Schedule for: 24w5249 - Mathematical Analysis of Soft Matter

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

I will give an overview of some commonly used physical models of liquid crystals and their defects, alongside example experimental systems that these models are used to describe. I will especially emphasize the assumptions and approximations made by these models, the cases and scales at which they are expected to break down, and how successful they have been in describing the experiments. I will close with a perspective on a few selected modeling challenges that remain important open questions.

We will present some basic models and  problems concerning a representative soft matter system, the nematic liquid crystals. We will aim to present and motivate in simple terms mathematical issues that the models generate and how mathematicians approach them, focusing on the possible relevance of these studies from a physics perspective.

Knots of vortex lines within physical fields were postulated to behave like particles already starting from Gauss and Kelvin, and recently topological order and phases represent an exciting inter-disciplinary research frontier [1]. I will describe knotted vortices that emerge in the physical order parameter fields of chiral liquid crystals. A combination of numerical modeling and nonlinear optical imaging uncovers the internal structure and topology of individual vortex knots and the various hierarchical organizations that they form via different reconnections. I will discuss their stability in molecular and colloidal liquid crystals of different symmetries and will show how low-voltage electric fields can switch between different types of behavior. Finally, I will discuss how this emergent paradigm of dynamic knotted meta matter could allow for imparting new designable material properties and physical behavior [2-4]. 1. I. I. Smalyukh. Rep. Prog. Phys. 83, 106601 (2020). 2. C. Meng, J.-S. Wu, and I. I. Smalyukh. Nature Materials 22, 64-72 (2023). 3. H. Zhao, J.-S. B. Tai, J.-S. Wu, and I. I. Smalyukh. Nature Physics 19, 451-459 (2023). 4. J. Peixoto, D. Hall, D. J. Broer, I. I. Smalyukh and D. Liu. Advanced Materials 36, 2308425 (2024).

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Focal conic domains, are defects characteristic of layered liquid crystal phases. Their association can built flowers where petals are the ellipses of the Dupin cyclides involved in these defect. We report here the observation of focal conic flowers in cholesteric droplets sessile on a glass surface and surrounded by glycerol. The observation of the droplets in different directions helps to solve the 3D architecture of the flower. The effects of the droplet size and of the pitch value are also reported.

Smectic liquid crystals are a phase of matter in which the constituent molecules tend to align locally parallel to one another and to arrange themselves in layers. Experimental evidence shows that the configuration of the layers in smectic films may be rather complex, possibly with defects - that is, localised regions of sharp change in the orientation of the layers. Defects may occur at isolated points, along lines or surfaces. In this talk, we discuss a free-discontinuity variational problem for smectic A liquid crystals in two dimensions, set in the space SBV. We focus on a specific form of the energy functional, which penalises dislocations of the layers along the defects and is lower semicontinuous, so that minimisers exist. The talk is based on joint work with John M. Ball (Heriot-Watt University, Edinburgh and Hong Kong Institute of Advanced Studies) and Bianca Stroffolini (Università Federico II, Napoli).

Area minimization among a suitable class of 2D surfaces is the basic variational model describing the interfaces in films/foams. In this talk we will discuss a modification of this paradigm in which surfaces are replaced with regions of small but positive volume. The model captures features of real films/foams, such as Plateau borders, that cannot be described by zero volume objects. We will also discuss the PDE approximation of this problem via the Allen-Cahn equation and its relation to Plateau's laws, which govern the possible singularities.

Liquid crystal elastomers (LCE) are soft materials that combine the elasticity of rubber and the ordering of liquid crystals. Under external stimuli, they exhibit complex morphing including out of plane actuation: twisting, bending, folding, etc. Via numerical experiments implementing finite element elastodynamics, we study the stimulus-responsive shape morphing of these materials. In the spirit of kirigami (the Japanese art of cutting and folding paper), we investigate the light-driven actuation of samples custom-cut to specific geometries. The microstructure in our LCE kirigami include in-plane liquid crystal defects with a preset topological charge coexisting with splay or twist in the liquid crystal director along sample thickness. We demonstrate the actuation of a variety of samples, including the fluttering of a bio-mimetic elastomer butterfly. Our numerical studies are in remarkable agreement with experimental results and demonstrate a fascinating actuation behavior arising from the interplay between microstructural topology, macroscopic geometry, and stimulus-response.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A ferroelectric nematic liquid crystal is formed by achiral molecules with large dipole moments. Its orientational order is universally described as unidirectionally polar, which raises the question of how the structure avoids a strong depolarization field and splay deformations which bring about a bound charge. We demonstrate a rich plethora of polarization patterns (1-3) that form in confined ferroelectric nematics not constrained by crystallographic axes. Domain walls adopt the shapes of conic sections, separating domains with uniform and circular polarization (2). When a flat ferroelectric nematic slab is anchored only at one bounding plate, its ground state is optically active, with left- and right-hand twists of polarization (3). Although the helicoidal deformations and defect walls separating domains of opposite handedness increase the elastic energy, the twists reduce the electrostatic energy and weaken when the material is doped with ions. An externally applied electric field unwinds the helices and produces complex three-dimensional structures. The study shows that the polar orientational order of molecules could trigger chirality in the soft matter with no chemically induced chiral centers. References 1. A. A. Marchenko et al. Phys Rev Lett 132, 098101 (2024). 2. P. Kumar et al. Nature Communications 14, 748 (2023). 3. P. Kumari et al. Science 383, 1364-1368 (2024).

We discuss a recent result, obtained in collaboration with Federico Luigi Dipasquale, on the regularity of local minimisers of a prototypical free-discontinuity problem involving both a manifold-valued constraint on the maps and a variable-exponent growth in the energy functional. To be more precise, we work in 2D domains, with sphere-valued special functions with bounded variation, and the energy functional we consider is the sum of the integral of the p(·)-power of their approximate gradient and of the H^1-measure of their jump set. The approach we follow is remiscent of the one devised by Conti, Focardi, and Iurlano to prove existence of strong minimisers for the Griffith energy and it is divided in two steps. As a first step, we extend to this setting the Sobolev approximation results for special functions with bounded deformation and small jump set originally proven by Conti, Focardi, and Iurlano. In second place, we use this extension and a suitable adaptation of the classical blow-up technique due to De Giorgi, Carriero and Leaci to prove the announced partial regularity theorem, avoiding truncation techniques.

Ferroelectric materials are utilized in various applications such as sensors, actuators, optical devices, and memory storage. The functionality of these materials hinges upon transitions between different polarization states. These transitions are mediated by the motion of domain walls. We develop the theoretical approaches needed to understand the structure and motion of ferroelectric domain walls in two distinct scenarios: crystalline materials, where such walls are Ising-like with atomic-scale widths, and in the recently discovered ferroelectric liquid crystal materials, where the domain walls have a continuously-varying polarization and the associated splay, twist, and bend elastic deformations. For solid state materials, noise plays a critical role in the microscopic dynamics and can be included within the Landau-Ginzburg-Devonshire modelling framework. On the other hand, in soft materials, the elastic and electrostatic energies compete to create large, stable solitonic structures.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We investigate minimizers of the Landau-de Gennes energy for liquid crystals in a three-dimensional slab, $\Omega$ x $(-\delta,\delta)$, where the cross section $\Omega$ is a simply connected smooth domain in the $\mathbb{R}^2$.  The minimizers are required to be tangential on the top and bottom faces and to satisfy prescribed uniaxial boundary conditions on the lateral surface of degree $d/2$ where $d \in \mathbb{N}$.  We analyze the patterns of these minimizers, including estimates on the nature, number, and location of their defects; and how they depend on the degree of their boundary conditions, the Landau-de Gennes parameter $\epsilon > 0$, and the slab thickness $2 \delta$.  In particular, assuming that $\delta=\epsilon^{\alpha}$, we show that in the thin slab case $1/2< \alpha < 1$, energy minimizers have $d$ defects each of degree $1/2$ for epsilon sufficiently small.  But in the thick case $0 < \alpha < 1/2$, they have $d/2$ defects of degree 1 if $d$ is even; whereas if d is odd, they have (d-1)/2 defects of degree 1 and one defect of degree $1/2$. This is joint work with Daniel Phillips.

After a general introduction about liquid crystals and their singularities, I present a variational convergence result based on the Landau-de Gennes model describing the Saturn ring effect around an immersed particle with homeotropic and an external magnetic field. We will see how the energy concentrates on lines and surfaces close to, or on the particle surface, leading to a generalized Plateau problem. Some properties of the limit functional will be given and I discuss examples of minimizers for some chosen particle shapes.

The anisotropic Ginzburg-Landau system \[ \Delta u+\delta \nabla (\dv u) +\delta \curl^*(\curl u)=(|u|^2-1) u, \] for $ u\colon\mathbb R^2\to\mathbb R^2 $ and $ \delta\in (-1,1) $, models the formation of vortices in liquid crystals. We prove the existence of entire solutions such that $|u(x)|\to 1$ and $u$ has a prescribed topological degree $d\leq -1$ as $|x|\to\infty$, for small values of the anisotropy parameter $|\delta| < \delta_0(d)$. Unlike the isotropic case $\delta=0$, this cannot be reduced to a one-dimensional radial equation. We obtain these solutions by minimizing the anisotropic Ginzburg-Landau energy in an appropriate class of equivariant maps, with respect to a  finite  symmetry subgroup. This is joint work with M. Kowalczyk and P. Smyrnelis.

We examine minimizers for the Ball-Majumdar energy modeling a liquid crystal film defined on a bounded simply connected domain $\Omega$ in $\mathbb{R}^2$. The minimizers  satisfy boundary conditions on the lateral edge that induce defects (disclinations) within the film. We determine the defect structures, liquid crystal patterns and estimates on the energies for minimizers as the elasticity parameter tends to zero. In particular, we show that away from the defect sites, the limit pattern is a local minimizer for a planar Frank energy. This is joint work with Patti Bauman.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

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. We prove that (1) every minimal surface is transformed into a minimal surface by a bending-neutral deformation and (2) given two minimal surfaces, there is a bending-neutral deformation that maps one into the other. Thus, all minimal surfaces have indeed a universal bending content. The lecture will show how these kinematic concepts pave the way to a natural, variational theory of soft shells.

In recent years, the analysis of Ginzburg-Landau type energy functionals has been successfully extended by Ignat and Jerrard to the setting of tangent vector fields defined on a smooth closed surface. Here we discuss two extensions of these results. With an eye towards the modeling of nematics deposited on a surface, the first generalization entails replacing the vector field by a complex line bundle, a setting that includes Q-tensors, The second project involves a tangent vector field sitting on a surface with a singularity, namely a cone. This is joint work with Christian Cofoid, Dmitry Golovaty, Alberto Montero and Etienne Sandier (in various combinations).

I will present recent results on modeling of ferroelectric nematics using a Ginzburg-Landau-type model with anisotropic elastic constants. We show that the singular structures exhibited by the minimizers of this energy contain both point and line singularities and describe these structures for particular examples of domains/boundary conditions. This is a joint work with Priyanka Kumari, Oleg Lavrentovich and Peter Sternberg.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Like many other active fluids, the dynamics of active nematics is bistable when confined to narrow channels: they flow spontaneously at a preferred speed in one randomly selected direction. But how do active nematic flows self-organize in microfluidic networks? In this talk, we shed light on this uncharted territory by reporting recent experiments with the microtubule-based active nematic (AN). We focus on small-sized open networks with inputs and outputs connected to the external, unconfined AN film that acts as a thermal reservoir. Following a bottom-up approach, we first examine the behavior in open individual channels and characterize how channel geometry —specifically length, width, and wall patterning —controls the flow rate. Next, we study bifurcation networks, where the topology creates a conflict between mass conservation and the bistability of active flows: the three channels cannot sustain flow at the preferred speed simultaneously. For symmetric geometries, we find that, contrary to theoretical conjectures and recent experiments with polar fluids, the flow splits at the junction. However, we demonstrate that it is possible to direct the flow along a particular path by introducing geometrical asymmetries in the bifurcation network. Finally, we show how the design rules derived from the observation of bifurcations can be applied to create a functional logical gate circuit.

Liquid crystals combine fluidity with long range orientational order. Their anisotropic physical properties are at the origin of technological applications, and result in fascinating problems when energy minimization conflicts with topological constraints due to boundary conditions, often involving the formation of topological defects. Liquid crystals can be easily driven away from equilibrium with electric, magnetic or flow fields. Recently, active liquid crystals have been discovered and characterized. They are intrinsically out of equilibrium, with constituents that transform chemical energy into flows and mechanical work. Examples are often found in nature, such as bacteria colonies or eukaryotic cell epithelia, and, recently, in artificial preparations, such as in-vitro reconstitutions of cytoskeletal proteins. In this presentation, I will focus on our experiments with the active material forms through the combination of microtubules and dimerized kinesin molecular motors. With the assistance of a depleting agent, microtubules and motors associate, forming thick elongated fibers that, in the presence of adenosine triphosphate, become extensible and prone to buckling instabilities. In the presence of a water/oil interface, the fibers condense in a quasi-two-dimensional layer where long-range orientational order emerges. This results in a two-dimensional active nematic, with the proliferation of self-propelling +1/2 defects and passively-advected -1/2 defects, in a seemingly disordered regime that has been called active turbulence 1 . We take advantage of the influence exerted by the oil rheology on the structure and dynamics of the active nematic to replace the usual isotropic oil with a thermotropic liquid crystal. The anisotropy of the passive liquid crystal provides us with a handle to control active flows using an external magnetic field coupled to the passive mesogen. When the latter is in the lamellar smectic-A phase, and under a homogeneous in-plane magnetic field, oil viscosity is orders of magnitude higher along the magnetic field that perpendicular to it. This results in an alignment transition of the active nematic along the easy flow direction, and turbulent active flows become quasi-laminar, with a well-defined spatial arrangement 2,3 . Since aligned extensile fibers are prone to bend distortions, accumulated active stress is frequently released through sudden, localized outburst of transversal distortions and flows, which quickly dissipate. Interestingly, when the active nematic layer is laterally confined inside narrow channels, transversal instabilities are prevalent and encompass the whole system. Although the active turbulence regime seems to have been reestablished, a steady state flow pattern emerges in the form of a lattice of antiparallel vortices that originate in the self-propelled defect motion. Moreover, the spatial distribution of active filaments has developed a density pattern complementary to the vortex lattice. On the other hand, the hydrodynamic coupling leads to the formation of new patterns in the passive mesogen due to local flow alignment following drag exerted by the active layer. We have explored this response for different passive mesophases: nematic, cholesteric, and smectic A. The spatiotemporal patterns are essentially defect-free, which opens the perspective of studying a defect-free active turbulent system. 1 Doostmohammadi, A., Ignes-Mullol, J., Yeomans, J. M. & Sagues, F. Active nematics. Nature communications 9, 3246 (2018). 2 Guillamat, P., Ignes-Mullol, J. & Sagues, F. Control of active liquid crystals with a magnetic field. Proceedings of the National Academy of Sciences of the United States of America 113, 5498-5502 (2016). 3 Bantysh, O., Martínez-Prat, B., Nambisan, J., Fernández-Nieves, A., Sagués, F. & Ignés- Mullol, J. First Order Alignment Transition in an Interfaced Active Nematic Fluid. Phys Rev Lett 132 (2024).

The behavior of topological defects in active nematics are intrinsically coupled to their macroscopic flows. Therefore, a better understanding of the complicated interactions between defects in these systems is essential to realizing their potential applications. In this talk, using a recently developed approximation scheme for the kinematics of nematic defects [1] and recent calculations for the flow field generated by defects in active nematics [2], I will present an approximate, analytical expression for the velocity of a topological defect in the presence of other defects in two-dimensional active nematics. The velocity may be decomposed into contributions from the coulomb interaction between defects, advection from flows generated by active stress, and the deflection of defects in shear flow. Focusing on the interaction between just two defects yields insights into phenomena observed in continuum numerical simulations and experiments. Finally, I will discuss our current efforts to numerically simulate a “particle model” for an active nematic using the expression for defect velocities. [1] C. Schimming & J. Viñals, Proc. R. Soc. A 489: 20230042 (2023). [2] J. Rønning, C. M. Marchetti, M. J. Bowick, & L. Angheluta, Proc. R. Soc. A 478: 20210879 (2022).

We examine a non-local diffuse interface energy with Coulomb repulsion in three dimensions inspired by the Thomas-Fermi-Dirac-von Weizs\"{a}cker, and the Ohta-Kawasaki models. We consider the corresponding mass-constrained variational problem and show the existence of minimizers for small masses, and the absence of minimizers for large masses. Finally, we explore the asymptotic behaviour of the energy as the mass goes to zero. This is joint work with Profs. Xin Yang Lu and Jun-Cheng Wei.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

In active nematic liquid crystals, topological defects drive chaotic flows in the bulk, yielding a positive Lyapunov exponent. While confined geometries of active nematics have been shown to be able to produce ordered flows, little is known about the types of periodic motion these flows permit for topological defects. To explore different active steady states, we simulate an active nematic system using active Beris-Edwards nematodynamics with curved boundary walls and a tunable winding number around the boundary to fix the net topological charge. We find that there are several ordered flows attainable that produce periodic motion of topological defects. Using tools from braid theory, we show connections between these defect motions andthe production of topological entropy, and specifically, that these ordered defect motions produce more chaotic mixing than the well-known “topological chaos”. We provide an analytical and numerical understanding of the emergence of these ordered flows and discuss their stability for higher net topological charges. Our findings suggest routes to controllable bulk active flows and stable self-mixing patterns realizable in future experiments.

Defects in smectic liquid crystals, owing to the breaking of translational symmetry, exhibit intricate topological behavior compared to their nematic counterparts. Using tools from knot theory, we study the ability of screw dislocations, edge dislocations, and smectic disclinations to form knots and links. Remarkably, we find that it is not always possible to knot smectic defects without introducing other defects into the system. We provide a lower bound on the number of point defects required to form a smectic dislocation or disclination of any given knot type. These extra point defects can be interpreted as focal conic-like structures, where the knot and point defects play the role of the ellipse and hyperbola, respectively. Our work uncovers a deep connection between smectic liquid crystals and modern topics in Morse-Novikov/knot theory.

In this talk I will discuss the 3D Landau-de Gennes model for nematic liquid crystal with an external magnetic field around an immersed particle. I will show that if we impose strong planar anchoring on the surface, then in the large particle limit, and assuming a weak magnetic field, we can characterize the defects that occur. We will see that only point defects can occur for minimizers in this regime.

On curved surfaces, liquid crystal topological point-defects are attracted to Gaussian curvature of the same sign. We show that this coupling extends to the endpoints of disclination lines of 3D nematic liquid crystals arising in a hybrid-aligned system with a double-undulated boundary with homeotropic anchoring and an opposite flat boundary with degenerate planar anchoring. We explore the properties of the multistable disclination lines exhibited by this system. Using Landau-de Gennes numerical modeling, we investigate how the Gaussian curvature of the boundary surface influences the winding character of the disclination lines in the liquid crystal bulk. The winding characters of the disclination lines have been observed to vary rapidly along the defect contours. We calculate the rotation vector to describe local defect winding, and use energetic arguments to understand the heterogeneity of the multistable disclination landscape. This is joint work with Mohamed Amine Gharbi and Daniel A. Beller.

Ferroelectric nematic liquid crystals are formed by achiral molecules with large dipole moments. Their three-dimensional orientational order is described as unidirectionally polar. We demonstrate that the ground state of a flat slab of a ferroelectric nematic unconstrained by externally imposed alignment directions is chiral, with left- and right-handed twists of polarization. Although the helicoidal deformations and defect walls that separate domains of opposite handedness increase the elastic energy, the twists reduce the electrostatic energy and become weaker when the material is doped with ions. This work shows that the polar orientational order of molecules could trigger chirality in soft matter with no chemically induced chiral centers.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

How is the emergent behavior of non-equilibrium systems different from equilibrium ones, what new collective motion and pattern formation can be found in living as opposed to simply “active” systems? This talk will combine large scale simulations of a variety of flocking models and a theoretical framework -- Toner-Tu hydrodynamics -- meant to describe flocking organisms as a type of active polar aligning matter. I will first argue that in a standard numerical model of flocking particles the aligning interactions of coarse grained representations of flocking agents correspond to nonreciprocal forces, i.e., forces that violate Newton’s third law. These forces lead to novel terms in the hydrodynamic description of these simulations, and I will discuss how these novel terms lead to new collective dynamics at large scales. Finally, I will connect these ideas to both long-standing puzzles in the field and speculations about actual flocking and schooling behaviors observed in nature.

A couple of modified preparations of the usual AN microtubule/kinesin system will be presented, and their potentialities explored. First, a photosensitive version of the active nematic will be briefly described that enables to study the response of the system to different patterns of distributed activity. In the second formulation, presented in more detail,  we report on our recent work with a hybrid version of the AN system after mixing the proteinic material with a light-polymerizable component. Two different scenarios will be commented. First, we report how a spatially extended illumination collapses the nematic order  through substrate friction and leads to a biphasic active fluid. Second, by employing specifically illuminated motifs, we are able to estimate for the first time rheological parameters of the active nematic material.

Active nematics are a class of liquid crystals driven out-of-equilibrium by the intrinsic activity of their rod-like constituents. In bulk, global nematic order is destabilised by the coupled feedback between nematic deformations and active flows, resulting in a steady-state population of half-integer nematic topological defects. This talk will demonstrate that despite turbulent-like flows, there exists a strong cross-field constraint between motile +1/2 topological defects and viscometric surfaces, which are contours of the flow where vorticity and strain-rate balance. Through experiments of microtubule-kinesin based active nematics and nematohydrodynamic simulations, we establish the importance of these contours as paths for +1/2 defect dynamics, which determine the handedness of defect trajectories and the sites of defect pair creation and annihilation. We show through a series of models that the constraint is maintained through an interdependence of viscometric surfaces, +1/2 defects, and line-like structures of nematic bend deformation. These results identify that active nematic defects should not be viewed as solitary points but are one component of mesoscale nematic structures, which suggests potential new avenues for exploring topological dynamics.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Liquid crystals are the basis of the modern display industry because of their unique properties. Yet, liquid crystalline ordering occurs in systems beyond displays and across length scales: from molecular to colloidal assemblies. Controlling the structuring of liquid crystals across these length scales remains an open challenge. Geometrical constraints can generate patterns and defects – localized, “melted” regions of disorder that can minimize the total elastic distortion in the system. In this talk, I will present the formation of defects within molecular and colloidal liquid crystals that are induced through geometrical frustration. I will begin by presenting work on a molecular, chiral liquid crystal confined to a spherical shell, with the use of microfluidics. I will then present experiments where surface-active colloids are patterned at the liquid crystal-water interface. I will then end by surveying ongoing experiments in my group that probe the role of confinement for structuring larger-scaled, colloidal liquid crystals, such as cellulose nanocrystals and silica nanorods. These organizing principles provide insight on pattern formation in anisotropic elastic materials, across length scales, the mechanisms of which can be leveraged for designing new materials.

We consider the problem of minimising the (simplest) Landau-de Gennes (LdG) energy in two-dimensional discs, under axial symmetry, a physically relevant pointwise norm-constraint in the interior, and radial anchoring on the boundary. The goal is to study the uniaxial or biaxial character of minimisers. We show that the latter depends crucially on the value of a parameter $\lambda \geq 0$ appearing in front of the potential and penalising biaxiality. For $\lambda$ large, minimisers are uniaxial. As $\lambda$ decreases, biaxiality is less penalised and a threshold $\lambda_* > 0$ is met at which uniaxial and biaxial minimisers coexist. Below $\lambda_*$, all minimisers are biaxial. For all biaxial minimisers, {\em complete biaxial escape} occurs. The cornerstone of the argument consists in an {\em energy gap} between {\em small} and {\em large} maps in the associated minimisation problem for the Dirichlet integral (i.e., for $\lambda = 0$). Here, a map is called {\em small} if it does not escape the spherical cap containing the image of the boundary data, and {\em large} otherwise. The energy gap is made fully explicit by describing the set of optimal maps in both the small and the large case. A major difficulty in the analysis lies in dealing with a possible lack of compactness in minimising sequences. This problem arose in a natural way in the framework of a broader investigation, carried out in a joint work with Vincent Millot and Adriano Pisante, of qualitative properties of LdG minimisers in 3D cylinders.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.