Langmuir 1992,8, 2907-2912
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Spinodal Decomposition in Colloidal Systems, with and without Shear Flow, Described on the Basis of the Smoluchowski Equation Jan K. G. Dhont,**+ Alexander F. H. Duyndam,t and Bruce J. Ackersod Van't Hoff Laboratory for Physical and Colloid Chemistry, University of Utrecht, Padualaan 8, 3584 CH Utrecht, The Netherlands, and Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078 Received April 13,1992. In Final Form: July 13, 1992 The spinodal instability and the initial decomposition kinetics of colloidal systems, with and without a simple shear flow, are described on the basis of the Smoluchowski equation, 'the Liouville equation for Brownian systems". Without shear flow,upon formallyexpandingcertainwave vector dependentfunctions, the Cahn-Hilliard form for the effectivediffusion coefficient is recovered. Also the usual thermodynamic criterion for spinodal instability follows from this Smoluchowskiequation approach. It is argued that, on application of a shear flow, the location of the spinodal curve no longer coincides with the cloud-point curve. Furthermore, the decompositiondynamics is highly anisotropic (alreadyfor such small shear rates for which the deformation of the pair-correlation function can be neglected),in a way that is in agreement with experiments on a binary fluid mixture. 1. Introduction
Colloidal systems may be regarded as large time- and length-scaleanaloguesof atomic/molecularsystems,except that their transport properties are modified by hydrodynamic interactions, and their potential interactions are medium averaged. Therefore, much can be learned about many-particle atomic/molecular systems via the study of colloidal systems, which are experimentally much more easily accessible since time and length scales are large. Perrin was amongthe first to realize that colloidalparticles are nothing but heavy "supermolecules" which obey the same statistical mechanics as any other type of atom/ molecule. For example, he emphasized the applicability of the equipartition principle and Boltzmann statistics to colloidal systems. Observational consequences of this viewpoint, such as Brownian motion and the barometric height distribution, were used by Perrin to determine Avogadro's number in various independent ways.l The consistencyof the thus obtained estimates for Avogadro's number was then taken as evidencefor the molecularorigin of Brownian motion and a strong support for the statistical theory of matter. Although the majority of the scientific communityin 1900already recognizedthe reality of atoms/ molecules,this work settled the issue of atomic/molecular reality definitely. Perrin mainly studied noninteracting systems,although a few years after the above-mentionedwork was published he reports experimentson charged and interactingsystems. It took a long time before the extension of Einstein's diffusion equation was established to include such potential interactions, as well as hydrodynamic interactions. Actually,the derivationof this extended diffusionequation (which is usually named after Smoluchowski) is based on the Liouville equation for the composite system of fluid molecules and heavy particles, thus treating the heavy particles simply as big (but still thermal) molecules, on the same footing as the fluid molecule^.^*^ This Smoluof Utrecht. S t a b University. (1) Perrin, J. Die Brownse Bewegung und die wahre Ezistenr der Molekiile, Sonderausgabe aIuI Kolloidchemische Beihefte; Verlag von Theodor Steinkopff: Dresden, 1910; pp 1-83. t University t Oklahoma
chowskiequation may indeed be regarded as 'the Liouville equation for colloidal systems". On the basis of the Smoluchowskiequation, Batchelor was the first to extend Einstein's predictions for noninteracting systems, which were used by Perrin, to include interactions in leading order. From then on, the Smoluchowskiequation has been widely used in the now rapidly developing statistical mechanics of colloidal systems. Specifically, transport properties have been calculated for systems, at densities and temperatures in the stable region of the phase diagram (for references, see for example the books of Russel4and Van de Ven5). Like the Liouville equation, however, the Smoluchowski equation is equally valid for metastable and unstable systems. In this paper we describe the spinodal instability and the initial decompositionkinetics for systems in the unstable state, with and without shear flow, on the basis of the Smoluchowskiequation. Without shear flow, the results of thermodynamic approaches are recovered (except for certain algebraically decaying pair potentials). The Smoluchowski equation approach leads quite straightforwardly to an accurate expression for the wave vector dependent effective diffusion coefficient in t e r m of the pair-correlation function and pair-interaction potential. Moreover,the effect of shear flow can be included in a natural way in the Smoluchowskiapproach. This is difficult in the context of a thermodynamic type of approach, since it is not clear how to include a (simple) shear flow in a thermodynamic description of the system. Several attemps to construct thermodynamic theories for sheared systems have been made, however.8-10 These issues are not yet settled. Since the Smoluchowski equation approach does not rely on any thermodynamic (2) Murphy, T. J.; Aguirre, J. L. J. Chem. Phys. 1972,57, 2098. (3) Deutch, J. M.; Oppenheim, I. J. Chem. Phys. 1971,64,3547. (4) Russel, W. B.; Saville,D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: New York, 1989. (5) van de Ven, Th. G. M. Colloidal Hydrodynumics;Academic Press: London, 1989. (6) Evans, D. J.; Hanley, H. J. M. Phys. Lett. A 1980, 80, 175. (7) Hanley, H. J. M.; Evans, D. J. J. Chem. Phys. 1982, 76, 3225. (8) Keizer, J. J. Chem. Phys. 1985,82, 2751. (9) Rovug, K. D.; Hanley, H. J. M. Znt. J. Thermophys. 1986,7,877. (10) Nettleton, R. E. Nuouo Cimento 1988, IOIB, 52.
Q743-7463/92/24Q8-2907$03.00/~ Q 1992 American Chemical Society
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2908 Langmuir, Vol. 8, No. 12, 1992 considerations, there is actually no need here to go into this matter. For colloidal systems, a further limitation of a thermodynamic approach is that it is unclear to what extent hydrodynamic interactions modify the demixing kinetics. The Smoluchowski approach does allow for a calculation of these modifications, although in the present paper we will not discuss these. Throughout this paper we neglect hydrodynamicinteractions between Brownianparticles. Also,the effect of gravity can be considered; again we will not discuss this here. This paper is organized as follows. In the next section we describe the Smoluchowski equation. Section 3A containsan analysisof the Smoluchowskiequation without shear flow, some numericalresults, and a comparison with the Cahn-Hilliard theory and the thermodynamic criterion for spinodal instability. Section 3B contains an analysis of the influence of a shear flow on the initial decomposition kinetics, some numerical results, and a comparison with experimental results on a binary fluid mixture. Section 4 contains some concluding remarks. More details on all this can be found in ref 11. 2. Smoluchowski Equation
Due to a separation of the time scales on which, on the one hand, the phase space coordinates of the fluid particles and the momentum coordinates of the Brownian particles fluctuate and, on the other hand, the position coordinates of the Brownian particles fluctuate,it is possible to derive a Markovian equation of motion for the probability density function of the position coordinates of the Brownian particle^.^^^ Neglecting hydrodynamic interactions between the Brownian particles and adding a term to include a shear flow yields (a/at)PN(rl,...,rN;t) =
N
where PNis the probability density function of the position coordinates ri of all the NBrownian particles in the system, DOis the Stokes-Einstein diffusion coefficient, /3 = l/k~jT, with k g being Bolzmann's constant and T the temperature, 4 is the total potential energy of the assembly of the Brownian particles, and I' is the velocity-gradient tensor, for which we will take
r=+(i
i)
(2.2)
+
+
D&V*p(r;t)Sdr' [VV(r,r')lp(r',t) g(r,r';t) (2.3) where Vis the pair-interaction potential (4 is assumed to be pairwise additive). In (2.3) we used the definitions (11) Physica A, in press.
(l/M)p(r;t)p(r';t) g(r,r';t) = S d r 3... S d r NPN(r,r',r3 ,...,rN;t) (2.4) where g is the pair-correlation function. The last term in (2.3) is the contribution to the (divergenceof) the particle flux due to the totalaverage force on a particle at r exerted by all the other particles. As the spinodal curve is approached from the stable side in the phase diagram, the interaction term (lastterm in (2.3)) increases in amplitude and cancels partly against the free diffusion terms in (2.3) (the first two terms in (2.3)). The dynamics of theoneparticledensity,for certain smooth spatially varying components of the one-particle density, will "critically slow down". As w i l l be seen, an effectivewave vector dependent diffusion coefficient,Dtt, that can be associated with (2.3) becomes small for smaller wave vectors in comparison to DO. As the spinodal is crossed, P f f ( k ) becomes negative, rendering the system unstable. Now the interaction term is larger and opposite in sign to the free diffusion terms in (2.3), which pulls particles into regions with a higher density; Cahn and Hilliard called this phenomenon "uphill diffusion". This is the mechanism that initiates the system decomposition. To obtain a closed equation of motion for the one-particle density, the pair-correlation function in (2.3) must be expressed in terms of the one-particle density. This will be discussed in the next section.
3, Time Evolution of the One-Particle Density A. Without Shear Flow. To describe the influence of inhomogeneity in the one-particle density on pair (P = (r + r'V2, R = correlations, we transform (r,r') r - r') or (r,R). For a homogeneous system, g(r,rW is a function of R only. For inhomogeneous systems,g(r,r';t) =g(t,R,t) will also depend on the average location t of the points r and r'. A smooth variation of the one-particle density resulta in a smooth variation of g with P for any fixed R. Next we anticipate, consistent with final results, that near the spinodal curve only long wavelength variations of the one-particle density are very slow whereas small wavelength variations remain as fast as in a perfectly stable system. The fast small wavelength dynamics of the oneparticle density is the mechanism through which the paircorrelation function attains its equilibrium form, throughout the initial stages of the decomposition, over small distances R = Ir - r'l. In particular
-
g(r,r';t) = go(R,[pl)
where is the shear rate. This is a shear flow in the r direction, increasing in the y direction. Integration of (2.1) over r 2 , r3, ...,r N yields an equation of motion for the one-particle density (r = r d (a/at)p(r;t) = DoV2p(r;t)- V.rrp(r;t)
(l/N)p(r;t) = S d r 2... Sdr, PN(r,r2, ...,rN;t)
R IR,
(3.1)
where R, is the range of the pair-interaction potential, and go is the equilibrium pair-correlation function; the square brackets [pl are used to denote functional dependence of g on p. (3.1) is the statistical equivalent of the classic thermodynamiclocal equilibrium assumption. Of course, the long-ranged part of g, R >> R,, may be very different from the equilibrium form go. Right after the quench into the unstable part of the phase diagram, longranged correlationsdevelopup to a point, time t = 0, where a cancellation of terms in (2.3) occurs, resulting in the initial stage of the (long wavelength) decomposition of the system. At each instant during (initial) decomposition one can use (3.1) in (2.31, since hereg is multiplied by VV, so thatg needs only be known for R IR,. A cloeed equation of motion for p to describe the initial decomposition kinetics is now obtained from knowledge of the variation
Langmuir, Vol. 8, No. 12, 1992 2909
Spinodal Decomposition in Colloidal Systems
I
.05
0
I
I
.10
.15
-(W2
I
Figure 1. Effective diffusion coefficient as a function of (kAI2 for A/d = 1/2and 4 = 0.148. A is the width of the square well, d is the core diameter, and 4 is the volume fraction of Brownian particles. The temperature where W ( k10) is zero is the spinodal temperature. Values for Be are (e is the depth of the square well)
(A) 1.3393,(B)1.3274, (C)1.3158,(D)1.2931,(E)1.2821,(F) 1.2605,and (G)1.2397.
of go(R,[pl),for R IR,, with small amplitude long wavelength variations of the one-particle density, 6p. A good approximation, to within about l o % , is”
RIR, where
6p(r;t)= p(r;t)- p (3.3) with p = N / V the mean density. Notice that 6 p in (3.2) is taken in the point f = (r rY2. In (3.2) initial inhomogeneities are neglected, so that (3.2) is only meaningfulfor describingvariations Gp(r;t>O)>> bp(r;t=O). The further analysis of the Smoluchowski equation as presented in this paper is therefore only valid in the unstable region of the phase diagram. Transforming(2.3) according to (r,r‘) (r,R),linearization with respect to Gp(r;t)/p,which is allowed for the initial stages of the decomposition process, and Fourier transformation with respect to r yields
+
-
(a/at)bp(k;t)= a f f ( k ) k2sp(k;t) where Dffis the effective diffusion coefficient
(3.4)
0
Figure 2.
.05
.10
-(W2 Effective diffusion coefficient mdtidied with the
square of the wave vector as a function of and the te-mperature 1/@c for A/d = 1/2 and 4 = 0.148. This is the k-dependent factor that appears in the exponent that solves the equation of motion for Gp(k;t). The dots indicate the minimum value for W ( k ) k *for which Sp(k;t)increases most rapidly. The dashed lines are projections of this minimum. The dotted lines indicate p i t i v e values of the effective diffusioncoefficient,which are of no physical relevance, since the analysis of the equation of motion is only valid up to O(bp(k;t=O)/p). (A) to (G)refer to temperatures given in the legend to Figure 1.
attractive potential (depth -e and width A) superimposed on a hard-core potential (range d ) . As the temperature is lowered, at some given density,Deff(k=O)is seen to become zero; this is a point on the spinodal curve. The spinodal curve in the ( p ,)‘2 plane marks the transition from (meta)stable to unstable against infinitesimally small and infinite wavelength density variations. As the temperature is decreased more, Deff(k) becomes negative,up to some small k value. For larger k values, Deff(k)=DO; this is consistent with our anticipation used to derive the “localequilibrium” closure (3.1). The demixing rate is, according to (3.41, -Dff(k)k2. For the same temperature as in Figure 1,the demixing rates are plotted in Figure 2. The minima in these curves correspondto the most rapidly growing oneparticle density wave vector component. The Cahn-Hilliard result for initial spinodal decomposition kinetics is recovered by expansion of the j functions in (3.5) in a power series of (kRl2,and interchanging the integration and summation
where
with j(x) =
x cos x
- sin x
x3
(3.8)
(3.6)
(3.5) is derived here, assuming that V is a differentiable function of R. Inclusion of the (nondifferentiable) hardcore contribution to V gives rise to an additional term in (3.5) (see ref 11 for details). Figure 1 gives numericalresults for ZPff(k),using a third virial coefficient approximation for go with a square well
and
These equations may be used to identify the phenomenological coefficienta appearing in the Cahn-Hilliard
2910 Langmuir, Vol. 8, No. 12,1992
Dhont et al.
theory." It is clear from (3.7) that the system is unstable only if du(p)ldp < 0. This is the classic thermodynamic criterion for spinodal instability, since r ( p ) in (3.8) is precisely the osmotic pressure. Notice that Deff(k=O)= Do/3du(p)ldp. This is precisely the k = 0 component of the well-known expression Dol S(&)for the initial slope of the density autocorrelation function on the Brownian time scale.12 Here, S ( k ) is the static structure factor. Our analysis, however, is valid on a much larger time scale, on which the stable larger wave vector density variations are relaxed. Only the time evolution of the slowamplitude increase of unstable modes is described here. The two expressions (3.5) and Do/S(k) for the effective diffusion coefficient can therefore only agree at k = 0 and on the spinodal curve. The interchange of integration and summation used to derive (3.7) is not allowed for algebraically decaying pair potentials V(R) R-"; n = 1, 2, ... The series obtained for these potentials contains divergent coefficients. Some leading order in kd 0 coefficientsin a correct expansion are De" --(kd)-2 for n = 1, --(kd)-l for n = 2, and In ( k d ) for n = 3. For these potentials the square gradient approximation fails. Notice that for n I 3 Deff(k-.O) diverges, but that the physically significant parameter Deffk2is well-defined, also as k 0. For more details, see ref 11. B. With Shear Flow. The pair-correlation function is distorted by a shear flow. Since, however, g need only be known for R IR,, this distortion may be neglected for short-ranged deeply attractive pair potentials. The interparticle forces will then dominate over the shear forces (for details, see ref 11). For the colloidal systems studied by Rouw et al.,13 for example, the attractive part of the pair potential extends only over 1 nm (the core diameter of these particles is -50 nm). For such systems interparticle forces dominate over the shear influences, for distances R I R, over which the pair potential acts. We can therefore use the identical closure relation (3.2) here, with go being the zero-shear pair-correlation function, provided one is not too close to the spinodal curve. The same procedure as described in the previous section, with the inclusion of the shear term in (2.1) leads to
-
-
-
0.380 1
I
I
0.350
~1 0
0.1
0.3
0.2
0.4
@ Figure3. Spinodal curve for the squarewell interaction potential with 4 d = 0.05. 4 ie the volume fraction of colloidal particles. The dot indicates the temperature and density for which the influence of a shear flow on the decomposition kinetics ie calculated.
kI2+ x 2 + k t (3.12) k2 For = 0, Deff(k;t(+)reduces to the effective diffusion coefficient Deff(k)without shear, as it should. Note that there is an explicit + and t dependence in the initial time prefactor in (3.11). It thus appears that it is not just the effective diffusion constant (3.12) that describes the decomposition dynamics, but also the kz dependence of initial structural properties of the suspension (in a light
scattering experiment, this would be the static structure factor right after the quench). Neglect of the shear-rate dependence of the paircorrelationfunction neglects the shift of the spinodalcurve with the shear rate. The spinodal curve /3 B(p) at which the system changes from (metalstable to unstable is = 0. From (3.12) it is clear implicitygiven by Deff(k=O;tl+) that Deff(k=Qtl+)= Deff(k=O),where the dependence of go is neglected. Hence the y dependence of go entirely determines the location of the spinodal curve. Moreover, according to (2.3), the shift of the location of the spinodal curve is related to the distortion of go(R;+) for distances R IR,, since go is always multiplied by dVldR. For these distances the distortion is regular, and one expects that the shift of the spinodal is regular in This is different for the cloud-point curve, where the turbidity diverges due to the long-ranged character of the pair-distribution function. For these large distanceagois distorted singularly by a small shear flow. Hence, on applying a shear flow, the spinodal curve and the cloud-point curve no longer coincide,as in the case without a shear flow. Experimental evidence for this has been obtained for a binary polymer fluid mixture.14 This is discussed in ref 11. Numerical results for DBff(k;tl*/)are obtained, as in the case where no shear flow is present, for a third virial coefficient approximation for go with an attractive square well pair potential, but now with a smallwidth A compared to d , so that the distortion of go by the shear flow can be neglected against interparticle forces for R IR,. For Ald = 0.05, the spinodal is given in Figure 3. The dot marks the point in the unstable region for which numerical values of the effectivediffusion coefficientare given here. Figure 4 gives effective diffusion demixing rates -Deffk2as a function of ( k l ,kz),left column, and (&I,&3), right column, for several times +t. Clearly, the demixing process is markedly anisotropic,ina waythat is significantlydifferent in the ( k l , k2) and ( k l , k3) planes. For &I = 0, Deff(k;tl+) = DBff(k;tl+=O)= Deff(k);that is, all cuts of the figures collected in Figure 4 through the plane kl = 0 are the same. Without a shear flow, according to (3.4) and (3.5), the most rapidly growing wave vector is independent of
(12) Ackerson, B. J. J. Chem. Phys. 1976,64, 242; 1978,69, 684. (13) Rouw, P. W.; Woutersen, A. T. J. M.; Ackerson, B. J.; de Kruif, C. G. Physica A 1989,156,876.
(14) Silbrbrg, A. Interfacial Tension and Phase Separation in two polymer-solvent systems. Thesis, University of Bade, Johannesburg, South Africa, 1952.
-
(3.10)
where Deffis given in (3.5). The solution of this equation of motion is exp(-Deff(k;tl+)k2t) Gp(k;tl+) = ~p(k,,k2++klt,k3;t=0) (3.11) with
+
+
+.
Langmuir, Vot. 8, No. 12,1992 2911
Spinodal Decomposition in Colloidal Systems 1.6
Deff
-lo5-
(klA)2t
DO
I
1.2
0.8
0.4
0
i
2
3
102klA
Figure 5. -Dffk2tas a function of kl, with 122 = 0 = k3,for several times. The left set of curves is for a sheared system with shear rate 0.05 s-l. The dotted lines, connecting the dots of both sets of curves, mark the location of the most rapidly growing wave vector. The right set of curves is for an unsheared system. Here the most rapidly growing wave vector is constant in time. k
k3=0
k2=0
Figure 4. - D f f k 2as a function of (kl, k2), left column, and of (k1, k3), right column. Only negative values of De” are shown. Beneath the rectangular grids, Deffis positive. The maximum value for - D f f ( k A ) 2 / Dis~29 X lo*. All graphs are on the same scale. All cuts along k l A = 0 are identical.
time (as long, of course, as the nonlinear terms in the equation of motion for p are small in comparison to the linear terms). This is different if the shear flow is applied, as can be seen in Figure 5. The most rapidly growing wave vector in the (kl, 0, 0) projection is changing its position toward smaller values right from the start. We are not aware of any experimental results on decomposition kinetics under shear in colloidal systems. There are however experiments on a binary fluid mixture,15-18isobutyric acid and water. Although these experiments extend to times somewhat beyond the linear regime (as evidenced by the time dependence of the position of the most rapidly growing wave vector without shear), it is possible to compare characteristic trends. The observed phenomena in these binary fluid experiments are in completeagreementwith our predictionsfor colloidal systems. In making a comparison, care must be taken to translate the geometry (kz,ky, k,) in refs 15-18 to our geometry (kl, k2,k3). First, the present theory predicts no influence of a shear flow during the initial stage in those directions where kl = 0, in agreement with the mentioned experiments. For example, in ref 18 it is found that the (15) Chan,C. K.; Perrot, F.; Beysens, D. Phys. Rev. Lett. 1988,61 (4), 412. (16) Perrot, F.; Cahn,C. K.; Beysens, D. Europhys. Lett. 1989,9,65. (17) Cahn,C. K.; Perrot, F.; Beysens, D. Phys. Rev. A 1991,43 (4), 1826. (18) Baumberger, T.; Perrot, F.; Beysens, D. Physica A 1991,174,31.
location of the most rapidly growingwave vector (0, 0,ks) is shear independent. From ref 15, “In the direction perpendicular to the flow and the shear, shear seems to have little effect on the growth.” This is in agreement with (3.11) and (3.12), since taking kl = 0 in these expressionsyields the expressionsfor the phase separation kinetics without shear. Another striking result is the difference in the developing anisotropy in the (121, ka, 0) and (kl, 0, 123) projections, as shown in Figure 4. All characteristic features of Figure 4 are also seen in the experiments described in refs 15-18. Furthermore, (3.11) and (3.12) predict a .;/tscaling; such a scaling is explicitly mentioned in ref 18. This scaling is only true when the .;/ dependence of the pair-correlation function is unimportant. OtherwiseDeffwould also depend on via a Peclet number. A more detailed discussion of the .;/ dependence of the pair-correlation function and comparison with the experiments is given in ref 11. In conclusionwe may saythat all essentialfeatures found experimentallyfor the binary fluid mixture, related to the anisotropy of demixing, are correctly described by the Smoluchowski equation results. 4. Concluding Remarks
The Smoluchowski equation provides a relatively straightforward basis for the description of spinodal instabilities and decomposition dynamics in the early stages,both without and with shear flow. One might think of extending this approach in two directions: late stage spinodal decomposition and crystallization kinetics. In both cases, the nonlinear contributions to the equation of motion (2.3) are essential. Furthermore, it is probably much more difficult to obtain a proper closure relation for the pair-correlation function. Still,for crystallization, (3.1) probably remains usable, as the metastability is a long wavelength phenomenon, since it involves many particles and the time scales of density fluctuations vary proportionally to k2,so that small wavelength dynamics is fast in comparisonto long wavelength dynamics,also in (meta)stable systems. For the problem considered in the present paper, the separation in the time scales on which long and short density wavelengths develop is more pronounced, since, in addition, the proportionality factor to k2 (=the
2912 Langmuir, Vol. 8,No.12,1992
effective diffusion coefficient) is small for small k. For late stage spinodal decomposition, a closure for g in (2.3) is needed in a situation of steep gradients in the density. One could now perhaps coarse grain (2.3)over distances comparable to the late stage interfacial thickness, to obtain an "effective" equation of motion in terms of "macroscopic" quantities which describe the interfacial properties pertaining to the late stage time evolution. These are much more difficult problems than initial spinodal decomposition, but may nonetheless be worthwhile to pursue on the basis of the Smoluchowski equation. There is a need for predictions on the distortion of the pair-correlation function (or, equivalently, the structure factor) due to shearing motion, which include higher order correlations. Theories on the distortion of the structure factor for low concentrations predict a zero distortion perpendicular to the flow direction. Experiments on sheared suspensions close to the spinodal curve, however, show a large distortion in just that region in k space,19in contrast to suspensions at a temperature far into the stable region of the phase diagram, which are distorted in a way that is in qualitative agreement with low density predictions. Lacking such theoretical predictions, nothing quantitative can be said about the shift of the cloud-point curve and the spinodal on applying a shear flow, except (19)Woutersen, A. T.J. M. To be published.
Dhont et al.
that the former is singularly shifted whereas the latter is regularly shifted. Here we considered only spherically symmetrical particles. For more complex geometries, like rigid rodlike particles, the effect of a shear flow on the location of the spinodal curve related to the isotropie-ieotropic phase transition is complicated by the particle alignment effect of the flow. On the one hand, the a l i m e n t destabilizes the system, as pair interactions between the rods vary roughly like l/lsin 71 (where 7 is the angle between the directors of the two rods), but on the other hand the alignment stabilizes the system, as the average segment distance increases. Still, a Smoluchowski equation approach is likelytobe feasible for, for example, rigid rodlike systems. Acknowledgment. This work is part of the research program of the Foundation for Fundamental Research of Matter (FOM) with financial support from the Netherlands Organization for Pure Research (ZWO). B.J.A. acknowledgeasupport for thiswork by the National Science Foundation, Division of Materials Science-Low Temperature Physics-through a grant numbered DMR8802801. Some of this work has benefited from several discussions with Professor H. N. W.Lekkerkerker.
