength depen- dence of the turbidity of the solution.The second objective is a preliminary attempt to classify the structures of casein protein aggregates in fractal terms. The caseins, a family of phosphoproteins, make up ca. 80% of the total protein of cow’s milk. They are four distinct proteins, a,], aS2, p and K , differing both in amino-acid composition and degree of phosphorylation. In normal milk ca. 95% of the casein exists in the form of large spherical aggregates or micelles, with radii ranging from 20 to 300 nm, but averaging ca. 100 nm. Establishing the structure of the micelle is central to our understanding of the stability (or instability) of processed milk products.To this end we have studied the aggregation of individual caseins, particularly a , ] , induced by the addition of Ca2+, and proposed a mechanism which highlights the importance of the charge of the aSl-Ca2+ Some of these measurements are extended to longer reaction times in this study, and the resulting aggregate structures examined from a fractal viewpoint. Since these reactions proceed through to ultimate precipitation, the products cannot be regarded as micellar. Neverthe- less, study of their properties may prove fruitful in understanding why the micellar aggregates of milk stop growing. The first measurements made using the turbidimetric fechnique were, however, on aggregates of casein micelles which had been destabilized by the addition of ethanol.As proteinaceous entities, the casein micelles are remarkable for their stability: centrigu- gation, boiling, cooling or drying of milk apparently do not destroy micellar integrity.’ This stability does have limits, however, and, on the one hand, in certain manufacturing processes, e.g. cheesemaking, the micelles are deliberately rendered unstable by limited proteolysis of K-casein, whilst on the other, certain long-life products that have been subjected to high heat treatments do show gradual changes in storage, leading eventually to failure. For the manufacture of cheese and yoghurt, it is necessary to understand the nature of the destabilization, however induced, and its consequent influence on the aggregate or curd structure if the yield and quality of the product are to be optimized.I: has Seen established that in both rennet and ethanol-induced coagulations the first stage involves the loss of a steric-stabilizing sheath.”lo The later stages of the reactions are of equal importance and some preliminary results are presented here. Experimental Materials The preparation of skimmed milk and defined size fractions of micelles obtained by successive centrifugation and resuspension of pellets in milk ultrafiltrate have been described previously” as has the procedure for separating pure bovine a,,-casein from acid-precipitated whole casein.12 In all of the experiments described, the aggregation reactions were induced in a buffer of 0.02 mol dm-3 imidazole/HCl, pH 7.0, which also contained 0.05 mol dm-3 NaCl.Calcium chloride hexahydrate was used as the source of Ca2+, concentrations of which were standardized by titration with EDTA using a calcium-ion selective selectrode (Radiometer, Copenhagen). All chemicals were AnalaR grade, with the exception of imidazole, which was recrystallized from ethanol.D. S. Horne 26 1 Methods Photon Correlation Spectroscopy PCS measurements of micellar and aggregate hydrodynamic radii were made using a Malvern K7027 correlator in conjunction with the PCSlOO spectrometer fitted with a Spectra Physics He-Ne laser (model 124B). The correlator was generally operated in the logarithmic mode, and the Malvern software was modified to print scattered light intensity as well as particle size and sample polydispersity. Unless otherwise stated, measurements were made at an angle of 90" and a temperature of 20°C.For PCS measurements of aggregation by ethanol 2mm3 of milk or resuspended micelles were diluted into 2 cm3 filtered buffer solution (<0.1 p m pore diameter, Millipore) which also contained 5 mmol dmP3 Ca2+ and an appropriate level of ethanol. Since each measurement required only a few seconds of data accumulation but 20 s of subsequent calculation, the time dependence of aggregate growth could be followed accurately in the early stages only for relatively slow reactions. In a few instances intensity was recorded as a function of scattering angle at the end of such kinetic runs. Turbidity Measurements These were carried out in a Cary 219 spectrophotometer equipped with wavelength and cell programming facilities or, for the asl aggregation studies, in a Cary 118 instrument.In order to calculate wavelength dependence of micellar or aggregate turbidity it was necessary to correct the raw turbidity values first for inter-cuvette variations from the recorded instrument baseline and next for any residual turbidity due to fat globules which had not been removed entirely by skimming and subsequent filtration of the milk (Whatman GF/A). The latter correction factor was directly measured as fat globule turbidity following dissociation of the micelles in a milk sample by addition of sufficient 10% w/w EDTA solution to completely sequester the Ca2' content of the solution. These background turbidities were subtracted from the raw turbidities measured in aggregating micellar systems, and the gradient of the linear double logarithmic plot of micellar turbidity us.wavelength was computed. Such measurements were repeated at frequent intervals over several hours during the progress of the aggregation reaction. Generally wavelengths scans of turbidity were recorded between 800 and 400 nm limits. For the aggregation reactions of a,,-casein induced by Ca2+ addition, the procedure described by Horne and Dalgleishs was adopted, recording solution turbidity at a fixed wavelength as a function of time of reaction. For each a,,-Ca2+ combination a series of separate runs was carried out over the wavelength range 400-800 nm at 50 nm intervals. By normalizing these to time t = 0, the gradient of the double log plot, log r us. log A, was computed as a function of reaction time.In most instances these rapid reactions were limited to a total duration of 10 min. Theory The turbidity, 7, of a suspension of particles is a measure of the reduction in intensity of the incident beam due to scattering. The turbidity is defined by T = 1 In ( I o / I ) where 1 is the scattering path length and lo and I are the intensities of the incident and transmitted beams, respectively. In terms of the optical density, A, of the suspension, r = 2.303A. For monodisperse particles of molecular weight M, concentration c and with refrac- tive index close to that of the solvent, the turbidity is given by3 r = HcMQ (3)262 where Structure of Aggregated Casein Systems H =327~~n‘($)*/3NA‘.Here N is Avogadro’s number and dn/dc is the specific refractive-index increment. at all angles 8, and is given by The function Q, results from internal interference of light scattered by the particle where P( 6 ) is the particle form factor, more properly written as P ( q r ) . For spherical and homogeneous particles of radius r, this factor is given by )* 3 P ( q r ) = (3 (sin qr - qr cos qr) . For a fractal object the light scattered at angle 6 is modified by multiplication of the monomer form factor by a structure factor S(q),13 which describes the spatial arrangement of the elementary scatterers or monomers within the aggregate. The turbidity particle dissipation function, Q, for a fractal object is then assumed to be equivalently modified as The theory of scattering relates S ( q ) to the pair-particle distribution function g ( r ) by the Fourier transform sin qr S(q)=1+4qb ~ g ( r ) - l ~ r * - - - - - d r i: 4‘ (7) where 4 = N / V, the number density of particles or individual scatterers in the system.T e i ~ e i r a ’ ~ has shown that in a real situation when the scaling region is finite and bounded by both upper and lower cut-off lengths, the introduction of an exponentially decaying cut-off function, exp( - r / 6) produces a pair-distribution function which can be transformed analytically to give the structure factor, S ( q ) , as where T ( x ) is the gamma function, a is the monomer radius, 5, the cluster size and D the fractal dimension. Qualitatively this result has the expected form.At large q, qa >> 1, it gives S ( q ) = 3. and the scattering will be dominated by the form factor of the individual spheres. At intermediate values of q, for 1/(<< q<< 1/a, S ( q ) has the limiting form S ( q ) cc ( qa)-”, the power-law dependence often used to define the fractal dimension, D. It is clear from the way it is obtained that this applies only in an intermediate q region for which both inequalities qa << 1 and qy >> 1 apply. Remembering that q = (4rrn/A) sin0/2r, direct substitution of S ( q ) az into eqn (6) would lead to an expression for the turbidity with a power-law dependence on wavelength A, and an exponent directly related to D. However, in performing the integration over all scattering angles, 6, we are perhaps sampling regions of q-space outwith the range of validity of the simple power-law expression for S ( q ) .Hence it is necessary to substitute the full expression for S ( q ) and integrate numerically to give turbidity as a function of wavelength for a fractal object.D. S. Home 263 2.6 2.2 P 1.8 1.4 i n I I I I 1 1 I 1 I I 2000 2500 I .u 0 500 1000 1500 cluster radius/nm Fig. 1. The structure factor exponent /3( = d log Q/d log A ) as a function of cluster radius, l, for a series o f input fractal dimensions from 1.7 to 2.5. Plots are shown for two monomer sizes, 5 (- - -) and 100 (-) nm. or Note that from eqn (3) we obtain the derivative d log ( n z ) dlogT dlogQ d log A d log A d log A - = 4 - p - y d log r - 4 - ys- d log A * p=-- dlog Q d log A Tabulated values of the specific refractive index increment and the coefficients of its wavelength dispersion equation show little variation between protein^.'^ These together with knowledge of the wavelength dependence of n, permit calculation of the y term in eqn (9a).A value of -0.2 was found appropriate for measurements extending over the wavelength range 400-800nm. Our experiments yield linear plots of log T against log A. From their gradient and the use of eqn (96), an experimental value can be obtained for the exponent p. First of all, however, we must calculate Q and obtain p from its wavelength dependence, enabling us to explore the relationship of this exponent to the input fractal dimensions. Calculations were carried out over the experimental wavelength range 400-800nm for a range of monomer sizes and fractal dimensions.The lower limit of integrations was set at 2" to allow for an instrument acceptance angle of this value. The results plotted in fig. 1 show p as a function of cluster radius, 6, for two monomer radii, 5 and 100nm. These monomer radii exemplify the radius of an individual casein molecule and of a casein micelle, respectively. All double log plots were of very high linearity with correlation coefficients >0.9999 for 5 nm monomers and >0.999 for all but the smalle5t clusters involving 100 nm monomers. Except for the smallest clusters, where P ( q ) vs. S ( q ) interactions were likely to be significant and where the form of S( q ) may not be truly appropriate, the value of the exponent p showed little dependence264 Structure of Aggregated Casein Systems 3.01 I I I I I I 2.8 l A .3 2 .6 - 3 2 . 4 - i I I I I 28 30 32 34 36 [ethanol] (%v/v) Fig. 2. The fractal dimension of micellar aggregates produced as a results of destabilization at the ethanol levels plotted. The fractal dimensions were determined from power-law growth of hydrodynamic radius (A), from power-law dependence of the scattered light intensity on scattering angle (O), and from power-law dependence of solution turbidity on wavelength (0). The last results are the mean of from 4 to 11 separate measurements. on the monomer radius. In all cases the exponent tended asymptotically towards the input fractal dimension as the cluster grew. Thus we would anticipate that for micro- metre-sized aggregates, the slope of the log7 us.logA plot should reflect the fractal dimension of the cluster. This asymptotic behaviour is akin to that demonstrated by Teixeira13 in calculations of the structure factor S ( q ) , where he showed that the exponent there was only equal to the fractal dimension for very large values of 5. Results and Discussion Ethanol-induced Aggregation of Casein Micelles For monomer particles as large as casein micelles, light scattering can be used to derive the fractal dimension of an irreversibly formed aggregate in three possible ways: (i) if the hydrodynamic radius exhibits power-law growth kinetics, the exponent is 1/ DH , (ii) from the power-law dependence of the angular scattered intensity in static light scattering, 10cq-”1 and, as suggested here, (iii) from the power-law dependence of solution turbidity on wavelength.The other method, double log plotting scattered intensity us. hydrodynamic radius, is not applicable because the aggregates rapidly grow larger than q-’ and we observe a constant scattered intensity while the measured hydrodynamic radius continues to grow, similar behaviour to that observed by Rarity and Pusey in the salt-induced aggregation of polystyrene spheres.lS To limit effects of micellar polydispersity, as found in a normal sample of skimmed milk, these aggregation studies were carried out on a defined micellar size fraction, pellet 4,11 resuspended in milk ultrafiltrate. Initial rate studies of the ethanol-induced aggregation of micelles showed the reaction to become diffusion-limited at concentrations of ethanol >29’/0 v/v for our reaction conditions.In the reaction-limited regime, however, between 27 and 29% Y/V, power-law growth kinetics were still observed for the hydrodynamic radius, with an exponent giving a fractal dimension of 2.2-2.3 (fig. 2). At ethanol levels 26% and below, log RH was non-linear with log t over the entireD. S. Home - 265 0.18 2.51 I I I I I I I 1 2.41 2.31 2.2 2.1 I € € A A A A A 10.12 1.94 I I I I I 1 I 10.06 0 4 8 12 16 t / l O % Fig. 3. The turbidity exponent ,L3, defined by eqn (9a), as a function of reaction time for an aggregating micellar system (pellet 4) in 30'/0 v/v ethanol, illustrating its approach to an asymptotic value as the cluster grows and sediments.Corresponding values of the solution turbidity (A), measured at wavelength 600 nm, are also shown. time span of the reaction. The value of D plotted is obtained from the gradient at long reaction times. These reaction mixtures were then left for several hours, after which the angular dependence of their light-scattering intensity was measured. Power-law dependence in wavevector, q, was observed giving exponents D I , as plotted in fig. 2. For the turbidity measurements calculations predict that the exponent p [eqn (9a)l should tend asymptotically to the fractal dimension D as the aggregate size increases. Fig. 3 shows the time dependence of the value of p in an aggregating pellet 4 system with 30°% v/v ethanol. The error bars on each point are those obtained for the gradient in the linear regression calculations of log T on log A.The time dependence of the solution tubidity at 600 nm (fig. 3) indicates that marked sedimentation is occurring in this system beyond ca. lo4 s reaction time (ca. 170 min). This settling appears to take place without change in p, since beyond this point p is approximately constant with an average value of 2.33k0.03. This is close to the value of DH (2.29+0,07) and of D, (2.2 i 0.2) measured under similar conditions. Moreover, from measurements of RH us. t, we can extrapolate to estimate cluster sizes of ca. 2-3 p m at this reaction time, sizes where p is calculated to have values close to the fractal dimension. In situations, therefore, where the particles are large enough to begin sedimenting, the turbidity method is yielding an exponent close to the fractal dimension measured by other light-scattering techniques.It seems reasonable to accept that this limiting value is also another effective measurement of the fractal dimension of the aggregate, D,. Further measurements of D, at other ethanol concentrations confirm this observation (fig. 2), giving 5, values close to Dk, and 5, in all cases where comparative measurements were made, except at low EtOH levels where DH values do not conform to model behaviour. It can also be seen here that varying the ethanol concentration between 27 and 34% v/v produces little systematic change in D, an average value of 2.21 kO.08 being obtained Over the range. Only at 26% ethanol was the value consistently less than 2.1. In this instance, because the aggregation was so slow, cluster size may not have grown266 Structure oj’ Aggregated Casein Systems sufficiently to enable the power-law dependence of turbidity to reach its limiting value.In this system therefore the fractal dimension remains unchanged when we move from the reaction-limited regime at low ethanol levels to diff usion-limited aggregation at higher concentrations. This value of 2.21 for the effective fractal dimension is somewhat higher than that predicted by model simulations of diff usion-limited cluster-cluster aggregation [ D(3 -- d ) = 1.75]16,37 or reaction- limited cluster-cluster aggregation [ D(3 - d ) = 2.0j1’ but is lower than that obtained in simulations of diffusion-limited particle-cluster aggregation [ D(3 - d ) = 2.5]’9-”.Experimentally results for aggregate fractal dimensions have been obtained for several systems. Gold colloids aggregated by addition of pyridine gave D = 1.7 * 0.1 in the diffusion-limited regime” but 2.0 f 0.05 for reaction-limited clusters,’” a value now revised to 2.2.24 Aggregation of colloidal silica particles has been studied by a combination of small-angle X-ray and light scattering, and a fractal dimension of 2.12 f 0.05 found for the In a more extensive study using light scattering, Aubert and Canne1126 have shown that slow aggregation of colloidal silica yielded clusters of D = 2.08 f 0.05 but rapid aggregation could give clusters with D = 1.75 * 0.05 or 2.08 * 0.05, although the clusters with D = 1.75 always restructured to give D = 2.08. Because the turbidity technique requires the production of a very large cluster to ensure limiting-slope behaviour has been attained, it is possible that a restructuring similar to that observed with colloidal silica is occurring during the ethanol-induced aggregation of casein micelles.Some evidence that this may be so can be deduced from the temporal behaviour of the turbidity exponent p as it approaches its limiting value. Fig. 1 demonstrates how the calculated value of p varies as the cluster size R increases. Replotting this behaviour on a semi-logarithmic scale, it is readily seen that p increases linearly with logR within reasonable error limits over a substantial range of R, i.e. we can write p = a log R + C. Over this same range of R, we experimentally observe power-law growth kinetics in micellar aggregation with R cx t”Dl+ or log R = l/DH log t+e.Combining these two equations leads to a prediction of a power-law depen- dence of /3 on reaction time with an exponent a / & , the constant a being calculable for the input fractal dimension, DH . These calculations show that in the fractal dimension range 2.2-2.3, a plot of p vs. log t should be linear (r>0.99) with a gradient of 0.25 to 0.27 (500 < R/nm < 2000). Experimentally we observe such linear behaviour but over the ethanol range 27-36% v/v where D-2.2, an average value of 0.35k0.05 was obtained for the gradient. In other words the exponent p is increasing faster than a mechanism invoking simple cluster-cluster aggregation would predict.Further specula- tions along these lines must await a full analysis of the kinetics of the aggregation and a more extensive study of the structural behaviour of the aggregates. Calcium-induced Aggregation of a,,-Casein Aggregation through to precipitation can be induced in a,,-casein solutions by the addition of calcium ions. These bind to the protein reducing its intrinsic negative charge sufficiently to allow electrostatic repulsive forces between monomers to be overcome and coagulation initiated.5 The time-course of the molecular-weight growth during the aggregation apparently involves two stages, the first consisting of a slow but increasing rate of growth of weight-average molecular weight and the second of a linear growth of molecular weight according to a Smoluchowski mechanism.‘ In this second stage the Smoluchowski rate constant was found to be exponentially related to the square of the net charge of the Ca-casein complex in monomeric Our present interest is in the structure of these random aggregates.Fig. 4 gives details of the temporal dependence of the exponent p derived from the power-law dependence o f the solution turbidity on wavelength. Results are shown for a range of calcium levels from 7.5 to 9 mmol dm--3 over which the aggregation is reaction-limited.D. S. Home 2.5 2.3 2.1 267 - - - 0 P 0 O I' o o 0 ' A ? 1.5j I I I I I I I I I I 100 200 300 400 500 6 00 l / S Fig. 4. The growth of the turbidity exponent, p, as a function of time after initiation of aggregation in four Ca"-a,,-casein systems.The points shown are for mixtures containing 8.5 mg cmP3 asl- casein and Ca'+ concentrations of 7.5 (a), 8 (A), 8.5 (U) and 9 (0) mmol dmP3. In all cases p continued to increase with time and showed no apparent sign of reaching an asymptote for any Ca2' level. We have no parallel measurement of hydrodynamic radius in this instance and although turbidity was levelling, appreciable sedimentation was not evident in these limited-duration experiments. Hence it is difficult to judge whether cluster size would be within the range where the exponent /3 would define an effective, limiting fractal dimension. The final values of p, particularly at higher Ca2' values, are much higher than those obtained in micellar aggregation, but are close to the vallie of D obtained in a study of aggregation in the immunoglobulin system ( D = 2.56 f 0.3).27 It is also possible that in this aggregation of a,,-casein we again have a system which is annealing, continuously rearranging and compacting itself to give an ever-increasing fractal dimension.This is also suggested by the linear behaviour of the exponent p with log t [fig. 5 ( a ) ] . This shows no signs of curvature at long times, possibly because we may still be in the size region where p is linear with log R. Its gradient, however, ( p us. log t ) is greater by a factor of 2 than that predicted from calculations of p vs. log R coupled with assumptions of Smoluchowski aggregation of clusters, suggestive of a more complex structure-building mechanism in the aggregate.This mechanism is apparently under the influence of factors which also control the early stages of the reaction. As already mentioned, in this aggregation reaction the time course of molecular-weight growth can be characterized by a lag stage followed by a second stage where molecular weight growth is linear with time.6 The coagulation time or length of this lag stage depends on the concentrations of both ca.sein and Ca2+ and can be determined by extrapolating the linear portion of the molecular-weight growth curve back to monomer molecular weight, effectively zero turbidity. It has previously been d e m ~ n s t r a t e d ~ ' ~ that reaction profiles generated for various a,,-Ca2' combinations can be superimposed if molecular weight is plotted against a reduced time t / t,, where t , is the coagulation time.This was taken to indicate that both phases of the reaction, the lag phase and the Smoluchowski phase, were controlled by similar factors. Indeed it can be shown that Smoluchowski slope and coagulation time are inversely related to one another. By observing the initial stages of the aggregation in these a,,-Ca2+ combinations we were able to determine their respective coagulation times. When the268 Structure of Aggregated Casein Systems 0 1.8 [ A 0 0 1.60 1.4 I I I i I I I 1 I I I I I I I 2.0 2.2 2.4 2.6 2.0 0.5 1 .o 1.5 2.0 log 1 1% ( t / f,) Fig. 5. ( a ) Turbidity exponent, p, as a function of the logarithm of the reaction time for the four cu,,-Ca*+ systems of fig. 4. The same symbols apply. ( b ) p as a function of the logarithm of reduced time ( = t / t , ) , where t, equals the coagulation time of the reaction for the same four a,,-Ca2+ mixtures.Coagulation times, measured as described in the text, were as follows: for 7.5 mmol dm-3 Ca2+, t, = 23 s, 8 mmol dm-3 15.6 s, 8.5 mmol dm-3 11.8 s and 9 mmol dmP3 7.3 s. exponent p is plotted against log ( t / &), the previously parallel lines collapse onto one another [fig. 5 ( b ) ] . p therefore scales with coagulation time, and the structure develop- ment in the aggregate is apparently controlled by similar factors to those governing monomer interactions and pair-particle potentials. Aggregation of Casein Micelles induced by Rennet Proteolysis Yet another reaction mechanism is demonstrated by casein micelles undergoing proteb- lysis by rennet.The resulting curd formation is the first stage in the cheese-making process. The aggregation follows cleavage of a specific peptide bond in K-casein, loss of this macropeptide destabilizing the micelle and inducing coagulation. In milk it has been demonstrated that of the K-casein must be proteolysed before substantial rates of aggregation occur,** the time required for this reaction producing a lag-phase overall. Information on the structure of the aggregate produced has obvious practical implications. A short series of experiments was therefore carried out to assess the applicability of fractal concepts and provide a further test of the usefulness of the turbidimetric technique for measuring fractal dimension. In this series 8 mm3 of milk was diluted into 3 cm3 of the standard imidazole buffer system with 5 mmol dm-' Ca2+ and the reaction initiated by the addition of rennet solution.Turbidity was scanned as a function of wavelength at suitable time intervals and the exponent /3 derived from the gradient of the double-log plot. As shown in the inset to fig. 6, p approaches an asymptotic value of 2.4. This is more clearly seen in the main body of fig. 6, where p plotted against log t is markedly non-linear and deviates downwards as the reaction progresses. Experience of the calculated behaviour of p vs. log R suggests ihat this indicates the presence of very large aggregates in the system, a view supported by the onset of sedimentation of the particles around this time as shownD.S. Horne 2.5. 2.0- P - 1.5 269 2 . 5 - P 2.0 - I I I I I I 0 0 0 o o o O O 0 0. '2@8 0 0 0 A 0.5 1.0 1.5 2.0 ti lo4 s e A A a A a A A 0 A A 4l 1.0 I 1 I I I I I I I I 2.0 2.5 3.0 3.5 4.0 4.5 log 1 Fig. 6. @, Turbidity exponent, p, as a function of the logarithm of time after addition of rennet solution to 8 mm3 skimmed milk diluted into 3 em3 standard buffer with 5 mmol dmP3 Ca2+. Corresponding changes in solution optical density (A) are also given, manifesting a drop as the aggregates settle out of solution. The inset shows p as a function of real time, illustrating its asymptotic behaviour. by the sharp drop in solution turbidity (fig. 6). Having fulfilled the criterion of cluster size and observed the asymptotic approach of the exponent p to a limiting value, it seems plausible to regard this limiting value as an effective fractal dimension for the micellar aggregate. At a value of 2.4 this aggregate apparently has a fractal dimension higher than that found in ethanol-induced aggregates.Its approach to this dimension is also faster than that observed in the ethanol reaction, but much more work on the growth kinetics of the rennet-induced aggregate is required before a complete picture of structural build-up will emerge. References 1 G. Oster, J. Colloid Sci., 1947, 2, 291. 2 P. Debye, Ann. Phys., 1915, 46, 809. 3 P. Doty and R. F. Steiner, J. Chem. Phys., 1950, 18, 1211. 4 B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1982). 5 D. S. Horne and D. G. Dalgleish, Znf. J. Biol. Macromol., 1980, 2 , 154. 6 D. G. Dalgleish, E. Paterson and D. S. Horne, Biophys. Chem., 1981, 13, 307. 7 D. S. Horne, Int. J. Biol. Macromol., 1983, 5 , 296. 8 D. F. Waugh, in Milk Proteins, ed. H. A. McKenzie (Academic Press, New York, 1971), vol. 2, pp. 3-85. 9 P. Walstra, V. A. Bloomfield, G. J. Wei and R. Jenness, Biochem. Biophys. Acta, 1981, 669, 258. 10 D. S. Horne, J. Colloid Interface Sci., 1986, 111, 250. 11 D. S. Horne and D. G. Dalgleish, Eur. Biophys. J., 1985, 11, 249. 12 W. D. Annan and W. Manson, J. Dairy Res., 1969, 36, 259. 13 .I. Teixeira, in On Growth and Form, ed. H. E. Stanley and N. Ostrowsky (Nijhoff, Dordrecht, 14 G. E. Perlmann and L. G. Longsworth, J. Am. Chem. SOC., 1948, 70, 2719. 15 J. G. Rarity and P. N. Pusey, in On Growth and Form, ed. H. E. Stanley and N. Ostrowsky (Nijhoff, 16 P. Meakin, Phys. Rev. Lett., 1983, 51, 1119. 17 M. Kolb, R. Botet and R. Jullien, Phys. Reu. Left., 1983, 51, 1123. 18 M. Kolb and R. Jullien, J. Phys. (Paris) Lett., 1984, 45, L977. Netherlands, 1986), pp. 145-162. Dordrecht, Netherlands, 1986), pp. 218-221.270 Structure of Aggregated Casein Systems 19 T. A. Witten and L. M. Sander, Phys. Rev. Lett., 1981, 47, 1400. 20 T. A. Witten and L. M. Sander, Phys. Rev. B, 1983, 27, 5686. 21 P. Meakin, Phys. Rev. A, 1983, 27, 604. 22 D. A. Weitz and M. Oliveria, Phys. Rev. Lett., 1984, 52, 1433. 23 D. A. Weitz, J. S. Huang, M. Y. Lin and J, Sung, Phys. Rev. Lett., 1985, 54, 1416. 24 P. Dimon, S. K. Sinha, D. A. Weitz, C. R. Safinya, G. S. Smith, W. A. Varaday and H. M. Lindsay, 25 D. W. Schaefer, J. E. Martin, P. Wiltzius and D. S. Cannell, Phys. Rev. Lett., 1984, 52, 2371. 26 C. Aubert and D. S. Cannell, Phys. Rev. Lett., 1986, 56, 738. 27 J. Feder, T. Jossang and E. Rosenqvist, Phys. Rev. Lett. 1984, 53, 1403. 28 D. G. Dalgleish, J. Dairy RPS., 1979, 46, 653. Phys. Rev. Lett., 1986, 57, 595. Received 27 th November, 1986

ISSN:0301-7249    

DOI:10.1039/DC9878300259

出版商:RSC    

年代:1987

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Faraday Discuss. Chem. Soc., 1987, 83, 271-285 The Dynamics of Colloidal Particles suspended in a Second-order Fluid Shirley J. Johnson and Gerald G. Fuller* Stanford University, Stanford, California 94305, U.S.A. The transient flow response of colloidal particles in sheared suspensions has been examined with linear dichroism measurements. Data are reported for dilute suspensions of spheroidal haematite ( a - Fe,O,) particles in both a Newtonian and a non-Newtonian Boger fluid consisting of 1000 ppm poly- isobutylene in low-molecular-weight polybutene. Haematite samples with average axis ratios of 2.66, 3.61 and 5.82 were used in the studies. Flow reversal and flow start-up experiments were conducted for shear rates up to 10 s-' using both Couette and parallel-plate flow devices.The experimental data for suspensions in the non-Newtonian fluid are in qualitative agreement with predictions from the theory for particle motion in a second-order fluid. The particles experienced orbit drift and tended toward alignment along the vorticity axis of the flow. Also the period of rotation of the particles relative to that in the Newtonian suspending fluid increased with increasing shear rate. The predictions do not explicitly account for particle geometry, so experimental trends with axis ratio cannot be compared directly with the theory. In addition, Brownian motion is not considered by the theory, and we expect that the presence of Brownian motion in our suspensions strongly contributed to the steady-state distribution of particle orientation. There is considerable interest in the microdynamics of flowing suspensions, because the bulk properties of a suspension are intimately related to the orientation of the suspended particles.The motion of particles in Newtonian suspending fluids is well documented, but particle motion in non-Newtonian suspending media is less well understood. Theo- retical studies of particle motion in non-Newtonian fluids are limited to analyses that model the suspending media by the second-order fluid constitutive equation. Experi- mental evidence supporting these analyses was minimal, in large part because of the lack of an adequate technique capable of extracting sufficient information from transient flow experiments. In this paper, we report results of rheo-optical experiments of sheared colloidal suspensions in a pseudo-second-order fluid.The optical technique of linear dichroism is employed to probe non-intrusively the particle dynamics in transient simple shear flow. Previous Theoretical and Experimental Investigations Particle motion in a Newtonian liquid was analysed theoretically by Jeffery,' who solved the creeping-flow equations for a freely suspended, rigid ellipsoid in a Newtonian liquid undergoing simple shear flow. He found that the particle rotates in a time-dependent, periodic orbit about the vorticity axis of the flow. In the absence of randomizing effects such as Brownian motion, the particle is predicted to remain in the same orbit indefinitely. Jeff ery's theory is supported by extensive experimental investigations, conducted primarily by Mason and coworker^.^-^ The motion of a particle in a non-Newtonian fluid is much more difficult to analyse because of the non-linearity of the constitutive equations describing non-Newtonian 27 1272 Particle Motion in a Second-order Fluid X Fig.1. Coordinate system for a particle having its axes oriented at 8 and 4. Simple shear flow is in the xy plane. fluid dynamics. Mathematical complexity has limited detailed theoretical calculations to perturbation expansions about the Newtonian limit. The suspending fluid is modelled as a Rivlin-Ericksen second-order fluid,5 which has a constant viscosity and a first normal stress difference that is quadratic in shear rate. The model is valid for rheologi- cally slow flows where Re << We << 1.Re is the Reynolds number (the ratio of hydrodynamic forces to viscous forces) and We is the Weissenberg number (the ratio of normal stress to shear stresses). The particle is assumed to be rigid, freely suspended and non-Brownian. Leal6 derived equations to describe the motion of a slender rod in a second-order fluid undergoing simple shear flow, and B r ~ n n ~ ' ~ derived analogous equations for a near-spherical spheroid and a rigid tridumb-bell. All three of these analyses predict that a prolate particle drifts to an equilibrium orbit where its major axis is aligned along the vorticity axis of the flow. There is no available theory to describe the motion of arbitrarily shaped particles in a sheared second-order fluid, but Brunns39 derived a two-parameter model applicable to any axisymmetric particle.The particle orientation is described by the Euler angles 6 and 4 as shown in fig. 1. The angular velocities of the particle are given by the time derivatives of 8 and 4 : (1) %= G( dt 1 + r: (sin2 4 + rz cos' 4 ) -iGH, sin' 6 sin (44) (3) where G is the shear rate and re is the particle axis ratio. The parameters H , and H2, whose magnitudes are assumed to be much less than unity, depend on the fluid elasticity and particle geometry. The equations reduce to Jeffery's results for particle motion in a Newtonian suspending fluid when H , and H- are zero. Experimental investigations of particle motion in non-Newtonian suspending fluids are limited to several studies by Mason and coworkers, which were performed priorS.J. Johnson and G. G. Fuller 273 to publication of the theoretical results of Leal and Brunn. Both viscoelastic and pseudoplastic suspending fluids were used, none of which corresponded to the second- order fluid constitutive model. Mason et al. photographed non-Brownian particles and manually measured their time-dependent orientations. The findings were of a qualitative nature, but it was observed that, under certain conditions, a prolate particle drifted to alignment along the vorticity axis. Brownian particles in sheared suspensions are not readily studied by visual observa- tion because of their small size. We have recently developed an optical technique capable of monitoring the dynamics of colloidal particles in sheared polymeric 1iq~ids.I~ Our technique measures the linear birefringence and linear dichroism caused by flow- induced anisotropy in the suspension microstructure.In a previous publi~ation,'~ we demonstrated that the particle dynamics can be monitored through dichroism, while the polymer dynamics can be monitored through birefringence. For the purpose of the present discussion, we emphasize the particle dynamics as observed through the dichro- ism measurement. The linear dichroism of a sheared suspension is directly related to the orientation of the suspended particles. With reference to fig. I , dichorism can be measured with the incident light propagating either along the z-axis (using a Couette flow cell) or the y-axis (using a parallel-plate flow cell). One measures the net anisotropy of the particles' projection into the plane perpendicular to the direction of light propagation.For a particle oriented along a unit vector (2, j, Z), the dichroism measured when the z-axis is the viewing axis is given by A n: = M [ (22 - Y-j)' - 4( %j>Z] 'I2. (4) When the y-axis is the viewing axis, we have An;= M I ( S - Z ) l . ( 5 ) M = ; ~ n , N k ' ( a : - as) (6) The constant M is given by15 where n, is the refractive index of the suspending fluid, N is the number of particles per unit volume, k is the wavenumber (27r/h), and a l and aZ are the principal polarizabilities of the particle. The angular brackets ( ) symbolize averaging over all particle orientations. The orientation angle, CP, of the dichroism in the xy plane is given by tan (2CP) = 2(2j)/(Z2 -yp).(7) This angle corresponds to the net orientation angle of the particles with respect to the flow direction. In the xz plane the orientation angle, 9, of the dichroism is given by tan (2V) = 2(3.?)/(32 - ZZ). (8) The moment (25) is zero by symmetry, so V is either 0" (when 2% >- ZZ) or 90" (when 22 < 55). The dichroism signal of a sheared suspension in a Newtonian fluid exhibits oscilla- tions reflecting the periodic Jeffery orbits of the particles. l 3 , I 6 Polydispersity in particle axis ratio causes the oscillations to be damped owing to phase mixing of the orbits. The resulting steady-state dichrosim is a consequence of the net alignment of the particles along the flow direction. Thus the net orientation angle, @, is damped to a value of 0".As described above, the theoretical analysis for a prolate particle suspended in a second-order fluid predicts that the particle drifts to alignment along the vorticity axis (the z-axis in fig. 1 ) . In this case the dichroism An: observable in a Couette flow experiment is expected to decay to zero because rods observed end-on project a circuIar (isotropic) image. The measurable quantity in a parallel-plate experiment is274 Particle Motion in a Second-order Fluid 0.0 r 0 . 6 2 $ 0 . 4 a 0 . 2 45 35 2 5 15 5 - 5 -1 5 -25 -35 - 45 r t 0 1 2 3 4 5 6 0 1 2 3 4 5 6 t / ?; t / ?. 0 . 5 - 1 I 1 I I 1 I 0 1 2 3 4 5 6 t / T Fig. 2. Theoretical predictions for a sheared suspension in a second-order fluid with parameters H , = H2 = 0.02. The particles are non-Brownian with a random initial orientation.A Gaussian distribution in axis ratio was assumed with an average of 3.0 and a standard deviation of 0.2. The abscissa is time divided by T, where T is the Jeffery period for a particle of the average axis ratio. Part (a) shows predictions for the dichroism and orientation angle in the xy plane, simulating results of a Couette flow experiment. Part ( b ) shows the prediction for the measurable quantity in a parallel-plate experiment. M is given by eqn ( 6 ) . An; cos (2*), where !?! is the orientation angle of the dichroism in the xz plane. For suspensions in Newtonian fluids, V is always 0", so the measured quantity is simply An;. In weakly non-Newtonian fluids, the particles rotate initially as in a Newtonian suspending fluid (*=O"), but eventually attain a net alignment along the vorticity axis (!P = 90").Thus the measured signal is expected to change sign. We solved eqn (2) and ( 3 ) numerically to quantify predictions for the dichroism and orientation angle according to eqn (4), (5) and (7). The calculation represented a polydisperse suspension with random initial orientation and no Brownian motion. Polydispersity was introduced as a Gaussian distribution in axis ratio with an average of 3.0 and a standard deviation of 0.2. The initial condition of random orientation wasS. J. Johnson and G. G. Fuller 27 5 0 . 3 r: * N 0.2 a --. 0.1 0 0 0.5 1 1.5 2 tl T 4 5 35 2 5 15 5 -. * - 5 -1 5 - 2 5 -35 -4 5 -0.8 I I I I 0 0.5 1 1.5 2 t / f Fig. 3. Theoretical predictions far a sheared suspension in a second-order fluid with parameters H , = H2 = 0.35.The particles are non-Brownian with a random initial orientation. A Gaussian distribution in axis ratio was assumed w$h an average of 3.0 and a standard deviation of 0.2. The abscissa is time divided by T, where T is the Jeffery period for a particle of the average axis ratio. Part ( a ) shows predictions for the dichroism and orientation angle in the xy plane, simulating results of a Couette flow experiment. Part ( b ) shows the prediction for the measurable quantity in a parallel-plate experiment. M is given by eqn ( 6 ) . achieved by spacing 560 particles evenly across the surface of a unit sphere. A shear rate of 5 s-l was chosen, and arbitrary values of H1 and H2 were assumed. Representative results are shown in fig.2 and 3, where part ( a ) of each figure simulates a Couette flow experiment and part ( b ) simulates a parallel-plate experiment, The predictions shown in fig. 2 are for H , = H2 = 0.02, so the non-Newtonian terms in eqn (2) and (3) are quite small. The dichroism An:' exhibits damped oscillations that tend toward zero as the particles drift toward the vorticity axis. The net orientation276 Particle Motion in a Second-order Fluid angle damps to 0" as it does when a Newtonian suspending fluid is employed. The quantity An; cos ( 2 q ) also shows damped oscillations and changes sign as expected. Fig. 3 shows predictions from calculations that incorporated much larger non- Newtonian influences, as here HI = H2 = 0.35. The oscillations characteristic of Jeff ery orbits are no longer apparent.The particles quickly drift out of their original orbits and tend toward the vorticity axis of the flow as indicated by the small An: The net orientation angle adopts a steady-state value that is negative as a result of an asymmetric rotation of the particles; the particles rotate out of the aligned state more slowly than they rotate into it. The quantity A n t cos ( 2 9 ) is negative because the net orientation of the particles is along the vorticity axis (V = 90'). In general, we found that as H1 and Hz were increased from zero, the damping of the oscillations became more pronounced. When HI and H2 were made large (but still less than unity), only a single peak was exhibited, as in fig. 3 ( a ) .Intermediate HI and H2 values gave rise to an increased period relative to the Jeffery period. Increasing the shear rate for a given H, and H2 also resulted in an increased period. Other quantities constant, the dichroism for particles with higher axis ratios decayed more rapidly. These results are described in more detail el~ewhere.'~ Cohen and coworkers'' have recently extended the analysis for the motion of a slender rod in a second-order fluid to include Brownian diffusion effects. The orientation distribution that is ultimately attained arises from a compromise between the fluid elasticity force and the randomizing force of Brownian diffusion. Non- Brownian slender rods in a second-order fluid align along the vorticity axis,6 while the preferred orientation for slender rods with weak Brownian diffusion in a Newtonian fluid is essentially along the flow direction." As a compromise between these two cases, the resulting steady state orientation distribution function for slender rods with weak Brownian rotations in a second-order fluid is essentially bimodal.There is a very sharp peak in the distribution function along the vorticity axis with a smoother peak along the flow direction. The population of particles oriented along the vorticity axis increases with increasing elasticity. The analysis for a general spheroidal particle with weak Brownian motion suspended in a second-order fluid has not yet been done. Rheo-optical Experiments and Discussion Spheroidal haematite (a-Fe,O,) particles of three different axis ratios, designated as particles A, B and C, were prepared according to the method of Ozaki et aLzo Trans- mission electron microscopy was used to characteristize the physical dimensions of the particles, and typical micrographs are shown in plate 1.Studies of the optical properties of the particles are reported elsewhere.21 There are very few fluids that are adequately described by the second-order fluid constitutive equation. A class of fluids commonly called Boger approximate second-order fluid behaviour over a limited range of shear rate, although they are a o r e accurately described by the Oldroyd B constitutive e q ~ a t i o n . ~ ~ ' ~ ~ A Boger fluid is a dilute solution of high-molecular-weight polymer chains dissolved in a high-viscosity Newtonian solvent, and many different combinations are possible.Following the example of Prilutski and coworkers,2s we prepared a Boger fluid of 1000 ppm by weight polyisobutylene ( M = 2.7 x lo6) in low-molecular-weight polybutene (Parapol 950 kindly provided by Exxon Chemical Company). Fig. 4 shows steady-state rheological data collected on a Rheometrics mechanical spectrometer. The solution exhibits second-order behaviour for shear rates between ca. 1 and ~ O S - ' . The Weissenberg numbers range from 0.3 at a shear rate of 1 s-l to 4.3 at a shear rate of 10 s-l, so the condition We<< 1 is not strictly met. We did not attempt to measure the second normal stress difference, but other have found that the second normal stress difference of Boger fluids is essentially zero.Rheological data collected for the polybutene alone revealedFaraday Discuss. Chem. Soc., 1987, Vol. 83 Plate 1 Plate 1. Electron micrographs of haematite particles. The scale is the same for all three photo- graphs. Upper: particles A, which have an axis ratio of 2.66k0.14 and an average length of 0.315 pm; centre: particles B, which have an axis ratio of 3.61 *0.24 and an average length of 0.360 pm; lower: particles C, which have an axis ratio of 5.82k0.58 and an average length of 0.485 pm. S. J. Johnson and G. G . Fuller (Facing p . 276)S. J. Johnson and G. G. Fuller 5.. I - n 2- v 3 - M I ..I n F 30 - 2 - - 1 - 0 277 K / x x 01 I I I I -0.5 0 0.5 1 1.5 log G Fig. 4. Rheological data for 1000 ppm polyisobutylene in polybutene as a function of shear rate G at a temperature of 23 "C.r] is the viscosity ( x ) and N , is the first normal stress difference (0). A line of slope 2 is shown. that it has a constant viscosity comparable to that of the polyisobutylene solution and no measurable first normal stress difference. Thus the polybutene is Newtonian over the shear rate range of interest. Dilute suspensions (<800pprn particles by weight) of each of the three particle samples were prepared in both polybutene (PB) and in the polyisobutylene solution (PIB). The suspensions were placed in a sonic bath to break apart any aggregates. Linear dichroism experiments were performed over a range of shear rates from 0.1 to 10 s-' using both Couette and parallel-plate flow cells. When the Couette flow cell was employed, the temperature was held at 25 "C by circulating water through the inner cylinder.Experiments using the parallel-plate device were conducted at ambient tem- perature (ca. 23 "C) in a thermostatically controlled room. Flow reversal experiments provide a convenient way to check the reversibility or memory of the particle orbits. In a Newtonain fluid, owing to the linearity of the creeping flow equations, it is expected that the particles retrace their orbits exactly when the flow is reversed. If there are non-linear contributions from Brownian rotations or a non-Newtonian suspending fluid, the particles experience memory impairment on flow reversal.28 Typical data from a flow reversal experiment for a suspension in PB are shown in fig. 5 ( a ) . The dichroism and net orientation angle exhibit the characteristic damped oscillations of a polydisperse suspension.When the flow is reversed, the oscillations are mirrored as a result of the reversal of the particle orbits. The initial condition is not recaptured exactly owing to irreversible effects such as Brownian motion. Data for flow reversal experiments for the particles suspended in PIB are shown in fig. 6. These plots demonstrate that the PIB has a marked effect on the particle orbits. Flow reversal demonstrates that there was severe memory impairment at a shear rate of 5 s-' and complete memory loss at a shear rate of 10 s-'. These data establish that the particles quickly drift out of their initial orbits. To explore the orbit drift more thoroughly we performed experiments of very long flow duration.Orbit drift was not observed for suspensions in PB, as demonstrated by typical data from a Couette flow experiment shown in fig. 5 ( b ) . The oscillations in the dichroism damp to a steady-state value, and the steady-state net orientation angle is278 Particle Motion in a Second-order Fluid 2 45 30 15 1.5 % N E P 1 8 0 2 -15 -30 0 - 45 0.5 0 3 6 9 12 0 3 6 9 12 t l s r l s 8 6 S 2 - 45 30 15 t o 8 -15 0 50 100 150 200 0 50 100 150 200 t l s t l s Fig. 5. Typical results of Couette flow experiments for suspensions in polybutene (Newtonian). ( a ) Dichroism and net orientation angle for particles A in a flow reversal experiment at a shear rate of 10 s-'. The arrows indicate where the flow was stopped, s, and reversed, r. ( b ) Dichroism and net orientation angle for particles C in a flow start-up experiment at a shear rate of 7.5 s-'.The arrow indicates where the flow was stopped. along the flow direction. When the flow is stopped, the dichroism smoothly decays as the particles relax by Brownian motion, and the net orientation angle remains at 0". Parallel-plate experiments with the particles in PB duplicated the dichroism measure- ments collected using the Couette flow cell as expected. The suspensions in PIB behaved as though the suspending fluid were Newtonian for shear rates < ca. 1 s-', but for higher shear rates, significant orbit drift was observed. Fig. 7 shows results of Couette flow experiments for suspensions in PIB. The dichroism still displays damped oscillations (compressed near the origin on this timescale), but there is a decay in the signal over a rather long timescale. This is barely apparent for particles A, but is easily discernible for particles B and C.The decay in the dichroism results from the shortened projection of the particles in the xy plane as they tend toward alignment along the vorticity axis. The alignment of the particles is incomplete, however, as the dichrosim does not decay to zero. The steady-state angle for particles B and C is 0"; on average, the particles spend more time oriented in the flow direction. The steady-state angle for particles A is negative, probably as a result of an asymmetric rotation about the vorticity axis. Data collected from parallel plate experiments for suspensions in PIB are shown in fig.8- The signal decays for all three samples, indicating that there is drift in the particle orbits toward the z-axis. The signal does not change sign for particles C, suggestingS. J. Johnson and G. G. Fuller 279 2 r 1.5 0.5 0 2 1.5 -0.5 45 30 15 e o 8 -15 -30 -45 I' s r r s r 45 I s r t - -301. 1 . I . I . I -45 0 0 5 10 15 20 0 5 10 15 20 t l s t / s Fig. 6. Dichroism and net orientation angle collected in Couette flow experiments for a suspension of particles A in the PIB solution. The experiments consisted of a flow-reversal sequence with the arrows indicating where the flow was stopped, s, and reversed, r. ( a ) Shear rate of 5 s-'. ( b ) Shear rate 10s-'. that the degree of alignment along the vorticity axis is less complete for particles C than for particles A and B, where An;cos ( 2 q ) does change sign.An interesting feature of the data is the abrupt change when the flow is arrested, which is especially evident in the plots of and An: cos (2!P). This indicates that the particles experience torque when the flow is stopped. The timescale for this relaxation phenomenon corresponds to the timescale associated with the PIB molecules, as observed through birefringence measurements (not shown). The effect of shear rate is summarized in fig. 9. We measured the period of oscillation of the dichroism and compared it to the period for the same particles suspended in PB. Fig. 9 ( u ) shows the percentage increase in the period as a function of shear rate, The steady-state dichroisrn data for the Couette and the parallel-plate experiments are shown in fig.9 ( b ) and ( c ) , respectively. The ordinate for these plots is the ratio of the maximum dichroisrn (measured as the height of the first overshoot peak) to the steady-state dichroism of a suspension in PIB, normalized by the same ratio for a suspension in PB. By plotting the data in this way, we eliminate the need to know the exact concentration of each suspension, which is very difficult to measure. Non-Newtonian influences of the suspending fluid are more pronounced at higher shear rates, and definite trends are observed for each suspension. Increasing the particle axis ratio, however, does not result in consistent trends. Brunn' stated that particle dynamics in a second-order fluid280 Particle Motion in a Second-order Fluid L ------ - 1 .I . I . l 2 1.5 I ( a > S t 45 S t 30 15 r o 8 -15 0.5 -30 - 45 0 2 45 r I A\ 30 15 1.5 L O 8 - 15 t - 3 0 -45 c 1 . 1 . 1 . 1 . 1 0 50 .lo0 150 200 t l s 6 45 S # 30 15 4 > N E a r 2 L L- L O IQI -15 - 30 -4 5 0 0 5Q 100 150 200 t l s 0 50 100 150 200 t / s Fig. 7. Dichroism and net orientation angle collected in Couette flow experiments for suspensions in the PIB solution at a shear rate of 7.5 s-'. The arrow indicates where the flow was stopped. ( a ) Particles A, ( h ) particles B and (cl particles C. depend on the exact geometry of the particle, not simply on an equivalent axis ratio. These data lend support to that hypothesis. The degree of particle alignment along the vorticity axis is not as complete as the theory predicts (see fig.2 and 3), even at the highest shear rates studied. We expect that the effect of Brownian motion, which was not considered in the original theory, is largely responsible for this incomplete alignment. Particle shape effects and the importance of Brownian motion are coupled, however, and these could not be varied independently.S. J. Johnson and G. G. Fuller 28 1 -2 0 u 50 100 150 200 t / s S t 0 50 100 150 200 t / s S t -1 0 50 100 150 200 t l S Fig. 8. Results of parallel-plate experiments for suspensions in the PIB solution at a shear rate of 7.5 s-'. The arrow indicates where the flow was stopped. ( a ) Particles A, ( b ) particles B and (c) particles C. The theory for Brownian spheroids in a second-order fluid not being available, we rely on Cohen's results for slender rods to aid in interpreting our data.The possibility of a bimodal orientation distribution function suggests that our data may be qualitatively represented as in fig. 10. These schematic drawings represent contours of constant orientation probability as projected onto the xy and xz planes for those shear rates where significant orbit drift occurred. The dashed ellipses correspond to steady-state projections for suspension in a Newtonian fluid.282 1 - 0.5 m c z E ' 0 . w -0.5 Particle Motion in a Second-order Fluid - . 0 ' I I I I 0 3 6 9 12 G/s-' ( b ) 1.2 1 . 1 m 1 G 3 0.9 ' 0.8 0.7 Oa6 0.5 0 6 3 6 9 12 GIs-' *\ - 1 ' I 1 1 I 3 6 9 12 GIs-' Fig. 9. Comparison of the data collected for suspensions in PIB and in PB as a function of shear rate.???e syrnbc!~ represent data co!!ecteb f ~ r A, partic!es A; W, prtic!es !3 and a, particles C. (a) Percentage increase in the period of oscillation in the dichroism, An"=, relative to that for the same suspension in PB. (b) Results of Couette flow experiments. The ordinate is the ratio of maximum dichroism to steady-state dichroism for suspension in PIB normalized by the same ratio for suspensions in PB. (c) Results of parallel-plate experiments. The ordinate is the same as in part (b).A S. J. Johnson and G. G. Fuller B C 0 0 Y Y z z 283 Fig. 10. Estimated contours of constant orientation probability as projected onto the xy plane (based upon Couette flow data) and the xz plane (based upon parallel-plate data) for suspensions in PIB at steady state.The dashed ellipses represent similar projections for the suspensions in the Newtonian fluid. The particles themselves (A, B and C) are represented at the top. Data collected from the Couette flow experiments were used to sketch the orientation probability as projected onto the xy plane shown in fig. 10. The steady-state projection for particles A in PIB onto the xy plane is lengthened from the projection in a Newtonian fluid, as the ratio of peak to steady-state dichroism is higher in PIB than in PB. Furthermore the ellipse is oriented slightly off the flow direction, because the steady-state net orientation angle is negative. The ratio of peak to steady-state dichroism for suspensions of particles B and C in PIB is lower than that for these particles in PB.This suggests that a significant number of particles drifted toward the vorticity axis, which effectively shortens their projections onto the xy plane. Therefore the ellipses representing the projected orientation probabilities for particles B and C have a smaller eccentricity than the dashed ellipses that signify their projections when a Newtonain suspending fluid is employed. The projections of constant orientation probability onto the xz plane shown in fig. 10 are adapted from the parallel-plate experiments. The signals for both particles A and B changed sign, thus indicating that a higher density of particles were aligned along the vorticity axis than along the flow direction. Hence we speculate that the projected orientation probabilities have small lobes along the x-axis and larger lobes along the z-axis.The signal for particles C decayed but did not change sign, so .the projection has a larger lobe along the x-axis than along the z-axis. The trend with axis ratio for the data collected from the Couette flow experiments is not easily correlated with the data collected from the parallel-plate experiments. For example, the projection of the orientation probability onto the xy plane for particles C is less anisotropic than that for either particles A or B, so one might anticipate that particle C would exhibit the greatest degree of alignment along the vorticity axis. Yet the dichroism from the parallel-plate experiment did not change sign for particles C, suggesting that a higher proportion of particles are oriented along the x-axis.Unfortu- nately, the dichroism experiment yields information on the moments of the distribution284 Particle Motion in a Second-order Fluid function and cannot yield the exact details of this function. A complementary technique, small-angle light scattering, has been developed in our l a b ~ r a t o r y , ~ ~ and it has the potential of extracting the entire orientation distribution function. Studies of particle motion in non-Newtonian suspending media are currently being conducted with this technique. In comparing our results to the predictions, we note that the theoretical assumptions were not met exactly. Boger fluids only approximate the second-order fluid model, and the fluid that we used had Weissenberg numbers on the order of one, not We<< 1 as the theory specifies.Furthermore, the particles are not perfectly monodisperse in size, and contain surface imperfections. Also polymer adsorption onto the particle surfaces undoubtedly occurs which may affect the particle dynamics. We expect that this effect is small, however, since trends in the data as velocity gradient is decreased tend smoothly toward the results obtained for suspensions in the Newtonian fluid. Brownian motion of the particles is probably the largest contributing mechanism to discrepancy between the theory and our experimental data, because Brownian rotations have a randomizing effect that was not incorporated into the original theory. Conclusions The linear dichroism data collected for sheared colloidal suspensions in a Boger fluid are generally in qualitative agreement with theoretical predictions for particle motion in a second-order fluid.The particles experience orbit drift toward the vorticity axis of the flow, although the alignment is not as complete as the theory predicts. Trends with particle axis ratio were not obvious, suggesting that specifying an equivalent axis ratio is insufficient for predicting particle motions in non-Newtonian suspending fluids. For particles of a given axis ratio, increasing the shear rate resulted in a greater degree of orbit drift and also an increased period of rotation relative to the Jeffery period. As the available theory for spheroidal particles does not consider Brownian rotations, we expect that Brownian motion exhibited by the particles in our experiments is largely responsible for discrepancies between our data and the predictions. Theoretical analysis, similar to that done by Cohen et al.for slender rods, is needed for Brownian spheroids in a second-order fluid to more closely represent our experimental conditions. We gratefully acknowledge Dr A. Huang of Rheometrics Inc. for collecting the rheologi- cal data on the Rheometric mechanical spectrometer. This research was supported by a grant from IBM and the National Science Foundation grant CPE 8412647. References 1 G. B. Jeffery, Roc. R. SOC. London, Ser. A, 1922, 102, 161. 2 H. L. Goldsmith and S. G. Mason, in Rheology Theory and Applications, ed. F. R. Eirich (Academic 3 A. Okagawa and S. G. Mason, J. Colloid Interface Sci., 1973, 45, 330. 4 A. Okagawa and S. G. Mason, Can. J. Chem., 1977, 55, 4243. 5 G. Astarita and G. Marrucci, Principles of Non-Newtonian Fluid Mechanics (McGraw-Hill, London, 6 L. G. Leal, J. Fluid Mech., 1975, 69, 305. 7 P. Brunn, Rheol. Acta, 1979, 18, 229. 8 P. Brunn, J. Fluid Mech., 1977, 82, 529. 9 P. Brunn, J. Non-Newtonian Fluid Mech., i980, 7, 271. Press, New York, 1967), vol. 4, p. 85. 1974), p. 198. 10 F. Gauthier, H. L. Goldsmith and S. G. Mason, Kolfoid 2. Z. Polym., 1971, 248, 1000. 1 1 F. Gauthier, H. L. Goldsmith and S. G. Mason, Rheol. Acta, 1971, 10, 344. 12 E. Batram, H. L. Goldsmith and S. G. Mason, Rheol. Acta, 1975, 14, 776. 13 S. J. Johnson, P. L. Frattini and G. G. Fuller, J. Colloid Interface Sci., 1985, 104, 440. 14 S. J. Johnson and G. G. Fuller, Rheol. Acta, 1986, 25, 405. 15 M. Stoimenova, L. Labaki and S. Stoylov, J. Colloid Interface Sci., 1980, 77, 53.S. J. Johnson and G. G. Fuller 285 16 P. L. Frattini and G. G. Fuller, J. Colloid Znterface Sci., 1984, 100, 506. 17 S. J. Johnson, PhD. Thesis (Stanford University, 1987). 18 C. Cohen, B. Chung and W. Stasiak, Rheol. Acta, in press. 19 L. G. Leal and E. J. Hinch, J. Fluid Mech., 1971, 46, 685. 20 M. Ozaki, S. Kratohvil and E. MatijeviC, J. Colloid Interface Sci., 1984, 102, 146. 21 S. J. Johnson and G. G. Fuller, J. Colloid Interface Sci., in press. 22 D. V. Boger, J. Non-Newtonian Fluid Mech., 1977, 3, 87. 23 D. V. Boger and H. Nguyen, Polym. Eng. Sci., 1978, 18, 1037. 24 R. J. Binnington and D. V. Boger, J. Rheol., 1985, 29, 887. 25 G. Prilutski, R. K. Gupta, T. Sridhar and M. E. Ryan, J. Non-Newtonian Fluid Mech., 1983, 12, 233. 26 M. Keentok, A. G. Georgescu, A. A. Sherwood and R. I. Tanner, J. Non-Newtonian Fluid Mech., 1980, 27 K. P. Jackson, K. Walters and R. W. Williams, J. Non-Newtonian Fluid Mech., 1984, 14, 173. 28 A. Okagawa, G. J. Ennis and S. G. Mason, Can. J. Chem., 1978, 56, 2815. 29 A. J. Salem and G. G. Fuller, J. Colloid Interface Sci., 1985, 108, 149. 6, 303. Received 12th December, 1986

ISSN:0301-7249    

DOI:10.1039/DC9878300271

出版商:RSC    

年代:1987

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1987, 83, 287-295 Static and Dynamic Light-scattering Study of Solutions of Strongly Interacting Ionic Micelles Laura Cantu” Dipartimento di Chimica e Biochimica Medica, Universita di Milano, 20133 Milano, Italy Mario Corti and Vittorio Degiorgio Dipartimento di Elettronica-Sezione di Fisica Applicata, Universitci di Pavia, 271 00 Pavia, Italy Dilute solutions of ionic micelles formed by biological glycolipids (ganglio- sides) have been investigated at very low ionic strength by static and dynamic light scattering. With no added salt the effect of electrostatic interactions is so strong that both the scattered intensity and the measured diffusion coefficient are different by an order of magnitude with respect to the values obtained for weakly interacting micelles.The electric charge of the micelle is derived by a fit of the experimental structure-factor data which uses the hypernetted chain approximation for the radial distribution function. The dependence of the diffusion coefficient on the ionic strength is partially explained by considering, in addition to hydrodynamic interactions, effects due to the non-negligible size of the Na+ and C1- ions. The measured intensity correlation function deviates from exponential behaviour at extremely low ionic strength. In the last few years many paFers have discussed the behaviour of interacting Brownian If we confine our attention to scattering experiments performed with globular particles, we find that the list of systems studied so far by light or neutron scattering includes large colloidal particles (polystyrene latex sphere^),^,' small protein^,"'^ micel- less-’’ and microemu1sions.12 The interpretation scheme for static scattering data is well established: the intensity scattered by the Brownian particles, I , ( k ) , can be expressed as where A is a constant, c is the P ( k ) the particle form factor the existence of interparticle function g ( r ) as follows:* S( k ) = I,( k ) = AcMP( k ) S ( k ) ( 1 ) particle concentration, M the particle molecular weight, and S ( k ) the structure factor which takes into account interactions.S ( k ) is related to the radial distribution 1 + 4vck-I [g( r ) - l]r sin kr dr. (2) I The function g ( r ) can be calculated once the pair interaction potential V ( r ) is known. Depending on the specific system under investigation, various approximations have been used to derive g ( r ) : the dilute gas approximation,* the mean spherical approxima- tion (MSA),” the rescaled MSAI3 and the hypernetted chain a p p r o ~ i m a t i o n .~ ” ~ ” ~ The main problem in a real system is to assign precisely the potential V ( r ) . From this point of view; a dilute collection of e!ectrically charged identical particles in a low ionic strength solution, where the dominant contribution to V ( r ) comes from the Coulomb repulsion and short-range interactions can be neglected, is a very convenient system to investigate. As we show in this paper [see also ref. (lo)], solutions of ionic amphiphiles can be good candidates for such a study, provided that their critical micelle concentration is sufficiently low.287288 Light-scattering Studies of Ionic Micelles The dynamics of interacting Brownian particles represents a very difficult theoretical problem, because the calculation of the dynamic structure factor F(k, t ) has to take into account not only V ( r ) , but also hydrodynamic interactions. In most cases6-'' the measured F(k, t ) is an exponential with time constant (k2D)-', where D is a collective diffusion coefficient, but in the most general situation'-3 the shape of F(k, t ) may be non-exponential. It has been recently pointed out that an approach which treats counterions and coions as point charges is not completely satisfactory when the colloidal particlcs are rather small, as it happens with micellar solutions.As far as the calculation of S ( k ) is concerned, it is possible to treat both the macroions and the small ions in a consistent way, taking into account both their charges and sizes, and postulating their mutual interactions as bare Coulomb interactions in a dielectric m e d i ~ m . ~ ~ . ' ~ An approximate treatment of the effect of small ions on the diffusion coefficient D has also been pre~ented.'~ In this paper we report light-scattering measurements performed on ionic micelles of biological glycolipids in very low ionic strength aqueous solutions [some of our results have appeared in ref. (lo)]. Our experiments are similar to those performed by Cannell and coworker^,^ who studied dilute solutions of a globular protein, bovine serum albumin (BSA), at very low concentrations of added salt.Our system, however, differs fundamentally with respect to solutions of macromolecules or inorganic colloids, because the micelle is a spontaneous aggregate, its electric charge is unknown a priori, and the system may present an intrinsic polydispersity. The static and dynamic light- scattering data are compared with calculations based on the HNC approximation. In particular, we derive the fractional charge of ganglioside micelles, and we discuss the effect of hydrodynamic interactions and of the ion size on the dynamics. The measured correlation function was found to be non-exponential at extremely low ionic strength, unlike the behaviour observed in the experiment of ref. (7). It is possible that the observed non-exponentiality is due to the intrinsic polydispersity of micellar solutions.Experiment Gangliosides are anionic amphiphilic glycolipids occurring in neuronal plasma mem- branes. The materials used in the present investigation have been prepared as sodium salts by Tettamanti and coworkers. Detailed information about nomenclature, properties and preparation procedure can be found in ref. (18) and papers quoted therein. We have used the gangliosides GM1 and GM2 (both containing one sialic acid residue) and GDla (containing two sialic acid residues). These gangliosides are known to form micelles above a critical micelle concentration < lop6 mol drnp3. The solutions were admitted to the scattering cell through Teflon tubings and a 0.2 p m pore size Nucleopore filter mounted in a Teflon holder.The solutions were prepared with doubly distilled and degassed water. The cell was accurately precleaned, and was flushed with large amounts of pure water before each measurement in order to eliminate possible ionic impurities present in the cell. All measurements were performed at 25 "C. The light-scattering apparatus includes a 514.5 nm argon laser, and a Langley-Ford digital correlator. Both scattered intensity and correlation function measurements were made at 90" scattering angle. With the GM1 solution at the lowest ionic strength, data were also taken at the angles 20 and 160". The aggregation number rn and the size of ganglioside micelles, as found in previous experiments," are reported in table 1. The effect of NaCl addition to the ganglioside solution was studied at a fixed ganglioside concentration.We show in fig. 1 the behaviour of the scattered intensity as a function of the ionic strength for GM1 solutions at a concentration of 0.5 and 1 mmol dm-3. Similar results have been obtained with GM2 and GDla solutions. TheL. Cantu, M. Corti and V. Degiorgio Table 1. Aggregation number m, hydrodynamic radius R, electric charge Q, and fractional ionization (Y of ganglioside micelles" ganglioside m R/nm Q CY GM1 302 5.9 48 0.16 GM2 529 6.6 100 0.19 GDla 226 5.7 60 0.13 The estimated experimental uncertainties are: 10% for m, 2% for R and 10% for Q. 289 t v 1 I I I I I l l l I I 1 I I l i l l I I l l 0 c!l 1 ia ionic strength/mE Fig. 1. Structure factor S of 0.5 (e) and 1 (V) mmol dm-3 GM1 solutions measured as functions of the ionic strength I at 25 "C and scattering angle of 90".The full curves represent theoretical results calculated for a micellar charge Q = 48 electronic units. reported values are normalized by the intensity scattered from the ideal solution which was derived by measuring the scattered intensity at various GM1 concentrations in 30 rnniol Om-' NaC! solutions and extrapolating to zero micelle concentration. The ionic strength i s calculated as c,+iQc/rn [see eqn (6) below], where c, is the NaCl molarity, c is the amphiphile molar concentration and Q is the electric charge of the micelle in electronic units. In order to check whether the aggregation number of the ganglioside micelle depends on the ionic strength of the solution, we have measured the intensity of scattered light as a function of the GM2 ganglioside concentration at two distinct salt concentrations, 1 and 30 mmol dmP3.The results are shown in fig. 2. The ratio I , / c depends very strongly on c for the 1 mmol dm-3 NaCl solution because of the effect of the electrostatic interactions.* In the case of the 30 mmol dmP3 NaCl solution interactions are screened much more effectively, and, as a consequence, I,/ c is only weakly dependent on c. The extrapolation at zero concentration, which is proportional to the micelle aggregation number, is, however, the same for the two solutions. At low NaCl n;olarity I s ( k ) becomes appreciably dependent on the scattering angle because of the k-dependence of the structure factor. As an example, we show in fig.3 the behaviour of the scattered intensity as a function of the scattering angle for a 1 mmol dm-' GM1 solution with an added salt molarity around 1 mmol dm-' (unfortu- nately in this measurement we did not control very carefully the salt molarity). We see that the intensity of light scattered at 162" is 50% larger than that scattered at 22". The measured correlation functions were exponential in all cases, except for very low ionic strengths. We report in fig. 4 the results obtained with the same GM1 solution290 Light-scattering Studies of Ionic Micelles 5 h ? v u ... 4" 3 4 w c 1- c v . 1 -I ---"-i I I I I 1 I I I I I 1 0.2 0.4 0.6 0.8 ganglioside concentration/mmol dm-3 Fig. 2. Concentration dependence of the quantity I / c for GM2 solutions at 1 (V) and 30 (0) mmol dm-3 NaCI.7 O O 2 4 k / lo5 cm-' Fig. 3. Angular dependence of the scattered intensity for a 1 mmol dmP3 GM1 solution with ca. 1 mmol dm-3 NaCI. used for the static data of fig. 3. The deviation from exponentiality is more marked for the low-angle measurement. For sake of comparison, we also present in fig. 4 a correlation function measured at 90" in a 30 mmol dm-3 NaCl solution. The correlation functions obtained at a scattering angle of 90" for various NaCl concentrations, were analysed by the standard cumulant fit which gives the diffusion coefficient D and the relative variance u. The ratio Do/ D is shown in fig. 5 as a function of the ionic strength I for two distinct GM1 concentrations, Do being the diffusion coefficient of the individual GM1 micelle.Do is derived by measuring D at various GM1 concentrations and extrapolating to zero micellar concentration. In order to show how the deviation from exponentiality depends on the salt concentration, we have reported in fig. 6 the behaviour of the relative variance. Data Interpretation and Discussion Since micelles are spontaneous aggregates in dynamic equilibrium with monomers, it is not evident a priori that a micellar solution can be treated as a solution of rigid Brownian particles. However, the critical micelle concentration of gangliosides is known to be extremely OW,"^^^ so that we can neglect for all purposes the presence of ganglioside monomers in our solutions. Furthermore, we have verified, by extrapolating to zero micelle concentration the scattered intensity measured at different ionic strengths (seeL. Cantli, M.Corti and V. Degiorgio I I I I I l l l l I 1 I I 1 1 1 1 1 I I I 1 0 .1 0.01 29 1 Fig. 4. Normalized time-dependent part of the intensity correlation function measured for 1 mmol dmV3 GMl solutions. e, 1 mmol dm-3 NaCl, 8 = 22"; V, 1 mmol dmC3 NaCl, 6 = 162"; 0, 30 mmol dme3 NaCl, 8 = 90". 1 I I I I I I I I I 1 1 I I I I I I I I I l i Fig. 5. The quantity Do/D plotted us. the ionic strength for 0.5 (0) and 1 (V) mmol dm-' GM1 solutions. The full curves represent theoretical results calculated as described in the text.292 Light-scattering Studies of Ionic Micelles I i I I l l l l i I I I I i l l 1 i / I I I l l l i l i '\ \ \ \ I I I 1 I l l 1 I I I 1 I l I ! l I I I I I ! I l l I 1 0.1 1 10 ionic strength/ mE Fig.6. The relative variance plotted us. the ionic strength for 0.5 (0) and 1 (V) mmol dmP3 GM1 solutions. The dashed curves are drawn only to guide the eye through the data. fig. 2), that the micelle aggregation number does not depend, within the experimental uncertainties, on the added salt concentration. This latter result may at first seem surprising, since for many common ionic amphiphiles, such as the alkyl sulphates, the aggregation number is very sensitive to the concentration of added salt. However, the head group of a ganglioside is very bulky. This implies that the head-group interactions in the micelle are mainly determined by steric rather than electrostatic contributions. It is interestifig to m t e in this cmtext that by adding a few oxyethylene units to the head group of alkyl sulphates, the micelle aggregation number becomes far less sensitive to the salt content of the solution.20 Since the micelle size and shape do not change with the ionic strength, we can conclude that the quantity plotted in fig.1 is indeed the static structure factor S ( k ) evaluated at k = 2ko sin 6/2, where ko = 2 ~ n / A , n is the index of refraction of the solution and 6 = 90" in our experiment. We have obtained the theoretical structure factor S ( k ) by using the HNC approxima- tion for g ( r ) . Our computer program is a copy of that used by Cannell et al.' The pair interaction potential consists of a hard-core repulsion plus the screened Coulomb potential: where a is the radius of the particle, Q is the electric charge, E is the dielectric constant of water and K is the inverse Debye-Huckel screening length.We assume in this paper that ganglioside micelles can be considered as spherical particles. As discussed in detail elsewhere," the aggregation numbers are too large for a spherical shape, but the micelles are still globular, with an axial ratio between 1.5 and 3. We have taken a as equal to the measured hydrodynamic radius which is given in table 1 . The parameter K depends on both the NaCl concentration c, and the amphiphileL. Cantu, M. Corti and V. Degiorgio 293 concentration c. Its expression can be derived by starting from the Poisson-Boltzmann equation: 02+ = p / E (4) where 9 is the electric potential and p is the local charge density expressed as 3 p = 1 Njzj exp ( -zje@/kT) j = 1 ( 5 ) where Nj is the concentration of ions of species j and zj is their charge in electronic units.Labelling micelles as species 1, counterions as 2 and coions as 3, we can write N1 = c / rn, N2 = Qc/ m + c, , N3 = c, , z1 = Q and z2 = z3 = 1. By assuming that e@/ kT << 1 << Qe+/kT, we find that the inverse Debye-Huckel length is given by K~ = ( 2 e 2 / ~ k ~ ) ( @ / r n + CJ. ( 6 ) Eqn (6) shows that the Debye-Huckel length depends on both salt and micelle concentra- tion and remains finite at zero salt concentration. The continuous curves in fig. 1 show the theoretical structure factor calculated by using the best-fit value of the micellar charge, Q = 48. We have reported in table 1 the best-fit values of Q and of the fractional ionization a = Q/(zrn), where z is the valence of the ganglioside monomer.We note that the micelles formed by monovalent anions (GM1 and GM2) present a value of a larger than that of the GDla micelle, which is formed by divalent ions; i.e. the counterion condensation is fractionally larger when the surface charge density is larger. The experimental data shown in fig. 1 seem to indicate that the micellar charge decreases when the ionic strength is very small (<0.5 mmol dm-3. We have also observed this behaviour with the other gangliosides. In principle, several effects might influence the low ionic strength fit: ( a ) the solvent (or the solute) may contain some residual salt which makes the effective ionic strength larger than calculated; ( b ) the theoretical S( k ) might change if the finite size of small ions is taken into a c ~ o u n t ; ’ ~ * ’ ~ (c) the HNC approximation may begin to fail at very low ionic strength.By inspection of fig. 1 we see that a residual salt concentration of 0.1 mmol dm-3 would be needed to explain our data only with effect ( a ) . Since this figure is much larger than the estimated upper limit in the residual salt, we infer that effect ( a ) cannot explain our results. Recent calcula- tions16 show that effect ( b ) is negligible in our situation and, anyway, would go in the opposite direction because it tends to overestimate the micelle charge. Furthermore, the available information7 does not indicate any failure of the HNC approximation in the range of interest for our calculations.We conclude therefore that the discrepancy between theory and experiment is due to the fact that the micellar charge depends, at very low ionic strength, on the salt concentration. A plausible explanation for such dependence is that the strongly repulsive intermicellar forces may enhance ion condensa- tion on the micelle surface. Fig. 1 shows that deviations are larger when the micelle concentration is higher. This is also consistent with an explanation in terms of inter- actions. Unfortunately all the theoretical treatments of which we are aware neglect interparticle interactions. We hope our data will stimulate new theoretical work in this field. We now discuss the dynamic data. The full curves in fig.5 are derived by inserting the calculated g ( r ) into the expression derived by Ackerson21 which considers hydrody- namic interactions at the Oseen level, by using for Q the best-fit value derived from the static data, Q=48, and by applying the correction factor proposed by Belloni and Drifford” to take into account the non-negligible size of small ions. As discussed in more detail in ref. (lo), the trend is correct, but the agreement is only qualitative. Finally, we briefly comment on the data presented in fig. 4 and 6. Since explicit theoretical results concerning the full shape of the correlation function are not available,294 Light-scattering Studies of Ionic Micelles we do not know whether our results are compatible with the theory of strongly interacting monodisperse particles.However, the fact that exponential correlation functions were measured by Cannell and coworkers’ with a monodisperse protein solution in a similar range of ionic strengths suggests that the observed non-exponentiality is due to the intrinsic polydispersity of the micellar solution. The analysis of the results appears rather difficult because the decoupling approximation proposed by Weissrnan,** who first described such a polydispersity effect, does not seem to apply to our case. The fact that the non-exponentiality is stronger at low angles (see fig. 4), where the scattered intensity is weaker (see fig. 3), is consistent with the polydispersity hypothesis, as discussed by Pusey in ref. (2). Conclusions We have reported in this paper static and dynamic light-scattering experiments on a dilute solution of small strongly interacting Brownian particles.The main results are the following. ( a ) We have derived the electric charge of the ganglioside micelles by a best-fit procedure which uses the hypernetted chain approximation for the calculation of the radial distribution function. Our data suggest that the electric charge decreases when the ionic strength is reduced to very small values, probably because of the strong repulsive intermicellar interactions. (6) The diffusion-coefficient data are not described satisfactorily by theories which take into account hydrodynamic interactions, but treat the solvent as a continuum. The agreement is improved if the effect of the finite size of small ions is included. ( c ) The correlation function measured at extremely low ionic strength is not exponen- tial.This seems to be a specific feature of micellar solutions which are intrinsically polydisperse, and could perhaps be exploited to measure small polydispersities in dilute solutions. We are very grateful to D. S. Cannell for kindly giving us the computer program used to calculate S ( k ) and D( k ) . We thank G. Tettamanti and coworkers for the preparation of the gangliosides. This work was supported by grants of the Italian Ministry of Public Education, and by Progetto Finalizzato Chimica Fine e Secondaria of the Italian National Research Council (CNR). L.C. thanks Fondazione Hoechst for financial support. References 1 2 3 4 5 6 7 8 9 10 11 Light Scattering in Liquids and Macromolecular Solutions, ed.V. Degiorgio, M. Corti, and M. Giglio (Plenum Press, New York, 1980). P. N. Pusey and R. J. A. Tough, in Dynamic Light Scattering and Velocimetry: Applications of Photon Correlation Spectroscopy, ed. R. Pecora (Plenum Press, New York, 1985). W. Hess and R. Klein, Adv. Phys., 1983, 32, 173. J. C. Brown, P. N. Pusey, J. W. Goodwin and R. H. Ottewill, J. Phys. A , 1975, 8, 664. F. Griiner and W. Lehmann, in ref. (1). P. Doherty and G. B. Benedek, J. Chem. Phys., 1975, 61, 5426. D. S . Cannell, in Physics of Amphiphiles: Micelles, Vesicles and Microemulsions, ed. V. Degiorgio and M. Corti (North-Holland, Amsterdam, 1985), p. 202; D. G. Neal, D. Purich and D. S . Cannell, J. Chem. Phys., 1984, 80, 3469. M. Corti and V. Degiorgio, J. Phys. Chem., 1981, 85, 711. D. F. Nicoii and R. B. Dorshow, in Physics of Amphiphiles: Micelles, Vesicles and Microemulsions, ed. V. Degiorgio and M. Corti (North-Holland, Amsterdam, 1985), p. 429. L. Cantu, M. Corti and V. Degiorgio, Europhys. Lett., 1986, 2, 673. J. B. Hayter and J. Penfold, J. Chem. SOC., Furaduy Trans. 1, 1981, 77, 1851; D. Bendedouch, S-H. Chen and W. C. Koehler, J. Phys. Chem., 1983, 87, 2621; R. Triolo, J. B. Hayter, L. J. Magid and J. S . Johnson, J. Chem. Phys., 1983, 79. 1977.L. Cantu, M. Corti and V. Degiorgio 295 12 A. Vrij, E. A. Nieuwenhuis, H. M. Fijnaut and W. G. M. Agterof, Faraday Discuss. Chem. SOC., 1978, 13 J. P. Hansen and J. B. Hayter, Mol. Phys., 1982, 46, 651. 14 D. W. Schaefer, J. Chem. Phys., 1977, 66, 3980. 15 G. Nagele, R. Klein and M. Medina-Noyola, J. Chem. Phys., 1985, 83, 2560. 16 L. Belloni, J. Chem. Phys., 1986, 85, 519. 17 L. Belloni and M. Drifford, J. Phys. (Paris) Letr., 1985, 46, 1183. 18 G. Tettamanti et al., in Physics of Amphiphiles: Micelles, Vesicles and Microemulsions, ed. V. Degiorgio 19 L. Cantij, M. Corti, S . Sonnino and G. Tettamanti, Chem. Phys. Lipids, 1986, 41, in press. 20 C. Minero, E. Pramauro, E. Pelizzetti, V. Degiorgio and M. Corti, J. Phys. Chem., 1986, 90, 1620. 21 B. J. Ackerson, J. Chem. Phys., 1976, 64, 242; 1978, 69, 684. 22 M. B. Weissman, J. Chem. Phys., 1980, 72, 231; see also P. N. Pusey, in ref. (1). 65, 101; A. M. Cazabat and D. Langevin, J. Chem. Phys., 1981, 74, 3148. and M. Corti (North-Holland, Amsterdam, 1985), p. 607. Received 27th November, 1986

ISSN:0301-7249    

DOI:10.1039/DC9878300287

出版商:RSC    

年代:1987

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1987, 83, 297-307 Brownian Motion of a Hydrosol Particle in a Colloidal Force Field Dennis C. Prieve and Foo Luo Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, U.S.A. Frederick Lanni Department of Biological Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, U.S.A. Total-internal-reflection microscopy (TIRM) was used to monitor Brownian fluctuations in separation between a 10 pm polystyrene sphere and a glass plate when the two bodies are separated by a fraction of a micrometre of aqueous solution. Preliminary results clearly show evidence of double-layer repulsion which is weakened by increasing the ionic strength. Potential- energy profiles were obtained with a resolution of 2.5 nm.Although this falls short of the atomic resolution obtained with the crossed-mica cylinder technique for measuring colloidal forces, TIRM allows one of the interacting bodies to have colloidal dimensions. Besides measurement of colloidal and hydrodynamic forces, other applications include the study of hindered diffusion of the particle near a flat boundary and reversible adsorption of a single particle onto a plane surface. Van der Waals and double-layer forces play an important role in the aggregation of colloidal particles” and their adsorption onto For example, the van der Waals attraction between like particles accounts for the inherent instability of dispersions of lyophobic particles; their ultimate fate is to aggregate. Double-layer repulsion between bodies of like charge can delay the inevitable by retarding the rate of aggregation or adsorption.Moreover, the depth of the potential-energy well into which particles adsorb determines the equilibrium constant for adsorption. Several techniques have been developed to probe colloidal forces directly.*-” The most widely used technique is that of Israelachvili and ad am^.^'^ Cleaving mica sheets along crystal planes, they obtain surfaces which are smooth at the atomic scale. Gluing the mica on polished glass cylinders, which are placed at right angles and then pushed together with a known force, they measure the equilibrium separation by interferometry as a function of the force. With this technique, they can determine separation distances down to contact with an accuracy of *0.2nm.Israelachvili and Adams’ found that the double-layer repulsion measured in dilute KN03 solutions (< 10 mol m-’) decayed exponentially with separation distance as expec- ted. The decay lengths determined from these measurements were found to be within 4% of the Debye length calculated from the solution composition. At much higher salt concentrations (ca. 1000 mol m-’), adsorption of hydrated cations again gave rise to a strong repulsion through hydration forces.12 This same technique has 1 7 been applied to the study of solvation (oscillatory) forces in non-polar liquids,’” the hydrophobic interaction of mica surfaces bearing a monolayer of CTAB,I4 and the steric interaction of surfaces bearing adsorbed polymer chains.” developed a technique to determine the interac- tion between a single colloidal sphere and a flat plate separated by a glycerol-water Recently, Prieve and 297298 Brownian Motion of a Hydrosol Particle solution.They obtained the potential-energy profile from the equilibrium distribution of separation distances using Boltzmann’s equation. The instantaneous separation distance was deduced from the measured translation speed of the sphere in linear shear flow along the wall. When pure glycerol was used in place of the glycerol-water solution, they also observed a flowrate-dependent hydrodynamic repulsion. On the basis of the creeping flow equations, whose solution was used to deduce the instantaneous separation distance, no hydrodynamic force is expected to arise between the sphere and the plate in linear shear flow.’* Moreover, inertial forces, which are neglected in the equations for creeping flow, should be less important in the more viscous fluid where hydrodynamic repulsion was observed.Prieve and Bike’9 suggest that a repulsive force, which they call ‘electro- kinetic lift’, can arise from the streaming potential developed as a result of sliding motion between the sphere and the plate. Whatever the cause of the flowrate-dependent force, the failure of the creeping flow equations to predict it casts doubt on predictions by the same equations of the relation between the measured speed of the sphere and its distance from the wall. Another technique, which is independent of the hydrodynamics, is needed to measure the separation distance.In this paper we use total-internal-reflection microscopy (TIRM) to measure the instantaneous distance separating a sphere and a plate when the sphere undergoes Brownian motion. The intensity of light scattered by the sphere in an evanescent wave is measured and translated into separation distance. Scattering also results from surface defects. Indeed, TIRM is a term introduced by Temple,” whose used the technique to inspect optical surfaces for damage. As such, TIRM is similar to surface-contact microscopy,2’ which has been used to study the movements of fibrocytes,22 and to total-internal-reflection fluorescence, which has been successfully used to size free viruses,23 to measure the chemisorption of fluorescent molecules on a liquid/solid interfa~e’~ and to observe the organization of cells growing on glass substrate^.^^ Total-internal-reflection Microscopy When a ray of light strikes a planar interface from the more optically dense medium at an angle of incidence greater than some critical angle, f?,, total internal reflection results.Although all light energy is ultimately reflected back into the more dense medium, there is an optical disturbance in the less dense medium which takes the form of an evanescent wave. Unlike the incident beam, whose electric field magnitude is stationary across surfaces of constant phase (plane wavefronts), the evanescent wave decays exponentially with distance from the interface. The intensity of the evanescent wave, which is proportional to the square of the electric field magnitude, also decays exponentially with the distance h from the interface:26 L ( h ) = U O ) exp (-PW p = (4~/A,)(sin’ 6,-sin’ 1 9 , ~ ) ” ~ where ( 1 ) sin 6, = n2/ n , where A , is the wavelength of light in the more dense medium, n is index of refraction, 0 is an angle measured from the normal to the interface, and where the subscripts 1 or 2 refer to the medium on the incident side or opposite side of the interface, respectively. Note that the decay length for the exponential ( p ’> can be comparable to the wavelength of the incident light.D.C. Prieve, F. Luo and F. Lanni 299 If a third medium (having refractive index n,> n2) is brought through medium 2 near to the interface and if 8, is less than the critical angle for media 1 and 3, the evanescent wave in medium 2 gives rise to a plane wave in medium 3.Since light energy will now be transmitted across both interfaces, this situation is called ‘frustrated’ total internal reflection. When medium 3 is a colloidal particle, we can think of this transmitted light energy as scattering of the evanescent wave by the particle. Chew et ~ 1 . ~ ’ extended their theoretical analysis of Lorenz-Mie scattering of plane waves by a dielectric sphere to elastic scattering of evanescent waves. Assuming that back-scattering of the evanescent wave and reflection of the scattered wave at the interface causes no significant perturbation of the field at the location of the particle, this extension takes the form of an analytic continuation of their earlier result onto the complex plane of wavenumbers.Results are reported for the incident wave polarized both parallel to the plane of incidence and perpendicular to it. F0.r either type of polarization, the scattered electric field has the following asymptotic for1 1 in the far-field limit ( r + a): where k is the magnitude of the wavevector in medium 2 and Bl,ml are expressions containing vector spherical harmonics. Note that the intensity of the scattered light, which is proportional to the product of Esc and its complex conjugate, has the same dependence on the sphere-plate separation distance as the intensity of the unscattered evanescent wave. In particular, where Ixc( h, a) is the integral of the scattering intensity over some cone of solid angle corresponding to the numerical aperture of the microscope objective.Thus the scattering intensity is exponentially sensitive to the separation distance with a decay length p-’ comparable to the wavelength and adjustable uia the angle of incidence. Measurement of Interaction Energy Consider a single sphere which has settled through a less dense fluid and resides near the flat bottom of the container. If both the sphere and the flat plate forming the bottom of the container bear surface charges of sufficient magnitude and of like sign, then intimate contact between them is prevented by double-layer repulsion. Under the influence of the opposing forces of gravity and the electrical double layer, the particle will assume an equilibrium position near the plate. Through Brownian motion, the sphere randomly samples different locations around this equilibrium over the course of time.Let the probability of finding the particle at a location between h and h + dh be denoted by p ( h)dh, where h is the shortest distance between the sphere and the plate. Sampled over a sufficiently long interval of time, the probability density p ( h ) will be a Boltzmann distribution: p(h’l = A exp [ - 4 ( h ) / k T ] ( 3 ) where $ ( h ) is the total potential energy of the particle when it is at elevation h, k is Boltzmann’s constant, T is the absolute temperature and A is a normalization constant chosen such that p ( h ) dh = 1. Although the particle’s mobility and diffusion coefficient depend on h, this dependence does not alter the equilibrium distribution given by eqn (3).17 Thus, by observing the distribution of elevations, p ( h ) , assumed by the sphere over a long period of time, the potential energy of the sphere can be deduced as a function of its location relative to the plate.300 Brownian Motion of a Hydrosol Particle Expected Contributions to 4 ( h ) Besides double-layer repulsion and gravity, which were mentioned above, van der Waals attraction between the sphere and the plate also contributes to the interaction when the separation is small enough.For a macroscopic sphere (as opposed to a single atom) at a fixed finite distance from a plate, the van der Waals force between the sphere and plate is proportional to a, whereas gravity is proportional to a3, where a is the radius of the sphere.28 Thus for a large sphere, gravity will dominate the van der Waals attraction, except at very small separations.For simplicity in this preliminary study, we will neglect van der Waals interactions. As it turns out, the most probable location of the 10 p m sphere used in the present work is of the order of 30-60nm from the plate. Since this separation distance is also several times larger than the Debye length, we can estimate the double-layer potential energy between a spherical particle and a flat plate using the linear superposition and Derjaguin approximations: B=16&a __ tanh ( - e i u l ) tanh (*) 4kT 4kT for a 1 : 1 electrolyte, where 1/ K is the Debye length, and iu2 are the Stern potentials of the particle and the plate, e is the dielectric constant, e is the protonic charge, and a is the sphere's radius.This approximation for the double-layer interaction is expected to be valid when Ka >> ~h >> 1. The gravitational contribution is G = ( 4 / 3 ) r a 3 ( A p ) g where Ap is the difference in density between the particle and the fluid and g is the gravitational acceleration. Adding eqn ( 4 ) and ( 5 ) yields the total potential energy: This function has a single minimum at a separation distance, h l , which is given by ~ h , =In ( K B I G ) . (7) This expression can be used to eliminate B from eqn (6). After rearranging, eqn (6) becomes: where x = K ( h - h , ) is the displacement from the most probable separation distance normalized with respect to the Debye length. As a sample calculation, consider the case in which a = 5 pm, 1/ K = 3 nm, $, = G2 = 50 mV, and Ap = 0.05 g Eqn (7) yields ~ h , = 12.Owing to the logarithm in eqn (7) this result is not very sensitive to changes in the assumed values of the parameters. Thus the technique is expected to probe double-layer interactions at separations for which the degree of overlap of opposing double layers is slight and where eqn (4) should be applicable.D. C. Prieve, F. Luo and F. Lanni 301 Experimental Sample Preparation Solutions containing polystyrene latex microspheres were prepared by diluting a 10% stock dispersion (Duke Scientific, Palo Alto, CA) in which the particle diameter was 10.04~0.06 pm. In addition to NaCl in various concentrations, 5 mol dm3 of SDS was included to inhibit sticking of particles to the glass plate.Clean microscope slides were rinsed sequentially with dichlorodimethylsilane, methanol and distilled water, then allowed to dry in air. Using a mixture containing equal amounts of Vaseline, lanolin and paraffin, a circle of wax was formed on upper surface of the microscope slide, which was large enough to contain 2 cm3 of the solution. Using immersion oil, the bottom side of the microscope slide was optically coupled to a prism 25 x 36 x 25 mm. Apparatus A 2 mW helium-neon laser ( A o = 632.8 nm) serves as the light source for the scattering. After reflection from a mirror mounted on a rotating stage, the laser beam passes through one side of the cubic prism ( n , = 1.5222), is internally reflected off the top surface of the microscope slide and passes out the other side of the prism.The angle of incidence at the microscope slide ( Oi = 70.94" for the data reported here) is controlled by rotating the mirror while the polarization is perpendicular to the plane of incidence (s-polarized). The entire apparatus is mounted on an optical bench with vibration isolation supports. A 40x water-immersion objective is submerged in the aqueous solution ( n2 = 1.3330) contained in the well on the microscope slide and then focused on particles near the bottom. A Zeiss Universal microscope equipped with a Zeiss photometer 01 was used to measure the intensity of scattering from a single sphere. Light incident upon the photomultiplier tube first passes through a 0.5 transmission filter to reduce the signal and a changeable aperture stop to reduce the background noise.Besides the eyepieces mounted below the image-plane aperture, the microscope was equipped with a single ocular mounted above the aperture to permit centering the particle in the aperture. After conversion of current to frequency, the output of the photometer was fed to a digital correlator (Malvern, type K7023) operating in the probability density mode. In 3ms intervals over 5 or lOmin, the correlator averages the frequency, which is proportional to the current from the photomultiplier tube, and increments the count by one in the appropriate bin of a digital storage device containing a total of 100 bins. In this way, a histogram of the sampled photocurrent is accumulated. Procedure During scattering measurements, the room was darkened.Ca. 2 cm' of solution contain- ing the particles was placed on the microscope slide. Ca. 10min was allowed for the particles to settle to the bottom. The field of view was adjusted until a single sphere appeared which was then centred under the aperture using the ocular. At higher salt concentrations, some of the particles were motionless and apparently stuck to the glass plate. Such particles were not examined further. About once per minute, the collection of scattering intensity was suspended briefly to recentre the particle in the aperture. If the particle drifted outside the aperture (even partially}, the results accumulated to that moment were discarded and the run was restarted after recentring the particle. To eliminate variations among washed slides, one microscope slide was used for all the measurements.After results at one salt concentration were obtained, the solution was removed from the well using a pipette. The well was then flushed several times3 02 Brownian Motion of a Hydrosol Particle using the next solution (higher concentration) before refilling. The total residence time in the well of any one solution was kept below 30 min to avoid any significant increase in salt concentration due to evaporation. Data Analysis Background photocurrent was generally much less than the current due to scatter from a microsphere. Also, because the photon count rate was generally much faster than the inverse relaxation time of the Brownian motion, the photocurrent was treated as an analogue variable and scaled down by current-to-frequency conversion to average over shot noise. The histograms of scattering intensity were therefore used directly to obtain the histogram of particle-plate separation distance.Scattering intensities can be translated into separation distances using eqn (2). We can eliminate the pre-exponential factor by dividing through by the most probable intensity. Solving for the separation yields: where Isc( h,, a) is the most probable scattering intensity and h2 is the separation distance corresponding to the most probable scattering intensity. In the limit of an infinite number of observations, the shape of the histogram of intensities, N ( I ) , corresponds to the shape of the probability density function for scattering intensity, P( I ) . To convert this into the probability density function for separation distance, we note that the probability, p ( h ) dh, of finding a particle at a separation distance between h and h + dh is the same as the probability, P(I)dI, of observing a scattering intensity between I and I + d I : where I = Isc(h, a).Since dp/dh = 0 does not imply d P / d l = 0, the most probable separation distance, 11, , is generally not equal to the separation distance, h,, correspond- ing to the most probable scattering intensity. Given p ( h ) , the potential profile can be deduced from Boltzmann’s eqn (3). The normalization constant, A, can be eliminated by dividing through by p at some convenient location, say p ( h , ) . The ratio, p ( h ) / p ( h , ) , can then be evaluated by applying eqn (2) and (10): p ( h ) = P ( I ) d l / d h (10) Approximating the ratio of probability densities for scattering as the ratio of the number of observations, the local potential energy can be estimated from: where N ( I ) is the number of observations of scattering intensity in the interval ( I , I + A I ) and N [ I ( h , ) ] is the maximum o f N ( I ) .Results and Discussion Effect of Ionic Strength Fig. 1 shows the histograms of scattering intensities separately obtained for each of five solutions having different ionic strengths. Although the histograms overlap to some extent, the average intensity clearly increases with ionic strength. Since the refractive index of the solution changes very little (from 1.3330 to 1.3340) over this range of salt concentration, such a large increase in scattering clearly indicates a decrease in sphere- plate separation with ionic strength.D.C. Prieve, F. Luo and F. Lanni Oe20 r 303 0.1 s 6 z g 0.10 W 0.05 0.00 scattering intensity (arb. units) i Fig. 1. Five histograms of scattering intensities for a 10 p m polystyrene sphere in five solutions differing in ionic strength (see table 1). The increase in scattering intensity with ionic strength occurs as a result of Debye screening. Table 1. Properties of solutions specific ionic debye calcd" obsd solution [ NaCl] conductance refractive strength length h, I J h J no. /molm-3 /mS index /molm-3 /nm /nm (arb. units) 1 0 0.350 1.3330 4.89 4.35 44-56 21 2 5 0.850 1.3330 8.82 3.24 34-43 51 3 20 2.29 1.3332 21.3 2.08 23-28 96 5 100 10.7 1.3340 103 0.95 11-14 4 50 5 9 0 1.3333 54.6 1.30 15-18 a h, is calculated from eqn (7) using assumed values of q!! ranging from 25 mV to a.A decrease in separation distance with increasing ionic strength is expected through Debye screening of the double-layer repulsion between the sphere and the plate. A weakening of double-layer repulsion means that the sphere must approach the plate more closely before double-layer repulsion will have the same magnitude as gravity. Although ionic strength can also affect van der Waals attraction, Lifshitz theory predicts a weakening of attraction through ionic screening of the low-frequency contribu- t i ~ n . ~ ~ This would result in an increase in separation distance and a decrease in scattering intensity, which is contrary to our observations. Similarly, adding salt will reduce the gravitational attraction by increasing the density of the fluid.Since only double-layer forces can produce the trend observed in scattering intensity with ionic strength, we conclude that double-layer repulsion can be studied with TIRM. Table 1 summarizes the properties of the five solutions used for the data in fig. 1. Ionic strength was estimated from the measured specific conductance using equivalent conductances reported for NaCl and SDS (below the critical micelle concentration). Owing to the presence of 5 mol m-3 of SDS in each solution, the ionic strength generally exceeds the concentration of NaC1. The difference in the two concentrations is expected to decrease with increased ionic strength as a result of micellization (recall3' that the c.m.c.for SDS is reduced from 8.1 mol m-' in de-ionized water to 1.4 mol mP3 in 100 mol m-3 NaCI).304 Brownian Motion of a Hydrosol Particle Also shown in table 1 are values of the most probable separation distance, h , , calculated from eqn (7) and the most frequently observed intensity of scattering. In the absence of complete characterization of the electrical properties of the polystyrene and glass surfaces, a range of h , is reported corresponding to surface potentials between 25 mV and infinity. Owing to the logarithm in eqn (7), the result is not very sensitive to the assumed value. To estimate the sensitivity of the scattering intensity to separation distance, we can compare (for example) the first two solutions in table 1.Assuming the surface potentials are independent on the ionic strength, eqn (7) predicts that the separation distance will decrease by 24% whereas the intensity of scattering more than doubles. Thus TIRM seems to provide a very sensitive measure of separation distance. Using eqn (2), the ratio of scattering intensities can be used to estimate the change in h,. Eqn (1) yields a value for the critical angle of 8, = 61.13" and an evanescent decay length of p-' =93.01 nm. 'Then, eqn (2) predicts that, between the first two solutions in table 1, h, changes by 83 nm, which is considerably larger than the 10-13 nm change expected in h , . Although there is no reason to expect h , to be equal to h2 [see discussion following eqn (lo)], the potential energy profile determined below indicates that the differences in the two is considerably smaller than the discrepancy above. Other explanations of this discrepancy include back-scattering of the evanescent wave, which was neglected in eqn (2), or colloidal forces in addition to those considered in eqn (6).Reversible Adsorption of the Spheres Three of the five histograms in fig. 1 are unimodal and have a shape which agrees, at least qualitatively, with that predicted by eqn (3) and (6). However, the remaining two histograms are either irregular or clearly bimodal. Moreover, at these high ionic strengths, a significant fraction of the spheres appeared to be immobilized in the sense that no lateral Brownian motion could be observed by normal viewing through the microscope.The particular spheres chosen for scattering measurements appeared to be mobile, at least at the outset of the 10 mill interval. At such high ionic strengths, the separation distance might become small enough that van der Waals attraction becomes significant compared to gravity. If so, more than one minimum can be expected in the potential-energy profile. One of these minima, corresponding to a very small separation distance, is probably the energy well into which particles reversibly adsorb to become temporarily immobilized. Assuming this to be the case, any appreciable energy barrier separating the two minima would significantly reduce the rate of diffusion between the minima, thereby increasing the time required for an equilibrium sampling of accessible energy levels.Perhaps 10min of sampling (which, with pauses for re-centring the particle in the aperture and recording of intermedi- ate results, requires 30 min) is not sufficient time to obtain the equilibrium probability density at high ionic strength. In future experiments, we intend to increase the sampling time until the shape of the histogram converges. However, this will require some modification of the apparatus to prevent evaporation. With the present apparatus, evaporation over periods longer than 30min were observed to cause a noticeable decrease in volume of liquid on the slide. For the present, we must be content with the possibility that TIRM might also be capable of studying the reversible adsorption of colloidal particles in a 'secondary' minimum.Potential-energy Profile Each of the histograms in fig. I. can be converted into a potential-energy profile. Using eqn (9), the scattering intensities can be translated into displacements from h,; usingD. C. Prieve, F. Luo and F. Lanni 305 I-. U W a I h 0 0 0 Fig. 2. Potential-energy profile deduced from the histogram of scattering intensity observed in solution 3. The solid curve is the prediction of eqn (8). eqn ( 1 l), the relative frequency of observing an intensity within a particular interval can be translated into potential energy relative to that corresponding to the most probable scattering intensity. Fig. 2 gives the potential-energy profile thus obtained using the histogram for solution 3. For comparison, the theoretical prediction of eqn (8) is shown as the solid curve.The following discussion applies equally well to solutions 1 and 2, except that the latter show that h2 - h , can be as large as 5 nm. A linear relationship is evident in both the experiments and the predictions of eqn (8). Of course, at large enough separation distances, colloidal forces are expected to be negligible compared to gravity which corresponds to a linear relationship between potential energy and separation distance. The slope of this line is GILT: the net gravitational force which causes the latex sphere to settle to the bottom of the solution. Agreement between the slope of the experimental data in the linear region and that of the solid curve is remarkable in that no adjustable parameters have been used either to compute the solid curve or to interpret the experimental data.Indeed, the two slopes could be made equal if the value assumed for the specific gravity of polystyrene were changed from 1.05 to 1.07. This close agreement tends to confirm the validity of eqn (8), and its predecessor eqn (2), which were used to interpret scattering intensity in terms of separation distance. By the same token, the poor agreement between the experimental points and eqn (8) at small separations, where double-layer repulsion is expected to dominate gravity, suggests that the contributions to eqn (8) from colloidal forces are not correct. One possibility is that van der Waals forces are not negligible. Another possibility is that the expression used to predict double-layer repulsion, which was based on linear superposition and Derjaguin’s approximation, is not valid.Spatial Resolution Note that the tic marks on the abscissa of fig. 2 correspond to a displacement of 0.01 p m or 10 nm. Near the minimum in that potential-energy profile there are four data points in one 10nm interval. This implies that spatial resolution for these experiments is 2.5 nm. This is approximately the resolution of a good scanning electron microscope (SEM). However, unlike the SEM, we have achieved this resolution in an aqueous3 06 Brownian Motion of a Hydrosol Particle environment and, at the same time, we can follow dynamic processes such as the Brownian motion of our spherical particle. This is not the best resolution we can obtain. From eqn (9), a displacement of 2.5 nm corresponds to change in scattering intensity of 2.7%.We believe our photomultiplier can resolve smaller changes in light intensity. Instead, the resolution is determined by the limited number of bins available to accumulate the histogram. We are currefitly refining our data acquisition system so that we can remove this restriction. In any case, the angle of incidence can also be increased to reduce the decay tength p-’ for the evanescent wave [see eqn (1 >] which will reduce the displacement corresponding to a given fractional change in intensity. In short, there are several avenues available to refine the spatial resolution which is already comparable to that of SEM. Some Applications of TIRM Although the results presented here must be considered preliminary, we have demon- strated that TIRM can be used to study double-layer repulsion and possible van der Waals attraction.These preliminary results suggest that eqn (4) might not correctly predict the double-layer interaction although K a and K h are very large. If this discrepancy persists upon closer scrutiny, this observation could be significant, especially in light of Pashley’s’’ experiments which verify eqn (4) with respect to h at low ionic strengths when the 10 pm sphere is replaced by a 1 cm cylinder. Even if this discrepancy proves to be an aberration, we should be able to provide an experimental test of Derjaguin’s approximation for the dependence on particle radius. Another application is the study of hydrodynamic forces exerted on the sphere in linear shear flow along the plate. In fluids having low conductivity, Alexander and Prie~e’~.’’ have observed a repulsive force whose magnitude increases with shear rate.Prieve and Bike” speculate that the force has an electrokinetic origin. Moreover, in the proximity of a wall, the mobility both in the tangential and normal directions will be reduced.31332 TIRM permits the observation of such effects using colloid particles. Finally, TIRM can be used to study steric interactions resulting from adsorbed polymer on either the sphere or the plate. By choosing a solvent system that is isorefractive with respect to’the polymer, the scattering intensity will not be affected by the presence of the polymer. Financial support for part of this work came from the National Science Foundation.We also thank Prof. Gary D. Patterson for the use of the correlator. References 1. E. J. W. Verwey and J. Th. G. Overbeek, Theory of the Stability of Lyoplzobic Colloids (Elsevier, Amsterdam, 1948). 2 H. Reerink and J. Th. G. Overbeek, Discuss. Furaday SOC., 1954, 18, 74. 3 R. H. Ottewill and J. N. Shaw, Discuss. Faruday Soc., 1966, 42, 154. 4 E. Ruckenstein and D. C. Prieve, J. Chem. Soc., Furaday Trans. 2, 1973, 69. 1522. 5 J. A. FitzPatrick and L. A. Spielman, J. Colloid Inferface Sci., 1973, 43, 350. 6 G. E. Clint, J. H. Clint, J. M. Corkill and T. Walker, J. Colloid Interface Sci., 1973, 44, 121. 7 D. C. Prieve and M. M. J. Lin, J. Colloid Interface Sci., 1980, 76, 32. 8 J. N. Israelachvili and G. E. Adams, Nature (London), 1976, 262, 774, 9 J. N. Israelachvili and G. E. Adams, J. Chem. Soc., Furaduy Trans. 1, 1978, 74, 975. 10 F. W. Cain, R. H. Ottewill and J. R . Srnitham, Furuduy Discuss. Chem. Soc., 1978, 65, 33. 11 Y. I . Rabinovich, B. V. Derjaguin and N. V. Churaev, A h . Colloid Interfuce Sci., 1982, 16, 63. 12 R. M. Pashley, J. Colloid Interfuce Sci., 1981, 83, 531. 13 H. K. Christenson, J. Chem. Ph,Fx, 1983, 78, 6906. I 4 J. N. Israelachvili and R. M. Pashley, .I. Colloid Inter-uce Sci., 1984, 98, 500. 15 J. N. Israelachvili, M. Tirrell, J. Klein and Y. Almog, Macromolecules, 1984, 17, 304. 16 D. C . Prieve and B. M. Alexander, Science, 1986, 231, 1269.D. C. Prieve, F. Luo and F. Lanni 307 17 B. M. Alexander and D. C. Prieve, Langmuir, in press. 18 A. J. Goldman, R. G. Cox and H. Brenner, Chem. Eng. Sci., 1967, 22, 653. 19 D. C. Prieve and S. G. Bike, Chem. Eng. Commun., in press. 20 P. A. Temple, Appl. Opt., 1981, 20, 2656. 21 E. Ambrose, Nature (London), 1956, 178, 1194. 22 E. J. Ambrose, Elcper. Cell. Res. Suppl., 1961, 8, 54. 23 T. Hirschfeld, M. J. Block and W. Mueller, J. Histochem. Cytochem., 1977, 25, 719. 24 N. L. Thompson, T . P. Burghardt and D. Axelrod, Biophys. J., 1981, 33, 435. 25 F. Lanni, A. S. Waggoner and D. L. Taylor, J. Cell Biol., 1985, 100, 1191. 26 S. G. Lipson and H. Lipson, Opt. Phys. (Cambridge University Press, New York, 2nd edn., 1981). 27 H. Chew, D-S. Wang and M. Kerker, Appl. Opt, 1979, 18, 2679. 28 J. N. Israelachvili, Intermolecular and Surface Forces (Academic Press, New York, 1985). 29 V. A. Parsegian, In Physical Chemistry: Enriching Topics j'rom Colloid and Surfnce Chemistrj., ed. H. Van Olphen and K. J. Mysels (Theorex, La Jolla, 1975), p. 27. 30 J. N. Phillips, Trans. Fnraday Soc., 1955, 51, 561. 31 H. Brenner, Chem. Eng. Sci., 1961, 16, 242. 32 A. J. Goldman, R. G. Cox and H. Brenner, Chem. Eng. Sci., 1967, 22, 637. Rewiued 8 th Decrmher, 1986

ISSN:0301-7249    

DOI:10.1039/DC9878300297

出版商:RSC    

年代:1987

数据来源: RSC

摘要:

Furudaq, Discuss. Chem. Soc,, 1987, 83, 309-316 Radiation Pressure as a Technique for Manipulating the Particle Order in Colloidal Suspensions Bruce J. Ackerson* and Aslam H. Chowdhury Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, U.S.A. Radiation pressure can be used to manipulate the particle organization in collodial suspensions. A single laser beam focussed into a sample will draw in particles and push them in the direction of the light propagation, if the index of refraction of the particles is larger than that of the medium. A single laser beam can be used to trap and move individual particles or to concentrate many particles into a clsse-packed two-dimensional layer on the cell wall. When two laser beams are crossed in the sample, a set of intensity fringes is produced.Particles become trapped in these fringes and execute essentially one-dimensional Brownian motion along the fringes, being limited by both the cell wall and the radiation field. Highly charged particles will also tend to align along the intensity fringes, producing a particle density wave. However, the strong interaction between the particles can lead to further organization: a sample which exhibits amorphous order in the absence of the fringes will exhibit solid-like order in the presence of the fringes. In this paper results will be presented for the structures observed for these different radiation field geometries. Static and dynamic light scattering techniques’ have proved enormously useful in extending our understanding of Brownian motion.’ In these techniques the incident light is considered to have no effect on the motion of the suspended particles.However, light can influence particle motion. Both photophoretic and radiation pressure forces exist. Photophoretic forces3 result from the absorption of radiant energy and the uneven heating between the particle and solvent such that convection develops in the solvent and drives particle motion. On the other hand, radiation pressure forces4 result from the elastic scattering of incident radiation. The change in photon momentum is absorbed by the particle-solvent system and drives particle motion. Which force dominates in a particular system depends on a variety of parameters (including particle size, composi- tion, index of refraction, thermal conductivities, etc.) and is not yet well understood for photophoretic forces. It is an active area of research given the present interest in ‘nuclear winter’.5 Ashkin has investigated these forces for pm-sized spheres in vacuum, where a focussed laser beam is used to trap and suspend particles against the force of gravity.6 This experiment demonstrates the strength of radiation pressure forces in the absence of photophoretic forces.Ashkin further investigates these forces for particles suspended in water, where the transparent spheres in a transparent medium are subject to strong radiation pressure forces and much weaker photophoretic forces.’ In this work we make use of the radiation pressure forces to induce order in suspensions of pm-sized poly- styrene spheres. We investigate the induced order by scattering and direct observation techniques. Experimental The samples used in these experiments consist of aqueous suspensions of Dsw 1.09 p m 3 09310 Particle Order in Colloidal Suspensions diameter polystyrene spheres or 0.98 p m diameter polystyrene spheres kindly supplied by S .Hachisu. The Dow spheres were used directly in the sample cell without deioniz- ation, had a small Debye screening length and interacted significantly only near contact. The Hachisu particles are highly charged and samples were deionized using a mixed strong acid-strong base ion-exchange resin. These particles have a large Deb ye screening length and interact significantly at distances of several pm. In scattering experiments, ion-exchange resin is introduced into the sample cell to maintain low ionic strength.The sample cell consists of two closely spaced quartz optical flats. The spacing may be varied from contact to several hundred pm. For spacings greater than a few hundred microns we can observe convection due to heating by the incident beam. The narrow spacing quenches convection at the input powers of interest to us. The gap between the quartz flats is often adjusted to form a wedge such that particles are completely excluded at one end, but several particle layers exist at the other. In this way we can usually find a region where the sample is two-dimensional, a single layer of particles is trapped between the flats. We have performed experiments in both the ‘three’- dimensional and ‘two’-dimensional regid ns.For direct observation of particle order we have used a microscope cover slip on a quartz flat to confine the particles. This accommodates the short working length of high-power oil immmersion objectives, but is much less controllable experimentally. The incident laser radiation ( A = 488 nm) is directed nearly normal to the plane of the sample and focussed to a waist of ca. 100pm diameter. The maximum incident power per area used is ca. 5 x lo6 W mW2 per beam and corresponds to fairly low laser output powers when concentrated by a focussing lens. Two different geometries are employed. One consists of simply focussing a single beam in the sample. The other consists of splitting the beam into two equal-intensity beams which are focussed and crossed in the sample plane (see fig.1). The crossing angle can be varied between 5 and 13” and is bisected by an axis which is normal to the sample plane, so each beam is nearly normal to the plane of the sample. The crossed-beam geometry produces a set of intensity interference fringes which are in the plane of the sample and normal to the plane of the crossed beams. Both these geometries produce radiation pressure forces which push the particles along the direction of the incident beam(s) and into the most intense regions. This can be understood qualitatively by considering a spherical particle (refractive index n =r 1.59) in water ( n = 1.33) as a positive lens. A ray, incident on the particle to the left of an axis running parallel to the ray and through the centre of the sphere, will be deflected to the right through the focus of the lens.This change in direction corresponds to a change in photon momentum. To conserve momentum, the particle moves in a direction which has components parallel to the incident ray and to the left (the illuminated side of the particle). Ashkin7 has shown that a diverging lens (an air bubble in water) will move out of the most intense regions, as expected. Quantitative estimates of the radiation pressure require an analysis which takes into account the wave nature of the light and ~cattering.~ For an incident power per area of lo7 W m-2 the radiation pressure force parallel to the beam on a 1 p m diameter particle in water is ca. 5 x N or equal to ca. 5 times the force of gravity on the particle in vacuum.The force transverse to the beam is equal to this parallel force when the intensity doubles in a distance of ca. 1 p m transverse to the propagation direction. Thus the incident radiation can act as an externally applied force field on the particle and can be treated theoretically as an external potential in which the particle is trapped. Scattering is used as one tool to investigate the inter-particle order. We have used a self-scattering technique because of its relative simplicity. Here the incident radiation both orders the particles and scatters from them. The scattered light is observed on a screen placed from 10 cm to 1 m from the sample, opposite the incident beam, and in a plane parallel to the plane of the sample (see fig.1). The scattered intensity patternFaraday Discuss. Chem. SOC., 1987, Vol. 83 Plate 1 Plate 1. The scattered intensity distribution shown in ( a ) is produced by a single focussed laser beam in a dilute sample of highly screened particles of diameter 1.09 p m which form a close-packed two-dimensional array shown in ( b ) . The scattered intensity distribution shown in ( c ) is produced in the crossed-beam geometry by highly screened particles aligned in rows along the intensity fringes as shown in ( d ) . B. J. Ackerson and A. H. Chowdhury (Facing p. 3 10)Faraday Discuss. Chem. SOC., 1987, Vol. 83 Plate 2 Plate 2. The scattered intensity distributions ( a ) - ( e) are for two-dimensional layers of weakly screened colloidal particles. In ( a ) a single beam illuminates the sample and produces a Debye- Scherer ring concentric with the incident beam, indicating an amorphous, fluid-like ordering. In ( 6 ) - ( e ) two beams are crossed in the sample.The intensity maxima which lie on a horizontal line passing through the two beam stops are produced by particles aligned in rows parallel to the intensity fringes. The other intensity maxima are produced by inter-row particle registration. The ratio of the fringe-spacing to the average particle spacing is 1.31 ( b ) , 1.04 (c), 0.97 ( d ) , 0.94 ( e ) , and 0.88 (f).B. J. Ackerson and A. H. Chowdhury 311 lens sample plane (fringes 1 detection iy plane Fig. 1. Experimental geometry for the crossed-beam configuration. See text for a discussion. may be viewed, photographed or video-taped.A (possible) disadvantage of this tech- nique occurs in the crossed-beam experiments, where two beams scatter from the ordered sample. At a given point of the viewing screen the scattered light from both beams mixes coherently. The resulting intensity is the absolute square of the sum of these two electric fields. For finite-size particles these electric fields must include the phase shifts and attenuation due to internal interference (e.g. particle form factor effects). The mixing of the two scattered beams can be constructive or destructive. The intensity of a given point on the viewing screen can increase or decrease immediately upon blocking one of the beams, before the particle structure decays. A novei use of this property may be in the construction of optical logic circuits.8 Results and Discussion Plate 1 ( a ) and ( b ) present the real-space structure and the self-scattered diffraction pattern that results when a single laser beam is focussed into a sample containing the Dow sample at large gap-spacing.Particles which diffuse into the beam are centred in the beam and are pushed onto the cell wall. A close-packed two-dimensional array of particles results even though the bulk sample contains a low concentration of particles (< lo9 particles ~ m - ~ ) . There is no preferential orientation of the two-dimensional lattice, as can be observed by blocking the beam long enough for the particles to diffuse away and then unblocking the beam to allow a new 'crystal' to form. Plate 1 ( c ) and ( d ) present the real-space structure and the self-scattered diffraction pattern for the Dow particles in a cross-beam geometry.The crossing angle (in the sample) is 6 = 9.6" such that the fringe spacing is ca. 2.18 pm. The fringe spacing is small enough that single, well defined strings of particles form, aligned along the intensity312 Particle Order in Colloidal Suspensions fringes and cell wall. These strings of particles form a diffraction grating which scatters the incident radiation into a series of intensity maxima directed into a line which passes through both incident beams. The intensity maxima of the diffraction patterns produced by each incident beam coincide. However, a given intensity maximum will consist of a coherent mixing of two different orders (separated by one order), one from each beam.For finite size particles, there is also a phase introduced owing to the particle form factor which varies with scattering angle. For 2 p m diameter spheres and 6 = 5.6": we have observed* the low-order intensity maxima to increase in intensity immediately upon blocking one of the incident beams. This is a result of the particle form factor being positive at a value of the scattered wavevector k equal to that for the first-order intensity maximum and negative for the second-order intensity maximum. The intensity potential well is sufficiently deep for the results presented in plate 1 ( c ) and i d ) that a particle requires several k H T of energy to jump to an adjacent fringe. Furthermore, the fringe-separation is larger than the range of the inter-particle interaction so that there is little correlation between particles in adjacent fringes.The result is a collection of particles which execute one-dimensional Brownian motion. Owing to the small screening length, the particles in a row only interact with their nearest neighbours in the row and do not have enough energy to exchange places with other particles confined to the same fringe. Thus we have a model Tonks gas system for which exact statistical-mechanical solutions are known.' The scattering structure factor has been calculated for hard rods and more general potentials." It has the features seen in plate 1, where the broad intensity band above and below the line of intensity maxima are due to the regular spacing of particles within a fringe.As the particles become more concentrated in a fringe, the intensity bands become sharper and more intense. The bands shift to larger scattering angles as the particle separation within a fringe decreases. Of course, the fringes have been populated by radiation pressure forces, pushing particles in from the bulk solution, such that particle concentration within a fringe is much greater than the bulk solution. The particie pressure along a fringe is balanced by the radiation pressure force due to the weak intensity gradient along the fringe. If the fringe-spacing is increased and the potential well becomes broader, then each fringe can accommodate two or three rows of particles. Such structures have been observed in both real-space images and in self-scattered diffraction patterns.Measurements of the magnitude of the intensity maxima in the self-scattered light from the induced particle grating have been made as a function of the input power per area for thsse systems." These data can be explained using a theory of Rogovin et al.,12 if we account for the coherent mixing of the two scattered beams and for the finite size of the particles in both the radiation pressure force and in the scattering. We now discuss results which utilized more strongly interacting microspheres in deionized aqueous suspension. The incident beams are directed through the cell where a single monolayer of particles occupies the gap. Because the radiation pressure pushes the particles closer to the 'downstream' wall, it is possible to have an infiltration of particles from the sample surrounding the target volume.Care is taken to find a target region where the scattered intensity is stable. In unstable samples we have observed intensity maxima to increase as the particles reorganize in response to the applied radiation field, followed by a decrease in intensity as the local particle density changes (either increasing or decreasing). In stable samples the intensity maxima increase and stabilize. Plate 2 ( a ) shows the scattered intensity distribution for a single laser beam passing through the sample containing a monolayer of particles. The scattering pattern shows a bright ring (Debye-Scherrer ring) concentric with the incident beam. The scattering angle flus = 8.7" for the maximum intensity of the ring corresponds to a k-space radius k = 1.3 x 18' m-', implying an average particle separation of a = 2.40 pm.The scattering pattern is characteristic of an amorphous or liquid-like order.B. J, Ackersan and A. H. Chawdhury 313 In concentrated two-dimensional colloidal liquids it is known from cross-correlation intensity fluctuation studies (CCIFS) that the local interparticle order is dominated by a six-fold symmety (local hexagonal close packing).I3 However, this structure does not extend for more than a few particle separations a, the dimensions of a fluid correlation region. Thus, the Debye-Scherrer ring can be thought of as a collection of uncorrelated scattering patterns from randomly oriented hexagonal closest-packed regions.Each region produces six intensity maxima on the ring, but the collection of several randomly oriented scattering patterns fills out the pattern to a smooth ring. When a second beam is directed through the sample such that it crosses the first beam, making an angle equal to 8 = 8.4", then a fringe pattern is produced with spacing 2.50pm, a length approximately equal to the average particle separation. Each fluid correlation region can accommodate both the inter-particle forces and radiation pressure forces by rotating until a local hcp (10) direction lies parallel to the fringes. In this way the particles align with the fringes but maintain their local hexagonal order. The result is that all fluid correlation regions become oriented in the same way to produce a macroscopic 'solid' order, as indicated by the scattering pattern shown on plate 2(c).Here, rather than having two Debye-Scherrer rings each concentric with one incident beam or the other, there results a hexagonal pattern of intensity maxima. The external periodic field has induced the colloidal sample to 'freeze'. Not all intensity maxima in plate 2(c) have the same magnitude. Those which lie on a line passing through the two incident beams are the most intense and result from particles aligned along the intensity fringes. This modulated particle density is directly stimulated by the radiation field, as in the case of 'non-interacting' particles. The other intensity maxima are of lesser magnitude and result from particle density modes indirectly coupled by interparticle interactions to the directly stimulated mode. The intensity distribution which replaces the Debye-Scherrer ring in plate 2( c ) has many features which are reminiscient of CCIFS results.I3 Here the autocorrelation configuration gives a larger correlation than the other maxima measured in a cross- correlation configuration.The autocorrelation corresponds with measurement of the directly stimulated mode and the cross-correlation to the measurement of the indirectly stimulated modes. These two techniques can be shown to be equivalent in the limit of small applied radiation fields.I4 The laser trapping method is a form of stimulated CCIFS spectroscopy. If the incident beam crossing angle is changed, the fringe spacing no longer accommo- dates the preferrred particle spacing and the local order becomes strained.Plate 2 ( d ) shows the same sample as in plate 2 ( c ) where the crossing angle is now 8 = 9.1" (fringe-spacing 2.32 pm). The real-space structure is a strained hexagonal close-packed Iattice, having a reduced particle separation perpendicular to the fringes and increased particle separation along the fringes. In plate 2(e) the angle is increased to 8 =r 9.3" (fringe-spacing 2.25 pm). Further change in the crossing angle cannot be accommodated by a strain of the local hexagonal order and the structure becomes more fluid-like (or chaotic), as indicated in plate 2 ( b ) and ( f ) by the increased Debye-Scherrer ring intensity for a crossing angles of 6.6" (fringe-spacing 3.16 pm) and 10.0" (fringe-spacing 2.10 pm), respectively.In a different sample it was possible to reduce the fringe-spacing to the point where a nearly two-dimensional square lattice was ob~erved.'~ Reconstruc- tion of the real-space lattice from these scattering patterns indicates that the particle density is constant within an experimental error of 5%. The degree of alignment of the particles with the intensity fringes in the directly stimulated mode can be studied by measuring the magnitude of a lowest-order intensity maximum in the scattering patterns. Fig. 2 show the magnitude of the first intensity maximum to the right of the right beam stop, as seen in plate 2 ( b ) - ( f ) , as a function of the crossing angle. Information concerning the indirectly stimulated modes is deter- mined from the magnitude of the first intensity maximum making an angle of cu.60"314 Particle Order in Coiloidai Suspensions 20 15 - rn Y .e 1 $ 10 v h Y r n .e 5 Y .c1 5 0 I I I I I I I 1 5 6 7 8 9 1 0 1 1 12 crossing angle/" Fig. 2. Magnitude of the intensity maxima for a fixed input intensity of 4 x lo6 W m-' as a function of the beam crossing angle for scattering from low-order directly stimulated modes (0) and from low-order indirectly stimulated modes (0). The error bars indicates the r.m.s. fluctuation in the xattered intensity. The maximum reponse occurs for values of the spacing ratio slightly smaller than unity. The intensity scale factor is lo3 times greater for the directly stimulated modes compared to the indirectly stimulated modes. with respect to the horizontal about the right beam stop.The position of this intensity maximum is difficult to resolve for 0<7.5" and the detector is positioned on the Debye-Scherrer ring, making an angle of 60" with respect to the horizontal for these angles. In both measurements the maximum intensity occurs at a fringe-spacing/average particle separation ratio slightly less than unity. For this series of measurements the average particle separation is 2.68 pm or ODs = 7.8". The intensity decreases for larger or smaller values of the ratio. Because the intensity maxima result from a coherent mixing of scattered light by the two incident beams from different density modes, a model for the density modes is needed to fit the data. This fit should take into account the amplitude of each mode, the form factor amplitude for the particles and the effect of finite particle size on the radiation pressure force." For this reason uncorrected data are presented at this time.The radiation pressure induced transition from an amorphous or liquid-like structure to a solid-like order has also been studied as a function of the input laser power. Both the directly stimulated density modes and the density modes indirectly coupled byI o3 102 .C C v 3 x c.’ ...1 v) 5 = 10’ .” 1 o‘ B. J. Ackerson and A. H. Chowdhury 106 5 x 106 10‘ 10’ 10‘ I I I 106 5xlC 315 intensity/W m-* Fig. 3. Magnitude of the intensity maxima as a function of the input intensity for scattering from low-order directly stimulated modes ( a ) and low-order indirectly stimulated modes (6).The error bars indicate the r.m.s. fluctuation in intensity. The solid lines have slope 3 and 3/2 in ( a ) and (b), respectively. The data for different fringe spacing/average particle separation values corre- spond to 1.28 (0), 1.05 (a), 0.87 (V), 0.82 (0) and 0.71 (A) and have been scaled for presentation. The relative magnitude of the intensity for different spacing ratios may be estimated from fig. 2. The average particle separation is 2.50 pm in all data shown. particle interactions, as described for fig. 2, have been monitored. Fig. 3 presents these results for different fringe-separation/average particle separation ratios, where the average particle separation is 2.50 pm. The directly stimulated mode shows a continuous increase in amplitude with increasing input power and has a nearly cubic dependence on the input power.These results are the same as those obtained for ‘non-interacting’ particles in the low input power regime before saturation effects are seen.” The indirectly stimulated modes also show a continuous increase in amplitude, but with a much lower power law dependence of three halves. These data suggest that we have a second-order transition from liquid-like order to solid-like order. This does not violate any general principles because the radiation pressure field is a symmetry-breaking field. Of course, the sample region contains ca. 2000 particles so would not exhibit a truly sharp first-order transition. A first-order transition may also have occurred at much lower input powers below our level of detection.However, a simple Landau theory has been presented which demonstrates both first- and second-order transitions for these systems, given that a first-order transition is possible in the absence of radiation fields in two dimensions.16 The support of the National Science Foundation, Division of Materials Research, Low Temperature Physics grant no. DMR 85-00704 is gratefully acknowledged. We have also profited by useful discussions with N. A. Clark, a gift of particles from S. Hachisu, the typing assistance of E. D. Ackerson and technical assistance from the Van’t Hoff Laboratory, Utrecht, The Netherlands, in the preparation of this manuscript.316 Particle Order in Colloidal Suspensions References 1 €3. J . Berne and R. Pecora, Dynamx Light Scatterzng (Wiley, New York, 1976). 2 P. N. Pusey and R. J. A. Tough, in Dvnamic Light Srattering-Application of Photon Correlation Spectroscqy, ed. R. Pecora (Plenum Press, New York, 1985), chap. 4. 3 M. Kerker, Am. Sci., 1974, 62, 92. 4 A. Ashkin, Science, 1980, 210, 1081. 5 R . P. Turco, D. B. Toon, T. P. Ackerman, J. B. Pollack and C. Sagan, Science, 1983, 222, 1283. 6 A. Ashkin and J. M. Dziedzic, Appl. Phys. Lett., 1976, 28, 333. 7 A. Ashkin, Phyp. Rev. Left., 1970, 24, 156; P. W. Smith, A. Ashkin and W. J . Tornlinson, Opt. Lett., 8 A. H. Chowdhury, F. Wood-Jezercak, B. J. Ackerson, M. A. Karim and A. A. S . Anwal, Opt. Lett., 9 A. Munster, Statistical Thermodynamics (Springer-Verlag, Berlin, 1969), vol. 1, p. 292 f; p. 374f 1981, 6, 284. submitted. 10 B. J. Ackerson, unpublished calculation. 11 A. H. Chowdhury and B. J. Ackerson, to be published. 12 D. Rogovin and S. 0. Sari, Php. Rev. A, 1985, 31, 2375. 13 N. A. Clark, B J. kckerson and A. J. Hurd, Fh;vs. Rev. Lerr., 1983, 50 1459; B. J. Ackerson and N. A. Clark, Farndajf Discuss. C'hem. Soc., 198.7, 76. 219. 14 B. J. Ackerson and A. H. Chowdhury, Rev. Meu. Fis., to be published. 15 A. H. Chowdhury and B. J. Ackerson, Ejects of Particle Interactions on Optical Signal Processing, 29th 16 A. Chowdhury, B. J. Ackerson and N. A. Clark, Phvr. Rev. Lett., 1985, 55, 833. Midwest Symposium on Circuts and System, Lincoln, Nebraska, 11-12 August 1986. Receiwd 8th December, 1986

ISSN:0301-7249    

DOI:10.1039/DC9878300309

出版商:RSC    

年代:1987

数据来源: RSC

摘要:

GENERAL DISCUSSION Dr K. Schatzel (University ojKiel, Federal Republic ofGerrnany) said: Just like other methods of Laplace inversion, the maximum entropy method definitely requires data measured over a very large span of lag-times, if reasonable resolution and reproducibility are to be achieved in the calculated distribution of relaxation times. PCS data in the paper were in fact covering up to 7 decades of lag-time. Such large ranges cannot be economically covered by traditional digital correlators with a linear channel arrangement. The data given in the paper were obtained by running a series of correlation measurements using different lag-times. Besides being a time consuming procedure, this scheme has two major shortcomings. First, there is a need for ‘matching’ adjacent correlation functions, i.e. the introduction of more or less arbitrary multiplicative constants needed to correct for statistical signal fluctuations on long time scales and, if not a great amount of overlap is provided between adjacent correlograms, to compensate the triangular weighting distortions due to finite channel width.’ Secondly, the limited accuracy of traditional correlators (typically 4 bit) requires prescaling of the photon count signal at large sampling times. This procedure introduces a significant amount of quantization noise2 in the far tail of the correlation function, and can only be overcome by the use of extremely large total measurement times. While the use of single clipping3 would relax this second problem, it would make the matching problem considerably more difficult and restrict the method to signals with Gaussian complex amplitude statistics.A novel correlator architecture was required to overcome these problems of uneven channel spacing, matching and excess quantization noise. Combining a new fully parallel processor design, which provides a significant increase in computational speed, with data buffering and a facility to contract adjacent samples,’ I was able to design a new generation of correlator~,~ which can cover more than 7 decades in lag-time (e.g. 1 p s to ca. 60 s) in one single truly real-time experiment. The whole lag-time range is evenly covered by 192 channels with both the sample time and the lag-time increment doubling every 8 channels (resulting in 23 different simultaneous sample times).The channel width increases from 4 x 4 bit to 16 x 16 bit processing, and a new normalization scheme considerably reduces noise (and hence total measurement time) for very large lag-time channels.’ The ability to measure photon correlation functions far beyond their final decay implies new demands on inversion programs. The common scheme of reducing second- order correlation functions to first-order by baseline subtraction and taking a square root cannot handle data points properly, which fall below the baseline due to residual noise. This problem seems to be overcome with some elegance by the maximum entropy method’s ability to use second-order correlation data directly. Another potential problem results from the dominance of signal fluctuations rather than photon noise in large lag-time As a result, we obtain highly correlated noise in adjacent channels and a non-diagonal covariance matrix for the residuals. A modification of the simple chi-squared test used in the maximum entropy program’ seems to be appropriate.1 K. Schiitzel, Inst. Phys. Con$ Ser., 1985, 77, 175. 2 K. Schatzel, Appl. Phys., 1980, 22, 251. 3 E. Jakeman, .I. Phys. A, 1969, 3, 201. 4 ALV-3000 User’s Manual, (ALV Laser GmbH, D-6070 Langen, F.R.G., 1985). 5 K. Schatzel and M. Drewel, J. Mod. Opt., 1987, in press. 6 E. Jakeman, E. R. Pike and S. Swain, J. Phys. A, 1971, 4, 517. 7 K. Schatzel, Opt. Ada, 1983, 30, 155. 8 K. Schatzel, Appl. Phys. B, 1987, 42, 193. 9 A. K. Livesey, P. Licinio and M. Delaye, J. Chem. Phys., 1986, 84, 5102.317318 Genera 1 Discussion Dr A. K. Livesey (DAMTP, Cambridge) replied: We certainly agree that such a correlator would significantly reduce the measurement time involved in measuring 7 decades. Indeed, simply having 192 channels instead of our 48 would cut the measure- ment time by a factor of 4. The matching of our successive time windows has a firm justification. We adjusted these (inaccurately known) experimental parameters until the highest entropy curve was found. If, as a result of using your correlator, these parameters are accurately calibrated between the banks of 8 channels then this restriction on their values can only improve the reconstruction. I would like to stress the importance of maximum entropy’s ability to handle the second-order correlation data directly.This enables the measurements to be taken out to very long decay times when the measured signal (after background subtraction) can go negative owing to the noise in the background. Since maximum entropy can handle such data correctly, weak long delay signals (perhaps due to a few clusters or impurities) can be correctly determined and will not bias the shape or estimate of polydispersity of the major peak. The program is quite capable (in theory at least) of increasing the sophistication of the test of acceptability of a spectrum by including the off-diagonal terms in the covariant matrix. Currently these are set ‘unmeasured’ ( i e . with infinite errors) and not set to zero. Thus our current estimates of the spectra will have less structure than could possibly be extracted.It is therefore ‘safe’. In our particular case, of course, the 15 or so separate sets of lag times used were measured at different times, so that the correlation between these groups would necessarily be zero and only the near-diagonal terms would be relevant. Prof. V. Degiorgio (Pavia, Italy) asked: Concerning the examples discussed in fig. 2 and 3 of your paper, what is the minimum accuracy required on a single channel of the measured autocorrelation function in order to extract the information about the position and relative amplitude of the three peaks? Dr Livesey replied: Typically I estimate that ca. lo6 counts or more are needed in about a hundred channels spread out in log(measurement time) to resolve three well separated peaks.This gives a signal-to-noise ratio running from 500: 1 to 1 to 10 (or less). However, at this poor signal-to-noise ratio (for QLS measurements) there would probably be a certain overlap of the peaks and the user will have to set his own limits on the integration under the peaks. Such overlaps are still apparent in our fig. 2 and 3 taken with ca. 2 x lo7 counts in 200-400 channels (depending on the temporal range of the d-ta). Nevertheless, the MEM graphs also give the number of distinguishable peaks, their width, size, asymmetry, whether there are any shoulders etc. directly and visually without recourse to any models. Prof. M. Fixman (Colorado State University) asked: Is your method a candidate for a general ‘black box’ Laplace inversion scheme, at least for the recovery of positive functions? Is it possible to state general criteria for such a ‘library’ routine that would permit a competitive test of the different methods? Dr Livesey replied: The maximum entropy programs run entirely automatically without any user intervention except to set the required background level (although even this could be automated). The technique is not strictly limited to positive functions since it can recover the difference between two positive functions.This has been successfully applied in chemical n.m.r. spectrum analysis.’General Discussion 319 To run the routine we require data in the form of triples; autocorrelator delay, measured value and estimated standard deviation of random noise on the measurement. We have analysed Laplace transform data ranging from only 15 points with 10% accuracy to over 500 points with better than 0.1% accuracy.We would be glad to take part in any competitive test of analysis methods and hope the test includes the analysis of data from complex spectra containing many, asymmetric broad peaks, as well as simpler unimodal spectra. 1 E. D. Laue, J. Skilling and J. Staunton, J. M a p . Reson., 1985, 63, 418. Dr P. N. Pusey (R.S.R.E., Malvern) remarked: A measure of the resolution of an inversion technique is its 'response' to an ideal signal. In your case, if you construct numerically a single-exponential correlation function with noise (corresponding to an experiment of finite duration on a monodisperse sample) what is the width of the peak in Laplace space returned by the maximum entropy method? With careful technique it is possible, when studying, say, nearly monodisperse polystyrene spheres, to obtain experimental values for the normalized second cumulant Q in the range 0.01 +0.01.Since for narrow distributions Q =: ( T ~ , where (T is the normalized standard deviation of the distribution (see my comment on Dr van de Ven's paper), these values correspond to an apparent size distribution having half-width at half-height of ca. 10%. Does maximum entropy achieve this when an appropriate noise level is chosen? Dr Livesey replied: A delta function can be recovered as a single peak with half-width at half-height of 10%, if 48 channels (linearly spaced in time) are measured to contain between 1 x 10' ( t - 0 ) and 5 x lo6 (t=m) counts.Dr D. A. Weitz (Exxon , Annandale, NJ) addressed Dr Horne: You are no doubt aware that the power-law dependence of scattered light intensity on wavevector, which yields the fractal dimension as its exponent, holds only over a restricted range of that wavevector, q. When q is small, intensity becomes independent of q. Turbidity encom- passes the results of scattering over all, or nearly all angles, some very small, and therefore I wonder if your results will not be influenced by part of your scattering coming from this low-q regime and not all of it taking place in the power-law-dependent range. I would also like to ask you what effect cluster polydispersity has on the measurement of the fractal dimension using your turbidity technique.Aggregation processes often produce cluster mass distributions which are power-law in form, N ( M ) 0~ M-'. A measurement of dt using the k dependence of the scattering intensity is insensitive to polydispersity provided 7 < 2. Does the same limit hold for this turbidity measurement? Further, how sensitive is the measure of df to the choice of the lower cut-off angle included in your calculation of the scattering when the clusters have a power-law p o 1 y di s per s it y '? Dr Horne replied: I am aware of the restricted range of validity of the power-law dependence equation you cite. My paper includes a series of calculations of the turbidity/wavelength exponent using for the structure factor of the cluster the equation proposed by Teixeira [ref. (13) and eqn (S)].Qualitatively this equation gives the expected behaviour you cite, tending to unity at high y, power-law dependence at intermediate q. and, as Teixeira demonstrates, tending towards the Ornstein-Zernicke result at low 9 when D = 2. The full equation, giving structure factor as a function of monomer radius, a, cluster size, (, and fractal dimension, D, was used in my calculations of turbidity. Fig. 1 of the paper shows the behaviour of the dissipation factor/wavelength exponent as the cluster size increases, demonstrating that its asymptotic value for large clusters is the fractal dimension. Experimentally I follow this exponent as a function of reaction time, finding that it too reaches an asymptotic value (fig. 3 and 6) by which time the aggregates are sedimenting.I am therefore of the opinion that the wavelength320 Genera 1 Discussion exponents measured are for clusters large enough for the product q,$ to be in the range where power-law dependence is expected at all angles and that the experiments provide a value for the cluster fractal dimension. As to the influence of cluster size polydispersity on the value of the measured fractal dimensions, I think it would be quite small. Looking at fig. 1, once again, if we consider the plot for a fractal dimension D = 2, the exponent, p, is already > 1.9 for clusters > 1250 nm in radius, the limiting value of ,$+ being 2. It is my intention to carry out further calculations, involving realistic cluster size distribution functions, to verify these speculations.Dr Rarity then added: In the limit of small monomer particles and large aggregates the turbidimetry technique you have used measures the same moment of the aggregate size distribution as in the conventional measurement of light scattering us. scattering vector. Martin’ has studied the effects of polydispersity on measured fractal dimension for this case. He concludes that within this range of polydispersity expected from non-gelling reaction-limited aggregation the measured dimension is the fractal dimension. 1 J. E. Martin, J. Appl. Crystalfogr., 1986, 19, 25. Prof. M. Fixman (Colorado State University) addressed Prof. Fuller: Is it possible to specify which aspect(s) of the behaviour of non-Newtonian fluids are illuminated by experiments of this kind? The different aspects that I have in mind might be classified as follows.( a ) Fluid motion: ( 1 ) non-locality in time (memory functions, (2) non-locality in space (or higher spatial derivatives) and (3) non-linearity. ( b ) Particle-fluid interaction: (4) boundary conditions at surface of particle. Does a non-slip boundary condition suffice to determine a solution? Is it physically correct? ( c ) Mathematical solutions for a given model: (5) are there notable approximations made in solving the problem, the accuracy of which is tested? There then followed informal discussion on the analogue of Stokes’ Law for a second-order fluid, a problem which had been addressed by L. G. Leal. Prof G. G . Fuller (Stanford University) said: The points raised by Prof. Fixman and the problems of Stokes’ law for a second-order fluid are closely related and we will respond to them simultaneously.The objective of this paper was to exaiiiiile the respmse of a colloidal particle to flowing, non-Newtonian suspending fluids, and the materials were chosen in order to access the influence of Brownian motion and to mimic, to the best degree possible, the rheological properties of the second-order fluid constitutive model employed in the only available theories. The class of Boger fluids is currently thought to offer the closest approximations to second order fluid behaviour but even these systems will fail in several important ways, the principal difference being that the polymeric Boger fluids have finite relaxation times whereas an ideal second order liquid is modelled to respond instantaneously. For this reason, non-linear constitutive models such as the Oldroyd B model would be more appropriate.With respect to promoting the influence of Brownian motion, the appropriate dimensionless group which must be kept in the vicinity of unity is [from ref. (18) in our paper] +,G2/2r~77nr where I), is the first normal stress difference coefficient, D, is the rotary diffusivity and all other symbols are defined in our paper. This criterion was met for all three particles used in our study and the principal conclusion obtained by Cohen et al. that the effect of Brownian motion is to induce alignment in the flow direction at lower shear rates was verified in our measurements. This point is most clearly brought out in fig.9(c) where the ordinate passes from positve values at low shear rates to negative values at high shear rates.Genera l Discussion 321 The available theory, however, cannot be expected to capture the full response of the suspensions which were studied, and memory effects in the fluid are readily apparent in the data following the cessation of flow. In fig. 7, for example, examination of the response of the average particle orientation angle reveals that the particles rotate significantly after the flow is stopped whereas the theory predicts this angle should remain constant as the particles disorient. The timescale for this rotation is proportional to the relaxation time of the fluid and not the rotational diffusion of the particles. These two timescales are separated by ca.2 orders of magnitude. Indeed, this dimensionless group D,T, where T = +,/a7 is the fluid relaxation time, specifies whether or not the rotational diffusivity can be considered to be independent of time in these systems. It would appear that the separation of timescales was sufficient in our experiments to make this assumption. The question of the choice of boundarj conditions at the interface of a coiioidai particle and a polymeric liquid are certainly important to the problem being addressed in our paper. However, the answer to this question would more easily be obtained by experiments which are designed to look specifically at the friction fact of a particle residing in polymeric liquids. We are currently carrying out such an investigation by measuring the rotational diffusivity using electric dichroism techniques.In the experi- ments discussed in the present paper, the length scales of the particles (ca. 0.5 pm) and the polymer chains (ca. 0.1 pm) are not dissimilar, and there may be situations where non-slip boundary conditions are not appropriate. However, at the low polymer con- centrations employed here, we suspect that this is not the case. Our data do indicate, however, that the rotational diffusivities were altered upon the addition of the small amount of polyisobutylene, even though this addition did not significantly increase the shear viscosity. This effect is most clearly brought out by comparing the relaxation of the dichroism for particle A in fig. 5 ( b ) (Newtonian case) and fig.7( a ) (the Bsger fluid). The particles clearly relax more quickly in the Boger fluid. Dr J. Penfold ( Rutherford-Appleton Laboratory ) said: During his Lennard-Jones lecture Prof. Oppenheim spoke of the complexities of Brownian motion for spherical particles, and he alluded to the greater difficulties of treating anisotropically shaped particles. I would like to draw your attention to some experimental work on anisotropi- cally shaped micelles. Other groups’ and myself in collaboration with John Hayter, Oak Ridge’ and Ed Stables and Phil Cummins, Unilever3 have made investigations by small-angle neutron scattering of rod-like micelies aligned by shear flow. Experimentally the situation is simpler than the situation so well described by Prof. Fuller for non- Newtonian fluids, alignment is in the flow direction and is dependent on the ratio of the shear gradient, G, to the randomising factor, the rotational Brownian diffusion, 0,.Much of the information obtainable from such measurements is not particularly relevant to this meeting, such as particle size, size distribution, structure, conformation, and flexibility; and I will emphasise only aspects of interest to a meeting on Brownian motion. Fig. 1 shows some representative data for a 0.04 mol dm-3 dodecyldimethyl ammonium chloride-4 mol dm-3 sodium chloride micellar solution at different shear gradients; and the alignment of the rod-like micelles in the flow direction ( Qil) is clearly demonstrated. For dilute non-interacting systems we can, using existing theories,2 obtain an ‘effective’ rod length from the shear dependence of the anisotropy.At higher shear gradients (2 10 000 s-’) polydispersity and the onset of turbulence perturb the interpreta- tion. Although in terms of interactions the systems investigated by us were dilute, in the language of Doi and Edwards4 they are in the entangled regime where C CK: 1/ L3, with C the micelle number density and L the rod length. At low values of shear gradient hindered rotational diffusion, due to entanglements, is observed. This is illustrated clearly in fig. 2, where the effective rod length increases dramatically at low shear322 General Discussion e ( i i ) - - 01, Fig. 1. Intensity contours for scattering from 0.04 mol dm-? dodecyldimethyl ammonium chloride- 4 mol dm-3 sodium chloride-deuterium oxide at T = 298 K.The Q range is 10.6 nm-' in both the QII and Q, directions. The shear gradient, G, is ( i ) 250, (ii) 5000 and (iii) 7500 s-'. gradients. At higher shear gradients [in this case ca. (5-10 000) x lo3 s-I] the rods are sufficiently well aligned that hindered rotation is not present. Fig. 3 shows the shear dependence of the effective diffusion coefficient, D,, obtained from such assumptions. The plateau at 5000 s-l corresponds to dilute free rotational diffusion. The form of the dependence is similar to that reported by Keller et al.' for polymer solutions. Evidence of rod flexibility in some systems has been observed3 and will further complicate the shear dependence of the rotational diffusion. We are not, at present, in a position to offer any theoretical interpretation to our data, but nevertheless feel it contains important information with regard to the rotational diffusion of anisotropically shaped particles.1 H. Thurn, J. Kalus and H. Hoffman, J . Phj~s. Clrem., 1984, 80, 3440. 2 J . 3. Hayter and J. Penfold, J. Phyx Chein., 1984, 88, 4589.General Discussion 3 23 0 0 0 0 i 0 L I 1 1 I 1 I I 0 2 L 6 8 1 0 1 2 l L shear gradient, G/ lo3 s-' Fig. 2. Effective rod length as a function of shear gradient for 0.03 mol dm-3 cetyltrimethyl ammonium bromide-0.4 mol dm-3 potassium bromide-deuterium oxide at T = 313 K. .' 800 8 I I I I I I I 1 0 2 L 6 8 10 1 2 - 1 1 , shear gradient, G / lo3 s-' Fig. 3. Effective rotational diffusion coefficient as a function of shear gradient for 0.03 mol dm-3 cetyltrimethyl ammonium bromide-0.4 MOI dm-3 potassium bromide-deuterium oxide at T = 313 K. 3 P.G. Cummins, J. B. Hayter, J. Penfold and E. Staples, J. Chem. SOC., Faraday Trans. I , in press. 4 M. Doi and S. F. Edwards, J. Chem. Soc., Faraday Trans., I , 1978, 74, 560. 5 J. A. Odell, A. Keller and D. T. Atkins, Macromolecules, 1985, 18, 1443.324 General Discussion Dr W. van Megen (R.M.I. T., Melbourne, Australia) addressed Prof. Degiorgio: In the interpretation of, what I assume, as small-k diffusion constants (fig. 5 of your paper), do you think that Oseen level hydrodynamics is sufficiently accurate? Approximate calculations of the k-dependent diffusion constant D( k) (as obtained from the initial decay of the intensity autocorrelation function) for dilute (4 5 0.1) charge stabilized dispersions, suggest that the contribution of hydrodynamic interactions to this quantity is small at intermediate and large k.However, at small k, D( k ) is very sensitive to both direct and hydrodynamic interactions.’ 1 1. Snook and W. van Megen, J. Colloid Interface Sci., 1984, 100, 194. Prof. Degiorgio replied: Our results agree completely with your remark that, at small k, D( k) should be very sensitive to hydrodynamic interactions notwithstanding the fact that particles are, on the average, quite far apart one from the other in our system. We have only a qualitative justification for the fact that our calculations take into account hydrodynamic interactions at the Oseen level: since g ( r ) is different from 1 in a range of distances r which is much larger than the particle size a, and since the terms which appear beyond the Oseen level are of the order ( a l r ) ” with n = 2 or larger, we suppose that in our case tne Oseen level is sufficiently ac- -u rat e.Dr P. N. Pusey (R.S.R.E., Malvern) then asked: Quite plausibly you attribute the slow decays in fig. 4 to polydispersity. Is it possible that the disagreement between experiment and theory seen at low ionic strength in fig. 1 reflects incoherent scattering also caused by polydispersity? Prof. Degiorgio replied: If we analyse our correlation functions in terms of a superposition of two exponentials, A , exp (- t / 7’) + A2 exp (- t / T2) with r2 >> T ~ , the observed ratio A , / A , is too small to explain the disagreement between theory and experiment.This would indicate that the cause for the disagreement is not polydispersity alone. On the other hand, the dependence of A 2 / A l on the salt concentration is not in agreement with the decoupling approximation, so that our conclusion should be taken with some caution. Dr 0. A. Nehme (Cavendish Laboratory, Cambridge) said: I have a cautionary tale for the dynamic-light-scattering experimentalists, following Dr Pusey’s comment on polydispersity values deduced from light scattering data. There are evidently many sources of noise that produce non-exponential, non-random contributions to the correlation function of supposedly nearly monodisperse systems. This will give an abnormally high value for the polydispersity. The origins of the non-random ‘noise’ could be mains electrical supply problems, thermal fluctuations and at long sample times, mechanical vibrations.We have been able to track down the non-random ‘noise’ and separate from the exponential decay by using the maximum entropy data analysis, and by thorough examination of the residuals. It is hoped that non-random ‘noise’ could be minimised experimentally, although it is advisable to examine the residuals of the fit before trying to obtain quantitative values for polydispersity. Dr Pusey commented: I do not think it is necessary to resort to sophisticated methods of data analysis such as maximum entropy to detect many of the artifacts which can arise in a PCS experiment. As mentioned in my comment on Dr Livesey’s paper, it is possible, although not easy, to take a carefully prepared nearly monodisperse suspension of particles and to obtain essentially a single-exponential correlation function (to withinGeneral Discussion 325 experimental error).It seems much better to discover and remedy experimental di ficul- ties during such a test of equipment and technique than to attempt to allow for it in subsequent analysis of data obtained from a more complicated system. Dr A. K. Livesey added: Maximum entropy itself cannot correct for these systematic effects often seen in poor experimental measurements, unless some correct model of their origin is supplied. However, since the data analysis is completely general and makes no assumptions about the shape of the correlation times, then any non-random residuals which appear during the data-analysis must have arisen from some experimental artifact.It is this confidence that the problem lies in the experiment and not the fitting procedure which enables the research worker to pursue subtle experimental artifacts and eliminate them at source. The experimental papers in this Discussion have demonstrated how well (or other- wise) they have fitted their data by displaying a graph of their calculated data superim- posed on their experimental data. Given the large dynamic range of the data, and the high precision to which QLS scatterers work, I suggest that such a representation cannot adequately display the small differences between these curves. I recommend that users wishing to report an 'excellent' fit to their data should publish a graph of their normalised residuals where Dy''d is the kth calculated datum point, Dibsd is the kth observed datum point and 0; is the (estimated) standard deviation of the kth measurement.These graphs should appear random (further statistical tests could be applied to check this visual appearance if wished) and have an average deviation from zero of ca. *1 with very few outliers lying beyond *3. Dr A. Philipse (Utrecht, The Netherlands) said to Prof. Degiorgio: You state that the non-exponentiality of the correlation functions at low ionic strength is due to polydispersity in micelle size. However, if K - ' is large V ( r ) [eqn (3)] only weakly depends on the micelle size so the influence of this polydispersity will be small. At high salt concentration V ( r ) is more sensitive to the micelle size.Therefore, I would expect the variance V in fig. 6 to slope upwards at high ionic strength instead of downwards. Since a weakly screened V ( r ) depends strongly on the particle charge Q, V in fig. 6 probably reflects polydispersity in Q. Its effect will be smaller if the charge is sufficiently screened, in accordance with your data. This polydispersity also explains why the HNC fit of S ( K ) fails at low ionic strength, whereas at high ionic strength a fit is possible for one value of Q. [In addition, if K - ~ increases the S ( K ) peak shifts to higher K-values, away from the value of K in your experiment. So (any) polydispersity will manifest intself more clearly.] Thus, it seems to me that charge polydispersity explains as much as your suggestion that Q changes at low salt concentration.Prof. Degiorgio replied: The expression of the correlation function for the case of arbitrary polydispersities and interactions is very complicated [see ref. (2) and (22) of our paper], but it seems to me that what is required in order to give a non-exponential correlation function is a polydispersity in scattering amplitude rather than in the electric charge. However, as your remark implicitly suggests, a polydispersity in aggregation number implies a polydispersity in the micellar charge. As a consequence, this would lead to a strong correlation between concentration fluctuations and number fluctuations, and would not allow one, in agreement with our observations, to consider the correlation function as the sum of two independent modes controlled by collective diffusion and self-diffusion, respectively.326 General Discussion 1 I 1 " 0 15 Fig.4. Scattered neutron intensity for a 3 mmol cmP3 GMl ganglioside solution in D20 with no salt (0) and with 100 mol dm-3 NaCl. (W). Prof. V. Degiorgio (Pavia, Italy) said: We have recently performed small-angle neutron-scattering measurements on the same solutions studied by light scattering by using the spectrometer D17 at ILL, Grenoble. I show in fig. 4 the intensity of scattered neutrons as a function of k. The high-salt measurement gives essentially the form factor of the globular micelle, whereas the no-salt curve gives the product between form and structure factor.I mention here two features of our neutron data: (i) the two curves are perfectly superposed at high k, thus confirming that the size of the ganglioside micelle is very insensitive to salt concentration; (ii) the no-salt curve presents a marked peak at a value of k which is given approximately by 2v/(average interparticle distance), i.e. the micellar solution begins to form a structure reminiscent of that of colloidal crystals. Dr A. R. Rennie (I.L.L., Grenoble, France) then commented on the small-angle neutron scattering data presented by Prof. Degiorgio in the discussion in fig. 4 of this section. These data confirm the view that the ganglioside micelles are extremely monodis- perse. A least-squares fit to his high salt data gives a polydispersity of <2%.Dr Pusey then said: You mentioned that your analysis of the neutron-scattering data of Cant6 et al. yielded a very small polydispersity for their micelles. How was this analysis performed? If we consider their fig. 6 and take the data at high ionic strength to apply to a non-interacting system, the value of second cumulant V== 0.06 implies a polydispersity (standard deviation/mean) of cr = V1'2 == 0.25. How does this compare with your findings? Prof. Degiorgio answered: We believe that some of the observed polydispersity is due to some disturbance which is difficult to eliminate because the signal is very low: the intensity scattered from the solution of 1 mmol drn-3 GMl with no salt is only a few times that scattered by pure water! We were mainly interested in the trend of the variance u as a function of the salt concentration.Dr M. La1 (Unilever Research, Port Sunlight Laboratory) turned to Prof. Prieve: The disagreement between the calculated and measured potential-energy profiles, presentedGeneral Discussion 327 in fig. 2 of the paper, has been attributed by the authors to the fact that the particle-wall van der Waals interaction has not been taken into account. Surely, it is a simple matter to include this component of the interaction (assuming that the Hamaker constant for the system is known) in their eqn (6) and see whether better agreement is achieved. Can the experimental technique used by the authors be adapted to study the dynamical behaviour of surface-confined particles? Prof. D. C. Prieve (Carnegie-Mellon University) replied: We have estimated the van der Waals potential using Lifshitz theory.This estimate suggests that the van der Waals contribution cannot be neglected at the separation distances expected in our experiments, despite significant weakening of the van der Waals force by retardation as well as Debye screening. A more quantitative comparison of theory and experiment must await evaluation of the parameters appearing in the theory by independent experiments. In more recent experiments, histograms of scattering intensity have also been obtained for particles which appear to have stuck to the plate. As expected, they are much brighter than free particles. Although the intensity of stuck particles also fluctuates, the fluctuations represent a smaller percentage of the mean than for free particles. Prof.B. U. Felderhof ( R . W. T.H., Aachen, Federal Republic of Germany) addressed Prof. Ackerson: I would like to remark that the pair correlation function may be calculated exactly, not only for a one-dimensional system of hard rods, but more generally for one-dimensional fluids with nearest-neighbour interactions. This was first shown by Salsburg et al,' See also the article by Fisher and Widom.' 1 2. W. Salsburg, R. W. Zwanzig and J. G. Kirkwood, 1. Chem. Phys., 1953, 21, 1098. 2 M. E. Fisher and B. Widom, J. Chem. Phys., 1969, 50, 3756. Prof. Ackerson replied: The calculation may be difficult to carry out explicitly. Prof. Felderhof continued: That may be. I would also like to ask you whether it would be possible to measure the velocity of a colloidal sphere driven by radiation pressure.For known friction coefficient that would allow one to find the force. This may be of importance in connection with the longstanding controversy on the form of the Maxwell stress tensor in a liquid. Prof. Ackerson: Such a measurement would be possible. Prof. Fixman interjected: Which controversy are you referring to? Prof. Felderhof answered: In the early days of special relativity different forms have been proposed for the form of the electromagnetic stress tensor in a medium. Different proposals have been made by Minkowski, Abraham, Einstein and Laub and by others. The controversy has been discussed in great detail by S. R. de Groot and L. G. Suttorp in their book Foundations of Electrodynamics.They come to the conclusion that the issue cannot be decided on macroscopic grounds and that a microscopic derivation is needed. They gave such a derivation and came to a new form of the electromagnetic stress tensor. Prof. Fixman: For such matters I refer to the book by Landau and Lifshitz! Prof. Felderhof: There you will find a particular form derived on the basis of macroscopic arguments, but it is not the only possible one.328 General Discussion Prof. Fixman: Does it. make any difference for non-relativistic effects? Prof. Feiderhof: It does. The choice of the stress tensor and of the electromagnetic momentum density determines the electromagnetic force density exerted on a fluid element. Prof. Aekerson then took up this point: I assume the dispute to which you refer is whether the form of the momentum density for electromagnetic waves in a material medium is ( D x B)/47rc (Minkowski form) or ( E x H)/47rc (Abraham form).Presum- ably one could measure the Doppler shift of light scattered by a particle which is being pushed by the radiation pressure forces of an intense laser beam. From the measured velocity and known drag coefficient a force is determined, and the two theoretical predictions may be tested. Alternatively, one may test the two forms by measuring the degree of alignment of particles in a given magnitude intensity fringe pattern (by scattering from the induced diffraction grating) as a function of solvent refractive index. However, there are many experimental difficulties to overcome, especially for measure- ments which be can be compared from sample to sample. So I do not know if the dispute can be resolved easily.Gordon' has given a detailed discussion of this problem with reference to colloids and fluid surfaces. He favours the Abraham form but states that 'laboratory experiments designed to demonstrate the nature of the true electromag- netic momentum in dielectric media may not be feasible'. 1 Gordon, Phps. Rev. A, 1973, 8, 14. Prof. J. A. McCammon asked: Tn the two-dimensional colloid with solid-like order, the particles are trapped in potential wells that arise from interactions among the particles as well as between the particles and the radiation field. Thermal displacement fluctu- ations away from the minimum-energy positions should affect the scattering intensities; cf: the Debye-Waller factors in other diffraction experiments.Is it possible to extract information on the interparticle potentials by analysing such intensity variations? Prof. Ackerson replied: In principle, it is possible. However, to do thermal diffuse measurments will require a change in our scattering technique. Presently we are in a self-scattering mode. Both incident beams which produce the intensity fringes also scatter from the particles. The observed scattering pattern is a coherent mixture of the scattering patterns produced by each beam. A region which is bright with either beam alone (for a given particle configuration) may be dark with both beams on, because the scattering amplitudes from each beam are out of phase at that point.One can utilize this effect to make optical logic gates with the desired binary output properties or to resolve the phase problem inherent in standard scattering measurements. However, to measure thermal diffuse scattering one should filter out the self-scattered light from the input beams and probe with another beam of a different wavelength. Prof. J. F. Brady (CaItech, Pasadena) remarked: You have worked with two- dimensional colloidal systems. Can you extend this work to three dimensions? Prof. Ackerson answered: We have found it relatively easy to work with two- dimensional systems, both experimentally and in interpreting the results. In three dimensions one begins to have difficulties with the light scattered by the particles, and the fringes degrade in quality as the sample becomes thick.There may be some geometries where the scattered light will reinforce the interference pattern, but we have not followed this up. On the other hand, we have observed (microscopically) the three-dimensional ordering of particles near a wall with applied intensity fringes. The ordering and particle motion was interesting but restricted to a few layers of particles. I think these restricted three-dimensional systems are worth studying further.General Discussion 329 Dr R. B. Jones (Queen Mary College, London) said: I would like to ask if the radiation pressure effect could be used in an optically polydisperse suspension to separate the more optically active particles from the less active ones. If such a procedure were possible it offers a mechanism for particle sorting and could serve as another means of establishing the initial conditions for an exchange or tracer-diff usion experiment. Prof. Ackerson replied to Dr Jones: The radiation pressure forces depend on the difference in index of refraction of the particle and the solvent as well as the particle size. A focussed beam in a suspension will preferentially draw in the larger and more optically dense particles (assuming the solvent refractive index is less than that of all the particles). Because our particles are optically homogeneous and size monodisperse, we do not seem to induce preferential motion in certain particles. However, it may be interesting to seed a suspension with a trace amount of larger or more optically dense particles, as you suggest, then use radiation pressure to form a grating of these particles. By monitoring the time decay of the grating, when the fringes are eliminated, the low-k long-time self- or exchange-diffusion constant could be measured. These would be measurements on a different scale in length and time from those discussed in the Bristol and Utrecht papers at this Discussion. Prof. N. Sheppard (University of East Anglia) asked: Can radiation pressure be used to move single colloidal particles? Prof. Ackerson replied: Yes, we have trapped single particles near the cell wall and translated them about by moving the laser beam. One should also be able to scan a beam throughout the sample, stopping briefly in certain selected positions. Particles will tend to accumulate at the selected positions. In this way any pattern (including a Penrose tiling) of colloidal particles may be achieved. Prof. E. R. Smith (LaTrobe Uniuersity, Australia) then asked: In your one-dimensional colloidal hard-rod system, do particles which are not nearest neighbours interact? If not, then we can calculate exactly the properties of the system in terms of the Laplace transform of the Mayer $function for the pair potential. This will allow calculation of potential parameters from experimental properties of the system. Prof. Ackerson: I am very interested in an potential for another problem that I am involved with. Prof. Smith: I’ll send you the answer!!

ISSN:0301-7249    

DOI:10.1039/DC9878300317

出版商:RSC    

年代:1987

数据来源: RSC

摘要:

List of Posters Linear Kinetic Theory of Interacting Brownian Particles Systems Warsaw, Poland B. Cichocki, University of Mobility Matrix for Two Spherical Particles with Hydrodynamic Interaction R. Schmitz, R WTH Aachen, Federal Republic of Germany Computer Simulations of Brownian Interacting Particles with Memory J. A. Padro, Uniuersity of Barcelona and E. Guardia and G. Sese, Universitaf Politecnica Catalunyu. Barcelona, Spain Criticai Fluctuations Measured in Organic Liquid Mixtures by Dynamic Light Scattering R. B. Flippen, E I du Pont de Nemours, Wi!mington, U.S.A. Simulation Methods for Diffusion-controlled Reaction Kinetics N. J. B. Green, University of Notre Dame, U.S.A. and M. J. Pilling and S. M. Pimblott, Uniuersity qf Oxford Hydrodynamic Interactions in Concentrated Solutions Computer Simulation of the Aggregation Process R.D. Mountain and F. Sullivan, National Bureau of Standards, Gaithersburg, U.S.A. The Structure of Ordered Colloidal Suspensions in a Sedimentation Equilibrium H. Versmold, RWTH Aachen, Federal Republic qf Germany Brownian Motion on a Potential Surface: Solvent Effects on Isomerization S. Okuyama and D. W. Oxtoby, University of Chicago, U.S.A. Study of Charged Interacting Brownian Particles by Static and Dynamic Light Scattering M. Drewel, University of Kiel, Federal Republic of Germany Aggregation in Interfacial Colloidal Systems J. C. Earnshaw and D. Robinson, The Queen's Uniuersity of Belfast Cluster Size Distributions during Salt-induced Coagulation of Polystyrene Microspheres M. L. Broide and R.J. Cohen, MIT and Haruard-MIT, Cambridge, U.S.A. Viscoelastic Properties and Diffusion Behaviour in Dispersions of Interacting Clay Particles J. D. F. Ransay, A E R E Harwell Interactions and Dynamics of Silica Colloidal Particles in Water at High Temperatures and Pressures. J. D. F. Ramsay, A E R E Harwell Dynamics of Concentrated a-Crystallin Dispersions: A Maximum-entropy Analysis Photon Corre- lation Spectroscopy Data P. Licino, M. Delaye and A. K. Livesey, Orsay, France Rotational and Bending Brownian Motions in Macromolecules as revealed by Polarised Pulse Fluorimetry J. C. Brochon, Universiti de Paris-Sud, Orsay, France and A. K. Livesey, DAMPT and MRC Cambridge Mass and Entropy Transport in Colloidal Suspensions D. Lhuillier, Universif; Paris 6, France Evanescent Wave Photon Correlation Spectroscopy to Study Brownian Diffusion close to a Wall Hydrodynamics Description of Suspensions of Brownian Particles Colin, Universidad Autonoma Metropolitana-Izta~ulapa, Mexico Lead-time Effects on Photon Correlation and Photon Structure Functions Stampa, University of Kiel, Federill Rcpublic of Germany Determination of Static and Dynamic Interactions in Non-aqueous Charge Stabilized Sus- pensions A, Philipse, M. penders and A. Vrij, Universitj, of Urrecht, The h'etherlands L. Belloni, CEN Saclay, France K. H. Lan, N. Ostrowsky and D. Sornette, Uniuersiy of Nice, France 0 . Alarcon Waess and Garcie- K. Schatzel and B.Transport on Brownian Suspensions under Shear G . Bossis, University of Nice, France and J. F. Brady, Caltech, Pasadena, U.S.A. Colloid Particles Interacting with Surfaces under Photon Excitation A. Pelad, Holon Technical Institute, Israel

ISSN:0301-7249    

DOI:10.1039/DC9878300330

出版商:RSC    

年代:1987

数据来源: RSC

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Index of Names* Ackerson, €3. J., 87, 106, 110, 225, 309, 327, 328, 329 Adler, P. M., 145, 229 Allison, S. A., 213 Ansell, G. C., 167 Bacquet, R. J., 213 Botet, R., 125 Brady, J. F., 96, 328 Brochon, J-C., 247 Cantu, L., 287 Chen, Z., 153 Chowdhury, A. H., 309 Cichocki, B., 91 Clark, A. T., 179 Corti, M., 287 Degiotgio, V., 1 11, 231, 287, 318, 324, 325, 326 Delaye, M., 247 Deutch, J. M., 1 Dickinson, E., 96, 167, 224, 227. 239, 240, 241, 243 Earnshaw, J. C., 226 Family, F., 139 Felderhof, B. U., 69,92, 93, 94, 97, 106, 107, 329, 233, 244, 327, 328 Fixman, M., 94, 99, 106, 199, 242, 244, 318, 320, 327, 328 Fuller, G. G., 271, 320 Green, N. J. B., 245, 246 Gregory, J., 228 Hentschel, H. G. E., 139 Horne, D. S., 109, 227, 231, 238, 259, 319 Johnson, S. J., 271 Jones, K. B., 69,45, 102, 243, 329 Jullien, R., 125, 226, 227, 228 Klein, R., 153 de Kruif, C.G., 59 Kuin, A. J., 229 Lal, M., 94. 179, 240, 242, 245, 326 Lanni, F., 297 Lekkerkerker, H. N. W., 59, 89, 99, 103, 240 Licino, P., 247 Lin, M. Y., 153 Lindsay, H. M., 153 Livesey, A. K., 97, 247, 3 18, 3 19, 325 Luo, F., 247 McCammon, J. A., 213, 241, 242, 243, 245, 246, 328 Markovic, l., 104 Mazur, P., 33, 93, 94, 95, 99 Meakin, P., 113, 153, 223, 224, 225 Medina-Noyola, M., 21, 87, 89, 92, 107 Mors, P. M., 125 Nehmi, 0. A., 324 Northrup, S. H., 213 Oppenheim, I., I , 83 Ottewill, R. H., 47, 97, 99, 104 Penfold, J., 321 Philipse, A., 325 Pimblot, S. M., 223 Prieve, D. C., 297, 327 Pusey, P. N., 47, 102, 108, 230, 319, 324, 325, 326 Ramsay, J. D. F., 236, 237 Rarity, J. G., 99, 111, 234, 320 Rennie. A. R., 326 Robinson, D. J., 126 Schumacher, G. A., 75 Schatzel, K., 317 Sheng, P., 153 Sheppard, N., 329 Smith, E. R., 193, 243, 244, 329 Underwood, S. M., 47 Van Megen, W., 47,96, 98, 101, 106, 324 Van Veluwen, A., 59, 97, 98, 99. I03 Van de Yen, T. G. M., 75, 108, 109, 110 Vrij, A., 59, 97 Watson, G. M., 179 Weitz, D. A., 109, 153, 228, 230, 231, 232, 233, 234, Williams, N. St. J., 47 237, 319 *The page numbers in bold type indicate papers submitted for discussion.

ISSN:0301-7249    

DOI:10.1039/DC9878300332

出版商:RSC    

年代:1987

数据来源: RSC

摘要:

GENERAL DISCUSSIONS O F T H E FARADAY SOCIETY/FARADAY DISCUSSIONS O F THE CHEMICAL SOCIETY Date 1907 1907 1910 191 1 1912 1913 1913 1913 1914 1914 1915 1916 1916 1917 1917 1917 1918 1918 1918 19'" 191Y 1919 1920 I920 1920 1920 1971 1921 1921 1921 1922 1922 1923 1523 1923 1923 1923 I924 1924 1924 I924 I924 1925 1925 1926 1926 1927 1927 I927 1928 1929 1929 I929 1930 Subject Voiume Osmotic Pressure Trans. 3" The Constitution of Water 6" Hydrates in Solution 3* High Temperature Work 7* Magnetic Properties of Alloys 8* Colloids and their Viscosity 9" The Passivity of Metals 9" Optical Rotatory Power 1 o* Laboratory 12" Refractory Materials I2* Pyrometers and Pysometry 13* The Corrosion of Iron and Steel 9" The Hardening of Metals 10; The Transformation of Pure Iron Methods and Appliances for the Attainment of High Temperatures in a I I Training and Work of the Chemical Engineer 13" Osmotic Pressure 13" The Setting of Cements and Plasters 14" Etectrical Furnaces 14" Co-ordination of Scientific Publication 14* The Occlusion of Gases by Metals 14" The Present Position of the Theory of Ionization 15" The Examination of Materials by X-Ray-s 15" The Microscope: Its Design, Construction and Applications 16" Basic Slags: Their Production and Utilization in Agriculture 16" Physics and Chemistry of Colloids 16" Capillarity 17* The Failure of Metals under Internal and Prolonged Stress 17* Physico-Chemical Problems Kelating to the Soil 17" Catalysis with special reference to Newer Theories of Chemical Action 17* Some Properties of Powders with special reference to Grading by Elutriation 18" The Generation and Utilization of Cold 18 Alloys Resistant to Corrosion I9* The Physical Chemistry of the Photographic Process 19 The Electronic Theory of Valency 13* Electrode Reactions and Equilibria 19 Atmospheric Corrosion.First Report 19" Investigation o n Oppau Ammonium Sulphate-Nitrate 20" Fluxes and Slags in Metal Melting and Working 20" Physical and Physico-Chemical Problems relating to Textile Fibres 20 The Physical Chemistry of Igneous Rock Formation 20* The Physical Chemistry of Steel-Making Processes 21 Photochemical Reactions in Liquids and Cases 21* Explosive Reactions in Gaseous Media 22" Physical Phenomena at Interfaces, with special reference to Molecular Orientation 22* Atmospheric Corrosion. Second Report 23* The Theory of Strong Electrolytes 23" Cohesion and Related Problems 24" Homogeneous Catalysis 24 * 75" Atmospheric Corrosion of Metals.Third Report 25* Molecular Spectra and Molecular Structure 26" Colloid Science Applied to Biology 26 Electrodeposition and Electroplating I6* Rase Exchange in Soils 20; Crystal Structure and Chemical Constitution334 Date 1931 1932 1932 1933 1933 1934 1934 1935 1935 1936 1936 1937 1937 1938 1938 1939 1939 I940 1941 1941 I942 I943 1944 1945 1945 1946 1946 I947 1947 I947 I947 1948 1948 1949 1949 1949 1950 1950 19S0 1950 1951 1951 I952 1952 1952 1953 1953 1954 1954 1955 1955 1956 1956 1957 1958 1957 I958 1959 1959 1960 1960 1961 1961 1962 1962 1963 Faraday Disciissions of the Chemical Society Subject Photochemical Processes The Adsorption of Gases by Solids The Colloid Aspect of Textile Materials Liquid Crystals and Anisotropic Melts Free Radicals Dipole Moments Colloidal Electrolytes The Structure of Metallic Coatings, Films and Surfaces The Phenomena of Polymerization and Condensation Disperse Systems in Gases: Dust, Smoke and Fog Structure and Molecular Forces in ( a ) Pure Liquids, and ( b ) Solutions The Properties and Functions of Membranes, Natural and Artificial Reaction Kinetics Chemical Reactions Involving Solids Luminescence Hydrocarbon Chemistry The Electrical Double Layer (owing to the outbreak of war the meeting was The Hydrogen Bond The Oil-Water Interface The Mechanism and Chemical Kinetics of Organic Reactions in Liquid The Structure and Reactions of Rubber Modes of Drug Action Molecular Weight and Molecular Weight Distribution in High Polymers (Joint Meeting with the Plastics Group, Society of Chemicai Industry) The Application of Infra-red Spectra to Chemical Problems Oxidation Dielectrics Swelling and Shrinking Electrode Processes The Labile Molecule Surface Chemistry (Jointly with the SocietC de Chimie Physique at Bordeaux) Colloidal Elect 1-01 ytes and Sol u t ions The Interaction of Water and Porous Materials The Physical Chemistry of Process Metallurgy Crystal Growth Lipo-proteins Chromatographic Analysis Heterogeneous Catalysis Physico-chemical Properties and Behaviour of Nuclear Acids Spectroscopy and Molecular Structure and Optical Methods of Investigating Electrical Double Layer Hydrocarbons The Size and Shape Factor in Colloidal Systems Radiation Chemistry The Physical Chemistry of Proteins The Reactivity of Free Radicals The Equilibrium Properties of' Solutions on Non-electrolytes The P'nysical Chemistry of Dyeing and Tanning The Study of Fast Reactions Coagulation and Flocculation Microwave and Radio-frequency Spectroscopy Physical Chemistry of Enzymes Membrane Phenomena Physical Chemistry of Processes at High Pressures Molecular Mechanism of Ratt Processes in Solids Interactions in Ionic Solutions Configurations and Interactions of Macromolecules and Liquid Crystals Ions of the Transition Elements Energy Transfer with special reference to Biological Systems Crystal Imperfections and the Chemical Reactivity of Solids Oxidation- Reduction Reactions in lonizing Solvents The Physical Chemistry of Aerosols Radiation Effects in Inorganic Solids The Structure and Properties of Ionic Melts Inelastic Collisions of Atoms and Simple Molecules High Resolution Nulcear Magnetic Resonance abandoned, but the papers were printed in the Transactions) Systems Published by Butterworths Scientific Publications, Ltd Cell Structure Volume 27 * 28 29 29* 30* E* 31* 32* 32* 33" 33* 34" 3 4* 35" 35* 35* 36* 37* 37* 38 39* 40* 41* 42* 42 A* 42 B Disc.I" 2 Trans. 43* Disc. 3 4* 5" 6 7" 8" Trans. 46* Disc. 9" Trans. 47* Disc. 10* I I * 12' 13 14 15 16* 17 1 8* 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33* 34 The Structure of Electronical&, Excited Species in the Gas Phase 35Dare 1963 1964 964 965 96 5 966 966 967 967 968 1968 1969 1969 I970 1970 1971 197 1 i 972 1972 1973 I973 1974 1974 1975 1975 1976 I977 1977 1977 1978 1978 I979 1979 1980 19x0 1981 I981 1982 1982 1983 1983 1984 1984 1485 1986 1986 1987 Furudajy Discussicrns of the Chemical Society Subject Fundamental Processes in Radiation Chemistry Chemical Reactions in the Atmosphere Dislocations in Solids The Kinetics of Proton Transfer Processes Intermolecular Forces The Role of the Adsclrbed State in Heterogeneous Catalysis Colloid Stability in Aqueous and Non-aqueous Media The Structure and Properties of Liquids Molecular Dynamics of the Chemical Reactions of Gases Electrode Reactions of Organic Cornpounds Homogeneous Catalysis with Special Reference to Hydrogenation and Bonding in Metallo-organic Compounds Motions in Molecular Crystals Polymer Solutions The Vitreous State Electrical Conduction in Organic Solids Surface Chemistry of Oxides Reactions of Small Molecules in Excited States The Photoelectron Spectroscopy of Molecules Molecular Beam Scattering Intermediates in Electrochemicai Reactions Gels and Gelling Processes Photo-effects in Adsorbed Species Physical Adsorption in Condensed Phases Electron Spectroscopy of Solids and Surfaces Precipitation Potential Energy Surfaces Radiation Effects in Liquids and Solids Ion-Ion and Ion-Solvent Interactions Colloid Stability Structure and Motion in Molecular Liquids Kinetics of State Selected Species Organization of Macromolecules in the Condensed Phase Phase Transitions in Molecular Solids Pho:oeiecirochemistry High Resolution Spectroscopy Selectivity in Heterogeneous Catalysis Van der Waals Molecules Electron and Proton Transfer Intramolecular Kinetics Concentrated Colloidal Dispersions Interfacial Kinetics in Solution Radicals in Condensed Phases Polymer Liquid Crystals Physical Interactions and Energy Exchange at the Cas-Solid Interface Lipid Vesicles and Membranes Dynamics of Molecular Photofragmentation Oxidation 335 Volume 36 37 38 39 40 41* 42" 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65* 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 * Nor available; for current information on prices, etc., of aoailable volumes, please contact the Marketing Oficer, Royal Society of' Chemistry, Burlington House, London Wi V OBN stating whether or not y o u are a member o f t h e Society.

ISSN:0301-7249    

DOI:10.1039/DC9878300333

出版商:RSC    

年代:1987

数据来源: RSC