GENERAL DISCUSSION Prof. M. Gordon (University of Essex) said : With reference to fig. 4 of the paper by Beltman and Lyklema, the gel point region is lost in the period of temperature equi- libration at 15” and 25”, but at 35°C gelation is slow enough to show the gel point, where G’ rises from zero, very close to the time origin. This rise from zero starts with zero slope (dG’/dt = 0) as predicted by the generalised statistical theory based on the Scanlan-Case definition of an active network chain. This generalisation covers the complete range of network formation from the gel-point onwards and supersedes previous approximations like chain-end corrections. An active network chain is a chain connecting two active junction points, i.e., two points from each of which ut least three independent paths can be traced through the network to the surface of the specimen.Three paths are required to fix the mean position of the junction in three- dimensional space and to prevent it relaxing after deformation of the sample. (Stat- istically, it is almost certain that any such independent path, once started, will lead to billions of connections to the surface, by virtue of the criticality of branching statistics). The slow initial build-up of modulus (dG’/dt = 0) is due to the inefficiency of cross-linking in producing active network chains, as just defined, just after gelation. This is also the reason for the high-order Ehrenfest transition in the reversible case (see my remarks to Edwards’s paper). As shown previously (ref. (19) of Burchard et al., this Discussion), the plot of modulus versus number of cross-links is always sigmoid.It is easily shown that a sigmoid shape is mandatory even if the number of crosslinks is replaced as abscissa by a time axis via the kinetics of cross-linking, provided the kinetics follows any positive reaction order : this is how I explain the bottom curve in fig. 5. The corollary that initially v increases more than proportionally to cCR squares with the remark by Beltman and Lyklema just below their very convincing fig 5. In view of the highly magnified temperature scale in fig. 5, could the slight upward drift of modulus with rising temperature displayed by the points indicate a reduction of modulus from energetic sources : i.e., the chains would appear to prefer a stretched configuration from the purely energetic standpoint ? Mr.H. Beltman and Prof. J. LykIema (Wageningen) said: In replying to Gordon’s interesting suggestion it appears expedient to distinguish two problems : (a) the actual gelling kinetics in the strict phenomenological sense, i.e., the growth of the number of contributing chain segments and cross-links as a function of time, and (b) the interpretation of the nature of the cross-links. Gordon’s remark concerns especially the first and amounts to a more generalized approach. If instead of more or less postulating the linearity G‘(v) as in our eqn (2) and attributing deviations to rate-influencing contributions of precursor-structures, as done by us, a more generalized G’(v) relation is used, the sigmoid shape can auto- matically be accounted for.We think that this is a worth-while suggestion deserving perhaps to be worked out further for our systems. However, we must not forget that there is at the same time much circumstantial evidence in favour of a definite precursor-structure effect. Part of the evidence is accumulated in our paper, but comparative work with other gelling agents (e.g., resorcinol) can also be mentioned. Such an effect belongs to category (b). There J. Scanlan, J. Polymer Sci., 1960, 43, 501 and L. C. Case, J. PoZymer Sci., 1960, 45, 397. 119120 GENERAL DISCUSSION is some danger that these chemically interesting features are overlooked if G'(t) is interpreted solely on the basis of the kinetics, suggested by Gordon. We tend to conclude that Gordon's suggestion is a valuable alternative, especially if it is so applied that the precursor-structure concept is not a priuri renounced.With respect to Gordon's remark concerning our fig. 5 we refer to our reply to Prim and Flory, who h.ave raised essentially the same point. Pro€ W. Prim (Syracuse University) said : Previous work on crosstinked swollen poly(vinylalcoho1) has shown that there is invariably an energy contribution to the elasticity as indeed one would expect in view of the conformational energetics.'. It seems therefore that your conclusion " energetic elasticity is excluded " should be somewhat modified. Prof. M. Gordon (University of Essex) said: A dynamic transition occurs in the simple system of molten linear polyethylene gelled by irradiation (fig.I). This figure, with data by Prof. Pechhold of Stuttgart, was previously published in Russian. The I I 50 100 dose/Mrad FIG. 1.-Measurements by W. Pechhold (Stuttgart) of the real part of the compression modulus of radiation cross-linked polyethylene as a function of dose in Mrad. x , Hz; n, lo-' Hz; A, 2.6 Hz ; A, 120 Hz; 0, 1.2 kHz; ., 12 kHz. The line Q is the theoretical asymptote, and it was found that at 400 Mrad measurements at all the frequencies shown lie on this line. The asymptotic line is drawn to point at the origin, because the pointy= 1 must lie very close to the origin. storage modulus yields the equilibrium modulus curve at very low frequency Hz). It has the typical shape I have repeatedly emphasised in this discussion. But at increasing frequency, the storage modulus G' (not G"!) goes through a maximum H.Abe and W. Prim, J. Polymer Sci. C, 1963, 2, 527. A. Nakajima and H. Yanagawa, J. Phys. Chem., 1963,67,654.GENERAL DISCUSSION 121 along the cross-link density axis. The maximum is seen to approach the gel point at high frequency ; the gel point is too close to the origin (zero radiation dose) to study this in detail. The mechanism behind the effect of frequency on G‘ is likely to involve the sol fraction : the sol fraction is mobile but its concentration falls rapidly towards zero after the gel point. Its molecular weight averages rapidly decline also. Since small sol molecules would require the application of high frequencies to contribute to the storage modulus, the whole shape of the diagram seems qualitatively explained in this way.Further work on this dynamic transition would be valuable. Prof. P. J. Flory (Stanford) said: It seems to me that the most significant con- clusion to be drawn from Beltman and Lyklema’s fig. 5 is that the temperature coeffi- cient of G’/Tis small. The chains in these networks are bound together by crystalline domains, and one might have expected the alteration of these domains with deforma- tion to enhance the elastic compliance, the more so the lower the temperature, with the results that G’/T should increase with temperature. The fact that both G’ and G” are nearly independent of the frequency argues against the appreciable occurrence of such processes at the small deformations here involved. On the other hand, as Prins points out, the elastic response of a poly(viny1 alcohol) network can scarcely be expected to be purely entropic. The conformational ener- getics of the PVA chains are such as to lead one to expect its energy to vary with elongation.Hence, for the rubber elastic deformation of a network of fixed structure, G’/T should change with temperature. The magnitude of this change must depend on the stereoregularity of the PVA. Mr. H. Beltman and Prof. J. Lyklema (Wageningen) said: We thank Prins and Flory for their remarks. With regard to the occurrence or not of an energetic com- ponent to the elasticity, we note in the first place that the very fact that energetic elasticity has been observed experimentally in other PVA-gels does not necessarily have a bearing on our results, since PVA-gels, prepared under different conditions may vary widely with respect to their viscoelastic properties. The gels Prins is referring to are concentrated (20-40 % PVA as compared with 2-6 % in our case) and have probably shorter chains between cross-links, so that the relative energetic contri- bution could be higher.Moreover, Prins et al. prepared their gels in an entirely different way, e.g., by spinning. Spinning has a strong ordering effect on PVA; even a spun network without a cross-linking agent can only be dissolved in boiling water whereas our gels all melt at about 50°C. It seems possible that the chains in spun PVA-gels have a greater energetic contribution to the elasticity than ours. The theoretical argument that the conformational energetics of the PVA-chain requires an energetic component of G’ remains, of course, valid.It would require a decrease of G’/T with increasing temperature. However, linear regression analysis of our data did not confirm this, even a slight increase was found. Not too much value must be attached to this, though, because the measuring stretch is short in view of the mentioned intrinsic difficult of “ melting ” of cross-links. Also Hirai has found a positive value for d(G’/T)/dT. In conclusion, there seems to be no reason to alter the view-point that the elasticity is mainly entropic, although we leave room for an (as yet unconfirmed) small energetic contribution. With respect to the presence of crystalline domains, see our reply to Franks, C .Bayer, Chem. Unserer Zeit, 1968, 2, 61. N. Hirai, Bull. Inst. Chem. Rex Kyoto Univ., 1955, 33, 21.I22 GENERAL DISCUSSION Dr. M. Sheriff (Ui?ii*crsitj? of London) said: It was established by Beltman and Lyklema that for the gel : PVA (4 %)-CR (2.4 %) aged for 200 h at 25”C, both G’(w) and G”(w) were independent of o over the range of to 102 rad s-l, and it was then assumed that both these parameters were frequency independent for all the gels investigated, and that the choice of co was a matter only of experimental convenience. This assumption is incorrect, and may lead to serious errors in the interpretation of results for the following reasons. The maximum time over which the gelling process was investigated was only 2 % of the ageing time for the 200 h gel, and any inferences as to the frequency behaviour of a gel made over such a wide time interval will be subject to considerable error.Furthermore, as the gel structure develops the viscoelastic response of the system will change from that of a viscoelastic liquid to that of a rubbery viscoelastic solid. It has been well established that, to a first approximation, for a viscoelastic liquid (corresponding to the terminal zone of the viscoelastic spectrum) G’ is proportional to frequency and G” is proportional to (frequency)2.L It follows, therefore, that in the initial stages of gelation the choice of frequency is not a matter of experimental convenience but one that can have a profound effect on the magnitude of G’ and G”. When the gel structure becomes substantial, e.g., at long times the viscoelastic res- ponse will correspond to that of a rubbery solid in the plateau zone, where G’ and G” should be independent of frequency. This behaviour was observed for the gel aged for 200 h.Another factor which might have an effect on the values of G’ and G” is whether or not the applied frequency interferes with the gelling process, as it has been observed that above certain frequencies there is serious interference in the development of a two- dimensional surface gel of Acacia senegal at the air-water interface2 Whether this phenomenon also occurs in the three-dimensional bulk gel has still to be investigated. Mr. H. Beltman and Prof. J. LykIema (Wageningen) said: Sherriff is correct in stating that there is some element of uncertainty involved in the extrapolation of the o-functionality, observed after completion of the gel, to the initial stage of gelation, where it is practically impossible to study the co-dependence.Although it is not feasible to justify our extrapolation rigorously, we can forward at least the following arguments in favour of it. (1) The independence of G and G” of o applies to any gel, even those with very few cross-links per unit volume, i.e., gels resemble other gels in an early state of gelati on. (2) The G’(co) and G”(02) dependences, referred to by Sheriff, reflect relaxation of cross-links during the measurement. No such relaxation is observed after completion of the gel. Assuming that the nature of the cross-links is the same in the various stages of the gelling process, no relaxation is to be expected in the initial stages.(3) For safety’s sake we choose o = 5.28 rad s-’, which is to the high side of the spectrum, thus maximally suppressing any relaxation. Sherriff’s second remark concerns the possible interference of the oscillatory deformation with the gelling process. There is no indication of that with our system : the deformations are far below the maximum admissible limit of linear viscoelasticity and intermittent interruption of the oscillations had no detectable effect on G’ nor on G”. We conclude that the measurement itself does not affect the cross-link density. Accepting in PVA-CR gels the absence of any influence of the measurement as correct, we nevertheless leave room for such an interference in other systems, notably J.D. Ferry, Viscoelastic Properties of Polymers (Wiley, New York. 2nd edn, 1970). M Sherriff and B Warburton, to be published.GENERAL DISCUSSION 123 in those exhibiting no linear viscoelasticity and in systems where the linear behaviour occurs only over a narrow range. Prof. F. Franks (Unilever) said : Beltman and Lyklema raise a number of inter- esting points, some of which are also touched upon in the papers by Harrison et al. and Eagland et al., in particular the requirement for a highly conformation-specific gel precursor. Whereas with gelatin this is almost certainly an intramolecular process involving collagen type triple helical segments, in the case of PVA this specific intra- molecular conformational arrangement appears to resemble closely the crystal struc- ture of anhydrous PVA.X-ray studies have shown that the diffraction pattern characteristic of the monoclinic lattice of crystalline PVA also develops in PVA solutions (in glycerol, ethylene glycol or water) on cooling. Thus no solvent swelling of the crystalline regions is observed.' The authors suggest that the cross-links arise from helical PVA regions but I do not know of any instance where carbon chains adopt such conformations. The results are more compatible with the formation of syndiotactic regions, such as occur in crystalline PVA. I do not think that too much importance can be attached to the difference in the G'(t) curves shown in fig. 4. Surely the type of behaviour observed for the 35" curve is the one to be expected, i.e., initially aG'/at > 0.Considering the moi- xular nature of the gel this is surprising and I wonder how much reliance can be placed on the truly horizontal nature of the plots, bearing in mind the small tempera- ture interval. The observed effects of NaCNS in suppressing gel formation makes me suspect the solvent involvement in the conformational rearrangement and/or gelling and this also points to a considerable enthalpic component. It would be interesting to ascertain whether the Hofmeister series can be applied to the effect of ions on the stability of the gel. Finally to study the function of Congo Red would it be possible to use low molec- ular weight oligomers of PVA? It might be expected that these would still exhibit the conformational transition to form microcrystallites without however giving rise to gels.This type of approach has been of use in studying the nature of the junction zones in carrageenan and analogues of ~t-gelatin.~ Fig. 5 is taken as evidence that the PVA gels are entropic. Mr. H. Beltman and Prof. J. Lyklema ( Wageningefz) said: Franks has made a number of points, the first of which concerns the possibility of a superstructure formation. Franks argues that this superstructure is not a kind of helix but a semi- crystalline region. At the outset we would bring to the fore that we did nowhere emphasize that the superstructure should be a helix; just for sake of argument the term " helical " has been invoked in the concluding section. Nevertheless, there are some arguments which favour a helix-like structure just as there are arguments against it.These have been mentioned in part in the paper but it might be advisable to mention some of them and some other ones again. We recall the blue colouring of (aged) PVA-solutions with 12, parallelling the blue colouring of amylose, and occurring with a rate that is comparable with the rate of (incipient) gelation. The promotion of this colouring by boric acid or by the G. Rehage, Kunststofle, 1963, 53, 605. K. A. Piez and M. R. Sherman, Biochem., 1970, 9, 4134, J. Horacek, J. Chern. Prumsyl., 1962, 12, 385. * D. S. Reid, T. A. Boyce, A. H. Clark and D. A. Rees, paper at this Discussion.124 GENERAL DISCUSSION introduction of F groups as well as its disappearance upon heating parallel these influences on gelation.All of this points too strongly towards the possibility of helix- like structures to be fortuitous, though it is not a definite proof. X-ray data, providing they are properly interpreted, support the concept of micro- crystalline regions. However, if these are due to syndioactic regions, as suggested by Franks, the vastly differing influences of differing gelling agents as Congo Red, resorcinol and borax are not easy to explain. The points raised by Franks with respect to our fig. 4 (behaviour of G’(t)) and 5 have been dealt with in our discussions with Gordon, Prins and Flory, to which we refer. We have not made a systematic study of the effect of electrolytes on the gelling. This could be an interesting topic for further investigation. Some work in this field has been reported by Dittmar and Priest (our paper, ref.(1)). Effects are observed only at high concentrations and only between different anions was it possible to discriminate ; not between (monovalent) cations. Moreover, NaCNS is effective in much Iower concentrations than those studied by Dittmar and Priest. This may point to a different mechanism, though not conclusively. For further information, NaCNS discolours the blue PVA + I, complex and prevents the autonomous gelling of a highly concentrated PVA-solution. Congo Red promotes the formation of the blue PVA + I2 complex, even in the presence of NaCNS. Also the rate of gelling of PVA/Congo Red is only slightly influenced by NaCNS. Our general conclusion is that a search for the physical interpretation of the blue colouring will presumably provide us with the clue for discrimination between the role of microcrystalline and helical structures.We have indeed found that low MW PVA’S are poor gel-formers. For example, a 3 % PVA-solution of M-25 000 did not give any detectable gelling though blue colouring still occurred with iodine. However, we do not think that this helps one to distinguish between microcrystalline regions or more helix-like structures. Franks’ last remark concerns the MW-effect. Dr. D. S. Reid (Unilever) said : What evidence have Beltman and Lyklema that the initial refluxing at 85°C produces complete dissolution of all components, and disruption of any preformed structures in the polymer? In previous work involving agarose, we found that complete dissolution was difficult.The only way to be sure of complete solution was to compare the results obtained for gelation temperature, etc., after differing pretreatments. Have the authors checked that their results are unchanged if they use a higher temperature for the refluxing stage. If the kinetics are altered, it would indicate that the initial treatment is insufficient to disperse all nuclea- tion sites. Mr. H. Beltman and Prof. J. LykIema ( Wageningen) said: Although the refluxing temperature was not varied, we did vary the refluxing time within certain limits and found the obtained gel properties unaffected by it. This applies to the stock solution of PVA as well as to the system after addition of Congo Red. The cloudiness disappears upon heating.For this reason only fresh PVA-solutions have been used. Upon standing the 10 % PVA solution becomes slightly cloudy. Dr. A. B. Fasina and Dr. R. F. T. Stepto (UMZST) said : The concept of an effective elastic functionality is an interesting one, but there are several points worthy of clarification. H. C. Haas and R. L. MacDonald, J. Polymer Sci. Al, 1972, 10, 1617.GENERAL DISCUSSION 125 fc is a parameter which can be evaluated experimentally (from swelling), or theoretically from crosslinking statistics. Because of network defects (free ends, closed loops, and entanglements), there is always the problem of the correspondence between chemical crosslinks and the number of EANC. However, in the statistical evaluation it is assumed that every monomer unit in the gel with more than two reacted functionalities makes a contribution.Will this assumption not result in differences between experimental and calculated values off, ? For stoichiometric polycondensations, fe apparently increases in value from 3 at the gel point tof(the chemical functionality) at complete reaction. However, it is not immediately obvious howf, can be used whenfitself is equal to 3. Isf, still a useful concept for such systems or should some other effective functionality be sought ? The concept of an effective chemical functionality (fc) has of course been used previously to characterise gelation in polycondensations. Here, f, is determined simply from the experimental value of a, and the gelation condition a, = l/(fc- 1). This concept is related only to the gel point, but it indicates that networks can form providedf, 2 2.This lower value of an effective functionality at gel is presumably because the sol fraction is included in the definition offc (but not in the definition offe) so thatf, as such cannot be used to describe the network. The approximate treatment of cyclisation used by DuSek, in which intra- molecular reaction is excluded in the gel, although justified by mathematical exped- iency, appears to be unrealistic. For sufficiently concentrated systems, and subject to the chain configurational properties of the reactants, it is known that cyclisation can be neglected in the sol fraction. However, intramolecular connections are an essential part of the gel structure and they will increase as the reaction proceeds from gelation to complete reaction.Dr. K. Dusek (Czechoslovakia) said : The effective functionality f, can be deter- mined from swelling provided x and v, are known. The present statistical evaluation fully covers only the free ends effect and in principle can be extended to include cyclization and trapped entanglements. The progress is dependent on the develop- ment of crosslinking statistics covering these effects. For ring formation it has been shown that the evaluation is still valid if one takes into account only intermolecular connections.2 When the chemical functionality is 3, thenf, is also 3 over the whole range on conversion. Fogiel’s fc is a corrected chemical functionality of crosslinks in the whole system, but fe is related only to the elastically effective crosslinks which exist in the gel; crosslinks in the sol as well as crosslinks in the gel that are not elastically effective are not counted.Cyclization in the gel has been discussed in detail in ref. (3). If one regarded the gel as a single giant macromolecule, all gel-gel reactions might be consideied as intramolecular, but it is not so with respect to the elastic properties of the gel. Essentially only loops formed within elastically active network chains between two crosslinks are to be counted as intramolecular and not contributing to the modulus, but even they gradually disappear when their segments become crosslinked. The fraction of elastically ineffective crosslinks engaged in elastically inactive loops is thus expected to be very small at high degrees of crosslinking. Fogiel, Macromolecules, 1969, 2, 581.K. DuSek, J. Polymer Sci. C, 1973, 42, 701. M. Gordon, T. C. Ward and R. S. Whitney, Pulpier Networks, ed. A. J. Chompff (Plenum, New York, 1971).126 GENERAL DISCUSSION Dr. D. J. Walsh (Manchester University) said: Could DuSek please enlarge on the physical meaning of an effective functionality of 3 in a system where the chemical functionality is 4. Also could he comment on what effect would have been observed in his calcula- tions if instead of using eqn (3), which is based on Flory’s expression for the free energy of deformation of a network, he had used one based on the expressions derived by James and Guth or any of the other alternative theories. Dr. I(. DuBek (Czechoslovakia) said : The concept of the effective functionality, fe, is related to the equilibrium elastic response of the network including swelling.In the network only such crosslinks that are a part of the gel and from which at least three independent paths issue to the surface of the sample are elastically effective (elastically effective crosslinks, EEC). Other crosslinks do not contribute to the number of elastically active network chains (EANC) although the number of their reacted functionalities may exceed the value of 2. In the vicinity of the gel point almost all of the few EEC are effectively three-functional because the chance of issuing additional paths to the surface (infinity) is extremely low. Further beyond the gel point this probability increases and so doesf,.Eventuallyf, approaches the chemical functionality for a network without free ends. The effective functionality of crosslinks in the network is thus a topological parameter, but its contribution to the free energy of deformation depends on the model. Following essentially the Flory-Wall approach one gets the result given, e.g., by eqn (8), but the procedure could be used also for different contributions due to stretching of EANC provided the additivity of the mixing and elastic terms is valid. Within the framework of the James-Guth theory, fe affects spatial fluctuations of crosslinks around their equilibrium positions and consequently may have an effect on the elastic modulus. Dr. B. Launay (Massy- France) said: Referring to the paper of Callaghan and Ottewill, I wonder whether the values of E calculated by eqn (6) and shown on fig.6, are really representative of a bulk modulus. As a matter of fact a bulk modulus is measured during the hydrostatic compression of a constant mass of a product; in your experience water is expelled from the gel. Do you think that E is nevertheless related to the bulk modulus of the gel? Prof. R. H. Ottewill (University of Bristol) said : Our experiments were carried out with a constant mass of montmorillonite in the compression cell and compression- decompression cycles demonstrated that the system was a reversible one. The displacements involved in the application of pressure were those in the “ lattice of montmorillonite plates ”, the number of units in the lattice remaining constant.The small displacement at each point corresponds to a particular volume fraction of the solid. I think it is therefore reasonable to assume that the figures obtained give a representation of the elasticity modulus of the montmorillonite gel-lattice. Dr. J. W. Goodwin and Mr. R. W. Smith (Bristol University) said: Electrostatic repulsive forces can result in gel formation in dispersions of spherical particles. This effect can be observed with small particle size polystyrene lattices during the dialysis procedure that is often used after emulsion polymerisation to remove electrolyte, surface active agents and reaction products. Polystyrene latices are dispersions of rigid spherical particles with a surface cliarge arising froin bound ionic groups. It has127 I I 1 1 I 4 0 5 0 6 0 7 0 interparticle separation Ho/nm FIG.1.-Shear modulus as a function of interparticle separation for polystyrene latex. 0, 1 x 1O-j mol dm-3 sodium chloride ; 0, 1 x mo1 dm-3 sodium chloride ; -, calculated values. I I I - 5.0 -4.0 -3.0 -2.0 log ionic strength FIG. 2.-Shear modulus as a function of ionic strength for polystyrene latex in 1 : 1 electrolyte at a volume fraction of 0.186 (Ho = 50 nm). 0, experimental values ; - , calculated values.128 GENERAL DISCUSSION been shown i * that monodisperse latices form ordered arrays and that the ordering process is the result of the net repulsive forces between the particles. The shear modulus of a polystyrene latex has been measured as a function of the volume frac- tion over a range of ionic strengths.The apparatus was built to the design of van O l ~ h e n . ~ It consisted of two parallel discs which were mounted on piezo-electric crystals. The separation of the discs could be varied fram 1 mm to 2 cm. A small amplitude shear wave was produced by feeding an electrical pulse to the lower crystal and movement of the upper disc was detected by the upper crystal. By displaying both signals on a twin-beam oscillo- scope and photographing the image, the time taken for the shear wave to traverse the distance between the discs was measured. For a Hookean material, the shear modu- lus, G, is related to the density of the suspension, p , and the propagation velocity of the shear wave, v, thus : G = v2p. A polystyrene latex was prepared by the emulsion polymerisation of styrene.The latex was extensively dialysed against distilled water, concentrated by evaporation and samples with different volume fractions were dialysed against sodium chloride solutions to give the chosen ionic strengths. Before each measurement, the suspen- sions were centrifuged to remove any aggregates and the polymer content of each suspension was determined by drying weighed samples to constant weight. The number average particle size was determined from electron microscopy to be 85.6 nm (lo3 particles were measured) and the electrokinetic potential (5) was found to be -50 mV. Some typical experimental data are shown in the figures. The surface to surface separation of the particles (Ho) was calculated from the volume fraction of the suspensions assuming hexagonal arrays.Fig. 1 shows that the shear modulus increases rapidly with decreasing interparticle separation over the range of volume fractions studied. It can be seen that an appreciable elastic modulus was observed at a sodium chloride concentration of mol dm-3 at a volume fraction as low as 0.18. The theoretical curves were calculated by assuming that only nearest neighbour interactions were important at the large separations used and that with the small displacements due to the simple shearing motion there was displacement between adjacent layers, but that displacement within a layer was negligible. The net restoring force per particle in each layer (li,) is : F, = Fi(cos Bi sin 4i) 1 where Fi is the net force between two particles along a line joining the centres, is the angle between that line and the x-z plane, while 4i is the angle between the x-axis and the projection of the line on the x-z plane.Now the shear modulus is related to the slope of the force-distance curve so for the regular array of particles : G = P. A. Hiltner and 1. M. Krieger, J. Phys. Chem., 1969,73,2386. L. Barclay, A. Harrington and R. H. Ottewill, KuZZuid Z. Z. PuZymere, 1972, 60, 966. R. W. Smith, MSc. Thesis (Bristol, 1973). H. van OIphen, C/ays and Clay Min., 1956, 4, 204. A. E. H . Love, Treatise on the Mathematical Theory of Elasticity (4th edn., London, 1934).GENERAL DISCUSSION 129 where a is the particle radius and Vtot is the total potential energy of interaction between adjacent particles calculated from the Derjaguin-Landau-Verwey-Overbeek theory of colloid stabi1ity.l The calculated curves of shear modulus as a function of inter- particle separation are of the same form as the experimental curves with the best agreement at sodium chloride concentration of mol dm-3. It is suggested that many electrostatically stabilised dispersions should have a measurable elastic modulus when the volume fraction is sufficiently high to produce large repulsive forces between particles due to the overlap of the electrical double layers. Prof. A. Silberberg (Israel) said: In a study of gel formation in well fractionated montmorillonite systems Dr. A. Posner and I (unpublished results) found that there was an optimum platelet size for gel formation which suggested to us that the inter- action in these systems was not simple and could involve edge to face interactions. Prof. R. H. Ottewill (University of Brisfol) said: It is not possible completely to rule out edge-face interactions and, indeed, our initial compression results suggest that these might well exist in some form in the original low concentration gel. Our contention is, however, that these are random contacts formed by thermal motion and that they are probably not a consequence of electrostatic interaction between the negatively charged faces and the positively charged edges. Although ion-exchange experiments indicate a small number of positively charged sites on the edges of the particles it must be remembered that the edge area is very small compared with the face area and that the surface charge density of the faces is very high. Our evidence is that in the sodium montmorillonite examined the edge electrostatic effects are screened by the long range electrostatic interactions of the faces. These arguments, however, could not be applied to kaolinites where edge-face interactions are very important. ' H. R. Kruyt, Cuffoid Science, Vol. 1 (Elsevier, Amsterdam, 1952). 57--E