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Thermodynamics of MacMillan's liquid crystal model

Hervé Le Dret, Annie Raoult

TL;DR

The paper provides a rigorous thermodynamic framework for bulk liquid crystal energies using an internal variable $\xi$ (the $Q$-tensor). By applying the Eulerian Coleman–Noll procedure, it derives constitutive laws that ensure the second law holds and specializes to MacMillan's internal-variable model with a de Gennes bulk energy, including detailed objectivity and traceless-derivative considerations. It presents explicit conditions (and a concrete de Gennes example) guaranteeing thermodynamic compatibility, and then extends the model to a quadratic-in-$d$ generalization, showing under what circumstances such extensions remain admissible and how to construct them. The results offer a principled pathway to design thermodynamically consistent, frame-indifferent bulk energies for liquid crystals and to extend them with controlled, dissipative dynamics.

Abstract

We study liquid crystal models with bulk free energy from the point of view of the second law of thermodynamics. We formulate these models as objective internal variable models. Examples of application are given for the de Gennes free energy.

Thermodynamics of MacMillan's liquid crystal model

TL;DR

The paper provides a rigorous thermodynamic framework for bulk liquid crystal energies using an internal variable (the -tensor). By applying the Eulerian Coleman–Noll procedure, it derives constitutive laws that ensure the second law holds and specializes to MacMillan's internal-variable model with a de Gennes bulk energy, including detailed objectivity and traceless-derivative considerations. It presents explicit conditions (and a concrete de Gennes example) guaranteeing thermodynamic compatibility, and then extends the model to a quadratic-in- generalization, showing under what circumstances such extensions remain admissible and how to construct them. The results offer a principled pathway to design thermodynamically consistent, frame-indifferent bulk energies for liquid crystals and to extend them with controlled, dissipative dynamics.

Abstract

We study liquid crystal models with bulk free energy from the point of view of the second law of thermodynamics. We formulate these models as objective internal variable models. Examples of application are given for the de Gennes free energy.

Paper Structure

This paper contains 9 sections, 13 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Thermomechanical background
  4. The Eulerian Coleman-Noll procedure with an internal variable
  5. MacMillan's internal variable liquid crystal model
  6. Model objectivity
  7. Traceless objective derivatives
  8. Compatibility of MacMillan's model with the second law of thermodynamics
  9. A generalization of MacMillan's model

Key Result

Proposition 4.1

The two laws of thermodynamics and the dynamics equation imply that i) the specific free energy density $\widehat{a}_m$ is only a function of $\theta$ and $\xi$, ii) the specific entropy density $\widehat{s}_m$ is only a function of $\theta$ and $\xi$ with iii) there is a constitutive law for the internal dissipation given by which satisfies the dissipation inequality Conversely, if the constit

Theorems & Definitions (40)

  • Proposition 4.1
  • proof
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4
  • Proposition 6.1
  • Proposition 6.2
  • proof
  • Proposition 7.1
  • proof
  • ...and 30 more