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From molecular model to tensor model of nematic liquid crystals through entropy decomposition

Baoming Shi, Dawei Wu, Lei Zhang, Pingwen Zhang

Abstract

In the mathematical modeling of nematic liquid crystals, a practical and physically reliable $\mathbf{Q}$-tensor model can be derived from Onsager's molecular model with the Bingham closure. However, this procedure leads to a singular entropy term that implicitly depends on $\mathbf{Q}$, creating both computational and theoretical difficulties. In this paper, we characterize this entropy contribution by splitting it into a singular but explicit leading term and an implicit but regular correction term, the latter of which is proven to be sufficiently regular to be accurately approximated numerically, for example, by neural networks. This yields a computationally convenient free energy that can be used for the computation of nematic liquid crystals. Our numerical experiments demonstrate that the resulting free energy can capture the isotropic-nematic phase transition as well as the free-boundary droplet configurations.

From molecular model to tensor model of nematic liquid crystals through entropy decomposition

Abstract

In the mathematical modeling of nematic liquid crystals, a practical and physically reliable -tensor model can be derived from Onsager's molecular model with the Bingham closure. However, this procedure leads to a singular entropy term that implicitly depends on , creating both computational and theoretical difficulties. In this paper, we characterize this entropy contribution by splitting it into a singular but explicit leading term and an implicit but regular correction term, the latter of which is proven to be sufficiently regular to be accurately approximated numerically, for example, by neural networks. This yields a computationally convenient free energy that can be used for the computation of nematic liquid crystals. Our numerical experiments demonstrate that the resulting free energy can capture the isotropic-nematic phase transition as well as the free-boundary droplet configurations.
Paper Structure (15 sections, 13 theorems, 125 equations, 5 figures, 1 table)

This paper contains 15 sections, 13 theorems, 125 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Entropy decomposition in dimension 2
  3. Bingham distribution in 2D
  4. Entropy decomposition
  5. Entropy decomposition in dimension 3
  6. Bingham distribution in 3D
  7. Entropy decomposition
  8. Proofs of lemmas
  9. Dimension reduction: embedding of the 2D model in the 3D model
  10. Neural network approximation to the correction term
  11. Numerical experiments
  12. Isotropic-nematic phase transition
  13. Droplet problem
  14. Conclusion
  15. Properties of the quasi-entropy

Key Result

Theorem 1

Let $\vb Q$ be the $d$-dimensional ($d=2,3$) tensor order parameter defined in Q-def, $f_{\vb Q}$ its Bingham closure and $S(\vb Q)=S[f_{\vb Q}]$ the molecular entropy from ons-fe-hom. Then, we have that where the correction term $\Delta S(\vb Q)$ is a Lipschitz continuous function on $\mathcal{Q}_p$ with a uniformly bounded gradient.

Figures (5)

  • Figure 1: The functions $S(q),\hat{S}(q)$ and $\Delta S(q)$ in the 2D Bingham closure.
  • Figure 2: Graph of $S(q)$ on its domain $\Delta_0$. Left panel: contour plots of $S$ as a function of $q$; right panel: gradient of $S$. The plane $\{q_1+q_2+q_3=0\}\subset\mathbb{R}^3$ is parametrized isometrically by $(\xi,\eta)\in\mathbb{R}^2$ with $\frac{\xi}{\sqrt{6}} [-1,-1,2]^T + \frac{\eta}{\sqrt{2}}[1,-1,0]^T.$ Visualization of $\nabla_0 S$ as a two-dimensional mapping within the $(\xi,\eta)$ plane uses alternating black and white contour lines to represent level curves of the logarithm of the magnitude of the function, and hue (in the order of the RGB colour wheel) for its argument. Projections of $q_1,q_2,q_3$-axes onto $\{q_1+q_2+q_3=0\}$ are plotted in the left panel.
  • Figure 3: Left panel: Accurate values of $\Delta S$; right panel: neural network approximation $\Delta S_{\rm NN}$. Both are plotted as functions of $q=\lambda(\vb Q)$ in the same way as Figure \ref{['fig:S']}. $\Delta S_{\rm NN}$ has a nice generalization beyond the physical domain (enclosed by white triangle), which ensures stable computation.
  • Figure 4: The numerically evaluated bulk energy \ref{['bing-bulk']} at uniaxial tensors \ref{['Q-uniax']} plotted against $s\in(-\frac{1}{2},1)$.
  • Figure 5: Droplets obtained from the gradient flow of \ref{['bingphf']}. Boundary of droplet (dark green) is the level set $\phi=0.5$. Colours denote biaxiality $\beta=1-6(\operatorname{tr}\vb Q^3)^2/(\operatorname{tr}\vb Q^2)^3$. White ellipsoids represent $\vb Q$-tensors, with principal axes parallel to the eigenvectors, and their lengths corresponding to eigenvalues in order.

Theorems & Definitions (25)

  • Theorem 1
  • Lemma 2
  • Lemma 3: abramowitz_handbook_2013fatkullin_critical_2005
  • proof
  • Theorem 4
  • Proposition 5
  • proof
  • Remark 1
  • Theorem 6
  • Proposition 7
  • ...and 15 more