Executive Summary
We continue working toward a robust and fundamental theory to predict the structure and dynamics of concentrated colloidal dispersions from the interparticle potential, assumed to be pairwise additive, and scalar functions describing the pair hydrodynamics. The former are generally available in approximate form from the colloid science and polymer physics literature. The latter we construct as interpolations between well established limits for the mean-field at large separations and lubrication near contact. Insertion of these into a Smoluchoski equation for the pair distribution function, with an approximate closure to account for couplings with a third particle through the pair potential, provides a basis for determining the non-equilibrium microstructure. From the microstructure we calculate transport propertics (e.g. low shear viscosity qO, frequency dependent shear modulus G’ and dynamic viscosity q’, long-time self-diffusion coefficient Ds”) and optical measures of the structure (eg. static structure factor and stress optical coefficient).
Papers now in press demonstrate the efficacy of the approach via comparison of the predictions without hydrodynamics with results from Brownian dynamics simulations for soft spheres and those with hydrodynamic interactions with experimental data for hard spheres. The theory is at least semi-quantitative with accuracy comparable (but not superior) to other approaches. For hard spheres of diameter d WC find finite d3Gi/kT with the lubrication approximation and --------- (52 = dimensionless frequency) with the discontinuous approximation, in agreement with definitive sets of data, qO/p within 50% of the experimental data and capturing the divergence as the volume fraction ----- c* 0.64, @m/Do conforming with the data for 4 zz 0.45 and qualitatively capturing the zero as ---- 0.64, static structure factors and stress optical coefficients for weak steady shear with the proper qualitative dependence on wave number and volume fraction.
This past year we have pursued a mechanistic study of spheres bearing grafted polymer layers, as studied extensively by Mewis in earlier IFPRI research. We adopt a simple mean field approximation for the interparticle potential between flat plates, apply the Derjaguin approximation to convert it to spheres, and implement Monte Carlo simulations to generate accurate equilibrium pair distribution functions. The high frequency limiting viscosity is calculated at low to moderate volume fractions by incorporating into a conventional effective medium theory the correction proposed by Bedeaux for short range hydrodynamic interactions, which are captured through pair hydrodynamic functions for uniformly permeable spheres in the literature. An asymptotic approach due to Acrivos and Frankel, together with the lubrication force between polymer coated spheres near contact, provides a robust limit at high volume fractions. The viscosity then diverges at a volume fraction that ranges from random close packing to unity, depending on the thickness of the grafted layer. Incorporation into our simple interpolations advanced for hard spheres, these provide sensible approximations for the hydrodynamic functions for polymer coated particles.
Calculations of the high frequency modulus d3GL/kT are now complete, demonstrating clearly the conditions under which one can extract the pair potential from this rheological measurement. Hydrodynamic interactions reduce the modulus significantly up to close packing of effective hard spheres, but when crowding compresses the layers significantly the moduli with and without hydrodynamic interactions differ only modestly. Comparison with the data indicates that the simple approximations for the hydrodynamic interactions appear to capture at least the qualitative features of the phenomena. To remove the error introduced through the approximation for the inter-particle potential, we employed instead measurements reported in the literature from a surface forces apparatus. However, these provided no better agreement, even at sufficiently high volume fractions that the remainder of the theory is demonstrably exact.
Now we are calculating the low shear viscosity, first with the simplified theory of Brady and ultimately with our full theory including the closure approximation. These will yield the low shear viscosity and linear viscoelastic response as a function of volume fraction and ratio of layer thickness to particle radius. To complete the project during the coming year we will assess the relative accuracy of the non-equilibrium theories and close with recommendations for those interested in relatively simple but qualitatively reliable means of calculating the effect of inter-particle potentials on rheology.