[Paper Review] Liquid crystal elastomers and phase transitions in rod networks

M. Carme Calderer, Carlos A. Garavito+1 more|arXiv (Cornell University)|Mar 25, 2013

Cellular Mechanics and Interactions24 references4 citations

TL;DR

This paper develops a continuum model for anisotropic polymer networks using liquid crystal elastomer theory, combining Landau-de Gennes nematic energy with polyconvex elastic energy to study phase transitions in rod-like systems. The model predicts non-monotonic stress-strain behavior and re-entrant order transitions under compression, matching Monte Carlo simulations and experimental data on actin cytoskeletons.

ABSTRACT

In this article, we construct and analyze models of anisotropic crosslinked polymers employing tools from the theory of liquid crystal elastomers. The anisotropy of these systems stems from the presence of rigid-rod molecular units in the network. We study minimization of the energy for incompressible as well as compressible materials, combining methods of isotropic nonlinear elasticity with the theory of lyotropic liquid crystals. We apply our results to the study of phase transitions in networks of rigid rods, in order to model the behavior of actin filament systems found in the cytoskeleton.

Motivation & Objective

• To develop a continuum model for anisotropic crosslinked polymers with rigid rod units, inspired by actin filament networks in the cytoskeleton.
• To analyze phase transitions in rod networks by combining nonlinear elasticity with lyotropic liquid crystal theory.
• To establish existence of energy minimizers using polyconvexity and deformation tensor analysis.
• To model the coupling between orientational order (via Q-tensor) and network shape (via L-tensor) in response to deformation.
• To validate the model against Monte Carlo simulations and experimental observations of cytoskeletal networks.

Proposed method

• Formulates a total energy as the sum of Landau-de Gennes nematic free energy and a polyconvex elastic stored energy function.
• Uses the anisotropic deformation tensor $ G = (L^{-1} F F^T L_0)^{1/2} $ as the argument of the elastic energy density $ w(X) = \hat{w}(G(X)) $, ensuring polyconvexity for existence of minimizers.
• Imposes a linear relation between the network shape tensor $ L $ and the nematic order tensor $ Q $, with $ Q = L - \frac{1}{3} \text{tr}(L) I $, ensuring shared eigenvectors.
• Applies variational methods from isotropic nonlinear elasticity to prove existence of minimizers under incompressible and compressible conditions.
• Employs a bulk free energy function $ f(Q) $ with physically motivated growth conditions to prevent unphysical infinite alignment energy.
• Numerically computes order parameter vs. density and stress-strain curves for varying rod aspect ratios and crosslinking parameters $ \chi $.

Experimental results

Research questions

• RQ1 How can a continuum model of anisotropic elasticity be constructed to capture phase transitions in rod networks with liquid crystal elastomer behavior?
• RQ2 What conditions ensure the existence of energy minimizers in compressible and incompressible anisotropic elastic systems with nematic order?
• RQ3 How does the coupling between network shape (L) and rod alignment (Q) influence the mechanical response and phase behavior?
• RQ4 To what extent does the model reproduce non-monotonic stress-strain responses and re-entrant order transitions observed in Monte Carlo simulations?
• RQ5 How do rod aspect ratio and crosslinking density affect the stability and transition profiles of nematic and isotropic phases?

Key findings

• The model predicts a non-monotonic stress-strain response for high-aspect-ratio rods (e.g., $ A_a = 80 $), with sharp changes coinciding with order parameter jumps.
• A re-entrant behavior in the order parameter is observed at low densities, where alignment increases after an initial decrease, due to elastic network constraints.
• For $ \chi = 10 $, the order parameter increases again at low densities, indicating that elastic coupling enhances alignment under expansion.
• Smaller aspect ratios (e.g., $ \chi = 0.5 $) lead to oblate phases with order parameters on the order of $ 10^{-2} $, consistent with red blood cell cytoskeletal networks.
• The model reproduces the three-stage density-order parameter dependence seen in Monte Carlo simulations: initial decrease, then increase, and a second increase at very low densities.
• Non-monotonic stress-strain curves emerge specifically when the order parameter undergoes sharp transitions, indicating volume changes due to reorganization of rod alignment.