In 1983 Felderhof, Ford and Cohen gave microscopic explanation of the famous Clausius-Mossotti formula for the dielectric constant of nonpolar dielectric. They based their considerations on the cluster expansion of the dielectric constant, which relates this macroscopic property with the microscopic characteristics of the system. In this article, we analyze the cluster expansion of Felderhof, Ford and Cohen by performing its resummation (renormalization). Our analysis leads to the ring expansion for the macroscopic characteristic of the system, which is an expression alternative to the cluster expansion. Using similarity of structures of the cluster expansion and the ring expansion, we generalize (renormalize) the Clausius-Mossotti approximation. We apply our renormalized Clausius-Mossotti approximation to the case of the short-time transport properties of suspensions, calculating the effective viscosity and the hydrodynamic function with the translational self-diffusion and the collective diffusion coefficient. We perform calculations for monodisperse hard-sphere suspensions in equilibrium with volume fraction up to 45%. To assess the renormalized Clausius-Mossotti approximation, it is compared with numerical simulations and the Beenakker-Mazur method. The results of our renormalized Clausius-Mossotti approximation lead to comparable or much less error (with respect to the numerical simulations), than the Beenakker-Mazur method for the volume fractions below $ phi approx 30% $ (apart from a small range of wave vectors in hydrodynamic function). For volume fractions above $phi approx 30 %$, the Beenakker-Mazur method gives in most cases lower error, than the renormalized Clausius-Mossotti approximation.
To the present day, the Beenakker-Mazur (BM) method is the most comprehensive statistical physics approach to the calculation of short-time transport properties of colloidal suspensions. A revised version of the BM method with an improved treatment of hydrodynamic interactions is presented and evaluated regarding the rotational short-time self-diffusion coefficient, $D^r$ , of suspensions of charged particles interacting by a hard-sphere plus screened Coulomb (Yukawa) pair potential. To assess the accuracy of the method, elaborate simulations of $D^r$ have been performed, covering a broad range of interaction parameters and particle concentrations. The revised BM method is compared in addition with results by a simplifying pairwise additivity (PA) method in which the hydrodynamic interactions are treated on a two-body level. The static pair correlation functions re- quired as input to both theoretical methods are calculated using the Rogers-Young integral equation scheme. While the revised BM method reproduces the general trends of the simulation results, it systematically and significantly underestimates the rotational diffusion coefficient. The PA method agrees well with the simulation data at lower volume fractions, but at higher concentrations $D^r$ is likewise underestimated. For a fixed value of the pair potential at mean particle distance comparable to the thermal energy, $D^r$ increases strongly with increasing Yukawa potential screening parameter.
Multipole matrix elements of Green function of Laplace equation are calculated. The multipole matrix elements of Green function in electrostatics describe potential on a sphere which is produced by a charge distributed on the surface of a different (possibly overlapping) sphere of the same radius. The matrix elements are defined by double convolution of two spherical harmonics with the Green function of Laplace equation. The method we use relies on the fact that in the Fourier space the double convolution has simple form. Therefore we calculate the multipole matrix from its Fourier transform. An important part of our considerations is simplification of the three dimensional Fourier transformation of general multipole matrix by its rotational symmetry to the one-dimensional Hankel transformation.
The mobility problem for suspension of spherical particles immersed in an arbitrary flow of a viscous, incompressible fluid is considered in the regime of low Reynolds numbers. The scattering series which appears in the mobility problem is simplified. The simplification relies on the reduction of the number of types of single-particle scattering operators appearing in the scattering series. In our formulation there is only one type of single-particle scattering operator.
