Nematic Bubbles and the Breaking of Spherical Symmetry

Abstract

The emergence of nematic order on deformable closed surfaces plays a pivotal role in the morphogenesis of active biological matter, such as the regeneration of Hydra. In this work, we present a continuum model that couples the two-dimensional Landau-de Gennes order tensor, describing in-plane nematic ordering, with the mechanics of a mass-conserving, deformable spherical shell. By investigating the isotropic-to-nematic phase transition driven by a reduction in temperature, mimicking the natural induction of nematic order in actomyosin fibres, we perform both linear and weakly non-linear bifurcation analyses. The onset of nematic ordering spontaneously breaks spherical symmetry, yielding distinct equilibrium morphologies governed by the shell's deformability. Axisymmetric configurations, featuring two +1 defects at the poles, emerge via a discontinuous bifurcation, resulting in a globally stable prolate shape, alongside a metastable oblate shape. Non-axisymmetric configurations, featuring four +1/2 defects arranged in a square, arise via a continuous bifurcation. Shell softness drives the first-order character of the transition, while in the limit of infinite stiffness all bifurcations become continuous. Integer defects strongly couple with local mass redistribution, manifesting as shell thinning or thickening, whilst half-integer defects induce no such local deformation. These findings provide a purely mechanical framework for understanding body-axis formation and defect-mediated morphogenesis in biological vesicles.

Figures (8)

• Figure 1: Nematic textures corresponding to the bifurcated equilibrium configurations for prolate (Left) and oblate (Right) shapes. Two $+1$ point defects are situated at the poles ($\vartheta = 0$ and $\vartheta = \pi$). The lateral insets illustrate the meridional cross-sections and three-dimensional renderings of the deformed shells alongside their associated nematic textures. Within the cross-sectional views, black dashed lines denote the reference sphere.
• Figure 2: Overview of the first-order bifurcation scenario. The blue and red lines refer to the equilibrium branches corresponding to $C_{0,-}^{\bar{\gamma}}$ and $C_{0,+}^{\bar{\gamma}}$, respectively, while the black line refers to the isotropic trivial solution $C_0 \equiv 0$. Dashed lines indicate metastable solutions, and solid lines indicate stable ones. Here, $\bar{\gamma}=8$, $k=1$, $\rho_0=0.1$, $r_0=1$, and $\bar{c}=2$. Consequently, $a_{\mathrm{cr}}^{\bar{\gamma}=8}=-0.67$, while $a^\star=-0.5$.
• Figure 3: 3D representations of the deformed shells and their corresponding nematic textures, illustrating the bifurcated equilibrium modes for $m=1$ and $m=2$. In the left-hand panel ($m=1$), two defects reside on the equator whilst the remaining two are situated at the poles; conversely, in the right-hand panel ($m=2$), all four defects lie along the equator. Furthermore, the configuration of the integral lines reveals that these defects carry a topological charge of $+1/2$. In the cross-sectional views taken along planes containing the defects, black dashed lines denote the reference sphere.
• Figure 4: Overview of the second-order bifurcation scenario. For $a \in (0, a^\star)$, the functional $\mathcal{F}$ attains a single minimum in $r^{m=i}$ from \ref{['eq:rm12_def']} at $r^{m=i} \equiv r_0$, corresponding to the isotropic branch, which is globally stable. A fold at $a = a_{\mathrm{cr}}^{\bar{\gamma}}$ gives rise to two additional non-trivial equilibrium solutions.
• Figure 5: Dimensionless energies $\mathcal{F}/(k\pi \rho_0)$ of the bifurcated configurations reported in Table \ref{['tab:modes_constrained']}. The blue line corresponds to the prolate solution associated with $C_{0,-}^{\bar{\gamma}}$, while the red line corresponds to the solution associated with $C_{0,+}^{\bar{\gamma}}$, which starts as prolate and subsequently becomes oblate. The orange line represents the squared configurations associated with the modes $m=1$ and $m=2$, which emerge at $a = a_{\mathrm{cr}}^{\bar{\gamma}}$ and share the same energy. Here, $\bar{\gamma} = 8$, with $a^\star = -0.5$ and $a_{\mathrm{cr}}^{\bar{\gamma}=8} = -0.67$. The associated complete bifurcation diagram is shown in the right panel.