Euler-Poincaré Approaches to Nematodynamics

Abstract

Nematodynamics is the orientation dynamics of flowless liquid-crystals. We show how Euler-Poincaré reduction produces a unifying framework for various theories, including Ericksen-Leslie, Luhiller-Rey, and Eringen’s micropolar theory. In particular, we show that these theories are all compatible with each other and some of them allow for more general configurations involving a non vanishing disclination density. All results are also extended to flowing liquid crystals.

Appendix: The Free Energy

We reproduce here the computation in [12] that shows how all terms in the Frank energy

can be rewritten in terms of the variables j=J(I−n⊗n) and γ, the latter being introduced through the invariant relation ∇n=n×γ. Thus, the explicit expression for the micropolar free energy Ψ(j,γ) of nematic media given in (4.4) will be written after computing the micropolar expression for each term in the Frank energy. Further details are given in [12].

Twist

Using n⊗n=I−j/J, we have

$$\mathbf{n}\cdot \nabla\times\mathbf{n}=-\mathbf{n}\cdot\boldsymbol{\gamma}( \mathbf{n})+\|\mathbf{n}\|^2\operatorname{Tr}(\boldsymbol{\gamma})= \frac{1}{J}\operatorname{Tr} ({j}{\boldsymbol {\gamma }})=\frac{1}{J} \operatorname{Tr} \bigl({j}{\boldsymbol {\gamma }}^S \bigr), $$

where γ(n)=γ _ia n _a , with a being the $\mathfrak{so}(3)\simeq\Bbb{R}^{3}$-index, and γ ^S denotes the skew part of γ, i.e., γ ^S=(γ−γ ^T)/2, where we see γ as a 3×3 matrix with components γ _ia .

Splay

We introduce the vector ${\vec{{\boldsymbol {\gamma }}} }_{b}=\epsilon_{abc}{\boldsymbol {\gamma }}_{ac}$, defined by the condition $\vec{\boldsymbol{\gamma }} \cdot\mathbf{u}=\operatorname{Tr}(\mathbf{u}\times\boldsymbol{\gamma})$, for all u∈ℝ³, where u×γ is the matrix with components (u×γ) _ia =(u×γ _i ) _a . We compute

where γ ^A denotes the skew part of γ, i.e., γ ^A=(γ−γ ^T)/2 and where we used the equality $\widehat{\vec{\boldsymbol{\gamma}}} =-2\boldsymbol{\gamma}^{A}$. The latter can be shown by noting that we have the equalities $\operatorname{Tr}(\widehat{\vec{\boldsymbol{\gamma}}}\hat {\mathbf{u}})=-2\vec{\boldsymbol{\gamma}}\cdot\mathbf{u}=-2\operatorname{Tr}(\boldsymbol{\gamma}\widehat{\mathbf{u}})$ for all u∈ℝ³.

Bend

For all u∈ℝ³, we have

so we get

$$\mathbf{n} \times ( \nabla \times \mathbf{n} )=\vec{\boldsymbol{\gamma }}- \frac{1}{J}\overrightarrow{\boldsymbol{\gamma } {j} } $$

and therefore

$$\big\|\mathbf{n} \times ( \nabla \times \mathbf{n} )\big\|^2=\biggl \Vert \frac{1}{J}\overrightarrow{\boldsymbol{\gamma } {j}} -\vec{\boldsymbol{ \gamma }}\biggr \Vert ^2=-2\operatorname{Tr} \biggl( \biggl( \frac{1}{J}({\boldsymbol {\gamma }}j)^A-{\boldsymbol {\gamma }}^A \biggr)^{ 2} \biggr). $$

Summing all the terms, we obtain that the Frank free energy is indeed given by the expression for Ψ provided in (4.4).