Surface Beris–Edwards Model on Curved Manifolds

• Surface Beris–Edwards model is a framework defining nematic liquid crystal dynamics on curved surfaces using a Q-tensor and incorporating both intrinsic and extrinsic curvature effects.
• It integrates a free-energy functional with elastic, bulk, and curvature-dependent terms to model defect dynamics and phase transitions on deformable substrates.
• By coupling Q-tensor evolution with surface hydrodynamics, the model predicts active stress-induced shape changes and morphogenetic instabilities in nematic films.

The surface Beris–Edwards model defines the hydrodynamics and orientational order dynamics of nematic liquid crystals on curved or deformable surfaces, generalizing the classical Beris–Edwards theory from flat geometries to two-dimensional manifolds with intrinsic and extrinsic curvature. Central to this framework is the Q-tensor order parameter, a symmetric traceless tensor encoding the nematic alignment, coupled to surface hydrodynamics under constraints of surface geometry. The model incorporates energetic, dissipative, and active (non-equilibrium) effects, and robustly describes phenomena such as defect dynamics, morphogenetic surface evolution, and phase transitions in nematic films on curved substrates (Nitschke et al., 2024, Nitschke et al., 2023, Bouck et al., 2022).

1. Free-Energy Functional and Geometric Setting

The fundamental free-energy functional for the surface Beris–Edwards model is constructed to encapsulate elastic, bulk, and curvature-dependent energetic contributions. For a two-dimensional surface $S$ with metric $g_{ij}$, mean curvature $H$, and extrinsic curvature tensor $K_{ij}$, the total free energy is given by

$$\mathcal{F}[Q,g]=\int_S \left\{ \frac{L}{2} \nabla_k Q_{ij}\nabla^k Q^{ij} + \frac{a}{2} Q_{ij}Q^{ij} + \frac{b}{3} Q_i{}^{j}Q_j{}^{k}Q_k{}^{i} + \frac{c}{4}(Q_{ij}Q^{ij})^2 + \frac{\kappa}{2}(H - H_0)^2 + \frac{\alpha}{2} K_{ik} K_j{}^k Q^{ij} \right\} \mathrm{d}A$$

where $Q_{ij}$ is the surface Q-tensor (symmetric, traceless), $L$ the elastic constant, $a, b, c$ the bulk Landau–de Gennes parameters, $\kappa$ the bending rigidity, $H_0$ the spontaneous curvature, and $\alpha$ controls extrinsic curvature-nematic coupling. This functional, and variants thereof, underlies both passive and active nematodynamics on surfaces, and forms the basis for variational derivations and numerical implementations (Nitschke et al., 2024, Nitschke et al., 2023, Bouck et al., 2022).

2. Q-Tensor Evolution and Objective Time Derivatives

Evolution of the Q-tensor couples advection, flow alignment, and molecular relaxation. The dynamical equation reads

$$\partial_t Q_{ij} + v^k \nabla_k Q_{ij} - S_{ij} = \Gamma H_{ij} - \zeta Q_{ij}$$

where $v^k$ is the surface flow velocity, $\Gamma$ the rotational diffusivity, $\zeta$ the activity coefficient (active nematics), and

$$H_{ij} = -L \Delta Q_{ij} + a Q_{ij} + b (Q^2)_{ij} + c (Q_{kl}Q^{kl}) Q_{ij} + \frac{\alpha}{2} K_{ik} K_j{}^k$$

defines the molecular field as the variational derivative of the free energy. The rate-of-strain $S_{ij}$ and vorticity $\Omega_{ij}$ define the generalized drag–alignment term, with the flow-alignment parameter $\xi$.

Two canonical objective derivatives for Q-tensor evolution are employed on surfaces:

• Material derivative $\dot Q = \partial_t Q + v^k\nabla_k Q$, physically corresponding to directors “frozen” into the surface flow.
• Jaumann (corotational) derivative $Q^\circ = \dot Q - [\nabla_S v, Q]$, correcting for local surface rotation.

Both are thermodynamically consistent and affect transient but not equilibrium behavior (Nitschke et al., 2023, Bouck et al., 2022).

3. Surface Stresses and Force Balances

Surface nematic dynamics are governed by a hydrodynamic force balance for the in-surface ($v^i$) and normal ($n^a$) directions. The total stress tensor comprises:

• Elastic stress: Ericksen stress includes the gradient elastic, bulk thermotropic, bending, and extrinsic terms.
• Viscous stress: Proportional to the surface rate-of-strain, incorporates flow-alignment effects.
• Active stress: In active nematics, $\sigma^{\text{active}}_{ij} = -\zeta Q_{ij}$, driving nonequilibrium phenomena.

The surface momentum equations are

$$\rho(\partial_t v_i + v^j\nabla_j v_i) = \nabla^j \sigma_{ij} + f^{\text{geom}}_i$$

with surface incompressibility $\nabla_i v^i = 0$. The normal force balance (“shape equation”) describes membrane deformation, with contributions from bending elasticity (Helfrich term), nematic-induced surface forces $\sigma^{ij} K_{ij}$, normal pressure, and active nematic forcing $-\zeta Q^{ij} K_{ij}$ (Nitschke et al., 2024, Nitschke et al., 2023, Bouck et al., 2022).

4. Curvature Coupling and Geometric Effects

Surface curvature enters the dynamics through multiple distinct mechanisms:

• Energetic: The free energy’s bending term $\frac{\kappa}{2}(H-H_0)^2$ and extrinsic coupling $\frac{\alpha}{2} K_{ik} K_j{}^k Q^{ij}$ directly penalize departures from preferred shape and encode curvature-order coupling.
• Dynamical: The molecular field, stresses, and defect-driven forces are curvature-dependent.
• Kinematic: The geometry of the surface affects objective time derivatives, transport, and boundary conditions for Q.
• Intrinsic vs Extrinsic: Models differentiate between Q-tensors anchored in the tangent plane (“surface-conforming”), as in biological membranes, and general 3D Q-tensor dynamics unconstrained by the surface (Nitschke et al., 2023, Bouck et al., 2022).

These couplings render nematic ordering and surface geometry mutually dynamic, supporting morphogenetic processes and topological defect dynamics unique to curved manifolds.

5. Active Stresses and Defect-Driven Dynamics

Inclusion of activity ($\zeta \neq 0$) fundamentally alters nematic evolution:

• Active force density: $f^{\text{active}}_i = -\zeta \nabla^j Q_{ij}$ splits into tangential components driving defect motion and normal components proportional to $-\zeta Q^{ij} K_{ij}$.
• Defect polarity: On a curved surface, both $\pm 1/2$ defects exert in-plane and out-of-plane forces, in contrast to flat geometries, where $-1/2$ defects are passive.
• Morphogenetic instability: Defect-induced normal forces generate shape changes and can initiate surface evolution mechanisms proposed for morphogenesis in biological systems (Nitschke et al., 2024).

6. Surface Phase Transitions: Sharp-Interface Limit

The sharp-interface limit of the Beris–Edwards system connects diffuse nematic-isotropic transitions to moving surface models:

• The Q-tensor evolution with small interfacial width ($\varepsilon$) leads, as $\varepsilon \to 0$, to nematic and isotropic domains separated by a moving interface.
• The surface Beris–Edwards model arises as the limit, with interfacial dynamics governed by mean curvature and balance of surface and bulk stresses:
  • $[u]\cdot n = 0$
  • $[\sigma n] = -\gamma H_s n$
  • $V_n = u \cdot n$
• Rigorous error estimates confirm convergence rates and validate the surface model as the asymptotic limit (Su, 2024).

7. Mathematical Structures, Model Specializations, and Applications

The Beris–Edwards framework on surfaces is developed in multiple related formalisms:

• 3D Q-tensor formulation: No anchoring, permits arbitrary nematic orientation relative to the surface (Nitschke et al., 2023).
• Surface-conforming (2D director) models: Q is projected tangentially, relevant for biological membranes (Nitschke et al., 2023).
• Onsager variational structure: Dissipative and reversible forces are consistently derived, ensuring energy dissipation and thermodynamic admissibility (Bouck et al., 2022).
• Numerical implementation: Surface finite elements, operator splitting for flow and order evolution, and consistent enforcement of geometry-dependent constraints (Nitschke et al., 2023).

Applications span biological morphogenesis, design of adaptive materials, and modeling of nematic films on complex substrates. The interplay of curvature, activity, and nematic order yields rich defect dynamics, shape instabilities, and emergent morphologies central to recent theoretical and experimental studies (Nitschke et al., 2024, Bouck et al., 2022).