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 interparticle 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 mean-field approximations that interpolate between physically valid near- and far- field limits, incorporating results for the short-time self-diffusion coefficient and the high frequency limiting dynamic viscosity, In particular, the theory calculates the non-equilibrium structure from a two-particle Smoluchoski equation with a hypemetted chain closure to account for three-body couplings through a pairwise additive potential and three different mean-field approximations for the hydrodynamics. Substitution of the structure into conventional expressions for the stresses and fluxes determines the transport coefficients.
The accomplishments to date include
(i) extensive comparison of predictions without hydrodynamic interactions with results for the low shear viscosity and long-time self-diffusion coefficient from Brownian dynamics simulations for soft spheres;
(ii) comparison of predictions with hydrodynamic interactions with experimental data for the low shear viscosity, nonequilibrium structure, long-time self-diffusion coefficient, and high frequency shear modulus for hard spheres;
(iii) calculations of the linear viscoelastic spectra of concentrated dispersions of hard and soft spheres with and without hydrodynamic interactions; and
(iv) calculations for shear thinning in the dilute limit for hard spheres. The accumulated results show quantitative or, at least, semi-quantitative agreement with data and simulations, suggesting success for the mean-field hydrodynamics but a tendency of the thermodynamic closure to over-estimate the total interparticle force attempting to restore equilibrium at high concentrations. The sensitivity of the response of hard spheres to the magnitude of the perturbation near contact implies that improvements in the hydrodynamics and the thermodynamic closures must be closely coupled. This work comprises the PhD dissertation which Robert A. Lionberger defended in December and is reported in this and pas1 Annual Reports as well as the papers listed below.
In addition WC have either completed or made significant progress toward
(i) extracting a more “user friendly”, approximate form of the theory that distributes the contribution from the thermodynamic closure between diffusion and interparticle force terms,
(ii) developing analogous approximations for the hydrodynamics in the presence of grafted polymer layers and short range attractions,
(iii) implementing Monte Carlo simulations to generate accurate equilibrium structures as the input for calculations with more complex pair potentials, and
(iv) calculating the non-equilibrium structure, long-time self-diffusion coefficient, low shear viscosity, and shear modulus for polymerically stabilized spheres. These objectives are being pursued by Stacey L. Elliott, with initial results to be reported at the International Congress on Rheology in August and in the next Annual Report, Our further goals of calculating the non-equilibrium structure, long-time self-diffusion coefficient, low shear viscosity, and shear modulus for adhesive hard spheres and addressing polydispersity through selected calcalations for binary mixtures and appropriate pre-averaging (over the size distribution) of the conservation equation governing the non-equilibrium structure will be undertaken after establishing a simpler approximate form of the theory.
This and the 1994 Progress Report focus on comparisons of the predictions with experimental data from hard sphere colloidal dispersions, which reveal the following: finite ------- with the lubrication approximation and --------- with the discontinuous approximation, with both in agreement with definitive sets of experimental data, ------ with either the lubrication or discontinuous approximation that conform within 20-30s with the body of experimental data and qualitatively captures the divergence as ----------- with the lubrication or discontinuous approximations that conform to the body of data for 4 I 0.45 and qualitatively captures the zero as ---------- non-equilibrium static structure factors, i.e. Fourier transforms of the non-equilibrium structure, for weak steady shear that exhibit the proper dependence on wave number but too small a magnitude with the lubrication or discontinuous approximations, and stress-optical coefficients in the low shear limit with the proper dependence on volume fraction and roughly the right magnitude but a size dependence inconsistent with the only set of data.
Underprcdiction of the non-equilibrium structure for hard spheres with hydrodynamic interactions and the low shear viscosities for soft spheres without hydrodynamic interactions suggests that the HNC closure over-estimates the interparticle forces driving the structure towards equilibrium. If corrected, the viscosities/self-diffusion coefficients predicted for hard spheres with the discontinuous and lubrication approximations for the hydrodynamic interactions would undoubtedly be too large/small. However, the physically more reasonable interpolation constructedwith the ADA then might prove more accurate. The predictions of the high frequency limiting viscosities lend considerable credibility to the simple, mean-field hydrodynamic approximations. None the less, deficiencies in the hydrodynamic models undoubtedly still exist, e.g. direct interactions with a third particle neglected in the conservation equation itself and the approximation of the conditionally averaged divergence of the relative velocity in the Brownian stress. These neglected terms could be estimated through Stokesian dynamics simutions.
Though imperfect the theory clearly provides robust semi-quantitative predictions for the structure and dynamics of concentrated dispersions with a variety of repulsive intcrparticle potentials. At present the only alternative is Brady’s approach, Though based on more serious, ad hat approximations, his theory is appealingly simple and, thus far, provides comparably accurate predictions for the transport coefficients, despite erring qualitatively and quantitatively on the non- equilibrium structure. We proceed now with reasonable confidence tha the current formulation and, perhaps, that of Brady’s should suffice for mechanistic studies involving complexities such as polydispersity, adsorbed or grafted polymer, or attractive interactions.
Publications R.A. Lionberger and W.B. Russel, “High frequency modulus of hard sphere colloids”, J. Rheology 38 1885 1908 ( 1994). R.A, Lionberger and W.B. Russel, “Effectiveness of closures for many-body forces in concentrated colloidal dispersions”, J. Chem. Phys, (submitted). R.A. Lionberger, Rheology, Structure, and DifSusion in Concentrated Colloidal Dispersions, PhD Dissertation, Department of Chemical Engineering, Princeton University, December 1995. A.A. Potanin and W.B. Russel, “Hydrodynamic interaction of particles with grafted polymer burshes and applications to rheology of colloidal dispersions”, Phys. Rev. E 52 730-7 (1995).