Tensorial Form of Leslie-Ericksen Equations and Applications

Abstract

We propose a system of dynamical equations coupling a tensor Q_ij related to the molecular order with the flow velocity, whose structure is very similar to the Leslie-Ericksen equations. In particular, the Leslie viscosities and the Frank elastic constants appear explicitly, which contrasts with constitutive equations derived from the microscopic Doi theory and using the full order parameter tensor S_ij. The description in terms of the tensor Q_ij may be seen as intermediate between the director of Leslie-Ericksen theory and the order parameter tensor S_ij of microscopic theories. With our approach we take advantage of the fact that the Leslie viscosities and the Frank elastic constants are well-established for several real systems (as for example 5CB, 8CB, or MBBA). Our equations, that become manifestly equivalent to Leslie-Ericksen equations when a director may be defined, are used in order to regularise the director field at the position of the field singularities in a simple way. Our objective is to simulate the macroscopic behaviour of nematics containing a large assembly of defects under several experimental conditions (under shear and/or magnetic field). Since we are essentially concerned by the macroscopic response, the exact structure of the defects is secondary, it is why we believe this approach to be pertinent in studying the dynamics of the texture induced by defects. The model is applied to a spinning nematic (5CB) subjected to a magnetic field, and a relaxation mechanism involving topological defects is evidenced.

Keywords:

• leslie-ericksen theory
• nematic
• relaxation
• spinning tube
• topological defect

Acknowledgments

This work was partly supported by the Fundação para a Ciência e a Tecnologia (Portugal) under research grant SFRH/BPD/8894/2002 to A. Véron, and contract C2007-443-CENIMAT-1.

Notes

¹σ ^e may be replaced by σ ^d in Eq. (17) since σ ^e – σ ^d is symmetric according to Eq. (12).

²It is a consequence of the Cayley-Hamilton theorem that states that every square matrix satisfies its own characteristic equation. For 3 × 3 matrices the theorem reads .