Active Liquid Crystal Continuum Model

• Active liquid crystal continuum models are frameworks that couple elasticity and energy-consuming processes to describe non-equilibrium dynamics in liquid crystals.
• They integrate variational principles with dissipative mechanisms using order parameters like the nematic tensor and displacement fields to capture complex mechanical behavior.
• The models successfully replicate experimental phenomena such as shape transformations and order parameter evolution, informing advances in soft robotics and material design.

An active liquid crystal continuum model is a theoretical and computational framework that describes the macroscopic dynamics of systems in which the orientational order and elasticity of liquid crystals are coupled with active, energy-consuming processes—such as those generated by embedded swimmers, external fields, or self-driven inclusions. These models combine classical liquid crystal elasticity, often in Lagrangian or Eulerian formalism, with additional terms reflecting the dissipative, out-of-equilibrium nature of active matter. They are grounded in variational principles with combinations of free energies, dissipation functions, and appropriate order parameters and have proven essential for predicting and understanding experimentally observed phenomena such as shape changes, instabilities, phase transitions, and self-organization in synthetic and biological liquid crystals.

1. Theoretical Foundations: Energetic and Variational Structure

Active liquid crystal continuum models usually employ a free energy functional built from multiple contributions:

• Elastic Free Energy: Derived from the deformation of a polymer network or nematic matrix and often cast in a nonlocal (integral) form to account for crosslink connectivity. The step length tensor L incorporates the orientational order, typically as $L = I + 2\mu Q$, with $Q$ the nematic order tensor.
• Nematic (Orientational) Free Energy: Based on Landau–de Gennes expansion, often using Maier–Saupe theory for the orientational interactions. Scalar and director fields ($S$, $n$) parameterize the nematic order.
• Volume-preserving Constraint: Enforced via a functional such as $$F_{\text{vol}} = (\Lambda/2) \int (J-1)^2\,d^3\alpha$$, with $J$ the deformation gradient determinant, to approximate incompressibility.
• Rayleigh Dissipation Functional: Encodes viscous losses in both mechanical and orientational degrees of freedom, via terms built from the rate-of-strain tensor $D$ and the time derivative of the order tensor $\dot{Q}$.

The equations of motion are derived by extremizing the action (kinetic energy minus free energy) including dissipation. The resulting governing equations are coupled and nonlinear, capturing dynamics of the material displacement, the scalar order parameter, and the nematic director; representative equations include: $$\rho_m \frac{\partial u}{\partial t} = \frac{1}{2\rho_c kT} \nabla_\alpha \cdot (L^{-1} F L_0) + \nabla_\alpha \cdot (\Lambda (J - 1) J F^{-T}) + \textrm{viscous terms}$$ and corresponding evolution laws for $S$ and $n$.

2. Order Parameters and Coupling

The central order parameters are:

• Material displacement field $x(\alpha, t)$: Mapping reference (Lagrangian) coordinates $\alpha$ to spatial positions.
• Scalar nematic order parameter $S$ and director field $n$: Together forming the order tensor $Q = S (3/2\, n n^T - 1/2\,I)$.
• Deformation gradient $F = \partial x/\partial \alpha$ and its determinant $J$.

The coupling between elasticity and orientational order is realized through the dependence of the step length tensor $L$ on $Q$ and the elastic free energy on both $x$ and $Q$. The orientational order feeds back to the mechanics, producing internal stresses and, when inhomogeneous, driving deformation and reorganization.

This coupled structure allows the model to describe phenomena where spatial variations in order parameter fields S and n result in mechanical stress gradients, driving macroscopic shape changes such as bending, buckling, or saddle-like deformations.

3. Dynamical Equations and Dissipation Mechanisms

The time evolution is described by a set of highly coupled, generally nonlinear partial differential equations:

• Momentum/velocity equation: Contains elastic stresses, volume constraint terms, and viscous dissipation (from both mechanical and orientational contributions).
• Order parameter equations: The $S$ field evolves by the balance of elastic, Landau–de Gennes, and dissipative (frictional) terms; the director $n$ evolves with a constraint to preserve unit length, slaved to elastic and dissipative dynamics.

Representative forms for the dissipation function per unit volume include: $$\mathcal{R}_d = \frac{1}{2} \gamma_1 D : D + \gamma_2 D : \dot{Q} + \frac{1}{2} \gamma_3 (\dot{Q} : \dot{Q})$$ where the viscosities $\gamma_i$ allow for tuning between different dissipative channels (mechanical, rotational, coupling).

The model supports not only equilibrium relaxation but can capture transient phenomena such as rapid deformations, slow recoveries, and pathway-dependent processes typical of non-equilibrium active systems.

4. Numerical Implementation: Spectral Methods and Integration Schemes

Numerical integration of the governing equations is challenging due to the complexity, stiffness, and required resolution:

• Spatial discretization: Chebyshev spectral methods and Gauss–Lobatto grids provide high accuracy for fields with non-periodic boundary conditions. Spectral differentiation matrices are used for efficient computation of spatial derivatives.
• Time integration: Implicit–explicit (IMEX) schemes split the equations into non-stiff (handled explicitly, e.g., second-order Adams–Bashforth) and stiff (handled implicitly, e.g., Crank–Nicolson) parts. This methodology provides stability and accuracy even for stiff dissipative systems.
• Linear solvers: For the velocity equation, large sparse linear systems are formed, typically solved iteratively (e.g., using GMRES).
• Boundary conditions and constraints: Realistic sample boundary conditions (e.g., free, anchored) are implemented to replicate experimental setups.

The numerical method allows for capturing the evolution of large-scale, inhomogeneous fields, tracking the internal distribution of S and n, and reproducing transitions between different mechanical morphologies.

5. Simulated Phenomena and Comparison to Experiments

The active liquid crystal continuum model reproduces experimental observations with high fidelity:

• Shape transformations: In box-shaped LCE samples, imposed temperature gradients drive the evolution of S, yielding stages of slight bending, rapid deformation, and slow relaxation toward saddle-shaped or bent configurations. Boundary conditions dramatically affect outcome (free vs. anchored surfaces).
• Internal order evolution: The vertical gradient in S resulting from the temperature gradient is quantitatively tracked and drives stress gradients across the sample; director field distributions show spatial and temporal heterogeneity matching observed textures.
• Dynamic pathway: The model predicts a sequence of deformations and order parameter reorganization that agrees with time-resolved experimental imaging.
• Order parameter tracking: Evolution of the order parameter tensor (both S and n fields) is consistent with experimental birefringence and polarization microscopy images.

A summary of observed results is shown in the table:

Experimental Condition	Simulated Deformation	Order Parameter Evolution
Free LCE slab, temp. grad	Saddle-shaped bending process	Vertical S gradient; n adjusts accordingly
Anchored edge	Asymmetric upward bending	Lateral/vertical S and n variation

6. Physical Significance, Applications, and Limitations

The active liquid crystal continuum model provides a predictive framework enabling:

• Quantitative modeling of complex, coupled shape and order dynamics in nematic LCEs under various loading, driving, and constraint configurations.
• Design and control of shape-changing elastomeric devices in soft robotics, actuators, and programmable matter.
• Insight into non-equilibrium phenomena including stress-driven phase transformations and the interplay between mechanics and orientational order.

Limitations include:

• Parameter estimation: Accurate simulation requires molecular-level input (chain statistics, relaxation rates, material constants).
• Higher-order phenomena: Effects such as defect dynamics, strong anisotropy, or beyond-Landau-order transitions are not fully captured; extensions to handle chirality, strong director distortions, or active flows require further development.
• Computational cost: High-fidelity spectral simulations with fine resolution are expensive in 3D settings.

The model establishes a robust foundation for simulating and interpreting experiments in active liquid crystalline solids and is extensible to a broad class of soft matter systems where coupled mechanical and orientational order dynamics are essential.