Numerical simulation of rodlike polymers in a uniaxial extensional flow: The distribution-function version versus the order-parameter version of Doi’s theory

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Abstract
The molecular theory of nematic liquid crystals, proposed by Doi in 1981, exists in two versions: the original kinetic equation written in terms of the orientational distribution function and an averaged approximation of the original kinetic equation written in terms of the order parameter tensor. In this study, the two versions, as well as a third (hybrid) approach by Edwards and Beris (1989), are compared. Due to the complexity of the original kinetic equation, only uniaxial extensional flow in a homogeneous monodomain is considered. The solution of the original kinetic equation without simplifications is obtained for the first time—using a finite difference scheme with an iterative procedure for an integral equation. The solution of the kinetic equation written in terms of the order-parameter tensor (a nonlinear algebraic equation) is obtained by Newton’s procedure. For the ordered phase, the order-parameter version provides acceptable approximation only in the low-concentration region, the extent of which decreases with increasing velocity gradient. The hybrid approach of Edwards and Beris (1989) provides no improvement over the order-parameter version; in the presence of a strong flow, it actually gives worse results at low concentrations.