Our objective is a robust and fundamental theory to predict the structure and dynamics of concentrated colloidal dispersions, including the shear viscosity, linear viscoelastic properties, and self diffusion coefficients. To achieve such the approach must handle
- three-body couplings that arise with pairwise additive inter-particle potentials and
- many-body hydrodynamics.
Our treatment is based on the classical configuration space, or Smoluchoski, approach which comprises a rigorous description of dynamics on the diffusion time scale. The couplings through the interparticle potential are approximated via nonequilibrium closures based on diagrammatic expansions and analogous to well-established equilibrium closures. Hydrodynamic interactions are embedded in resealings or lubrication approximations incorporating results for the short-time self-diffusion coefficient and the high frequency limiting dynamic viscosity. The primary accomplishments to date include
(i) comparison of predictions without hydrodynamic interactions with low shear viscosities and long-time self-diffusion coefficients from Brownian dynamics simulations for soft spheres and
(ii) comparison of predictions with hydrodynamic interactions low shear viscosities, nonequilibrium structure, and high frequency shear moduli from experiments with hard spheres.
The accumulated results show quantitative or, at least, semi-quantitative agreement with data and simulations, suggesting success for the hydrodynamic closure but some deficiency in the thermodynamic closure at high concentrations. This report focusses on the high frequency shear modulus mentioned in (ii) above, but includes in the appendices complete derivations of the stresses and the conservation equation governing the non-equilibrium structure. The mathematical analysis in the body of the paper establishes that the limiting shear modulus
- does not depend on three-body couplings through the interparticle potential, ?? is sensitive to the detailed two-particle hydrodynamics near contact, and
- is affected only in a mean-field sense by many-body hydrodynamic interactions.
Thus, the predictions solely test the hydrodynamic closures. The close agreement with experimental data for two different hard sphere dispersions with different high frequency asymptotes demonstrates the robustness of our relatively crude hydrodynamic closures and the ability of the non-equilibrium theory to resolve a perplexing, albeit esoteric, dilemma in the literature. In the coming year we intend to
- complete the assessment of the thermodynamic closure through detailed comparisons with results from simulations without hydrodynamic interactions,
- calculate the frequency-dependent viscoelastic moduli and shear rate dependent viscosity for hard spheres,
- extract a more “‘user friendly”, approximate form of the theory that distributes the contribution from the thermodynamic closure between diffusion and inter-particle force terms,
- develop analogous approximations for the hydrodynamics in the presence of grafted polymer layers and short range attractions, and
- calculate the non-equilibrium structure, long-time self-diffusion coefficient, low shear viscosity, and shear modulus for polymerically stabilized spheres.
The last two items will make contact with earlier measurements in Professor Mewis’ laboratory.