Modeling deformation and chaining of flexible shells in a nematic solvent with finite elements on an adaptive moving mesh

A micrometer-scale elastic shell immersed in a nematic liquid crystal may be deformed by the host if the cost of deformation is comparable to the cost of elastic deformation of the nematic. Moreover, such inclusions interact and form chains due to quadrupolar distortions induced in the host. A continuum theory model using finite elements is developed for this system, using mesh regularization and dynamic refinement to ensure quality of the numerical representation even for large deformations. From this model, we determine the influence of the shell elasticity, nematic elasticity, and anchoring condition on the shape of the shell and hence extract parameter values from an experimental realization. Extending the model to multibody interactions, we predict the alignment angle of the chain with respect to the host nematic as a function of aspect ratio, which is found to be in excellent agreement with experiments.

Figure 1

(a) Fluorescence microscopy images of quantum dot microshells [12] show subtly elongated shells that form chains, here at ${27}^{\circ }$ to the alignment axis of the nematic host. (b) Schematic of the computational domain. (c) Fully relaxed simulation with ${\mathrm{\Gamma }}_{\sigma }=20$, ${\mathrm{\Gamma }}_{\kappa }=0$, and ${\mathrm{\Gamma }}_W=100$, shaded to indicate the relative elastic energy of each mesh face. Two copies of the computational domain are depicted, one reflected about the $z$ axis, to assist visualization.

Figure 2

(a–c) Aspect ratio for simulated shells as a function of surface tension for ${\mathrm{\Gamma }}_{\kappa }=0$, 0.1, and 1.0. The shading of each data point tells the value of ${\mathrm{\Gamma }}_W$ as indicated by the labels in panel (a). The gray region denotes the 90th percentile of the range of aspect ratios observed in the experimental data, with the darker line indicating the median of 1.07. The inset plot in panel (c) shows the experimental aspect ratios as a function of average shell radius. Inset graphics visualize the shells for selected points with ${\mathrm{\Gamma }}_W=100$.

Figure 3

(a) An experimental crossed-polarizer image of a shell imaged close to the central plane. (b–d) Simulated crossed-polarizer images for three shells with different anchoring strengths (${\mathrm{\Gamma }}_W=10$, 50, and 100) and aspect ratios equal to the median aspect ratio observed experimentally. The director field superimposed over the simulated images shows stronger anchoring conditions leading to more curvature in the nematic field.

Figure 4

(a–c) Simulation snapshots of two shells with aspect ratio = 1.18 relaxing from ${\theta }_0={70}^{\circ }$ to ${\theta }_p={30}^{\circ }$. The shells repel initially, before finding their preferred alignment angle and attracting one another. (d) Preferred alignment angle ${\theta }_p$ as a function of shell aspect ratio for all of the simulated cases shown in Fig. 2. The black series contains cases with strong anchoring (${\mathrm{\Gamma }}_W\ge 50$) while the red series contains cases with weaker anchoring (${\mathrm{\Gamma }}_W<50$). The gray region denotes the range of aspect ratios and angles observed in the experimental data, with the darker lines indicating the medians of 1.07 and ${30}^{\circ }$.