Tracer-diffusion in colloidal mixtures: A mode-coupling scheme with hydrodynamic interactions

Abstract

In this work, we develop a general theoretical scheme to study tracer-diffusion in mixtures of interacting colloidal particles where the influence of solvent-mediated hydrodynamic interactions is also considered. Based on the many-body Smoluchowski diffusion equation, we derive in a first step an exact microscopic expression of the irreducible memory function ͑self-friction function͒ associated with the self-intermediate scattering function and with the mean squared displacement of a tagged particle. By applying a mode-coupling scheme ͑MCS͒ to the irreducible memory function, we obtain explicit expressions for the tracer-diffusion quantities in terms of partial static structure factors and hydrodynamic functions. The influence of hydrodynamic interactions ͑HI͒ is accounted for using a far-field expansion of the two-body hydrodynamic diffusivity tensors. For charge-stabilized colloids, this is a good approximation due to strong electrostatic repulsion between the particles. Various applications are discussed in order to illustrate the versatility of our mode-coupling scheme.

Microscopic Theories of the Rheology of Stable Colloidal Dispersions

“…Though agreement with experiment is promising, hydrodynamic interactions are not explicitly included. Very recent work modifies the mode coupling theory for direct application to Brownian systems with far-field hydrodynamic interactions [38,39]. Numerical solutions of the full mode coupling equations lead to predictions of the high frequency modulus [40].…”

Hydrodynamic and electrokinetic effects on the dynamics of charged colloids and macromolecules

“…A versatile method for calculating the long-time selfdiffusion coefficient of repulsive charged colloidal spheres, based on the Primitive Model where all ionic species are treated equally and individually as uniformly charged hard spheres immersed in a structureless solvent, and the many-body Smoluchowski diffusion equation, has been developed in a couple of papers by the present authors [2,37]. This method invokes a simplified mode-coupling scheme for mixtures of Brownian spheres [38,39,40,41], and it accounts for the long-distance part of the HI between all ionic species in the form of the so-called Rotne-Prager approximation [42]. The detailed derivation of this method, and a thorough discussion of the approximations involved in its derivation, has been given elsewhere [2,37] and will not be repeated here.…”

Dynamical Density Functional Theory with Hydrodynamic Interactions and Colloids in Unstable Traps

“…Apart from the stochastic Brownian motion of the colloidal particles due to their kicks with the solvent molecules, hydrodynamic interactions between colloidal particles arising from the induced solvent flow field are getting relevant for concentrated suspensions. It has been shown by experiments [3,4], computer simulations [5,6] and theory [7,8,9,10] that hydrodynamic interactions can lead to qualitatively different behavior in the bulk transport properties and in colloidal sedimentation as compared to simple Brownian motion valid at very low volume fractions.…”

“…In practice, however, such a predictive theory is hampered by the many-body nature of the problem and the long range of the Oseen mobility tensor which is the leading contribution for a colloidal pair. Explicit approaches have been worked out in the bulk for short-time and longtime diffusion coefficients [7,8,9], and for the viscosity [12]. There are also first investigations for colloids near walls and on interfaces [10,13], and for the nonequilibrium structure of colloids [14] but a general theory for an arbitrary and time-dependent inhomogeneous external potential is missing.…”