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