Long-time nonpreaveraged diffusivity and sedimentation velocity of clusters: Applications to micellar solutions

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Abstract
Calculations are presented for the long-time diffusivity and sedimentation velocity of associating colloids. Examples of the latter are micellar solutions and microemulsions. The analysis incorporates the role of reversible association-dissociation processes accompanying the physical-space transport of these clusters through the solution. This is accomplished without the need for preaveraging by transforming the association-dissociation processes into equivalent “size-space” diffusional processes, which are then embedded into the simultaneous physical-space transport processes occurring in three-dimensional space so as to obtain a four-dimensional convective-diffusion equation governing transport of the clusters in both the physical and size spaces. A generic “projection” scheme framework based on generalized Taylor dispersion theory is then applied to the problem, thereby reducing the four-dimensional transport equation to a coarse-grained three-dimensional physical-space convective-diffusion equation. Effects arising from the existence of a distribution of cluster sizes are accounted for in the latter formulation governing the mean transport process by the appearance of three coarse-grained phenomenological coefficients whose values depend inter alia upon the cluster-size distribution. These “macrotransport” coefficients include a mean sedimentation velocity vector arising from the action of external forces (if any), a mean molecular diffusivity dyadic, and an additional diffusive-type contribution to the diffusivity corresponding to a convective (“Taylor”) dispersivity. The latter contribution arises as a consequence of the spread in settling velocities of the differently sized clusters. The generic framework developed is illustrated by applications to two classes of micellar solutions: (i) solutions comprised of spherical micelles; (ii) solutions comprised of cylindrical or wormlike micelles (so-called “living polymers”). Each spherical micelle is modeled as an impenetrable rigid sphere, the radius of which is determined by its aggregation number. The living polymers are modeled by the Debye-Bueche theory, wherein a coiled macromolecular chain is regarded as a Brownian “spongelike” porous sphere through whose interior solvent percolates. Calculations of the resulting macrotransport coefficients, including their scaling relationships, are presented for both cases, and their physical significance discussed in terms of the underlying microscale physics. Possible applications and potential extensions of the generic framework are outlined.