We clarify a few issues that could possibly be misunderstood in our paper [J. Elast. 158(2), 25 (2026) https://doi.org/10.1007/s10659-026-10201-8 ].

Stretching, drilling, and bending are the independent deformation modes of a thin shell, each of which has an individual energy content. When the energy content of a mode vanishes, that mode is neutral. We characterize all neutral modes of deformation of minimal surfaces into minimal surfaces. A hierarchy is found among these: a stretching neutral mode (which is an isometry) is also drilling neutral, and a drilling neutral mode is also bending neutral. Thus, all isometries of a minimal surface are globally neutral and give rise to soft elasticity. More generally, all minimal surfaces can be classified relative to a reference one in terms of three energy contents, which can be given in closed form.

Three general modes are distinguished in the deformation of a thin shell; these are stretching, drilling, and bending. Of these, the drilling mode is the one more likely to emerge in a soft matter shell (as compared to a hard, structural one), as it is ignited by a swerve of material fibers about the local normal. We propose a hyperelastic theory for soft shells, based on a separation criterion that envisages the strain-energy density as the sum of three independent pure measures of stretching, drilling, and bending. Each individual measure is prescribed to vanish on all other companion modes. The result is a direct, second-grade theory featuring a bending energy quartic in an invariant strain descriptor that stems from the polar rotation hidden in the deformation gradient (although quadratic energies are also appropriate in special cases). The proposed energy functional has a multi-well character, which fosters cases of soft elasticity (with a continuum of ground states) related to minimal surfaces.

Minimal surfaces are ubiquitous in nature. Here, they are considered as geometric objects that bear a deformation content. By refining the resolution of the surface deformation gradient afforded by the polar decomposition theorem, we identify a bending content and a class of deformations that leave it unchanged. These are the bending-neutral deformations, fully characterized by an integrability condition; they preserve normals. We prove that: (i) every minimal surface is transformed into a minimal surface by a bending-neutral deformation; (ii) given two minimal surfaces with the same system of normals, there is a bending-neutral deformation that maps one into the other; and (iii) all minimal surfaces have indeed a universal bending content.

We study the equilibria of a photoresponsive nematic elastomer ribbon within a continuum theory that builds upon the statistical mechanics model put forward by Corbett and Warner (Phys. Rev. E 78:061701, 2008). We prove that the spontaneous deformation induced by illumination is not monotonically dependent on the intensity I. The ribbon's deflection first increases with increasing I, as expected, but then decreases and abruptly ceases altogether at a critical value Ie of I. Ie, which is enclosed within a hysteresis loop, marks a first order shape transition. Finally, we find that there is a critical value of the ribbon's length, depending only on the degree of cross-linking in the material, below which no deflection can be induced in the ribbon, no matter how intense is the light shone on it.

Nematic polymer networks are (heat and light) activable materials, which combine the features of rubber and nematic liquid crystals. When only the stretching energy of a thin sheet of nematic polymer network is minimized, the intrinsic (Gaussian) curvature of the shape it takes upon (thermal or optical) actuation is determined. This, unfortunately, produces a multitude of possible shapes, for which we need a selection criterion, which may only be provided by a correcting bending energy depending on the extrinsic curvatures of the deformed shape. The literature has so far offered approximate corrections depending on the mean curvature. In this paper, we derive the appropriate bending energy for a sheet of nematic polymer network from the celebrated neo-classical energy of nematic elastomers in three space dimensions. This task is performed via a dimension reduction based on a modified Kirchhoff-Love hypothesis, which withstands the criticism of more sophisticated analytical tools. The result is a surface elastic free-energy density where stretching and bending are blended together; they may or may not be length-separated, and should be minimized together. The extrinsic curvatures of the deformed shape not only feature in the bending energy through the mean curvature, but also through the relative orientation of the nematic director in the frame of the directions of principal curvatures.

When a material surface is functionalized so as to acquire some type of order, functionalization of which soft condensed matter systems have recently provided many interesting examples, the modeller faces an alternative. Either the order is described on the curved, physical surface where it belongs, or it is described on a flat surface that is unrolled as pre-image of the physical surface under a suitable height function. This paper proposes a general method that pursues the latter avenue by lifting whatever order tensor is deemed appropriate from a flat to a curved surface. To produce a specific application, we specialize this method to nematic shells, for which it also provides a simple, but convincing interpretation of the outcomes of some molecular-dynamics experiments on ellipsoidal shells.

Nematic shells are colloidal particles coated with nematic liquid crystal molecules which may freely glide and rotate on the colloid's surface while keeping their long axis on the local tangent plane. We describe the nematic order on a shell by a unit director field on an orientable surface. Equilibrium fields can then be found by minimising the elastic energy, which in general is a function of the surface gradient of the director field. We learn how to extract systematically out of this energy a fossil component, related only to the surface and its curvatures, which expresses a curvature potential for the molecular torque. At hyperbolic points on the colloid's surface, and only there, the alignment preferred by the curvature potential may fail to be a direction of principal curvature. There the fossil energy becomes bistable.

Due to recent advances in fast iterative solvers in the field of computational fluid dynamics, more complex problems which were previously beyond the scope of standard techniques can be tackled. In this paper, we describe one such situation, namely, modelling the interaction of flow and molecular orientation in a complex fluid such as a liquid crystal. Specifically, we consider a nematic liquid crystal in a spatially inhomogeneous flow situation where the orientational order is described by a second rank alignment tensor. The evolution is determined by two coupled equations: a generalised Navier–Stokes equation for flow in which the divergence of the stress tensor also depends on the alignment tensor and its time derivative, and a convection-diffusion type equation with non-linear terms that stem from a Landau-Ginzburg-DeGennes potential for the alignment. In this paper, we use a specific model with three viscosity coefficients that allows the contribution of the orientation to the viscous stress to be cast in the form of an orientation-dependent force. This effectively decouples the flow and orientation, with each appearing only on the right-hand side of the other equation. In this way, difficulties associated with solving the fully coupled problem are circumvented and a stand-alone fast solver, such as the state-of-the-art preconditioned iterative solver implemented here, can be used for the flow equation. A time-discretised strategy for solving the flow-orientation problem is illustrated using the example of Stokes flow in a lid-driven cavity.

Nematic shells are colloidal particles coated with nematic liquid crystal molecules, which may freely glide and rotate on the colloid's surface while keeping their long axis on the local tangent plane. Molecular dynamics simulations on a nanoscopic spherical shell indicate that under appropriate adhesion conditions for the molecules on the equator, the equilibrium nematic texture exhibits at each pole a pair of +1/2 defects so close to one another to be treated as one +1 defect. Spirals connect the polar defects, though the continuum limit of the interaction potential would not feature any elastic anisotropy. A molecular averaging justifies an anchoring defect energy that feels the geodesics emanating from the defect. All our observations are explained by such a geodesic anchoring, which vanishes on flat manifolds.

We derive expressions for the order parameter dependence of the viscosity coefficients of a biaxial nematic liquid crystal by comparing its dissipation function expressed in terms of directors with that expressed in terms of order tensors. The results enable us to identify the dominant flow viscosity coefficients and to compare their temperature variation according to their dependence on the dominant scalar order parameters. By considering different orientations of an external field, we identify three characteristic switching times corresponding to three rotational viscosities, and we estimate the ratio of the switching times of the primary and the secondary directors.

The existence of uniaxial liquid crystals comprising polar molecules, with all the dipoles aligned in a parallel pattern, is classically ruled out. Generally, there are two different avenues to a mean-field theory for liquid crystals: one is based on short-range, repulsive, steric forces, and the other is based on long-range, globally attractive, dispersion forces. Purely polar steric interactions have been shown to have the potential of inducing unexpected orientationally ordered states. In real molecules, anisotropies both in shape and in polarizability coexist; it has been shown that dispersion forces interaction can be combined with hard-core repulsion in a formal theory, based on a steric tensor. Starting from this, we build an interaction Hamiltonian featuring the average electric dipolar energy exchanged between molecules with the same excluded region. Under the assumption that the molecular shape is spheroidal, we propose a mean-field model for polar nematic liquid crystals which can exhibit both uniaxial and biaxial polar phases. By means of a numerical bifurcation analysis, we discuss the stability of the equilibrium against the choice of two model parameters, one describing the degree of molecular shape biaxiality and the other describing the relative orientation of the electric dipole within each molecule. We find only uniaxial stable phases, which are effectively characterized by a single scalar order parameter.

To appreciate the universal qualitative features of defect annihilation in nematic liquid crystals, we study how the viscous force of reorientational dynamics behaves under a transformation that reverses the sign of the defect's topological charge. As an illustration of our general results, we consider a class of point defects that were first studied by A. Saupe. The reorientational viscous forces acting on them differ dramatically from those acting on line defects.

We propose a simple phenomenological model for hydrodynamic defect dynamics in nematic liquid crystals, inspired by the Ericksen-Leslie theory. We identify the main forces that govern both fluid and defect motion and we comment on their symmetry. As shown for two annihilating disclinations, our model is predictive for arbitrary length scales and topological charges.

Classically, there have been two different ways to obtain mean-field theories for liquid crystals. One is based on short-range repulsive steric forces and the other on long-range attractive dispersion forces. In the former approach, it is the anisotropic shape of the molecules that leads to the anisotropic interaction, and in the latter it is the anisotropy of the molecular polarizability. In real molecules both causes of anisotropy can be expected to contribute to the effective interaction, and so it is desirable to assess the combined effect of anisotropic long-range attraction and short-range repulsion. Here we present an avenue to this end. We start from dispersion forces interactions and combine them with hard-core repulsions in a formal theory, whose crucial element is the steric tensor, a fourth-rank tensor depending on the anisotropy of the interacting molecules. This tensor can be determined analytically for a special class of molecular shapes.