Equilibrium order parameters of liquid crystals in the Landau-De Gennes theory

We study nematic liquid crystal configurations in confined geometries within the continuum Landau--De Gennes theory. These nematic configurations are mathematically described by symmetric, traceless two-tensor fields, known as \Qvec-tensor order parameter fields. We obtain explicit upper bounds for the order parameters of equilibrium liquid crystal configurations in terms of the temperature, material constants, boundary conditions and the domain geometry. These bounds are compared with the bounds predicted by the statistical mechanics definition of the \Qvec-tensor order parameter. They give quantitative information about the temperature regimes for which the Landau-De Gennes definition and the statistical mechanics definition of the \Qvec-tensor order parameter agree and the temperature regimes for which the two definitions fail to agree. For the temperature regimes where the two definitions do not agree, we discuss possible alternatives.Comment: Submitted to SIAM Journal on Applied Mathematic

Equilibrium order parameters of nematic liquid\ud crystals in the Landau-De Gennes theory

We study equilibrium liquid crystal configurations in three-dimensional domains, within the continuum Landau-De Gennes theory. We obtain explicit bounds for the equilibrium scalar order parameters in terms of the temperature and material-dependent constants. We explicitly quantify the temperature regimes where the Landau-De Gennes predictions match and the temperature regimes where the Landau-De Gennes predictions don’t match the probabilistic second-moment definition of the Q-tensor order parameter. The regime of agreement may be interpreted as the regime of validity of the Landau-De Gennes theory since the Landau-De Gennes theory predicts large values of the equilibrium scalar order parameters - larger than unity, in the low-temperature regime. We discuss a modified Landau-De Gennes energy functional which yields physically realistic values of the equilibrium scalar order parameters in all temperature regimes

Twisted rods, helices and buckling solutions in three dimensions

The study of slender elastic structures is an archetypical problem in continuum mechanics, dynamical systems and bifurcation theory, with a rich history dating back to Euler's seminal work in the 18th century. These filamentary elastic structures have widespread applications in engineering and biology, examples of which include cables, textile industry, DNA experiments, collagen modelling etc. One is typically interested in the equilibrium configurations of these rod-like structures, their stability and dynamic evolution and all three questions have been extensively addressed in the literature. However, it is generally recognized that there are still several open non-trivial questions related to three-dimensional analysis of rod equilibria, inclusion of topological and positional constraints and different kinds of boundary conditions

Nematic liquid crystals : from Maier-Saupe to a continuum theory

We define a continuum energy functional in terms of the mean-field Maier-Saupe free energy, that describes both spatially homogeneous and inhomogeneous systems. The Maier-Saupe theory defines the main macroscopic variable, the Q-tensor order parameter, in terms of the second moment of a probability distribution function. This definition requires the eigenvalues of Q to be bounded both from below and above. We define a thermotropic bulk potential which blows up whenever the eigenvalues tend to these lower and upper bounds. This is in contrast to the Landau-de Gennes theory which has no such penalization. We study the asymptotics of this bulk potential in different regimes and discuss phase transitions predicted by this model

Order Reconstruction for Nematics on Squares and Regular Polygons: A Landau-de Gennes Study

We construct an order reconstruction (OR)-type Landau-de Gennes critical point on a square domain of edge length $\lambda$, motivated by the well order reconstruction solution numerically reported by Kralj and Majumdar. The OR critical point is distinguished by an uniaxial cross with negative scalar order parameter along the square diagonals. The OR critical point is defined in terms of a saddle-type critical point of an associated scalar variational problem. The OR-type critical point is globally stable for small $\lambda$ and undergoes a supercritical pitchfork bifurcation in the associated scalar variational setting. We consider generalizations of the OR-type critical point to a regular hexagon, accompanied by numerical estimates of stability criteria of such critical points on both a square and a hexagon in terms of material-dependent constants.Comment: 29 pages, 12 figure

Multistability for a Reduced Landau--de Gennes Model in the Exterior of 2D Polygons

We present a systematic study of nematic equilibria in an unbounded domain, with a two-dimensional regular polygonal hole with $K$ edges in a reduced Landau--de Gennes framework. This complements our previous work on the "interior problem" for nematic equilibria inside regular polygons (SIAM Journal on Applied Mathematics, 80(4):1678-1703, 2020). The two essential model parameters are $\lambda$-the edge length of polygon hole and an additional freedom parameter $\gamma^*$-the nematic director at infinity. In the $\lambda\to 0$ limit, the limiting profile has a unique interior point defect outside a triangular hole, two interior point defects outside a generic polygon hole, except for a triangle and a square. For a square hole, the equilibrium has either no interior defects or two line defects depending on $\gamma^*$. In the $\lambda\to\infty$ limit, we have at least $[K/2]$ stable states differentiated by the location of two bend vertices and the multistability is enhanced by $\gamma^*$, compared to the interior problem. Our work offers new methods to tune the existence, location, and dimensionality of defects