Mobility of a Slender Object in Entangled Polymer Solution

Accurate physical modeling of heterogeneous complex fluids is paramount to understanding active and passive transport in biological and biomimetic environments. In the present paper we employ the phenomenological two-fluid model to derive frictional resistance of a rigid object moving through an entangled polymer solution. Depending on the interaction between the particle and the incompressible polymer network, we introduce four different types of boundary conditions involving velocity and elastic stress of the polymer at the particle surface. For strongly adherent polymer we show, by employing the no-slip boundary condition, that the generalized Stokes relation (GSR) holds for arbitrarily shaped particle, implying that its frictional resistance is the same as the hydrodynamic (Stokes) resistance, being linearly proportional to the bulk viscosity, η_b, of the entangled polymer solution. For particles pretreated to hinder polymer adhesion we showed, by assuming either direct or indirect (solvent-mediated) repulsive interaction between the particle and the polymer network, that GSR does not hold in general and that frictional resistance becomes mesh-size-dependent. In the latter case, approximate closed-form expressions for translational (i.e., longitudinal and transverse) frictional resistances of a slender rigid particle are obtained employing the classical Oberbeck's solution of the Stokes equations for prolate spheroid and the 2D solution of the Brinkman equations for an infinitely long cylinder. We demonstrate, in particular, that for high values of the bulk viscosity typical for biological and biomimetic gels η_b≳ 100η, with η being the solvent viscosity, the frictional anisotropy (i.e., the ratio of transverse to longitudinal resistances) could reach values >10, in qualitative agreement with experimental observations and numerical findings.