VOLUME 75, NUMBER 11 PHYS ICAL REVIEW LETTERS 1 1 SEPTEMBER 1995

Weitz and Ladd Reply: The authors of the previousComment [1] make the interesting observation that thehard-sphere pair distribution evaluated at contact, g(@),adequately represents the volume fraction (@)dependenceof both the inverse self-diffusion coefficient, Ds(@)/Do =

'(p), and the suspension viscosity, rI(@)/r)o = y(p);here Do = kpT /6rr rIoa is the self-diffusion coefficient ofan isolated particle of radius a, and go is the viscosity ofthe suspending quid. Physically, this observation suggeststhat long-range hydrodynamic interactions are not impor-tant; instead direct interactions between neighboring parti-cles dominate. However, there is abundant evidence thathydrodynamic interactions do determine the transport co-efficients in colloidal suspensions; experimental measure-ments of both the diffusion coefficient and the viscosityare quantitatively reproduced by theoretical and numeri-cal calculations based solely on hydrodynamic interactions[2] with no direct interparticle forces. Thus, while the ap-parent agreement of ~(P) with the data is appealing, theimplications of this observation for the underlying physicsmust be examined carefully.

The close correspondence between Ds(P)/Do and

rto/7J(@), noted in Ref. [1],can be understood in terms ofa mean-field theory. Because of the long-range hydrody-namic interactions, a diffusing sphere interacts with otherspheres as if they were a continuum Quid, stiffened by thesuspension viscosity, rJ(p), rather than the fiuid viscosity,

However, this mean-field approximation is not exact.Accurate numerical calculations of Dq(P) and rj(@) indi-cate that there is a distinct difference between them; theproduct Ds(P) rjo/Dorj(@) varies monotonically between1 at low volume fractions and 1.36 at @ = 0.45 [2]. Fur-thermore, a careful examination of the experimental dataindicates that there is a difference between Ds(@)/Do and

rJo/rJ(@) that is greater than the considerable experimen-tal scatter; this difference is better described by the nu-merical calculations then by a single function, ~ (P).

An important feature of the argument in the previousComment [1] is the presence of two distinct time scalesin a colloidal suspension. They are the viscous time,r, = pa / rtp (p is the fiuid density), which sets the timescale for the evolution of hydrodynamic interactions, andthe diffusion time, rD = a /Do, which sets the time scalefor changes in particle configuration. The key experi-mental result is that, for t, ~ t ~ 100t (( 7.~, the time-dependent self-diffusion coefficient at all volume frac-tions, H(t, @) = (AR (t))/6Dot, can be scaled to a sin-

gle master curve C(@)Ho(t/r(@)) [3]. The function Hodescribes the time-dependent self-diffusion coefficient ofan isolated Brownian sphere. The experimental data [3]scale most accurately when the amplitude is scaled to theself-diffusion coefficient [C(P) = Dq(@)/Do], and thetime scale is scaled to the inverse viscosity [r(@)/r, =rto/rt(p)]. The scaling obtained with a single function,either Dq(p)/Do, rjp/iJ (p), or some functions in between

[such as g '(P)] is clearly less good. Other experimentaldata, obtained for much shorter items (up to 4r, ) [4], wereinsufficient to distinguish this scaling from other possibili-ties. Self-diffusion data from computer simulations of hy-drodynamically interacting spheres [5] can also be scaledin the same way as the experiments. Unlike the exper-iments, the scaling of the computer data is improved ifboth the time and amplitude are scaled by a single coef-ficient [6]; however, this coefficient is Ds(@)/Do, ratherthan rto/rJ(p). These small discrepancies between sim-ulation and experiment are more pronounced at longerwavelengths, where the hydrodynamic interactions be-tween neighboring particles are more significant [7]. Wethink that these differences in apparent relaxation timemay actually reAect an extra relaxation mechanism in thelaboratory experiments that operates mainly at long wave-lengths and at short times. One possibility is additionalmomentum transfer caused by propagating sound waves[7], which was not included in the simulations. Anotherpossibility is that the short-time diffusion is affected by di-rect interparticle forces [6]; this would be consistent withthe suggestion of Cohen and de Schepper [1]. Neverthe-less, we emphasize that the transport properties of col-loidal suspensions are not determined primarily by directinterparticle interactions; however, they can be quantita-tively explained by hydrodynamic interactions alone.

This work was partially supported by NASA, the U.S.Department of Energy, and Lawrence Livermore NationalLaboratory under Contract No. W-7405-Eng-48.

D. A. Weitz' and Anthony J.C. LaddExxon Research and Engineering CompanyRoute 22 East, Annandale, New Jersey 08801

Department of Theoretical and Applied MechanicsCornell University, Ithaca, New York 14853

Received 18 April 1995PACS numbers: 82.70.Kj, OS.40.+j, 66.20.+d, 82.70.Dd

*Permanent address: Lawrence Livermore NationalLaboratory, Livermore, CA 94550.

[1] E. G. D. Cohen and I. M. de Schepper, preceding Com-ment, Phys. Rev. Lett. 95, 2252 (1995).

[2] A. J. C. Ladd, J. Chem. Phys. 93, 3484 (1990).[3] J.X. Zhu et al. , Phys. Rev. Lett. 68, 2559 (1992).[4] M. H. Kao, A. H. Yodh, and D. J. Pine, Phys. Rev. Lett.

70, 242I (1993)[5] A. J.C. Ladd, Phys. Rev. Lett. 70, 1339 (1993).[6] A. J. C. Ladd, H. Gang, J.X. Zhu, and D. A. Weitz (to be

published).[7] A. J. C. Ladd, H. Gang, J. X. Zhu, and D. A. Weitz, Phys.

Rev. Lett. 74, 318 (1995).

0031-9007/95/75(11)/2253(1)$06. 00 1995 The American Physical Society 2253