Three-dimensional formulation of curved nematic shells

TL;DR

This work shows that curved nematic shells require a full three-dimensional Q-tensor description to correctly capture orientational order in the presence of curvature, activity, and soft anchoring. It demonstrates that 2D surface reductions can introduce artefacts, such as a spurious second-order isotropic–nematic transition, and highlights curvature-induced couplings to order that are missed by purely tangential projections. The authors develop a comprehensive thin-film nematohydrodynamics framework for adhered curved shells, including weak anchoring and active stresses, and analyze stability on curved substrates like sinusoidal surfaces. Overall, the study provides theoretical grounds for using 3D Q-tensor formulations in curved nematic shells and offers a practical pathway to model their coupled nematic and hydrodynamic dynamics in biological and soft-matter systems.

Abstract

In soft matter, the phase of nematic liquid crystals can be made from anisotropic molecules in single component materials, or as a suspension of mesoscopic nematogens. The later offers more versatility in the experimental design of complex shapes, in particular thin curved shells, and is often found in biological systems at multiple scales from cells to tissues. Here, we investigate theoretically the transition from three-dimensional nematics to a two-dimensional description restricted to a tangent plane, using a mean-field approach. We identify a transition from first to second order isotropic-nematic transition in presence of weak tangential anchoring. Then, we clarify the conditions under which a two-dimensional description of thin nematic shells is relevant. Nonetheless, using the example of active nematic stress, we identify physical differences between two- and three-dimensional descriptions in curved geometry. Finally, we construct a thin film approximation of nematohydrodynamics for a nematic shell in contact with a curved substrate. All together, those results show that a tangential restriction of nematic orientation must be used with care in presence of curvature, activity, or weak anchoring boundary conditions.

Figures (5)

• Figure 1: Phase coexistence in the Maier-Saupe framework with soft tangential anchoring. (A) Sketch of a thin nematic shell (top left) and a nematic surface (bottom left) in the limit of vanishing thickness. (Orange lines) nematic director. Distributions of $\hat{\mathbf u}$-orientations at a surface point: planar-isotropic called z-oblate (top right), planar-biaxial called t-biaxial (middle right), planar-uniaxial called t-prolate (middle right) and normal-uniaxial called z-prolate (bottom right). (B-E) Equilibrium states of the Maier-Saupe potential with quadratic tangential anchoring. Normal nematic tensor component $Q_{zz}$ as a function of $T$ for $W=0$ (B), $W=0.01$ (D), $W=0.1$ (E). (Red) biaxial coefficient $\beta^2$. (C) Phase diagram of the isotropic-nematic states in the parametric plane $(T,W)$. (Dashed line) case $W=0.01$.
• Figure 2: Schematics of a thin nematic shell (light gray) adhered to a substrate (dark gray). At the surface of the substrate $\mathcal{S}$, surface coordinates $(u,v)$ with tangent vectors $(\mathbf b_u,\mathbf b_v)$ and normal vector $\hat{\bm\nu}$ are used. A level set of $\mathcal{S}$ is used with normal coordinate $n$, defining the surface $\mathcal{S}_{n}$ with the substrate at $n=0$. Each material point is described with the coordinates $(u,v,n)$. The free surface of the material is at $n=H(u,v)$ with interfacial tangent vectors $(\bm B_u,\bm B_v)$ and interfacial normal vector $\bm N$. Orange: nematic orientation at various spatial points.
• Figure A1: Landau-de Gennes free energy with quadratic tangential anchoring. (A,C) Normal nematic tensor component $Q_{\nu\nu}$ as a function of $a$ for $w=0.25$, with biaxial coefficient $\beta^2$ (red). $b=0.5$ (A), $b=3.5$ (C). (B,D) Phase diagram of the isotropic-nematic states in the parametric plane $(a,w)$. $b=0.5$ (B), $b=3.5$ (D).
• Figure A2: Phase diagram of the Landau-de Gennes free energy with quartic (A,C) or hexatic (B,D) tangential anchoring, as a function of $a$ and $w$. The case $b=0.5$ is shown on (A,B), and the case $b=3.5$ on (C,D).
• Figure A3: Phase diagram of the Maier-Saupe free energy with quartic (A) or hexatic (B) tangential anchoring, as a function of $T$ and $W$.