Neural-network approach to modeling liquid crystals in complex confinement

Finding the structure of a confined liquid crystal is a difficult task since both the density and order parameter profiles are nonuniform. Starting from a microscopic model and density-functional theory, one has to either (i) solve a nonlinear, integral Euler-Lagrange equation, or (ii) perform a direct multidimensional free energy minimization. The traditional implementations of both approaches are computationally expensive and plagued with convergence problems. Here, as an alternative, we introduce an $\mathrm{unsupervised}$ variant of the multilayer perceptron (MLP) artificial neural network for minimizing the free energy of a fluid of hard nonspherical particles confined between planar substrates of variable penetrability. We then test our algorithm by comparing its results for the structure (density-orientation profiles) and equilibrium free energy with those obtained by standard iterative solution of the Euler-Lagrange equations and with Monte Carlo simulation results. Very good agreement is found and the MLP method proves competitively fast, flexible, and refinable. Furthermore, it can be readily generalized to the richer experimental patterned-substrate geometries that are now experimentally realizable but very problematic to conventional theoretical treatments.

Figure 1

(a) A neurone fires (or not) on the basis of the stimuli it receives from other neurones. (b) An artificial neurone weights the inputs it receives and generates an output. It must be designed so as to produce the desired output.

Figure 2

Example of MLP with three units in the input layer, five units in the hidden (intermediate) layer, and two units in the output layer.

Figure 3

The HNW potential: the molecules see each other (approximately) as uniaxial hard ellipsoids of axes $({\sigma }_0,{\sigma }_0,\kappa {\sigma }_0$), but the wall sees a molecule as a hard line of length $L$, which need not equal $\kappa {\sigma }_0$. Physically, this means that molecules are able to embed their side groups into the bounding walls. Varying $L$ is therefore equivalent to changing the wall penetrability, which can be done independently at either wall.

Figure 4

Density ${\rho }^{*}\left(z\right)$ (top) and order parameter $Q_{zz}\left(z\right)$ (bottom) profiles for a symmetric film of HGO particles of elongation $\kappa =3$, for ${\rho }_{\mathrm{bulk}}^{*}=0.28$. The needle length is $L^{*}=0$ on both walls, inducing homeotropic anchoring (only one half of system is shown). Lines are from theory using the standard solution of the EL equation (solid) and our MNN (dashed), symbols are from simulation. This density lies in the isotropic phase.

Figure 5

Same as Fig. 4, but for ${\rho }_{\mathrm{bulk}}^{*}=0.35$. This density lies in the nematic phase.

Figure 6

Density ${\rho }^{*}\left(z\right)$ (top) and order parameter $Q_{zz}\left(z\right)$ (bottom) profiles for a symmetric film of HGO particles of elongation $\kappa =3$, for ${\rho }_{\mathrm{bulk}}^{*}=0.28$. The needle length is $L^{*}=\frac{1}{3}$ on both walls, inducing homeotropic anchoring (only one half of system is shown). Lines are from theory using the standard solution of the EL equation (solid) and our MNN (dashed), symbols are from simulation. This density lies in the isotropic phase.

Figure 7

Same as Fig. 6, but for ${\rho }_{\mathrm{bulk}}^{*}=0.35$. This density lies in the nematic phase.

Figure 8

Density ${\rho }^{*}\left(z\right)$ (top) and order parameter $Q_{zz}\left(z\right)$ (bottom) profiles for a symmetric film of HGO particles of elongation $\kappa =3$, for ${\rho }_{\mathrm{bulk}}^{*}=0.28$. The needle length is $L^{*}=\frac{2}{3}$ on both walls, inducing parallel anchoring (only one half of system is shown). Lines are from theory using the standard solution of the EL equation (solid) and our MNN (dashed), symbols are from simulation. This density lies in the isotropic phase.

Figure 9

Same as Fig. 8, but for ${\rho }_{\mathrm{bulk}}^{*}=0.35$. This density lies in the nematic phase. MNN results were calculated using two two hidden layers, $\eta =1.0$ and $\alpha =0.97$. The metastable profiles are for homeotropic alignment, the true equilibrium profiles are for planar alignment.

Figure 10

Density ${\rho }^{*}\left(z\right)$ (top) and order parameter $Q_{zz}\left(z\right)$ (bottom) profiles for a symmetric film of HGO particles of elongation $\kappa =3$, for ${\rho }_{\mathrm{bulk}}^{*}=0.28$. The needle length is $L^{*}=1$ on both walls, inducing parallel anchoring (only one half of system is shown). Lines are from theory using the standard solution of the EL equation (solid) and our MNN (dashed), symbols are from simulation. This density lies in the isotropic phase.

Figure 11

Same as Fig. 10, but for ${\rho }_{\mathrm{bulk}}^{*}=0.35$. This density lies in the nematic phase.