Viscoelasticity of suspensions of long, rigid rods

A microscopic theory for the viscoelastic behaviour of suspensions of rigid rods with excluded volume interactions is presented, which is valid in the asymptotic limit of very long and thin rods. Stresses arising from translational and rotational Brownian motion and direct interactions are calculated for concentrations up to (L/D) (with L the length; D, the thickness of the rods; and their volume fraction). It is argued that for very long and thin rods, contributions to the stress arising from hydrodynamic interactions vanish asymptotically with increasing aspect ratio relative to the single particle contribution. As will be discussed, this is supported by calculations of Shaqfeh and Fredrickson (Phys. Fluids A2 (1990) 7), although convergence to negligible hydrodynamics interactions with increasing aspect ratio is very slow (for aspect ratios larger than ≈50, the contribution of hydrodynamic interactions to the stress is at most ≈20%). It is argued that the pair-correlation function is in good approximation given by the Boltzmann exponential of the pair-interaction potential. The neglect of hydrodynamic interactions and the use of the Boltzmann exponential approximation for the pair-correlation function allows the microscopic evaluation of stresses in terms of concentration and the orientation order parameter tensor to within a Ginzburg–Landau expansion up to third order, without having to resort to thermodynamic arguments. The orientational order parameter tensor in turn is obtained from an equation of motion that is derived from the N-particle Smoluchowski equation. The resulting expression for the stress tensor and the equation of motion are similar to, but also in some respects significantly differing from, the well known theory due to Doi, Edward and Kuzuu. Analytic expressions are derived for linear and leading order non-linear, viscoelastic response functions. It is found that the zero shear viscosity varies linearly in concentration. The Huggins coefficient vanishes like the square of the shear-rate. Such a linear concentration dependence of the zero shear viscosity for very long and thin rods is also found in simulations by Claeys and Brady (J. Fluid Mech. 251 (1993) 443) and Yamane et al. (J. Non-Newtonian Fluid Mech. 54 (1994) 405) for the long rods, but is in contradiction with the Berry–Russel theory (J. Fluid Mech. 180 (1987) 475), where interactions are treated in an approximate, orientationally pre-averaged fashion. In addition, we find a Maxwellian frequency dependence of response functions at zero shear-rate. Highly non-linear viscoelastic response functions at higher shear-rates are computed numerically. Among other things, we find normal stress differences that do not change sign as a function of shear-rate and higher order harmonic response functions that are qualitatively different for the paranematic and nematic states