Transport properties of concentrated polymer solutions: Hydrodynamic screening theory of viscous friction in a suspension of spheres

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Abstract
A new hydrodynamic method for computing the transport properties of concentrated polymer solutions is proposed. The method is developed in detail for the case of N impermeable spheres of radius R suspended in a solvent of viscosity η0 and is applied in this paper to the calculation of the concentration, ρ0, dependent translational friction coefficient ζ[ρ0] of a typical sphere in the suspension. In the following paper the method is applied to compute the concentration dependent viscosity η[ρ0] of the suspension. The theory is based on the following key ideas. The microscopic steady state Navier–Stokes equation, which depends on the instantaneous N‐sphere configuration, is reduced by averaging to a coupled hierarchy of effective hydrodynamic equations which depend, respectively, on the coordinates of 0, 1, 2,..., etc., spheres. The first member of the hierarchy, which governs the macroscopic flow, involves η[ρ0] and an inverse screening length κ, which accounts for screening of hydrodynamic interactions in the suspension. The parameters κ and η[ρ0] are found to depend upon the difference between local and macroscopic velocity fields and field gradients near a sphere. This difference is determined by solving the second equation in the hierarchy. The second equation is rendered tractable by a closure of the hierarchy which is tantamount to assuming that the flow with one sphere fixed is governed by the same screened hydrodynamics which describes the macroscopic flow. This mean field closure gives self‐consistent equations for η[p0] and ζ[ρ0]. For the case that the spheres are modeled as point perturbers, solution of the self‐consistency problem yields at low concentrations ζ[ρ0] =6πη0R[1+4.5φ], where φ=4πρ0R3/3 is the volume fraction of the solute. For a realistic treatment of the finite size of the spheres which includes excluded volume effects and the effect of local velocity gradients, one finds at low concentrations ζ[ρ0] =6πρ0R[1+6.769 φ]. This result is in good agreement with the two sphere calculations of Pyun and Fixman, who find ζ[ρ0] =6πη0R[1+6.157φ]. At higher concentrations, the results for ζ[ρ0] are in fair agreement with results derived from sedimentation experiments. The discrepancy between theory and experiment is attributed to neglect of attractive forces between the spheres and neglect of the concentration dependence of the sphere–sphere correlation functions.