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