Landau-de Gennes Theory¶

Here we briefly overview the Landau-de Gennes theoretical framework for nematic liquid crystals implemented in open-Qmin. For a fuller explanation with references, please see Section 2 of our article in Frontiers in Physics.

The Q-tensor¶

Nematic liquid crystals are fluids with anisotropy, that is, properties that vary depending on direction. Nearby particles are oriented preferentially along a common direction, \(\hat n\), called the nematic director. Mathematically, a director is like a unit vector but with an added “head-tail” symmetry by which \(- \hat n\) represents the same physical state as \(\hat n \). Because the director may vary from place to place in the liquid crystal, we speak of a director field \(\hat n(\mathbf{r})\).

Partly to account for this \(\hat n = - \hat n \) symmetry, the nematic liquid crystal’s orientational state is better described by a symmetric, second-rank tensor (which in \(d\) dimensions is represented by a symmetric \(d\times d\) matrix). This is the nematic orientation tensor, or Q-tensor, \({\bf Q}\). In open-Qmin, the state of the nematic liquid crystal is represented by a value of the Q-tensor at each lattice site.

From director to Q-tensor¶

Because we are only interested in anisotropic properties of the liquid crystal, we subtract out the isotropic properties by making the Q-tensor traceless, \(Q_{xx} + Q_{yy} + Q_{zz} = 0\).

Uniaxial nematics¶

The simplest relation between \(\bf Q\) and \(\hat n\) is given by

\[ Q_{\alpha \beta} = \tfrac{3}{2} S \left(n_\alpha n_\beta - \tfrac{1}{3} \delta_{\alpha \beta}\right) \quad \text{(uniaxial limit)} \]

where \(\delta\) is the Kronecker delta and the scalar prefactor \(S\) is the nematic degree of order.

The above relation is only true in the uniaxial limit, in which the particles of the nematic have no orientational preference among the directions in the plane perpendicular to \(\hat n\).

Biaxial nematics¶

A more general expression for \(\bf Q\) allows for biaxial order \(S_B\) which favors one direction, \(\hat m\), in the plane perpendicular to \(\hat n \):

\[ Q_{\alpha \beta} = \tfrac{3}{2} S \left( n_\alpha n_\beta - \tfrac{1}{3} \delta_{\alpha \beta} \right) + \tfrac{1}{2} S_B \left( m_\alpha m_\beta - l_\alpha l_\beta \right) .\]

Here \(\hat l \equiv n \times \hat m\).

From Q-tensor to director¶

For a given \(Q_{\alpha \beta}\), the degree of nematic order \(S\) is recovered as the greatest eigenvalue, and the director \(\hat n \) is the corresponding eigenvector:

\[ Q_{\alpha \beta} n_\beta = S n_\alpha \]

Because \({\bf Q}\) is traceless, the remaining two eigenvalues must sum to \(-S\). The difference between these two eigenvalues becomes the degree of biaxial order \(S_B\). \({\bf Q}\) is diagonal in the basis \(\{ \hat n, \hat m, \hat l \}\), with diagonal elements

\[ {\bf Q}^{\text{(diag)}} = \mathrm{diag}\left(S, \tfrac{1}{2} (-S + S_B), \tfrac{1}{2} (-S-S_B) \right) \]

Landau-de Gennes free energy¶

open-Qmin conducts numerical minimization of the Landau-de Gennes free energy, which is a functional of the Q-tensor. Schematically:

\[ {\mathcal F}[{\bf Q}] = \int_V \left( f_{\text{bulk}} + f_{\text{distortion}} + f_{\text{external}}\right) dv + \sum_{\alpha} \int_{S_\alpha} \left( f^\alpha_{\text{boundary}}\right) ds. \]

The first integral is over the volume of the nematic; the second term is a sum of surface integrals over each boundary surface \(S_\alpha\). The energy density terms are described below.

Bulk free energy¶

The bulk free energy density describes the isotropic-nematic phase transition in the Landau paradigm, using rotational invariants of \({\bf Q}\):

\[ f_{\text{bulk}} = \tfrac{1}{2} A {\, \rm tr} ({\bf Q}^2) + \tfrac{1}{3} B {\,\rm tr} ({\bf Q}^3) + \tfrac{1}{4} C \left( {\rm tr}({\bf Q}^2) \right)^2 .\]

Here, \(A\), \(B\), and \(C\) are material constants with \(A\) depending on temperature \(T\) as \(A\propto (T-T^*_{NI})\), with \(T^*_{NI}\) the temperature below which the isotropic phase is unstable.

In the uniaxial limit, \(f_{\text{bulk}}\) becomes a polynomial in the nematic degree of order,

\[ f_{\text{bulk}} = \tfrac{3}{4} A S^2 + \tfrac{1}{4} B S^3 + \tfrac{9}{16} C S^4 , \]

which is minimized either by \(S=0\) (isotropic phase) or by

\[ S = S_0 \equiv \frac{ - B + \sqrt{ B^2 - 24 A C}}{6 C} \qquad \text{(preferred degree of order)} . \]

Distortion free energy¶

The distortion (or “elastic”) free energy density penalizes spatial gradients of \({\bf Q}\) as follows:

\[\begin{align*} f_{\text{distortion}} &= \tfrac{1}{2} L_1 \frac{\partial Q_{ij}}{\partial x_k} \frac{\partial Q_{ij}}{\partial x_k} + \tfrac{1}{2} L_2 \frac{\partial Q_{ij}}{\partial x_j} \frac{\partial Q_{ik}}{\partial x_k} + \tfrac{1}{2} L_3 \frac{\partial Q_{ik}}{\partial x_j} \frac{\partial Q_{ij}}{\partial x_k} \\ & \quad + \tfrac{1}{2} L_4 \epsilon_{lik} Q_{lj} \frac{\partial Q_{ij}}{\partial x_k} + \tfrac{1}{2} L_6 Q_{lk} \frac{\partial Q_{ij}}{\partial x_l} \frac{\partial Q_{ij}}{\partial x_k} . \end{align*}\]

Here, Einstein summation over repeated indices is implied, and \(\epsilon\) is the Levi-Civita tensor. Please pay close attention to how we define the \(L_i\) coefficients here, since other definitions are also commonly used in the literature.

Relation to Frank elastic constants¶

In the uniaxial limit, \(f_{\text{distortion}}\) maps onto the Frank-Oseen elastic free energy density,

\[\begin{align*} f_{\text{FO}} &= \tfrac{1}{2} \left\{ K_1 (\nabla \cdot \hat n )^2 + K_2 \left( \hat n \cdot (\nabla \times \hat n ) + q_0 \right)^2 \right. \\ & \qquad\; \left. + K_3 \left| (\hat n \cdot \nabla) \hat n \right|^2 + K_{24} \nabla \cdot \left[ (\hat n \cdot \nabla ) \hat n - \hat n (\nabla \cdot \hat n ) \right] \right\}, \end{align*}\]

where \(K_1\), \(K_2\), \(K_3\), and \(K_{24}\) are respectively the splay, twist, bend, and saddle-splay elastic constants, and \(q_0\) is the spontaneous chiral wavenumber in chiral nematics. The relations between these constants and the Landau-de Gennes distortion free energy density parameters in the uniaxial limit are:

\[\begin{align*} L_1 &= \frac{2}{27 S^2} (K_3 - K_1 + 3 K_2) \\ L_2 &= \frac{4}{9 S^2} (K_1 - K_{24}) \\ L_3 &= \frac{4}{9 S^2} (K_{24} - K_2) \\ L_4 &= -\frac{8}{9S^2} q_0 K_2 \\ L_6 &= \frac{4}{27 S^3} (K_3 - K_1) \end{align*}\]

Inverting these relations gives the following forms for the Frank-Oseen constants:

\[\begin{align*} K_1 &= \tfrac{9}{4} S^2 (2 L_1 + L_2 + L_3 - L_6 S) \\ K_2 &= \tfrac{9}{4} S^2 (2 L_1 - S L_6) \\ K_3 &= \tfrac{9}{4} S^2 (2 L_1 + L_2 + L_3 + 2 S L_6) \\ K_{24} &= -\tfrac{9}{4} S^2 (-2 L_1 - L_3 + S L_6) \\ q_0 &= -\frac{L_4}{2(2L_1 - S L_6)} \end{align*}\]

One-constant approximation¶

A common simplifying assumption for the Frank-Oseen free energy density is the “one-constant approximation”, sometimes called “isotropic elasticity”:

\[ K_1 = K_2 = K_3 = K_{24} \equiv K\]

along with \(q_0=0\). This assumption is a reasonable approach for many common molecular liquid crystals, where the elastic constants often have, at least, the same order of magnitude. Along with dramatically simplifying analytical approaches, the one-constant approximation also permits much faster computation with the corresponding choice of \(L_i\) coefficients:

\[ L_2 = L_3 = L_4 = L_6 = 0, \]

\[ L_1 = \frac{2}{9 S^2} K. \]

This leaves only one term in the distortion free energy density:

\[ f_{\text{distortion}} ^{(1)} = \tfrac{1}{2} L_1 \frac{\partial Q_{ij}}{\partial x_k} \frac{\partial Q_{ij}}{\partial x_k} . \]

External fields free energy¶

External electric fields \(\mathbf{E}\) and magnetic fields \(\mathbf{H}\) contribute the following free energy densities, respectively, calculated at each site in the nematic:

\[ f_E = -\tfrac{1}{2} \varepsilon_0 E_i \varepsilon_{ij} E_j, \qquad \varepsilon_{ij} = \varepsilon \delta_{ij} + \Delta \varepsilon Q_{ij} \]

\[ f_H = - \tfrac{1}{2} \mu_0 H_i \chi_{ij} H_j, \qquad \chi_{ij} = \chi \delta_{ij} + \Delta \chi Q_{ij} \]

where

\(\varepsilon_0\) is the vacuum permittivity
\(\varepsilon_{ij}\) is the material’s dielectric tensor
\(\varepsilon\) is the material’s dielectric constant (relative permittivity)
\(\Delta \varepsilon\) is the material’s dielectric anisotropy
\(\mu_0\) is the magnetic permeability of free space
\(\chi_{ij}\) is the material’s magnetic susceptibility tensor
\(\chi\) is the isotropic part of the material’s magnetic susceptibility
\(\Delta \chi\) is the anisotropic part of the material’s magnetic susceptibility

and \(\delta_{ij}\) is the Kronecker delta.

Boundary free energy¶

At boundary surfaces, liquid crystals typically experience an anisotropic surface tension, called “anchoring”, that depends on the relative orientations of the director \(\hat n \) and a certain special direction \(\hat \nu^\alpha\) picked out by a given point on the surface. (The superscript \(\alpha\) indexes the different boundary surfaces.)

open-Qmin has two categories of anchoring conditions: oriented and degenerate planar.

Oriented (including homeotropic) anchoring¶

Oriented anchoring conditions penalize the director’s deviations from a unique direction \(\hat\nu^\alpha\) preferred by the surface. One common example is “homeotropic” anchoring, in which \(\hat \nu^\alpha\) is the surface normal. Another is “oriented planar” ancohring, in which a particular direction in the surface’s tangent plane serves as \(\hat \nu^\alpha\).

For oriented anchoring, the surface free energy density at boundary surface \(\alpha\) takes the Nobili-Durand form:

\[ f_{\text{boundary}}^\alpha = W^\alpha_{\text{ND}} (Q_{ij} - Q_{ij}^\alpha) (Q_{ij} - Q_{ij}^\alpha), \]

where \(W^\alpha_{\text{ND}} > 0\) is the anchoring strength of surface \(\alpha\) and the surface-preferred Q-tensor is

\[ Q^\alpha_{ij} = \tfrac{3}{2} S_0 (\nu^\alpha_i \nu^\alpha_j - \tfrac{1}{3} \delta_{ij} ) .\]

Degenerate planar anchoring¶

Surfaces with degenerate planar anchoring disfavor director orientation along the surface normal \(\hat \nu^\alpha\), but exhibit no preference (i.e. a degeneracy) among the directions in the surface’s tangent plane. When modeling 3D nematics, it is not sufficient to use the above Nobili-Durand form with a negative anchoring strength. Instead, we use the Fournier-Galatola form:

\[ f_{\text{boundary}}^\alpha = W^\alpha_{\text{FG}} ( \tilde Q_{ij} - \tilde Q_{ij}^\perp ) ( \tilde Q_{ij} - \tilde Q_{ij}^\perp ) , \]

where \( \tilde Q_{ij} = Q_{ij} + \tfrac{1}{2} S_0 \delta_{ij}\), and \(\tilde Q_{ij}^\perp\) is the projection of \(\tilde Q_{ij}\) onto the plane perpendicular to \(\hat \nu^\alpha\),

\[ \tilde Q^\perp_{ij} = P_{ik} \tilde Q_{kl}P_{lj}, \]

using the projection operator \(P_{ij} = \delta_{ij} - \nu^\alpha _i \nu^\alpha_j \) .

Anchoring strength relation to Rapini-Papoular form¶

In terms of the director, both oriented and degenerate planar anchoring are often modeled with the Rapini-Papoular free energy density,

\[ - \tfrac{1}{2} W^\alpha_{\text{RP}} (\hat \nu^\alpha \cdot \hat n )^2, \]

with \(W_{\text{RP}}>0\) for oriented anchoring, and \(<0\) for degenerate planar anchoring.

Comparing to the Nobili-Durand and Fournier-Galatola forms, the anchoring strengths are related in both cases by

\[ |W_{\text{RP}}^\alpha| = 9 S_0^2 W^\alpha_{\text{ND,FG}}, \]

assuming \({\bf Q}\) is uniaxial with leading eigenvalue \(S=S_0\).

User Guide to open-Qmin

Landau-de Gennes Theory

Contents