Landau–de Gennes Model

• Landau–de Gennes is a continuum theory of liquid crystals that uses a symmetric, traceless tensor to represent local orientational order and phase transitions.
• The model integrates elastic penalties and bulk potentials, permitting asymptotic reductions to classical director theories and effective thin-film formulations.
• Its versatility extends to defect analysis, chirality incorporation, and free-boundary problems, offering practical insights for materials design and optimization.

Searching arXiv for recent and foundational Landau–de Gennes papers relevant to the requested encyclopedia article. The Landau–de Gennes model is a continuum theory of liquid crystals in which the local state is represented by a symmetric, traceless tensor $Q(x)$, and the free energy combines elastic penalties for spatial distortion with a bulk potential that selects orientationally ordered phases below a temperature threshold. In the cited literature, this framework appears in its classical nematic form, in asymptotic reductions to Oseen–Frank and thin-film models, and in modified versions that incorporate chirality, smectic layering, free boundaries, anisotropic elasticity, and homogenized microstructure effects (Majumdar et al., 19 Jan 2026, Golovaty et al., 2015, Canevari, 2015).

1. Order parameter, bulk potential, and vacuum structure

The basic Landau–de Gennes state space is

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$

The isotropic state is $Q=0$. A uniaxial state has the form

$Q=s\left(n\otimes n-\frac{I_3}{3}\right),$

where $n\in S^2$ is the director and $s$ is the scalar degree of order; biaxial states have three distinct eigenvalues (Majumdar et al., 19 Jan 2026, Bronsard et al., 2024).

In the standard quartic theory, the bulk potential is

$$f_b(Q)=\frac{A}{2}\operatorname{tr}(Q^2)-\frac{B}{3}\operatorname{tr}(Q^3)+\frac{C}{4}\big(\operatorname{tr}(Q^2)\big)^2,$$

with $A=\alpha_1(T-T_1^*)$ and $B,C>0$. In the three-dimensional setting summarized for the modified smectic model, the isotropic state minimizes $f_b$ if $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$0, whereas for $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$1 the minimizers are uniaxial with

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$2

This is the canonical thermotropic mechanism by which the bulk term selects the nematic well (Shi et al., 2024).

Several cited works use equivalent normalizations. In the vanishing-elasticity literature, the bulk density is written as

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$3

with $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$4 chosen so that $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$5, and the vacuum manifold becomes

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$6

Because $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$7, this vacuum manifold is diffeomorphic to $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$8, which is the topological source of half-integer disclinations in the classical nematic theory (Canevari, 2015).

The bulk structure can also be generalized beyond the quartic case. A sextic potential analyzed in three dimensions takes the form

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$9

with $Q=0$0 and $Q=0$1 chosen so that $Q=0$2. Its vacuum manifold is no longer uniaxial: $Q=0$3 where $Q=0$4 solves $Q=0$5. In that model, the well manifold is a compact smooth three-dimensional submanifold of $Q=0$6 diffeomorphic to $Q=0$7 (Wang et al., 2024).

2. Elastic sector and variational formulation

The simplest Landau–de Gennes energy combines the bulk potential with a one-constant elastic term,

$Q=0$8

or, equivalently, $Q=0$9 with $Q=s\left(n\otimes n-\frac{I_3}{3}\right),$0 (Canevari, 2015). This form underlies much of the asymptotic defect theory.

A broader three-constant elastic density is

$Q=s\left(n\otimes n-\frac{I_3}{3}\right),$1

and occurs both in thin-film reduction and in homogenization settings (Golovaty et al., 2015, Ceuca, 2020). A divergence-penalized variant adds

$Q=s\left(n\otimes n-\frac{I_3}{3}\right),$2

leading, in the small-particle exterior-domain problem, to the linear anisotropic Euler–Lagrange system

$Q=s\left(n\otimes n-\frac{I_3}{3}\right),$3

in $Q=s\left(n\otimes n-\frac{I_3}{3}\right),$4, with homeotropic data on the colloid and uniaxial far-field alignment (Bronsard et al., 2024).

The elastic sector can also be reformulated to improve well-posedness for disparate Frank constants. A quartic elastic density proposed within generalized Landau–de Gennes theory is

$Q=s\left(n\otimes n-\frac{I_3}{3}\right),$5

For $Q=s\left(n\otimes n-\frac{I_3}{3}\right),$6, this density is nonnegative and coercive, and the associated minimization problem is well posed; the construction avoids the unboundedness from below associated with certain cubic elastic theories (Golovaty et al., 2019).

Variationally, existence results in the cited literature rely on the direct method under coercivity and lower semicontinuity assumptions. In the chiral–smectic model, for instance, the admissible space is $Q=s\left(n\otimes n-\frac{I_3}{3}\right),$7, with $Q=s\left(n\otimes n-\frac{I_3}{3}\right),$8 and $Q=s\left(n\otimes n-\frac{I_3}{3}\right),$9, together with Dirichlet anchoring on a bounded simply connected Lipschitz domain. Under explicit inequalities on $n\in S^2$0, the energy is weakly lower semicontinuous and possesses a global minimizer (Majumdar et al., 19 Jan 2026).

3. Oseen–Frank correspondence and asymptotic limits

A persistent theme in the cited work is the reduction of Landau–de Gennes functionals to director theories when the bulk term strongly confines $n\in S^2$1 to the uniaxial well. In the chiral–smectic theory, the Oseen–Frank limit is obtained by rescaling $n\in S^2$2 with $n\in S^2$3 and letting $n\in S^2$4. Then $n\in S^2$5 converges to the uniaxial manifold

$n\in S^2$6

and the $n\in S^2$7-limit becomes an Oseen–Frank-type functional in $n\in S^2$8, coupled to the smectic scalar field. In that reduction, the Frank constants are

$n\in S^2$9

(Majumdar et al., 19 Jan 2026).

The quartic elastic theory achieves an exact four-constant Oseen–Frank reduction on orientable uniaxial states. For $s$0, the elastic density reduces to

$s$1

with the parameter correspondence

$s$2

The corresponding dimensionless family $s$3 $s$4-converges to $s$5, and minimizers converge strongly in $s$6 (Golovaty et al., 2019).

Other asymptotic regimes emphasize special elastic hierarchies. In the cholesteric–smectic model, letting the twist modulus $s$7 with $s$8 bounded forces

$s$9

so the limiting director is helical,

$$f_b(Q)=\frac{A}{2}\operatorname{tr}(Q^2)-\frac{B}{3}\operatorname{tr}(Q^3)+\frac{C}{4}\big(\operatorname{tr}(Q^2)\big)^2,$$0

Depending on $$f_b(Q)=\frac{A}{2}\operatorname{tr}(Q^2)-\frac{B}{3}\operatorname{tr}(Q^3)+\frac{C}{4}\big(\operatorname{tr}(Q^2)\big)^2,$$1, the limiting state is cholesteric $$f_b(Q)=\frac{A}{2}\operatorname{tr}(Q^2)-\frac{B}{3}\operatorname{tr}(Q^3)+\frac{C}{4}\big(\operatorname{tr}(Q^2)\big)^2,$$2, helical smectic $$f_b(Q)=\frac{A}{2}\operatorname{tr}(Q^2)-\frac{B}{3}\operatorname{tr}(Q^3)+\frac{C}{4}\big(\operatorname{tr}(Q^2)\big)^2,$$3, or SmC* $$f_b(Q)=\frac{A}{2}\operatorname{tr}(Q^2)-\frac{B}{3}\operatorname{tr}(Q^3)+\frac{C}{4}\big(\operatorname{tr}(Q^2)\big)^2,$$4 (Majumdar et al., 19 Jan 2026).

The same reduction logic appears in thin-film problems. When thickness and correlation length both vanish, the limiting energies contain leading-order perimeter terms and lower-order vortex terms, so the Landau–de Gennes theory simultaneously produces Allen–Cahn-type interface costs and Ginzburg–Landau-type singular structures (Novack, 2018).

4. Defects, singular sets, and convergence of minimizers

The defect theory of Landau–de Gennes minimizers is strongly shaped by the topology of the vacuum manifold and by the energy scaling regime. In two dimensions, for smooth boundary data valued in $$f_b(Q)=\frac{A}{2}\operatorname{tr}(Q^2)-\frac{B}{3}\operatorname{tr}(Q^3)+\frac{C}{4}\big(\operatorname{tr}(Q^2)\big)^2,$$5, minimizers in the low-temperature regime are maximally biaxial near singularities: the biaxiality parameter

$$f_b(Q)=\frac{A}{2}\operatorname{tr}(Q^2)-\frac{B}{3}\operatorname{tr}(Q^3)+\frac{C}{4}\big(\operatorname{tr}(Q^2)\big)^2,$$6

reaches the value $$f_b(Q)=\frac{A}{2}\operatorname{tr}(Q^2)-\frac{B}{3}\operatorname{tr}(Q^3)+\frac{C}{4}\big(\operatorname{tr}(Q^2)\big)^2,$$7, while $$f_b(Q)=\frac{A}{2}\operatorname{tr}(Q^2)-\frac{B}{3}\operatorname{tr}(Q^3)+\frac{C}{4}\big(\operatorname{tr}(Q^2)\big)^2,$$8, excluding isotropic melting in the core (Canevari, 2013).

In three dimensions and under the logarithmic energy scaling

$$f_b(Q)=\frac{A}{2}\operatorname{tr}(Q^2)-\frac{B}{3}\operatorname{tr}(Q^3)+\frac{C}{4}\big(\operatorname{tr}(Q^2)\big)^2,$$9

the one-constant quartic model produces line defects. There exists a closed set $A=\alpha_1(T-T_1^*)$0 of finite length such that $A=\alpha_1(T-T_1^*)$1 converges strongly in $A=\alpha_1(T-T_1^*)$2 to a locally minimizing harmonic map $A=\alpha_1(T-T_1^*)$3. The support of the limiting energy measure is $A=\alpha_1(T-T_1^*)$4, the interior part defines a stationary $A=\alpha_1(T-T_1^*)$5-varifold, the density is constant and equal to $A=\alpha_1(T-T_1^*)$6, and $A=\alpha_1(T-T_1^*)$7 is locally a finite union of straight line segments (Canevari, 2015).

Point defects have a different inner asymptotics. For global minimizers in three dimensions, if $A=\alpha_1(T-T_1^*)$8 is a singular point of the limiting harmonic map $A=\alpha_1(T-T_1^*)$9, then suitably chosen blow-ups $B,C>0$0 converge in $B,C>0$1 to a local minimizer on $B,C>0$2 whose tangent map at infinity is the standard hedgehog

$B,C>0$3

This implies that $B,C>0$4 is uniformly approximated by the Oseen–Frank minimizer outside an $B,C>0$5 neighborhood of the defect core (Geng et al., 2022).

The sextic theory changes the well topology and, with it, the defect taxonomy. In the bounded-energy regime, minimizers converge to locally minimizing harmonic maps into the biaxial vacuum manifold $B,C>0$6, and the singular set is locally finite. In the logarithmically divergent regime, the limiting line singular set $B,C>0$7 is closed, countably $B,C>0$8-rectifiable, and in compact subsets is a finite union of closed straight line segments; the limiting density takes the quantized values $B,C>0$9 or $f_b$0, where $f_b$1 (Wang et al., 2024).

Recent convergence refinements sharpen the classical vanishing-elasticity theory. For local minimizers with uniformly bounded energy and $f_b$2 norm, there exists a subsequence $f_b$3 strongly in $f_b$4, and the convergence is optimal in the sense that

$f_b$5

and compact $f_b$6. The same analysis yields the sharp bulk-energy rate

$f_b$7

with matching lower bounds in the hedgehog example (Fu et al., 20 Jul 2025).

5. Thin films, curved surfaces, and reduced geometries

When the domain is thin, Landau–de Gennes theories admit rigorous dimension reduction. For planar films occupying $f_b$8, with strong Dirichlet conditions on the lateral boundary and weak anchoring on the top and bottom, the rescaled energy $f_b$9 $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$00-converges to a two-dimensional functional $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$01 defined on

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$02

where $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$03 is the set of $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$04-tensors minimizing the leading-order surface energy. In the regime emphasized in that work, $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$05 encodes the requirement that the film normal $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$06 is an eigenvector of $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$07, and the limiting problem reduces, in suitable parameter regimes, to Allen–Cahn- or Ginzburg–Landau-type energies for an effective planar order parameter $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$08 (Golovaty et al., 2015).

A different thin-film scaling, in which both thickness and nematic correlation length vanish, yields a singularly perturbed family whose $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$09-limit consists of perimeter terms associated with the connected components of the zero set of an effective potential $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$10. In simply connected domains with planar anchoring and degree $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$11, minimizers satisfy the asymptotic law

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$12

so the reduced model combines an interface energy with a vortex contribution (Novack, 2018).

On curved substrates, the thin-film limit retains a nontrivial remnant of the normal derivative. The limiting surface energy is

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$13

where

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$14

This reduced energy depends explicitly on the surface normal and curvature. On a frustum, the equal-constant reduction leads to an effective director equation with curvature-induced preference for different winding numbers, and the minimizing class changes with the cone angle (Golovaty et al., 2016).

Surface Landau–de Gennes models derived from thin-film limits can preserve distinct three-dimensional features depending on how the normal eigenvalue is handled. In the general surface theory with tangential anchoring, the reduced tensor is decomposed as

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$15

and the surface energy contains the explicit curvature couplings

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$16

These terms align the in-plane director with principal curvature lines when $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$17, and different prescriptions for $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$18 produce either three-dimensional first-order or effectively two-dimensional second-order transitions (Nestler et al., 2019).

6. Smectic, chiral, and free-boundary extensions

The classical Landau–de Gennes tensor model can be enlarged by coupling orientational order to positional order. A modified Landau–de Gennes theory for Smectic-A introduces a real scalar $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$19 representing density deviation from the average molecular density. In that model, the authors prove existence and regularity of global minimizers in three dimensions, show that the theory can capture the isotropic–nematic–smectic phase transition as a function of temperature under suitable assumptions, and analyze stable smectic phases on a square domain with tangent boundary conditions. Their asymptotics for $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$20 and $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$21 relate layer number and thickness to the phenomenological parameters (Shi et al., 2024).

A more elaborate chiral theory couples the tensor $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$22 to a real-valued smectic modulation $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$23. Its total free energy is

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$24

where the layering term is

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$25

and the director–layer coupling is

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$26

The theory yields a complete sequence of temperature-driven symmetry-breaking transitions: cholesteric $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$27 helical smectic $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$28 SmC*, with explicit stability criteria and a pitchfork bifurcation at

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$29

(Majumdar et al., 19 Jan 2026).

Free boundaries can also be incorporated by coupling $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$30 to a phase field $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$31. In the diffuse-interface theory for nematic droplets in isotropic liquid, the energy is

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$32

with

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$33

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$34

and

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$35

The model admits global minimizers, satisfies a uniform maximum principle under explicit assumptions, and has a sharp-interface $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$36-limit in which the interfacial term blends perimeter and weak anchoring (Wu et al., 2024).

At still another level, a complex smectic order parameter $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$37 coupled to a director field $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$38 gives rise to surface smectic states. In the asymptotic regime $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$39 with $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$40, the ground-state energy satisfies

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$41

and the smectic order parameter localizes near the boundary where the reduced half-space energy density $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$42 is negative. This establishes the existence of a surface smectic state (Fournais et al., 2016).

7. Selected variants, reduced models, and materials-scale applications

Beyond the standard bulk–elastic theory, the cited literature develops several specialized variants that modify defect morphology, solution landscapes, or effective material coefficients.

Variant	Added structure	Reported consequence
Divergence-penalized exterior problem	$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$43	Saturn ring remains equatorial; biaxial region grows with $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$44 (Bronsard et al., 2024)
Cubic microlattice homogenization	Surface anchoring on a dilute scaffold	Effective bulk coefficients can be shifted, including the phase transition temperature (Ceuca, 2020)
2D elastic anisotropy	$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$45 term	Stable $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$46, $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$47, and $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$48 branches, with $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$49 favored for large $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$50 and large $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$51 (Han et al., 2021)
3D prism reduction	Reduced $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$52-tensor on polygonal cross-sections	3D critical points correspond to pathways on 2D solution landscapes (Han et al., 2022)
Functional RG	Flow of $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$53, cubic/quartic couplings, $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$54	First-order NI transition; $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$55 (Qin et al., 2018)

The divergence-penalized model is formulated on $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$56 with homeotropic anchoring at the colloid and a uniform uniaxial far field. In that linearized small-particle regime, the energy is convex, minimizers are unique, and numerical results show that increasing $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$57 only marginally decreases the Saturn ring radius while enlarging the biaxial region around the ring (Bronsard et al., 2024).

Homogenization of a connected cubic microlattice scaffold embedded in a nematic host produces a surface contribution $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$58 in the effective bulk energy. In the cubic-symmetry case, suitable surface anchoring can realize

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$59

so the effective coefficients $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$60 can be tuned through scaffold geometry and surface chemistry. In particular,

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$61

which shifts the characteristic transition temperature (Ceuca, 2020).

In reduced two-dimensional models, elastic anisotropy reorganizes the entire equilibrium landscape. On a truncated square with tangent boundary conditions, the energy

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$62

admits a symmetric critical point for every $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$63, which is globally stable for small $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$64. Numerical and asymptotic analysis reveal at least five classes of symmetric states: $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$65, $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$66, $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$67, and $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$68. The $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$69 branch becomes energetically preferable for large domain size and large $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$70 (Han et al., 2021).

A reduced three-dimensional prism model transfers this landscape logic into polygonal cross-sections. At the special temperature $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$71, the reduced functional

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$72

supports mixed three-dimensional critical points that correspond to paths between two-dimensional critical points on the cross-section. The numerical examples on cuboids and hexagonal prisms show explicitly how multistability in three dimensions can be tailored by two-dimensional solution landscapes (Han et al., 2022).

At a more coarse-grained statistical level, a functional renormalization-group treatment of the Landau–de Gennes model uses the flow of the effective potential $$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$73, together with running cubic and quartic couplings and anomalous dimension, to study the nematic–isotropic transition. The computed effective potential displays a first-order transition, and the resulting estimate

$$S_0:=\{Q\in \mathbb{R}^{3\times 3}: Q^T=Q,\ \operatorname{tr}Q=0\}.$$74

substantially improves on earlier values discussed in that work’s comparison (Qin et al., 2018). A plausible implication is that the Landau–de Gennes model is not only a continuum variational theory but also a viable starting point for fluctuation analyses that probe the limits of mean-field predictions.