Equivalent variational approaches to biaxial liquid crystal dynamics

TL;DR

This paper compares two variational models for biaxial liquid crystal dynamics, showing their equivalence under certain assumptions and extending the models to include dissipation and eigenvalue dynamics.

Contribution

It demonstrates the equivalence of QS and VK models via Hamilton's principle and introduces extensions for dissipation and eigenvalue evolution.

Findings

VK theory is a special case of QS model under rotational dynamics.

Inclusion of Rayleigh dissipation extends the models.

Eigenvalue dynamics are incorporated through scaling factors.

Abstract

Within the framework of liquid crystal flows, the Qian & Sheng (QS) model for Q-tensor dynamics is compared to the Volovik & Kats (VK) theory of biaxial nematics by using Hamilton's variational principle. Under the assumption of rotational dynamics for the Q-tensor, the variational principles underling the two theories are equivalent and the conservative VK theory emerges as a specialization of the QS model. Also, after presenting a micropolar variant of the VK model, Rayleigh dissipation is included in the treatment. Finally, the treatment is extended to account for nontrivial eigenvalue dynamics in the VK model and this is done by considering the effect of scaling factors in the evolution of the Q-tensor.