Landau–de Gennes Free Energy

• Landau–de Gennes free energy is a variational framework that uses a symmetric, traceless Q-tensor to describe the mesoscopic ordering in nematic liquid crystals.
• It integrates elastic gradient energy and a quartic bulk potential to capture isotropic–nematic phase transitions, spatial distortions, and defect structures.
• The model informs numerical and analytical approaches for device design, highlighting transitions from uniaxial to biaxial order and defect-driven switching mechanisms.

The Landau–de Gennes (LdG) free energy is a pivotal variational framework for describing nematic liquid crystals at the mesoscopic scale, representing local molecular orientation via a symmetric, traceless $3 \times 3$ tensor $Q(x)$. It generalizes mean-field models by accommodating spatial distortions, topological defects, and multiple modes of nematic ordering (uniaxial, biaxial, and isotropic states). The functional, established for domains such as 3D droplets and cuboids, integrates gradient elasticity and a polynomial (typically quartic) bulk potential that admits first-order isotropic–nematic phase transitions and captures the full complexity of defect cores and switching pathways.

1. Mathematical Structure of the Landau–de Gennes Free Energy

The classical LdG free energy on a domain $B(0,R_0) \subset \mathbb{R}^3$ is

$$I_{LdG}[Q] = \int_{B(0,R_0)} \left\{ \frac{L}{2} |\nabla Q|^2 + f_B(Q) \right\} dV$$

where $Q(x) \in S_0$ is a real, symmetric, traceless $3 \times 3$ matrix, and $L>0$ is the elastic modulus. The bulk free energy $f_B(Q)$ is typically quartic in the invariants of $Q$: $$f_B(Q) = \frac{A}{2} \operatorname{tr} Q^2 - \frac{B}{3}\operatorname{tr} Q^3 + \frac{C}{4}(\operatorname{tr} Q^2)^2$$ with material-dependent $A=A_0(T-T^*)$, $B>0$, $C>0$, corresponding to temperature $T$ and spinodal $T^*$. For boundary value problems with strong homeotropic anchoring, $$Q|_{\partial B} = s_+ (e_r \otimes e_r - \tfrac{1}{3}I)$$, $s_+$ given by

$$s_+ = \frac{B + \sqrt{B^2 + 24|A|C}}{4C} \quad (A < 0)$$

which determines the bulk minima representing maximally uniaxial nematic alignment (Henao et al., 2011).

2. Order Parameter, Uniaxiality, and Biaxiality

$Q$-tensor eigenvalue structure classifies nematic states:

• Uniaxial: $Q = s(n \otimes n - \tfrac{1}{3}I)$, $n \in S^2$, $s \in \mathbb{R}$; two equal eigenvalues, one distinct.
• Biaxial: three distinct eigenvalues.
• Isotropic: $Q = 0$.

The bulk potential $f_B(Q)$ has minima forming a vacuum manifold of uniaxial states $$\{Q = s_+(n \otimes n - \tfrac{1}{3}I), n \in S^2\}$$. The quadratic and cubic traces $\operatorname{tr}Q^2$, $\operatorname{tr}Q^3$ are sufficient invariants for phase identification and for formulating reduced models for special geometries (Henao et al., 2011, Shi et al., 2022).

3. Euler–Lagrange Equations and Minimizer Symmetry

Minimizers $Q$ of $I_{LdG}$ satisfy an elliptic matrix-valued PDE with the traceless constraint enforeced: $$L\,\Delta Q_{ij} = -A\,Q_{ij} - B \left(Q_{ip}Q_{pj} - \frac{1}{3}\delta_{ij}\operatorname{tr} Q^2 \right) + C\,Q_{ij}\operatorname{tr} Q^2$$ subject to symmetric $Q_{ij}$, $Q_{ii}=0$, and prescribed boundary $Q_{ij}=Q_{b,ij}$ on $\partial B$. For uniaxial and radially symmetric settings (hedgehog ansatz),

$$Q(x) = h(r) (e_r \otimes e_r - \tfrac{1}{3}I), \qquad r = |x|$$

the PDE reduces to a singular ODE for $h(r)$: $$h'' + \frac{2}{r} h' - \frac{6}{r^2} h = -A h + C h^3 - B \left( \frac{3}{2} h^2 \right)$$ with $h(0)=0$ (isotropic core), $h(R_0)=s_+$ (nematic boundary) (Henao et al., 2011).

4. Low-Temperature Regime: Uniaxial Symmetry and Instability

In the deep nematic (low-temperature) limit $|A| \to \infty$, the global minimizer among uniaxial configurations coincides (up to rotation in $O(3)$) with the radial hedgehog [Theorem 2, (Henao et al., 2011)]. However, for sufficiently low $T$, purely uniaxial minimizers cease to be globally optimal. Explicit biaxial perturbations, localized near the defect core, lower the energy, breaking uniaxial symmetry. This imposes a generic transition to biaxiality in the defect structure for $|A|$ large—a generic consequence for the topology and dynamics of nematic defects.

Summary of results:

• Symmetry theorem: Any global uniaxial minimizer must be the radial hedgehog.
• Non-existence theorem: For $|A|$ large, no purely uniaxial global minimizer exists (Henao et al., 2011). This is a rigorous demonstration of instability of the hedgehog defect in deep nematic regimes.

5. Numerical, Analytical, and Device Implications

Recent numerical schemes (hybrid saddle dynamics, Newton methods) in higher-dimensional and complex geometries (cuboids, prisms) facilitate computation of high-Morse-index critical points. These allow stacking of unstable 2D or uniaxial solutions into genuine 3D z-variant states, which can authorize new switching pathways and lower energy barriers for device operation (e.g., bistable displays, defect templates) (Shi et al., 2022). Analytical criteria determine exactly when hedgehog and related patterns are minimum-unique, and when they become unstable.

Device/design implications include:

• Multistability engineering: Tuning geometric and material parameters enables reconfigurable nematic devices based on LdG solution landscapes.
• Defect-driven switching: Uniaxial-to-biaxial transitions and z-variant critical points provide physical mechanisms for dynamic switching under external fields (Shi et al., 2022).
• Defect topology and architecture: Stacks of multi-layer critical points (e.g., D–B–D) enable sophisticated defect trapping and memory architectures.

6. Extensions: Bulk Potential Variants and Phase Transitions

The quartic form of $f_B(Q)$ enables first-order nematic–isotropic transitions but fails to capture deep biaxial phases. Higher-order (sixth) potentials have been analyzed to model robust bulk biaxiality at low $T$, modifying defect stability and promoting novel critical points in confined geometries (McLauchlan et al., 2023). This extension fundamentally alters the bifurcation landscape, yielding wider domains of stability for biaxial torus and split-core solutions, and shifting temperature thresholds for various defect regimes.

7. Summary Table: Key Landau–de Gennes Free Energy Ingredients

Term	Mathematical Form	Physical Role
Elastic energy	$\frac{L}{2} |\nabla Q|^2$	Penalizes spatial order parameter gradients
Bulk potential	$$\frac{A}{2}\operatorname{tr}Q^2 - \frac{B}{3}\operatorname{tr}Q^3 + \frac{C}{4}(\operatorname{tr}Q^2)^2$$	Drives phase transitions, uniaxial minima
Boundary anchoring	$$Q|_{\partial B} = s_+ (e_r \otimes e_r - \tfrac{1}{3}I)$$	Enforces nematic orientation at boundary
Uniaxial ansatz	$Q = s(n \otimes n - \tfrac{1}{3}I)$	Structure of defect cores, hedgehog profile
Symmetry breaking	Biaxial perturbations in $Q$	Instability of uniaxial minimizer at low $T$

The Landau–de Gennes functional provides a comprehensive, physically controlled, and mathematically rigorous tool for exploring nematic liquid crystal physics: phase transition mechanisms, defect structure, and device-oriented phenomena, with a complex landscape of symmetry-breaking and stability determined by the interplay of elastic and bulk potentials (Henao et al., 2011, Shi et al., 2022).