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Diffusion in a dilute polydisperse system of interacting spheres

Published online by Cambridge University Press:  20 April 2006

G. K. Batchelor
G. K. Batchelor
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
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Abstract

When m different species of small particles are dispersed in fluid the existence of a (small) spatial gradient of concentration of particles of type j is accompanied, as a consequence of Brownian motion of the particles, by a flux of particles of type i. The flux and the gradient are linearly related, and the tensor diffusivity Dij is the proportionality constant. When the total volume fraction of the particles is small, Dij is approximately a linear function of the volume fractions ϕ1, ϕ2, …s, ϕm, with coefficients which depend on the interactions between pairs of particles. The complete analytical expressions for these coefficients given here for the case of spherical particles are a linear combination of the second virial coefficient for the osmotic pressure of the dispersion (measuring the effective force acting on particles when there is a unit concentration gradient) and an analogous virial coefficient for the bulk mobility of the particles. Extensive calculations of the average velocities of the different species of spherical particles in a sedimenting polydisperse system have recently been published (Batchelor & Wen 1982) and some of the results given there (viz. those for small Péclet number of the relative motion of particles) refer in effect to the bulk mobilities wanted for the diffusion problem. It is thus possible to obtain numerical values of the coefficient of ϕk in the expression for Dij, as a function of the ratios of the radii of the spherical particles of types i, j and k. The numerical values for ‘hard’ spheres are found to be fitted closely by simple analytical expressions for the diffusivity; see (4.6) and (4.7). The dependence of the diffusivity on an interparticle force representing the combined action of van der Waals attraction and Coulomb repulsion in a simplified way is also investigated numerically for two species of particles of the same size. The diffusivity of a tracer particle in a dispersion of different particles is one of the many special cases for which numerical results are given; and the result for a tracer ‘hard’ sphere of the same size as the other particles is compared with that found by Jones & Burfield (1982) using a quite different approach.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 131 , June 1983 , pp. 155 - 175
Copyright
© 1983 Cambridge University Press

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References

Ackerson, B. J. 1978 Conditions for interacting Brownian particles. II J. Chem. Phys. 69, 684–690.Google Scholar
Batchelor, G. K. 1972 Sedimentation in a dilute dispersion of spheres J. Fluid Mech. 52, 245–268.Google Scholar
Batchelor, G. K. 1976 Brownian diffusion of particles with hydrodynamic interaction J. Fluid Mech. 75, 1–29.Google Scholar
Batchelor, G. K. 1982 Sedimentation in a dilute polydisperse system of interacting spheres. Part 1. General formulae J. Fluid Mech. 119, 379–408.Google Scholar
Batchelor, G. K. & Wen, C.-S. 1982 Sedimentation in a polydisperse system of interacting spheres. Part 2. Numerical results J. Fluid Mech. 124, 495–528.Google Scholar
Felderhof, B. U. 1978 Diffusion of Brownian interacting particles. J. Phys. A: Math. Gen. 11, 929–937.Google Scholar
Hanna, S., Hess, W. & Klein, R. 1982 Self-diffusion of spherical Brownian particles with hard-core interaction. Physic A 111, 181–191.Google Scholar
Hill, T. L. 1960 An Introduction to Statistical Thermodynamics. Addison-Wesley.
Jeffrey, D. J. 1973 Conduction through a random suspension of spheres Proc. Roy. Soc. Lond. A, 335, 355–367.Google Scholar
Jones, R. B. 1979 Diffusion of tagged interacting spherically symmetric polymers. Physica A 97, 113.Google Scholar
Jones, R. B. & Burfield, G. S. 1982 Memory effects in the diffusion of an interacting polydisperse suspension. Parts I and II. Physic A 111, 562–590.Google Scholar
Kops-Werkhoven, M. M. & Fijnaut, H. M. 1981 Dynamic light scattering and sedimentation experiments on silica dispersions at finite concentrations. J. Chem. Phys. 74 (3), 1618–25.Google Scholar
Landau, L. & Lifschitz, E. M. 1968 Statistical Physics. Pergamon Press.
Marqusee, J. A. & Deutch, J. M. 1980 Concentration dependence of the self-diffusion coefficient J. Chem. Phys. 73, 5396–7.Google Scholar
Mcquarrie, D. A. 1976 Statistical Mechanics. Harper & Row.
Pusey, P. N. & Tough, R. J. A. 1982a Langevin approach to the dynamics of interacting Brownian particles. J. Phys. A: Math. Gen. 15, 1291–1308.Google Scholar
Pusey, P. N. & Tough, R. J. A. 1982b Dynamic light scattering, a probe of Brownian particle dynamics Adv. Coll. Interface Sci. 16, 143–159.Google Scholar