Source-linked AI summary
Computing stable configurations of confined smectic liquid crystals with a deep variational framework
Yuchen Xie, Baoming Shi, Yucen Han, Lei Zhang
TL;DR
Stable configurations of confined smectics are difficult to compute because complex geometries, higher-order derivatives, and high-frequency layering must be resolved together. The paper introduces a deep variational framework that represents coupled mLdG order parameters on a regular reference domain, incorporates confinement through coordinate mappings, and uses a warmup penalty to guide layered states. It recovers stable and established smectic morphologies across diverse settings, including a chevron-like SmC state in a tangent-anchored sphere, with finite-difference relaxation confirming the persistence of recovered minima.
Problem
Computing stable confined-smectic configurations is challenging because complex geometries, higher-order derivatives, and strongly oscillatory layers must be resolved simultaneously.
Method
The DVF represents coupled mLdG order parameters on a regular reference domain, incorporates physical confinement through coordinate mappings, and uses a warmup penalty to promote high-frequency layering during early optimization.
Results
The DVF recovers stable layered configurations across diverse confinements, reproduces established morphologies, and predicts chevron-like SmC layers in a tangent-anchored sphere.
Takeaways & Limitations
The framework demonstrates applicability to stable smectic configurations across experimentally relevant confinement geometries and anchoring conditions.
Takeaways & Limitations
The study is limited to static states in fixed domains and a finite range of parameters and boundary conditions.
Abstract
from arXiv · showhide
Smectic liquid crystals are layered liquid-crystalline phases characterized by orientational order and periodic density modulation. Although their structures can be modeled using continuum theories, computing stable configurations remains challenging in complex geometries, particularly when the high-frequency density modulations associated with smectic layering should be resolved. We propose a deep variational framework (DVF) for computing these configurations within the modified Landau--de Gennes model, in which the coupled orientational and positional order parameters are represented on a regular reference domain while physical confinement is incorporated through coordinate mappings. A warmup penalty mitigates the spectral bias of neural networks toward smooth, nonlayered fields, enabling robust recovery of oscillatory smectic states. Comparisons with a neural-network baseline and finite-difference relaxation demonstrate the essential role of this penalty and the numerical stability of the resulting layered states. The DVF reproduces experimentally established smectic-A defect structures and layer morphologies across diverse confinement geometries and further predicts a chevron-like smectic-C state in a tangent-anchored sphere. Together, these results demonstrate the applicability of the DVF to computing stable smectic configurations across experimentally relevant confinement geometries and anchoring conditions.
I. INTRODUCTION
Confined smectics require simultaneous resolution of orientational order and oscillatory positional layering, making stable energy minimization difficult in complex geometries. The mLdG model couples these fields, while the DVF uses mappings and warmup guidance to compute layered configurations.
- Smectic structure combines orientational alignment with one-dimensional density modulation into layers that are sensitive to confinement, anchoring, and geometric frustration.
- Computing local energy minima is challenging because domains may be irregular or three-dimensional, the energy contains second derivatives of u, and smectic layers can be strongly oscillatory.The second derivatives produce a fourth-order Euler–Lagrange equation, while many layers require high numerical resolution.
- The DVF represents coupled order parameters on a regular reference domain, incorporates confinement through coordinate mappings, and uses a warmup penalty to recover layered states.The warmup promotes the prescribed wavelength during early optimization and is later removed so the final state is governed by the mLdG energy.
- The framework recovers established defect and layer morphologies across polygonal, spherical, irregular, and three-dimensional confinements, and predicts chevron-like SmC layers in a tangent-anchored sphere.Reported structures include corner-localized defects, bipolar textures, concentric layers, TFCD-like states, and geometry-dependent junctions or dislocations.
- The mLdG model represents orientational order with a symmetric, traceless tensor Q and layering with a scalar positional field u.The total energy includes nematic, smectic bulk, and orientational-positional coupling contributions.
- The phase-dependent coupling selects layer wavelength and director–layer geometry: SmA aligns the director with the layer normal, while SmC imposes a prescribed tilt angle.Both phases use layer spacing 2π/q, with SmC additionally selecting angle θ0 between director and layer normal.
III. DEEP VARIATIONAL FRAMEWORK
The framework addresses complex geometries and oscillatory smectic layers by combining reference-domain representations, coordinate mappings, Fourier-based networks, and wavelength-guided warmup optimization.
- The DVF adapts a neural variational representation to coupled mLdG order parameters on a regular reference domain.Physical confinement is incorporated through coordinate mappings, while a Fourier-based architecture and warmup penalty facilitate layered-state recovery.
A. Neural representation of the order parameters
The DVF represents smectic order parameters with a Fourier-based neural trial function on a regular reference domain and maps the resulting fields to physical confinement geometries.
- A neural network serves as a trial function for u and Q, with parameters optimized directly by minimizing the mLdG free energy.This replaces field minimization with finite-dimensional optimization over trainable network parameters.
- The Fourier architecture embeds coordinates, applies stacked Fourier layers, and projects features to the positional field and symmetry-preserving tensor representation.The projection enforces the required symmetry and tracelessness of Q by construction.
- A smooth one-to-one coordinate map pulls complex physical domains back to a regular reference domain for common numerical treatment.Energy integrals and derivatives are transformed using the mapping, Jacobian factors, and chain-rule conversions.
C. Free-energy minimization and warmup penalty
The DVF minimizes the mLdG free energy while using auxiliary regularization and a warmup penalty to avoid under-resolved artifacts and recover layered states. In square-confined SmA, the full DVF reaches a stable layered minimum that persists under finite-difference relaxation.
- Free-energy minimization: The neural formulation minimizes the normalized mLdG free energy with boundary anchoring and a small Euler–Lagrange residual regularization.The residual suppresses grid-scale configurations that have low sampled energy but are far from continuum stationarity, without replacing energy as the selection criterion.
- Warmup penalty: The warmup penalty promotes the prescribed smectic wavelength during early optimization, counteracting spectral bias toward smooth, nearly homogeneous positional fields.Its weight decays to zero, leaving the final state governed solely by the mLdG model.
- Validation and ablation: DRM and DVF without the penalty produce nonlayered WORS-type textures, while full DVF recovers the layered state with a D-like orientational texture.The square-confined SmA calculation uses λ2 = 30 under tangent anchoring.
- Validation and ablation: Finite-difference relaxation preserves the DVF solution’s layered D-like morphology and converges to a stationary state with a positive-definite Hessian.Differences between the DVF and relaxed states are localized in Q and phase-sensitive in the oscillatory field u.
A. Validation and SmA minima in square confinement
In square confinement, the warmup-penalized DVF recovers stable layered SmA minima, while increasing λ2 drives a transition from WORS-like nonlayered states to D-like layered states. Across irregular domains, the computed minima organize defects and layers according to boundary geometry.
- Square validation: At λ2 = 30, the full DVF recovers oscillatory layering and a diagonal D-like Q texture, unlike DRM and DVF w/o.The baseline and ablated network remain WORS-like and nonlayered.
- Square validation: The warmup penalty changes the optimization basin toward layered states, then is removed so the final configuration minimizes the original mLdG energy.This distinguishes basin selection from merely reducing the stationarity residual.
- Square validation: FD relaxation preserves the DVF's layered D-like morphology and reaches a stationary state with a positive-definite Hessian.The result supports the interpretation that DVF identifies a stable layered-minimum basin.
- Square confinement: As λ2 increases from 1 to 100, square minima transition from WORS-like nonlayered states through layering onset to D-like layered states without extended interior defect lines.At λ2 = 4.38, diagonal line defects begin reconstructing and layered modulation appears.
- 2D irregular domains: In irregular domains, polygons localize defects and layer dislocations near selected corners, while disks and spindles develop bipolar organizations with defects approaching spindle tips as narrowing increases.The triangle additionally contains an interior orientational defect and a layer triple junction.
C. SmA in 3D confinements
The computed 3D SmA minima show that geometry and anchoring organize orientational defects and layer morphology, producing distinct structures in cubes and spheres. The same framework also captures a surface-localized chevron-like SmC state and stable layered minima.
- Tangent anchoring: Tangent anchoring localizes cube distortions near edges and corners, while the sphere forms a bipolar texture with smoothly curved layers connecting antipodal regions.In the cube, interior layers remain comparatively regular; in the sphere, strongest distortions occur near the poles.
- Anchoring dependence: Homeotropic anchoring in the sphere produces approximately radial orientation and concentric onion-like smectic shells instead of the tangent-anchored bipolar stack.The comparison demonstrates markedly different layer organizations under the same spherical confinement.
- TFCD-inducing anchoring: Vertical anchoring at the cube top and radial alignment at the bottom confine reorientation to a near-substrate biaxial order-reconstruction zone.The upper domain retains nearly vertical orientation and horizontal layers, while the low-order core forms near the bottom surface.
- SmC in spherical confinement: In a tangent-anchored sphere, SmC retains the bipolar orientational organization but develops pronounced surface-localized chevron-like layer bending.The restructuring originates from the preferred director–layer tilt rather than an imposed change in equilibrium layer spacing.
- Computational framework: The DVF combines Fourier representation, coordinate mappings, and wavelength warmup, with finite-difference relaxation confirming recovered layered configurations as stable minima.The square-domain ablation identifies the warmup stage as essential in the tested regime.
- Conclusions: Across computed states, geometry and anchoring organize Q, while director–layer coupling links orientational organization to defect localization and layer morphology.Recovered structures include WORS-like, D-like, bipolar, TFCD-like, and chevron-like morphologies.
Appendix A: Nematic bulk potentials and nondimensionalization
The appendix specifies the dimension-dependent nematic bulk potentials and the nondimensionalization used in the computations. Two-dimensional calculations use a reduced Landau–de Gennes potential, while the energies are rescaled using the characteristic domain length and material parameters.
- Nematic bulk potentials: The 3D computations use the standard Landau–de Gennes bulk potential, whereas 2D computations use a reduced potential derived from the two-component tensor form.The reduced 2D form is the standard quartic potential used for two-dimensional nematic equilibria.
- Nematic bulk potentials: In 2D, tr(Q^3) = 0 and choosing A = −B^2/(2C) yields the reduced potential used throughout the 2D computations.The difference from the 3D potential comes from reducing the tensor itself, not merely the spatial domain.
- Nondimensionalization: Dimensionless energy is obtained by scaling coordinates with the characteristic length L and measuring free energy in units of KL^d−2.The appendix also introduces dimensionless parameters for λ2, a, c, q2, and B0.
- Nondimensionalization: The implementation minimizes a further normalized energy after the dimensional rescaling, with bars dropped in the main text and L_energy = E used for reported energies.For fixed rescaled parameters, the positive rescaling leaves energy minima unchanged.
Appendix B: Coordinate mapping and transformed derivatives
The coordinate-mapping formulation evaluates neural fields on a regular reference domain while incorporating irregular physical confinement through an invertible map and transformed derivatives. The Jacobian supplies first-order transformations, while map curvature contributes to second-order derivatives.
- Reference-to-physical mapping: An invertible map x = G(ξ) sends the regular reference domain [0, 1]^d to the physical confinement, allowing the network to be evaluated on a uniform grid.The forward map prescribes a one-to-one boundary correspondence and separates geometry handling from order-parameter representation.
- Geometric assumption: The mapping approach assumes that the Jacobian remains nonsingular at quadrature points so physical derivatives remain well-defined.This is the stated geometric assumption for mapped computations.
- First-order transformations: Physical first derivatives are obtained from the inverse Jacobian, while the same pullback transformation applies componentwise to Q and the volume element becomes dx = |det J| dξ.Mapped geometries use the uniform reference grid; square and cube domains can instead be evaluated directly on uniform physical grids.
- Second-order transformations: Second-order derivatives include the standard transformed Hessian and an additional term accounting for the local curvature of the coordinate map.This correction is required because the mapping is nonlinear.
- Variational implementation: Once the forward map is known, the derivative formulas provide first- and second-order physical derivatives from quantities defined on the reference grid.These formulas evaluate Q gradients and higher-order derivatives of u within the same variational framework.
1. Boundary conditions
Boundary conditions explicitly anchor Q on selected surfaces, while u remains unanchored and obtains its boundary behavior variationally. The framework supports tangent, radial, and face-specific anchoring targets.
- Geometric targets: For 2D tangent anchoring and spherical 3D homeotropic anchoring, targets use the local unit tangent t or outward radial normal ν, respectively.The tangent is applied edgewise on polygons, while ν is the sphere’s outward radial normal.
- Face-specific anchoring: The TFCD-inducing cube applies its anchoring penalty only on the top and bottom faces, leaving the four lateral faces unanchored.Its prescribed directions use the vertical unit vector ez on the top and the in-plane radial direction er on the bottom.
- Tangent anchoring: Tangent anchoring uses a degenerate planar surface loss that places a uniaxial director in the tangent plane without selecting an in-plane direction.This condition is used for the tangent-anchored cube and the SmA and SmC spheres.
- Boundary-condition structure: Q is prescribed on the anchored boundary subset Γbc, while neither u nor ∂νu is prescribed at the boundary.The positional field instead follows the natural boundary condition associated with the dependence of the energy on D2u.
2. Euler–Lagrange residual penalty
The Euler–Lagrange residual penalty measures stationarity of the network-generated scalar fields by penalizing residuals of the coupled orientational and positional equations. Its componentwise implementation is evaluated consistently for mapped geometries.
- Residual penalty: The residual regularization is the squared Euclidean norm of the componentwise Euler–Lagrange residuals.Equation (C3) converts the compact stationary operators into the residuals used in the code and in Table I.
- Interpretation: LEL measures stationarity with respect to the scalar fields produced by the network.For mapped geometries, the derivatives and integral are evaluated using the coordinate-mapping formulas in Appendix B.
- Phase-specific residuals: The residual system includes orientational and positional Euler–Lagrange terms for SmA, with a dimension-specific bulk contribution in 2D.The SmC formulation introduces an auxiliary scalar field before writing the corresponding residuals.
Appendix D: Implementation details
The implementation uses Fourier neural networks, double-precision grid evaluations, multiple random initializations, and fixed optimization and loss-weight settings. Three-dimensional runs additionally use temporary orientational guidance.
- Network and discretization: DVF computations use four Fourier layers with 64 hidden channels, GELU activation, and 16 retained Fourier modes in each direction.The fields and derivatives are evaluated in double precision on 129 grid points per direction in 2D and 65 in 3D.
- Optimization: AdamW uses a learning rate of 10^-3, weight decay of 10^-2, 5000 iterations in 2D, and 10000 iterations in 3D.The gradient norm is clipped at 1.
- Initialization: Each 2D setting uses 100 independent initializations, while each 3D setting uses 50 independent initializations.These repeated runs support exploration of multiple candidate configurations.
- Loss settings: All experiments fix the loss weights at wbc = 100 and wEL = 0.1.The warmup penalty uses a time-dependent weight, with parameters p and wmax specified separately for 2D and 3D.
- Warmup penalty: The warmup penalty parameters are (p, wmax) = (50, 0.01) in 2D and (80, 0.02) in 3D, using 32 sampled directions per iteration.The directional average in the penalty is estimated from independently sampled directions.
- Three-dimensional runs: Three-dimensional computations add temporary orientational guidance during the first 500 iterations.This guidance is analogous to that used for phases with narrow at-