Faraday Discuss. Chem. SOC., 1990, 90, 57-75 GENERAL DISCUSSION Prof. C. F. Zukoski (University of Illinois) addressed Prof. Gast: Is it necessary to consider how far ions move in half a cycle? The average drift velocity (U) of an ion can be calculated from (U)=oF, where F is the electrical force, eE, and o can be estimated from o = D / k g T . Then at lo6 V m-', (U) =r 3.8 cm s-l. For coated electrodes, no loss current can pass between the electrodes. As the relaxation frequency of the ions can be estimated from a/ toEo =: 2 x lo4 s-', ions can respond to a frequency of 3 x lo3 s-l. The question then is how far ions can move at a frequency of 3 x lo3 s-'. This can be estimated from the time for half a cycle (1.7 x s) and the average drift velocity of the ions. From these calculations one finds that the distance travelled at 500 V across 560 pm is ca.5.7 pm. Thus near the fields where the doublet concentration saturates, ions can move several particle diameters. As it is improbable that there will be any electrochemistry occurring at the electrodes, the field is unlikely to be uniform across the gap. Is it possible that the saturation in doublet concentration is due to relaxation of the field due to ions migrating to the electrode? Prof. A. P. Gast (Stanford University) replied: This is an interesting possibility that we had not considered. The characteristic length for ion diffusion within one cycle we calculate from l2 = D / o with an ion diffusivity of D = 1 x m2 s-' and a frequency of 3000 s-', is 0.6 pm, essentially one particle radius.From your equation for the drift velocity with ( U) = DeE/k,T we calculate a velocity of 0.038 m s-' for 1000 V mm-'. This provides a distance travelled in one half cycle, 2 / w = 1.7 x of 6.5 pm, a distance large compared to a particle but small compared to the whole gap. The saturation in electric field could have electrostatic origins such as the one you suggest, however, we have no evidence for large field gradients in our cell. We could only unambiguously test this hypothesis with a better cell design and further studies at higher frequencies. Prof. D. A. Saville (Princeton Uniuersity) said: We have studied the motion of small fluid globules in apolar liquids ( E d 10) at high field strengths (> 1000 V cm-') and find some evidence of field-induced charging. In such cases, the particle charge inferred from low-field-strength mobility measurements may not appreciably change at higher fields.Did you find any evidence of field-induced charging? Could such an effect be responsible for the 'saturation' shown in, e.g., fig. 3 and 4? Prof. Gast replied: Electric field-induced charging could provide an explanation for the saturation behaviour observed in our experiments. Note that the saturation did occur at similar field strengths for different volume fractions. Surely more experiments are in order to verify the mechanisms. It would be quite interesting to measure the electrophoretic mobility of our particles in strong fields to look for field-induced charging. One could perhaps make such a measurement by applying a high-voltage AC field with a DC bias.A better cell design would probably be required to avoid contributions from dielectrophoresis occurring under high field strengths. We have seen evidence for saturation effects in dielectrophoretic mobility measurements of glass beads in oils. Prof. S. P. Stoylov (Bulgarian Academy of Science, Sojia) asked Prof. Gast: First, in view of the numerous results which indicate a large induced-dipole moment of a non-dielectric (interfacial) nature for latex dispersions in aqueous media, is it not possible that the induced-dipole moment measured by you is of a similar nature? A way to 5758 General Discussion investigate this would be to make measurements at frequencies much lower or much higher than 3 kHz. Have you made such measurements? Secondly, have you measured the distribution of the electric field strength in your cell? You state that the Teflon tape covering your electrodes has the same relative permittivity as your solvent and that therefore the tape does not perturb your electric field.This will be correct only when the electric conductivities are equal. Electro-induced lattice interactions have been described by Vorobeva' and by Sokerov, Vorobeva and Stoylov' followed in the first case microscopically and in the second case by light scattering. 1 T. Vorobeva, Thesis, Moscow, 1967. 2 S. Sokerov, T. Vorobeva and S. P. Stoylov, Polym. Sci. Symp., 1974, 44, 147. Prof. Gast responded: We have no evidence for large contributions to our dipole moment due to interfacial effects.Owing to the good agreement with a dielectric dipole model we were not compelled to consider the interfacial dipole. It would be fruitful in the future to make measurements such as ours at a wide range of frequencies to determine whether this mechanism is important in low-relative-permittivity fluids. We found no difference between Teflon-coated and uncovered copper electrodes. We preferred the former simply because they provided a better seal with the cell walls and they reduced particle deposition on the electrodes. A simple calculation indicates only a minor perturbation to the electric field for a layer with relative permittivity so close to that of the solvent. We did not see any major distortions of the electric field across our cell. Dr B. Vincent (University of Bristol) said: As shown in fig.5 of your paper, with increasing particle volume fraction of the dispersion, the critical depth of the energy minimum (and hence the applied electric field) for the onset of dimer formation decreases. This is to be expected on thermodynamic grounds, in terms of the net energy/entropy balance in the system.' However, the experimental observation times were rather short (ca. 1 min). Are you sure that equilibrium has indeed been established with this timescale, because our experience of weak, reversible flocculation is that quite long timescales may be necessary to establish equilibrium? 1 J. A. Long, D. W. J. Osmond and B. Vincent, J. Colloid Interface Sci., 1973, 42, 545. Prof. Gast replied: Yes, the critical interaction energy to produce doublets increases with volume fraction as anticipated for a phase transition.Near this threshold we are observing equilibrium doublet formation; the formation of larger phases, such as those you have observed, could be much slower. As mentioned in the text, the time characteris- tic of this equilibrium, t,, = 6.rrq-a3/kgT exp (-umi,/kBT) is ca. 5 s near the onset of doublet formation. It is interesting to note, however, that the failure of the equilibrium model occurs at the same potential-energy minimum for each sample corresponding to an equilibrium time of ca. 500s. Prof. J. Lyklema and Dr P. C. van der Hoeven ( Wageningen University) commented: In the paper by Adriani and Gast the authors conclude, somewhat to their surprise, that in media of low relative permittivity, electrostatic repulsions may act beyond the range of stabilising layers.We have found similar phenomena. The stability of concen- trated dispersions of a number of inorganic solids and zeolites in pure non-ionic surfactants, in the presence of dodecylbenzenesulphonic acid, appeared to be entirely of electrostatic origin. Measurements to prove this included rheology, analyses of sediment structures and electrokinetics. Admittedly, in our systems the likelihood of electrostatic stabilisation is somewhat greater, because our relative permittivity wasGeneral Discussion 59 140 120 - > 100 - 'm -E 80 1 2 60 U z 40 20 -6 -5 -4 -3 -2 -1 0 log (% Ca octanoate) Fig. 1. Electrophoretic mobility against log (% calcium octanoate) for PMMA- PHS particles, diameter 614 nm, in dodecane.higher (ca. 6) and because of the presence of a 'candidate' charge-determining species. However, the repulsive forces must be substantial since systems containing over 20% of the solid phase can be stabilised. It may perhaps be concluded that electrostatic stabilisation in systems of low polarity is more common than is usually thought. Dr P. Bartlett (University of Bristol) said: I am surprised by the high electrophoretic mobility of 1.6 x m2 V-' s-I quoted by Prof. Gast for her PMMA spheres. I have made measurements on similar PMMA spheres of 670 nm in diameter in pure cis-decalin and find mobilities typically two orders of magnitude lower ( ~ 2 x lo-" m2 V-' s-I). I wonder if Prof. Gast could comment on the mobilities of her particles in pure hexane prior to the addition of sodium acetate? Prof.Gast responded: The particles had an electrophoretic mobility of the same order of magnitude before the addition of sodium acetate but it was difficult to reproduce. Freshly made samples had quite variable mobilities and conductivities presumably due to variable concentrations of ionic impurities. You will note that while the addition of sodium acetate helped to control the mobility, one sample had a higher ionic content and a lower mobility. Prof. R. H. Ottewill and Mr A. Schofield (University of Bristol) commented: This paper provides some nice results in the difficult field of electrical effects on colloidal particles in non-aqueous media. In laser electrophoresis studies on poly( methyl methacrylate) (PMMA) particles stabilised by poly( 12-hydroxystearic acid) (PHS) we have found the basic particles to have a weak positive charge.There has been some variation between samples and most examinations have been carried out in dodecane. The weak positive charge, however, can be considerably enhanced by the addition of small amounts of calcium octanoate. Fig. 1 illustrates some electrophoretic mobility results obtained as a function of calcium octanoate concentration. From these it can60 Genera 1 Discussion be seen that initially the electrophoretic mobility becomes more positive with increase in calcium octanoate concentration and then decreases. The positive charge is probably a consequence of adsorption of [Ca octanoateJ+ or Ca2+ by attraction to the ester linkages on the poly(l2-hydroxystearic acid) chains. The reason for the maximum is not completely clear but preliminary conductance measurements have provided some evidence for the self-association of calcium octanoate in this region.Small-angle neutron scattering measurements have also provided evidence for charg- ing of the latex particles on the addition of calcium octanoate.' 1 R. H. Ottewill, A. R. Rennie and A. Schofield, Prog. Colloid Polyrn. Sci., 1990, 81, 1. Prof. G. Frens (University of Delft) said: Could the effects which Prof. Gast has described be useful in the technology of electrophotography, where particles are attracted by, and deposited on, a charge pattern on a surface? The problem is to obtain deposits of more than one particle in thickness.For electrophotography one uses suspensions of electrically stabilised particles in dielectric liquids of low E.' The problem, as in electrophoretic deposition, is that particles in the second and further layers of the deposit must be fixed in their positions by some form of coagulation. Could the energy minima for two particles in an electric field be used to overcome their repulsion and give irreversible coagulation in the deposit? What field strength must be reached in the liquid to obtain such a result? 1 S. Stotz, J. Colloid Interface Sci., 1978, 65, 118. Prof. Gast replied: The interactions producing our particle chains are purely field- induced and thus vanish upon removal of the field. Such a field-induced dipolar interaction may, however, be sufficient to overcome a repulsive barrier and leave particles bound in a primary minimum after the field is removed.We have seen such effects with magnetically polarisable particles which can be made to remain in chains in the absence of the external field. Of course, the field strength required to produce such an effect will depend on the size of the repulsive barrier to be surmounted. The purely field-induced dipolar interaction energy scales with the square of the dipole moment as P2 U - 4.rr~~~,8a'li,T and to a first approximation the dipole moment scales with the electric field strength as while with multibody interactions an enhancement of the dipole strength over the isolated particle value is seen.' Thus, with knowledge of the stabilising forces in the suspension or with careful deposition experiments such a field-induced irreversible aggregation could be achieved.1 P. Adriani and A. P. Gast, Phys. Fluids, 1988, 31, 2757. Prof. G. Cevc ( University of Munich) (communicated): Looking at the experimental values for the chain number fractions as a function of volume fraction in the range of high peak-to-peak applied potentials (near the saturation region) I notice that the absolute numbers vary non-monotonically: i.e. for small q5 they decrease and for higher 4 they increase with the value of 4. Can you explain this phenomenon? Prof. Gast replied: We have looked at the number fractions of single particles where we believe the saturation phenomena to begin (by interpolating between available data points, not by extrapolation to higher fields).We find that the singlet volume fraction at saturation varies monotonically but not in the linear fashion that one would expectGeneral Discussion 61 if the experimental time was constant and the doublet formation time was tdb= 7rr]a3/4kgT. This deviation may be due to the fact that the limiting transport process may not be random diffusion but rather diffusion in a restricted space essentially transverse to the field. It is also useful to reiterate that the volume fraction 4 = 0.0028 sample did have a different ionic strength. Prof. Saville (communicated): The authors carry out experiments at a frequency of 3 kHz and argue that at this frequency the particles and their counter-ion cloud can no longer follow the imposed field.Accordingly, they ignore the dipole due to the particle charge and counter-ions. I believe that this dipole may be significant for the following reasons. One way of estimating a characteristic relaxation frequency is to use a timescale based on the distance ions must move to follow the imposed field, the particle radius plus Debye thickness, a + 1 / ~ , and the ion diffusivity, D. Then, as argued by DeLacey and White,' the characteristic frequency is approximately D ( u K ) ~ / u * ( 1 + U K ) ~ . For a 1 pm particle with U K =J 3 and D = m2 s-I, the characteristic frequency is slightly more than 2 kHz. Numerical calculations by DeLacey and White show that at such frequencies relaxation is not complete and a significant dipole still exists.Although estimates of this sort are imprecise, the operating frequency in these experiments is of the same order of magnitude as the relaxation frequency, so it appears that the induced dipole due to particle charge ought to be included in the analysis. For the problem under consideration the dipole due to particle charge, estimated as Q(a + K - ~ ) = C m, is larger than the polarization dipole due to differences in relative permittivity. If, as these estimates indicate, the particle charge-counter-ion cloud dipole is significant, what effect will this have on the theory of the equilibrium chain length? Will the agreement between theory and experiment be altered substantially? 1 E. H. B. DeLacey and L. R. White, J. Chem. SOC., Furuduy Trans. 2, 1981, 77, 2007. Prof.Gast replied: Your are correct in pointing out that the relaxation time for our ionic double layer (based on the estimates of ionic strength from conductivity) is comparable to the frequency of our alternating current field within numerical factors. Certainly it would be interesting to do such an experiment at much higher frequencies; however, our experimental arrangement precluded this. It is not clear how the dipole due to the distortion of the double layer contributes to our calculation of chain length and doublet equilibrium. As you point out, an upper bound on the magnitude of this dipole, Q(a + K - ' ) , is in the region of C m based on our estimate of the zeta potential and ionic strength. The dipole due to dielectric differences, p = 4 7 7 ~ ~ s , a ~ P E , is of the same order of magnitude, 7 .6 ~ for E = 1000V/560pm7 c0= 8.85 x C V-' m-', E, = 1.9, a = 0.50 pm and p = 0.162 for this system. If these estimates are correct, this dipole will indeed affect our predictions by adding an additional attraction between our particles. The effect of a double-layer dipole on our results depends on the field dependence of the dipole; this may be non-linear and even saturating at elevated field strengths. It would be interesting to learn, however, how two such distorted double layers interact upon close approach of the particles and how this dipole persists around a chain of particles. Surely the point-dipole approximations are no longer valid. It could be that the short-range attraction is not so severely altered thus providing the agreement between theory and experiment.Dr R. Buscall (ICI, Runcorn) (communicated): A number of years ago ICI provided Dr Block of Liverpool University with a sample of a similar latex in dodecane. Dr Block was performing dielectric spectroscopy on flowing polymer solutions and he wanted to compare the behaviour of a dispersion. The sample was found to show a very strong electrorheological response. This was unexpected and so several, larger62 General Discussion samples were synthesized for further study. These, however, failed to show any response whatsoever, even at very high field strengths. It was suspected that the response shown by the first sample may have been caused by some polar impurity, e.g. water. However, the addition of water and other polar liquids in trace amounts failed to produce an effect, as did copolymerisation of polar monomers with the MMA.Does Prof. Gast have any comments regarding what type of contaminant might produce such a response; is it correct to say that it would have to be surface-active and highly polarisable? Prof. Gast responded: The electrorheological response we find with the 1.0 pm poly(methy1 methacrylate) spheres is quite weak. As we show in the paper, the electro- rheological tendency is well described by material parameters expected for PMMA in hexane and thus requires no impurity to account for the effect. The fact that we do see a weak positive charge on the particles does, however, indicate the presence of an unknown impurity. The effect of charged impurities on electrorheology is not obvious, and, judging from the nature of this discussion, not fully understood or agreed upon.It is quite likely that residual surface-active impurities may influence such systems. The observation of a very strong electrorheological effect in such systems would, as you point out, imply the presence of a material having a very high relative permittivity. The addition of water, in general, may degrade the electrorheological effect by increasing the conductivity in the suspension leading to power consumption and heating. Prof. N. Ise (Kyoto Uniuersity) turned to Dr Dhont: Spinodal decomposition takes place when the second derivative of the free energy of the system with respect to concentration is negative. This implies that the concentration dependence of activity must be negative.Our previous measurements of various linear polyelectrolytes showed that the solute activity increased with increasing concentration. (For a convenient review, see ref. 1.) Furthermore, our measurements of ionic latex suspensions showed that the activity of the latex was practically independent of its own concentration.* We have never measured the activity of the silica particle suspension. Do you have experimental activity data for silica suspensions? 1 N. Ise, Ado. Polym. Sci, 1971, 7, 536. 2 M. Sugimura, T. Okubo, N. Ise and S. Yokogama, J. Am. Chem. Soc., 1984, 106, 5069. Mr M. H. G. M. Penders, Dr J. K. G. Dhont and Prof. A. Vrij (University of Utrecht) replied: The divergence of the measured structure factor at small wavevectors with decreasing temperature implies that the osmotic compressibility of the system of Brownian particles which scatter light, goes to infinity.This system of Brownian particles becomes unstable. A vanishing and, in the unstable region, negative reciprocal osmotic compressibility is equivalent to what you call ‘a negative concentration dependence of the activity’. Structure-factor measurements at small wavevectors thus probe the activity. We did not measure the osmotic compressibility using different experimental techniques besides light scattering. It is not possible to use electrochemical or isopiestic methods in our systems. Our systems are not of an ionic nature and the molecular weight of the particles is far too high to measure any change in the solvent activity in the concentration range we measured here.Prof. Ise (communicated ): You obtained the osmotic compressibility by extrapolating the structure factor S ( (I) to Q = 0, where Q is the scattering vector. This extrapolation is valid if the solution is homogeneous. There is direct experimental evidence available showing that solutions of ionic polymers and suspensions of ionic latex particles are not homogeneous. (See for example, ref. 1.) I presume that this is also the case with ionic silica particle suspensions. If so, there should be a sharp upturn of the structureGeneral Discussion 63 factor at very low Q regions, which would invalidate the extrapolation to Q = O . Therefore, I strongly suspect that the authors have been observing the spinodal decompo- sition. In this respect, I should draw the authors’ attention to similar phase-separation phenomena reported for ionic latex suspensions by Arora et aZ.* The height and width of the Bragg peak depend on three factors, namely the Debye- Waller effect, the paracrystalline distortion and the size of the ordered structures. Therefore, it is difficult to accept the authors’ assumption that rate of increase of the Bragg peak height is a measure of the crystallisation rate. According to our microscopy study on latex suspensions,’ the oscillatory motion of latex particles at lattice points (which corresponds to the Debye- Waller effect) is substantial at low latex concentrations and becomes less and less so as the concentration increases.It is also highly likely that the paracrystalline distortion becomes less and less pronounced with increasing con- centration. Thus it is not clear to me why the authors observed practically the same scattering intensity at t = 0 for three different concentrations (fig.5 , their paper) with the intensity of the 0.329gcm-’ sample reaching the largest value among the three concentrations afterwards. Could the sedimentation effect of the silica particles (radius, 1600 A) be ignored in these experiments? The authors observed crystallites with well defined sizes in solution. Similar structures have been reported for ionic latex particle suspensions by several author^.^ We inferred the existence of such localised structures for ionic polymer solutions on the basis of scattering experiments.’ The authors’ explanation of the existence of the crystallites in terms of repulsive interactions is not acceptable.How can the authors explain the formation of crystallites of definite sizes in solution, if the particles interact only via repulsion? How can the particles at the boundary stay there if they feel purely repulsive interactions from the particles inside? 1 N. Ise, H. Matsuoka, K. Ito and H. Yoshida, Furuday Discuss. Chem. Soc., this Discussion. 2 A. K. Arora, B. V. R. Tata, A. K. Sood and R. Kesavamoorthy, Phys. Rev. Lett., 1988, 60, 2438. 3 K. Ito, H. Nakamura, H. Yoshida and N. Ise, J. Am. Chem. Soc., 1988, 110, 6955. 4 A. Kose, M. Ozaki, K. Takano, Y. Kobayashi and S. Hachisu, J. Colloid Interface Sci., 1973, 44, 330; N. A. Clark, B.J. Ackerson and A. J. Hurd, Phys. Rev. Lett., 1983, 50, 1459; T. Yoshiyama and I. Sogami, Lungrnuir, 1987,3, 851; K. Ito, H. Nakamura, H. Yoshida and N. Ise, J. Am. Chem. Soc., 1988, 110, 6955. 5 N. Ise and T. Okubo, Acc. Chem. Res., 1980, 13, 303; N. Ise, Angew. Chem., Int. Ed. Engl., 1986, 25, 323. Mr Penders, Dr Dhont and Prof. Vrij responded: The silica systems used for the PICS measurements, presented in our paper, are not of an ionic nature and therefore the behaviour of these systems cannot be compared to the behaviour of the suspensions of ionic latex particles and solutions of ionic polymers you mention in your comment. The upswing of the structure factor at small wavevectors is a result of an increasing correlation length, or equivalently, due to an increase of the osmotic compressibility (just as in molecular or atomic systems).This upswing is monotonic as a function of the wavevector. With decreasing wavevector, it follows from SANS experiments’ that there is a monotonic increase of S(Q) for silica systems dispersed in benzene. As the temperature is lowered this increase is more steep, as it should be at the approach of the spinodal curve from the stable region in the phase diagram. In a decomposing system on the other hand, the intensity develops a maximum at very small wavevectors which increases with time, as described by the Cahn- Hilliard theory. Such a time dependence of the intensity was not observed in the experiments described in our present paper. 1 C. G. de Kruif, P. W. Rouw, W. J. Briels, M.H. G. Duits, A. Vrij and R. P. May, Lungmuir, 1989, 5,422. Dr Dhont continued: What is measured at t=O is the structure factor of the (metastable) fluid. From its primary maximum Bragg peaks start to grow. In your64 General Discussion remarks you seem to imply that there is a relation between the height of the primary maximum of the structure factor of the metastable fluid and the height of the measured Bragg peak after completion of crystal growth. This is certainly a complicated relation which is not yet fully understood: a statistical-mechanical theory for the kinetics of crystallisation is not available. Sedimentation of Brownian particles in the fluid phase is only noticeable after ca. 2 weeks (in samples which are dilute enough or so concentrated that no crystallisation occurs), and therefore plays no role during the crystallization process which takes at most ca.1.5 h. However, crystallites, once formed, do sediment during their growth up to a level that remains unchanged over more than a day or two, after which separated crystals start to coalesce without further noticeable change of the volume occupied by the crystalline phase. Thus, the nucleation kinetics are not affected by sedimentation processes, since sedimentation simply does not occur, but the details of crystal-growth kinetics might be influenced a little owing to sedimentation of crystals. As was mentioned above, the separated crystals which are grown out of the indepen- dently formed nuclei, start to coalesce after a day or two. The final thermodynamically stable state is therefore not the long-lived state occurring immediately after crystal growth stops, but is a state where a single large crystal is in coexistence with a fluid phase.This coalescence process is probably driven, in part, by gravity forces. The experiments described in the paper are all restricted to times where nucleation and subsequent crystal growth appear, before coalescence occurs. We disagree, totally, with Prof. Ise’s view that crystallisation can occur only in coexistence with a fluid phase for systems with an attractive component in the pair- interaction potential. The total force on a particle is not the sum of all the pair-potential interaction forces from the other particles in the system, as is incorrectly assumed in Prof.Ise’s question. There is an additional force on a particle j , which is called the Brownian force FLr, and is given by where PN is the probability density function of the position coordinates r l , . . . , r N of the N particles in the system, and V j is the gradient operator with respect to 5, the position coordinate of the j t h particle. The total force experienced by this particle is therefore where V is usually taken to be equal to a sum of pair-interaction potentials. The equilibrium form of PN, for a canonical ensemble, is obtained by minimising the free energy (with constraints pertaining to the canonical ensemble), and is, of course, the Boltzmann factor, Substituting this in eqn (2) gives, Fljj,, = 0; in equilibrium (4) as it should be. In his argument, Prof.Ise overlooks the Brownian force in eqn. (2), which amounts, in thermodynamic terms, to saying that the system will minimise its total potential (plus kinetic) energy, instead of its free energy. A particle on the boundary of a crystallite will not be pushed into the fluid phase, rendering an unstable crystallite, just because the potential energy of the system then becomes smaller. The potential energy is decreased in this event but the free energy is increased. In terms of forces, the total potential interaction force will tend to push a particle into the fluid phase, but the Brownian force will do the opposite; in equilibrium these two forces cancel, accordingGeneral Discussion 65 to eqn (4). We note here that computer simulations and density functional theory calculations do predict crystal-fluid coexistence for systems with a purely repulsive pair-interaction potential.From the thermodynamic point of view, crystallisation in these systems can be understood as follows. The total potential energy will probably increase on formation of a crystallite (with a higher overall density than the fluid phase) and therefore counteracts crystallisation. The entropy can be thought of as having two distinct contributions, a ‘configurational’ entropy, which decreases as a crystal is formed and which therefore also counteracts crystallisation, and a ‘translational’ entropy, related to the free volume which is available for each particle. In an ordered structure, the space available for each particle is larger than in a disordered structure. Therefore, this contribution to the entropy will favour crystallisation.At a high enough volume fraction in the ‘disordered’ fluid phase, the translational-entropy contribution to the free energy will be larger than the potential-energy and configurational-entropy contributions, so that crystals will be formed. The formation of crystals continues up to a point where the volume fraction of the remaining fluid is lowered such that the free energy per particle in the fluid phase equals that of the crystal phase. Prof. Ise (communicated): I was very much puzzled by the responses of Dr Dhont and co-workers for various reasons. While we are fully aware of the uncertainty about the ionic state of silica particles, the authors clearly reported in their paper, ‘the sample was slightly charged, giving rise to a screened Coulomb type of interaction potential. The Debye length is estimated to be eu.100 nm.’ Now, in the response to my comment, they have said that the samples are not of an ionic nature. These descriptions are contradictory. I would like to ask the authors how they came to these diametrically opposed conclusions. I know from our discussion in the meeting how the osmotic compressibility was determined. The reference to the SANS experiment is not relevant to the present argument. The point in question is how reliable the S ( Q ) value determined by the authors’ extrapolation can be. My question arises not only from the observation of inhomogeneity in highly charged polymeric systems mentioned in my previous comment but also from the observed sharp upturn of SAXS curves for low-charge-density ionomer films.’ Even though one takes into account the fact that the upturn was obtained in solid without solvent, the presence of a very small number of ionic groups might render inaccurate the extrapolation of S ( Q ) to Q = 0.Therefore it is necessary to estimate the osmotic compressibility by an independent method, if the authors are to claim that they are observing spinodal decomposition. (Note that the isopiestic method or solvent vapour pressure measurement can be used, in principle, whether the systems are ionic or non-ionic, while the electrochemical methods are of course not feasible in non-ionic cases.) It is not clear to me what the authors want to imply by ‘a relation between the height of the primary maximum of the structure factor of the metastable fluid and the height of the measured Bragg peak after completion of crystal growth’.All that I wished to emphasise was that the peak height is determined by the three factors according to the principles of scattering and cannot generally be a measure of the size of the crystallites only. Therefore the authors may be correct when the other two factors, the Debye-Waller effect and the paracrystalline distortion, are practically constant or do not become less enhanced during the crystallisation process. However can the authors be sure about this? The final point, on which the authors claimed to disagree totally with our view, was mentioned in my lecture. As far as I know, most of the computer simulation, for example, was carried out with a purely repulsive potential.The same calculation has not been carried out with a more realistic potential containing repulsive plus attractive com- ponents, except by Tata et uZ.* No one has yet demonstrated how satisfactory it is to take into consideration the attractive components. Certainly, the explanation advanced66 General Discussion by these authors and many others in terms of the repulsive potential and positive translational entropy increase is not the only possibility. 1 R. A. Register, A. Sen, R. A. Weiss and S. A. Cooper, Macromolecules, 1989, 22, 2224. 2 B. V. R. Tata, A. K. Sood and R. Kesavamoorthy, Prarnana-J. Phys., 1990, 34, 23; B. V. R. Tata, A. K. Arora and M. C.Valsakumar, submitted for publication; A. K. Sood, Solid State Physics, ed. H. Ehrenreich and D. Turnbull (Academic Press, New York, 1990), in press. Dr Dhont responded: It is clearly stated in our paper that the PICS measurements were performed on stearylsilica, which is non-ionic. The quoted lines, ‘the sample was slightly charged, giving rise to a screened Coulomb type of interaction potential. The Debye length is estimated to be ca. 100nm’, refers to TPM-silica, and this is certainly ionic. Certainly, besides the size of the crystals, the measured Bragg peak heights are functions of the mean-square displacements of particles in the crystal and possible defaults in the crystal structure. In a (semi-)quantitative interpretation of the data in terms of crystal growth rates one should try to take these things into account.Probably a cleaner quantity from which crystal growth rates can be determined is the integrated intensity of a Bragg peak. Prof. Ise’s point was that with only repulsive pair interactions no crystallisation can occur. The mentioned computer simulations do include only repulsive interactions and do show crystallisation. Prof. Ise (communicated): Your answer reminds me of an anecdote discussed by M. Eigen.’ The experiment (simulation) must be done without cutting the legs of cockroaches (in other words, without ignoring the attractive component), if the correct answer is to be sought. Thus I have to disagree with the authors. 1 M. Eigen and R. Winkler, Das Spiel, Naturgesetze sfeuern den Zufall (Piper, Munich, 1976), chap.17; R. Kimber and R. Kimber, Law of the Games (Penguin, New York, 1983). Dr Dhont replied: The anecdote of M. Eigen does not apply to this discussion: Prof. Ise’s view is that crystallisation cannot occur in systems with a purely repulsive pair- interaction potential, in contradiction with the experiment (simulation). Prof. W. B. Russel (Princeton University) addressed Dr Dhont: Your presentation introduced data on the number of crystallites, showing a maximum as a function of concentration. If your data on crystallisation rate are normalised, i.e. reduced to a rate per crystallite, does the maximum in fig. 6 of your paper persist? Dr Dhont replied: In order to obtain the growth rate of a single crystal, the crystallisation growth rate should be divided by the intensity of light scattered from a single crystallite of size ca.half that of the crystallites after completion of the crystallisa- tion process, and the number density of the crystals. The results shown in the presentation were obtained two months ago. These kinds of calculation are therefore yet to be performed. Prof. Russel continued: Fig. 2 shows the spinodal from Baxter’s model plus the limited calculations of Haymet for the fluid-solid transition.’ Continued to lower T this transition might transcend the fluid-fluid as the equilibrium process. The question is whether the decomposition observed leads to a fluid-like dense phase or a solid as established for other systems with short-range attractions. 1 N. A. Seaton and E. D. Glandt, J. Chem.Phys., 1987, 86, 468; Y. C. Chiew and E. D. Glandt, J. Phys. A, 1983, 16, 2599; S. A. Safran, 1. Webman and G. S. Crest, Phys. Reti. A, 1985, 32, 506; S. I. Smithline and A. T. S. Haymet, J. Chern. Phys., 1985,83, 4103.General Discussion 67 2.0 1.0 0.5 7 0.2 0.1 \ \ I / 1 I I I I I I 0.00 0.20 0.40 0.60 77 Fig. 2. Phase diagram for spheres with adhesion, T - ~ , and volume fraction, 7: spinodal and fluid-solid transition (-- - - ), percolation transition (-, * - ) . I Mr Penders replied: Our measurements have been performed in the metastable region of the phase diagram. It was ensured that all the samples rested in the metastable region only for a short time ( t = 2 0 s ) by moving the samples quickly from a ‘high’- temperature air bath to a ‘low’-temperature silicone oil bath in the neighbourhood of the spinodal curve (the duration of the ‘thermal pulse’ was ca.0.2 s). In this way it is possible to approach the spinodal curve very closely before phase separation takes place. During the measurements in the metastable region our samples stayed homogeneous and we observed no phase transitions like the percolation line described by Seaton and Glandt, Chiew and Glandt, Safran, Webman and Grest, and the fluid-solid transition described by Smithline and Haymet. The results shown in fig. 3 of our paper support support the fact that the experimentally determined spinodal is in reasonable agreement with the spinodal curve following from the adhesive hard-sphere model of Baxter. From these experiments performed in the metastable region there are no indications that the fluid-solid transition might transcend the transition determined by the spinodal line according to Baxter.In the unstable region the sample separates into two phases spontaneously. The question may then arise whether the dense phase is fluid-like or more glass-like. This may be temperature- and time-dependent. Further research is necessary to elucidate the behaviour of the separated phases in our silica dispersions. Mr Penders continued: In relation to the paper by Vrij and co-workers, in which the importance of solvation interactions is discussed, I wish to present some recent results of our work regarding solvation forces (M. H. G. M. Penders, A. Imhof and A. Vrij). In many theories of colloidal systems the solvent is often regarded as a continuous background.More recently, the finite size of the solvent molecules was taken into account by Hansen et al.’ and by our group.* By considering a binary mixture of large hard spheres (2) in a ‘solvent’ of small hard spheres ( 1 ) a system of colloidal particles in a solvent was simulated. Based on calculations of the osmotic compressibility of the large particles versus the volume fraction of the large particles it can be concluded that the effective repulsion between the large particles decreases owing to the presence of68 General Discussion -2.0 -‘*O- 0.0 0.2 0.4 0.6 4 Fig. 3. In 1/ S( Q = 0 ) versus c,b as a function of T~ for a binary mixture of large and small spheres (diameter ratio 160/1). The PY curve ( a ) represents the In l/S(Q = 0) versus 4 plot for monodis- perse hard spheres dispersed in a solvent which is regarded as a continuous background.T], = 00 ( b ) , 2.0 (c), 1.0 ( d ) , 0.6 ( e ) and 0.5 (f). small solvent particles. The use of such a hard-sphere solvent is not so realistic because attractive forces are totally neglected. The next step is to introduce one adhesive parameter to mimic, in an approximate way, the influence of attractive forces on the effective interactions between the colloidal particles. This will be accomplished by using the Baxter theory of adhesive hard spheres3 as shown by B a r b ~ y . ~ By introducing an attraction between two solvent particles, a ‘poor’ solvent can be modelled ( T , ~ < 00; T~~ = T~~ = 00; the reciprocal value of T , , ~ measures the adhesive strength between Q and p ) .The influence of T~~ on the compressibility [or S( Q = 0), the structure factor at zero wavevector, which is proportional to the compressibility] as a function of the volume fraction, 4, of the colloid particles (2) is presented in fig. 3 and 4. The volume fraction of the solvent particles ( 1 ) is 41 = @;( 1 - 4 ) with 4: = 0.4. From these figures it can be seen that the initial slope of the In [ 1/S( Q = O)] versus 4 plot decreases as TI1 decreases. This slope is proportional to the second virial coefficient, BZ, and is negative when T~~ < 0.5. At lower T~~ values ( T l l S 0.3) there exists a volume fraction, 4 (see fig. 4), at which the compressibility goes to infinity and phase separation may occur.A ‘good’ solvent can be modelled by increasing the attraction between small (solvent) and large (colloidal) particles ( T~~ < a; T~~ = T~~ = 00). The influence of T~~ on S ( = 0 ) as a function of a 4 is shown in fig. 5 . The initial slope of the In [ 1/S( Q = O)] versus 4 plot increases as T~~ decreases indicating that the effective repulsion between two large particles increases. At T~~ = 150 the plot matches the Percus-Yevick curve of hard spheres, where the solvent was treated as a continuous background. At lower T~~ values there is a larger effective repulsion compared to the PY curve. It can be concluded that the solvent quality can be altered by varying the interactions between small and large particles (simulating a good solvent) or varying the interaction between two solvent particles (simulating a poor solvent).General Discussion 69 0.0 n 0 -5.0 II 0 w v \ CI - c -10.0 -15.0 -20.0 ' I I 1 I I I 0.0 0.2 0.4 0.6 4 Fig.4. In l / S ( Q = 0) versus 4 at low T~~ values for a binary mixture of large and small spheres (diameter ratio 160/1). ( a ) PY, T~~ = 0.4 ( b ) , 0.3 ( c ) , 0.25 (d) and 0.20 (e). 4 Fig. 5. In 1/S( Q = 0) versus 4 as a function of T~~ for a binary mixture of large and small spheres (diameter ratio 160/1). (c) PY, 712 =20 ( a ) , 50 ( b ) , 200 (d), 500 (e) and co (f). Recently some structure-factor calculations have been performed. The results of these will be given in a following publication. 1 T. Biben and J. P. Hansen, Europhys. Lett., 1990, 12, 347. 2 A. Vrij, J. W. Jansen, J.K. G. Dhont, C. Pathmamanoharan, M. M. Kops-Werkhoven and H. M. Fijnautj, Faraday Discuss. Chem. Soc., 1983, 76, 19. 3 R. J. Baxter, J. Chem. Phys., 1968, 49, 2770. 4 B. Barboy, Chem. Phys., 1975, 11, 357.70 General Discussion Fig. 6. Model for two interacting core-5hell particles. A, and A, are the Hamaker constants for the core and the medium, respectively; A, is the mean Hamaker constant for the composite sheath. Dr Vincent said: The authors have chosen the Flory-Krigbaum segment-mixing model to account for the attractive interactions in poor solvency conditions between C1,-coated silica particles. I think this is unrealistic because the packing density of the C18 chains is too high to permit significant overlap. Moreover, I agree with Prof. Russel that an interaction requiring sheath overlap is unlikely to lead to vapour-liquid-type colloidal phase transitions, as reported.In an earlier paper’ we demonstrated that such phase transitions in these types of system could be interpreted in terms of longer-range van der Waals interactions between (non-contacting) particles. The segment-solvent interactions still play a significant role (in terms of the Flory x parameter) as shown in the following derivation* for the long-range van der Waals interaction ( VA) between two core-shell particles, as illustrated in fig. 6. As b --+ 0, the first term in the square brackets dominates. Also, designating A, and A, a? the Hamaker constants for the ‘polymer’ and solvent, respectively, A2 = A,,’ and A3 = [ + ( 1 - #,)A, 1: where 4, is the average segment volume fraction in the sheath.Hence, 1/2 2 Using a lattice model for the shell, and assuming only van der Waals forces between segments and solvent molecules (ie. no ‘associative’ bonds such as H bonds), then it may readily be shown2 that where z is the lattice coordination number ( z = 6 for a cubic lattice), and xu is the enthalpic part of x. Thus, For athermal mixtures (xu = 0) and/or &-+ 0, V, -+ 0, as expected. However, for x -xu = 0.5 and & --+ 1, as in the present case, VA( b) is significant. For example, for a = 3 0 n m a n d z = 6 , VA-13k,Tat b=0.3nmand 1.3kBTat b=3nm. 1 J. Edwards, D. H. Everett, T. O’Sullivan, I . Pangalon and B. Vincent, J. Chem. Soc., Faraday Trans. I , 2 B. Vincent, J. Colloid Interface Sci., 1973, 42, 262.1984, 80, 2599. Mr Penders replied: What we think in physical terms about the interactions between our silica particles in a solvent is carefully described in our paper.General Discussion 71 As a detailed understanding of the local liquid structure between particle surfaces is lacking, it is not possible at the moment to write down an expression for the effective pair potential between the particles from first principles. (Very recently, however, we made an attempt to take a first step in this direction: see the discussion contribution of Penders and Vrij. There it is shown, from first principles but with an approximate model, that colloidal particles that do not show direct interactions can, nevertheless, show indirect attractions, by the presence of a solvent consisting of attractive molecules.) As a compromise we have chosen a square well with a well depth that contains elements of an entropic and of an energetic nature, which balance each other at a temperature T = 8, i.e.V(r, T ) / k , T = - L ( r ) = 0; T > 8 We think that it will be possible to express the results of any detailed model in this form, when it is used near the temperature where the contribution of the surface layers to the second virial coefficient is zero, or 1; r2{ 1 - exp [ - V ( r, T ) / k , TI} dr = 0. For T = 8, we expect that the exponent in the integral will be proportional to [( 8/ T) - 13. This is corroborated by the experimental results shown in fig. 2 of our paper, which show an ‘exponential upswing’. The use of the expression in our paper for V ( r ) does not imply the use of the lattice model for polymer segments in solution.Also Flory and Krigbaum discuss that such a direct connection is not necessary, and they introduce instead of the lattice parameters jy and 1/2, the auxiliary parameters K and +, with 8 = TK/+. We prefer not to invoke macroscopic van der Waals-London forces at this level. It is not permissible, we believe, to apply these equations at microscopic distances of the order of 1 A. Further, contributions of solvation forces are neglected, as these are unrealistic for such small distances between the particle surfaces. Of course, for distances larger than the range of local solvation forces, one may use macroscopic van der Waals-London forces but the effects of these will be negligible for the small particles used here, which have only small differences in refractive index between the particle core and solvent.Dr Bartlett said: Could Dr Dhont comment on the structure of the crystalline phase formed by TPM-stabilised silica? His system might be expected to show a rather interesting crystal structure intermediate between f.c.c., the stable high-density solid phase for long-range (screened Coulombic) potentials,’ and the random stacked structure observed2 in sterically stabilised colloids where the potential is very short-ranged. 1 E. B. Sirota, H. D. Ou-Yang, S. K. Sinha, P. M. Chaikin, J. D. Axe and Y. Fujii, Phys. Rev. Left., 1989, 2 P. N. Pusey, W. van Megen, P. Bartlett, B. J. Ackerson, J. G. Rarity and S . M. Underwood, Phys.Rev. 62, 1524. Lett., 1989, 63, 2153. Dr Dhont replied: The scattering curves in the paper exhibit only the (111) Bragg peak of an f.c.c. and/or h.c.p. structure, owing to the fact that the form factor of the particles is extremely small beyond that peak. The particles are refractive index matched up to the third decimal point. Under slightly different conditions, two additional Bragg peaks are found, which are located at the (200) and (220) positions of an f.c.c. structure. The (200) peak is much less intense than the (220) peak, indicating that the structure72 General Discussion is some random mixture of f.c.c./h.c.p.-wise stacked hexagonally packed planes. The volume fraction at which the system crystallises (ca. 0.18) is still too large for a b.c.c.structure to be more stable than f.c.c./ h.c.p. structures. Prof. Cevc asked: In your text you mention that ultrasonication does not destroy the nuclei for your phase transitions whereas vortex mixing is devastating in this respect. Can you explain this, at first glance, counter-intuitive observation? Dr Dhont replied: An explanation would be that the disturbances created by the ultrasonic sound waves are of an amplitude and wavelength that matches with number- density fluctuations leading to the formation of crystallites. The sound waves could then be considered as an external force field that helps the system to find its way to the crystalline state. Dr Vincent turned to Dr Tadros: With regard to the depletion experiments, you indicate much better agreement between the values of Esep (calculated from T~ values) and values estimated from FSV theory, in the cases where HEC is the added polymer compared to when PEO is added.Since the polystyrene particles you used had grafted PEO chains on them, it would be more appropriate to use the soft-sphere model developed by ourselves,’ rather than the hard-sphere model implicit in the FSV theory. The soft-sphere model predicts weaker depletion interactions, especially in the case where the grafted and free polymers are of the same type, owing to some interpenetration of the two. However, when the two polymers are incompatible (e.g. PEO and HEC), the depletion interaction is stronger and closer to the hard-sphere case. This may account for the discrepancies you observe.1 A. Jones and B. Vincent, Colloids Surf.‘, 1989, 42, 113. Dr Th. F. Tadros ( I C I Jealotts Hill, Bracknell) replied: I agree with you that for our system with grafted PEO, we should take into consideration the interaction between free PEO and the grafted PEO chains. This, as I mentioned, explains the discrepancy between Esep calculated from T~ and Gdep calculated using FSV theory. As you men- tioned, HEC is probably incompatible with PEO and hence better agreement is obtained in this case using the hard-sphere theory. I am still not certain, however, that introducing the ‘softness’ of interaction suggested by Jones and Vincent will be sufficient to bring the experimental Esep value into line with theory. As you observe in table 2 of our paper, the discrepancy is very large and hence agreement with a modified theory is unlikely. Prof.Zukoski remarked: The G, data for charge-stabilised spheres are not in keeping with a lot of data published on charge-stabilised suspensions where the scaling theories of Goodwin et al. have been shown to be very accurate. I am concerned about the authors’ statement that there are no models accurate for G,. Dr Tadros responded: The agreement between experimental G, values and those calculated using the scaling theories of Goodwin et al., was obtained for relatively small latices. In this case, regular packing occurs as indicated by iridescence. In our system, the particles were larger (700 nm radius) and regular packing may not occur. In addition, with large particles contributions from van der Waals attractions at high 4 values may be considered.It is not, therefore, surprising that disagreement between GLxp and G&,eor is obtained. We are currently studying a latex system consisting of smaller particles to see whether the agreement can be restored.General Discussion 7 3 Prof. Zukoski then said: This is a comment referring to the use of strain sweeps to determine characteristic energies of strongly interacting systems. We have studied ordered charge-stabilised latex suspensions. Strain sweeps show plateau storage moduli up to strains of ca.0.04. For larger strains, the storage moduli decay. Defining a characteristic strain, yy, as the start of the decay we find that the product of Go and yy gives a measure of the static yield stress, as characterised by a plateau stress seen at low shear rates, and the stress at which the viscosity diverges, as determined from constant stress measurements. We find that these three measurements give the same yield stress over a wide volume-fraction range.Dr Tadros replied: Your analysis is quite interesting and would suggest a possibility of calculating the yield stress from dynamic measurements. We have many systems in our laboratories on which both dynamic and constant stress measurements can be made. We will certainly check whether such agreement is observed. Prof. Russel addressed Dr Tadros: Goodwin and Russel et al. have successfully correlated a rather substantial set a data for G’ with charged latices over a broad range of ionic strengths and volume fractions using the model represented by eqn (12) of your paper.Your data fall in a similar range of dimensionless separation (see fig. 7) but do not compare well with the rest. The reason might lie as easily with the strain amplitude and frequency dependence of your data as with the correlation. The data in fig. 2 of your paper are limited and perplexingly scattered. The correlation based on eqn (12) is demonstrated to be valid, with the maximum packing fraction as an adjustable parameter, by data from Mewis and co-workers yet to be published, but not by yours. With the weakly flocculated dispersions you do not demonstrate that: ( a ) the data is reproducible and free of history dependence; ( b ) the extrapolation to T~ is accurate; (c) T~ is linear in 4 as required by eqn (16) of your paper; or ( d ) the flocs break down to individual particles.Hence the calculations of Esep/ kB T have little credibility. For strongly flocculated dispersions the power-law dependence of G’ on 4 is commonly observed and the exponent is recognised to depend on the structure. However, I know of no evidence that n increases with the strength of attraction as asserted. So with the current state of theory, I suggest that rheological measurements can be reliably inverted to yield the interparticle potential only for closely packed particles. Dr Tadros replied: As I mentioned in the paper, we obtained only a small linear region at very low strains. This is due to the large particle size of our latex system. This, as you suggested, may explain the discrepancy between Gd,, and G:heor.We are currently carrying out experiments using latices of various sizes in order to establish the reason behind this discrepancy. I agree with you that the data shown in fig. 2 of our paper are limited. However, we have recently obtained more results using latices of various sizes. The general trend is still the same. Clearly if one uses the maximum packing fraction as an adjustable parameter, better agreement may be obtained. In response to your third point, ( a ) the data are certainly reproducible and free of history dependence; ( b ) the extrapolation to T~ is accurate since we obtained straight lines above jcrit; ( c ) and ( d ) the weakly flocculated structure is broken to individual particles as evidenced by the low qpl obtained which is very close to that in the absence of added free polymer.I agree with you that the power dependence of G’ on 4’ is related to the structure of the flocculated system. I have perhaps overstated in our paper that n depends on floc strength. This may not be the case and we are currently checking more systems.74 General Discussion 10' h E -8 lo-' * o o o o 0 * * * * * A I A A I 0 I 8' +. - lo-* * * ! lo-' loo 10' l o 2 4red Fig. 7. Correlation of static shear modulus for polystyrene latices in electrolyte solutions (Russel, Saville and Schowalter, Colloidal Dispersions (Cambridge University Press, Cambridge, 1989), p. 475. Finally, I agree with you that concordance between theory and experiment is obtained only for closely packed particles, which unfortunately is not the case with many practical systems.Mr A. E. Duisterwinkel (Derft University) commented: When you equate the Bingham yield stress, T ~ , to the interparticle interaction, you assume that all bonds are separated, i.e. T~ = NEsep. However, as shown by Prof. Ottewill for non-interacting particles and as assumed by Stein et al.' (for interacting particles), shear planes are formed at high shear. Thus, only 1/3 of the bonds must be separated, i.e. T~ = fNEsep. With this equation, a much better agreement between experiment and theory is found. 1 F. W. A. M. Schreuder and H. N. Stein, Rheol. Ada, 1987, 26, 45. Dr Tadros replied: I agree that shear planes can be produced at high shear rates, however, whether this occurs with weakly flocculated systems is difficult to ascertain. If one agrees with your picture then a factor of 1/3 for N would produce better agreementGeneral Discussion 75 with FSV theory for the PEO system. I will certainly consider your suggestion in future calculations. In addition, we will carry out experiments to ensure that layering occurs at high shear rates. Dr H. J. Ploehn (Texas A&M University) said: My question is primarily concerned with your interpretation of the linear viscoelastic behaviour of sterically stabilised suspensions. The dominance of the elastic modulus over the loss modulus for volume fractions greater than 0.50 is attributed to compression of the stabilising PEO chains. Is this an oversimplification of the physical situation? There is also an indirect effect: interactions of the stabilising layers hinder particle self-diffusion and increase the characteristic time for microstructural rearrangement in the suspension. When this time is shorter than the experimental time (the reciprocal of the frequency of the applied deformation), then the elastic modulus varies with frequency; when the suspension’s characteristic time is very long, then the elastic modulus becomes independent of frequency, at least for the ‘frequency window’ provided by the Bohlin VOR. Linear viscoelastic measurements of the modulus of hard-sphere silica suspensions’ and effective hard-sphere PMMA suspensions* show that the elastic modulus becomes greater than the loss modulus at particle volume fractions of ca. 0.50. This is similar to what you observe. Do your measurements indicate ‘softness’ in the interactions of the PEO layers, or can the increase in elastic modulus be ascribed simply to the increase in volume fraction beyond 0.50, with the PEO layer providing an effective hard-sphere interaction ? 1 J. C. van der Werff, C. G. de Kruif, C. Blom and J. Mellema, Phys. Rev. A, 1989, 39, 795. 2 W. J. Frith, Ph.D. Thesis (Catholic University, Leuven, 1986). Dr Tadros responded: I agree with you that the interpretation based solely on interpenetration and/ or compression of the chains is an oversimplification. As you stated, interactions of the stabilising layers hinder self-diffusion and that probably explains the lack of dependence of G‘ on frequency. Our results indicate ‘softness’ in the interactions of PEO layers, since they give large thicknesses (hydrodynamic) of the order of 10-20 nm. I would not, therefore, treat our particles as hard spheres.