An analytical approximation for the orientation-dependent excluded volume of tangent hard sphere chains of arbitrary chain length and flexibility

Abstract

Onsager-like theories are commonly used to describe the phase behavior of nematic (only orientationally ordered) liquid crystals. A key ingredient in such theories is the orientation-dependent excluded volume of two molecules. Although for hard convex molecular models this is generally known in analytical form, for more realistic molecular models that incorporate intramolecular flexibility, one has to rely on approximations or on computationally expensive Monte Carlo techniques. In this work, we provide a general correlation for the excluded volume of tangent hard-sphere chains of arbitrary chain length and flexibility. The flexibility is introduced by means of the rod-coil model. The resulting correlation is of simple analytical form and accurately covers a wide range of pure component excluded volume data obtained from Monte Carlo simulations of two-chain molecules. The extension to mixtures follows naturally by applying simple combining rules for the parameters involved. The results for mixtures are also in good agreement with data from Monte Carlo simulations. We have expressed the excluded volume as a second order power series in sin (γ ), where γ is the angle between the molecular axes. Such a representation is appealing since the solution of the Onsager Helmholtz energy functional usually involves an expansion of the excluded volume in Legendre coefficients. Both for pure components and mixtures, the correlation reduces to an exact expression in the limit of completely linear chains. The expression for mixtures, as derived in this work, is thereby an exact extension of the pure component result of Williamson and Jackson [Mol. Phys. 86, 819-836 (1995)].

The isotropic-nematic phase transition of tangent hard-sphere chain fluids—Pure components

“…The latter is defined as the set of all bond-and torsion angles ω ω ω of the molecule and the orientation ω ω ω of the overall molecular axis with respect to some fixed reference frame. Analogous to our previous work, 43 the molecular axis is defined as the axis around which the molecule has the smallest moment of inertia. The probability density to find any molecule in a configurationr r r is denoted by ρ(r r r), which is normalized as ρ(r r r)dr r r = N.…”

“…can be readily obtained from MC simulations of two chain molecules. 23,43 Thereby, due to the averaging, it is a function solely of the angle γ between the molecular axes, as…”

“…As discussed in a previous work, 43 it is convenient to introduce a dimensionless rigidity parameter, defined as the ratio of the total number of rigid bond angles and the total number of bond angles in a molecule:…”

“…In our previous work, 43 the orientation-dependent excluded volume V ex (γ ) was calculated for a wide range of different rod-coil molecules. An analytical approximation was developed by correlating the MC simulation results:…”

“…Onsager's second virial theory is extended to incorporate the effects of intramolecular flexibility using the method of Fynewever and Yethiraj. 23 In contrast to the original work of Fynewever and Yethiraj, the ensemble average of the pair-excluded volume that is needed to calculate the second virial coefficient is here obtained from an analytical approximation 43 instead of MC simulations of two chain molecules. Consequently, the resulting theory is computationally much more efficient, while, due to the high accuracy of the approximation, a similar accuracy is maintained.…”