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Liquid crystal droplets are of great interest from physics and applications. Rigorous mathematical analysis is challenging as the problem involves harmonic maps (and in general the Oseen-Frank model), free interfaces and topological defects which could be either inside the droplet or on its surface along with some intriguing boundary anchoring conditions for the orientation configurations. In this paper, through a study of the phase transition between the isotropic and nematic states of liquid crystal based on the Ericksen model, we can show, when the size of droplet is much larger in comparison with the ratio of the Frank constants to the surface tension, a $Gamma$-convergence theorem for minimizers. This $Gamma$-limit is in fact the sharp interface limit for the phase transition between the isotropic and nematic regions when the small parameter $varepsilon$, corresponding to the transition layer width, goes to zero. This limiting process not only provides a geometric description of the shape of the droplet as one would expect, and surprisingly it also gives the anchoring conditions for the orientations of liquid crystals on the surface of the droplet depending on material constants. In particular, homeotropic, tangential, and even free boundary conditions as assumed in earlier phenomenological modelings arise naturally provided that the surface tension, Frank and Ericksen constants are in suitable ranges.
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For any smooth domain $Omegasubset mathbb{R}^3$, we establish the existence of a global weak solution $(mathbf{u},mathbf{d}, theta)$ to the simplified, non-isothermal Ericksen-Leslie system modeling the hydrodynamic motion of nematic liquid crystals with variable temperature for any initial and boundary data $(mathbf{u}_0, mathbf{d}_0, theta_0)inmathbf{H}times H^1(Omega, mathbb{S}^2)times L^1(Omega)$, with $ mathbf{d}_0(Omega)subsetmathbb{S}_+^2$ (the upper half sphere) and $displaystyleinf_Omega theta_0>0$.
In this paper, we will establish the global existence of a suitable weak solution to the Erickson--Leslie system modeling hydrodynamics of nematic liquid crystal flows with kinematic transports for molecules of various shapes in ${mathbb{R}^3}$, which is smooth away from a closed set of (parabolic) Hausdorff dimension at most $frac{15}{7}$.
We consider a monomer-dimer system with a strong attractive dimer-dimer interaction that favors alignment. In 1979, Heilmann and Lieb conjectured that this model should exhibit a nematic liquid crystal phase, in which the dimers are mostly aligned, but do not manifest any translational order. We prove this conjecture for large dimer activity and strong interactions. The proof follows a Pirogov-Sinai scheme, in which we map the dimer model to a system of hard-core polymers whose partition function is computed using a convergent cluster expansion.
High-resolution ac-calorimetry has been carried out on dispersions of aerosils in the liquid crystal octyloxycyanobiphenyl (8OCB) as a function of aerosil concentration and temperature spanning the crystal to isotropic phases. The liquid-crystal 8OCB is elastically stiffer than the previously well studied octylcyanobiphenyl (8CB)+aerosil system and so, general quenched random disorder effects and liquid-crystal specific effects can be distinguished. A double heat capacity feature is observed at the isotropic to nematic phase transition with an aerosil independent overlap of the heat capacity wings far from the transition and having a non-monotonic variation of the transition temperature. A crossover between low and high aerosil density behavior is observed for 8OCB+aerosil. These features are generally consistent with those on the 8CB+aerosil system. Differences between these two systems in the magnitude of the transition temperature shifts, heat capacity suppression, and crossover aerosil density between the two regimes of behavior indicate a liquid crystal specific effect. The low aerosil density regime is apparently more orientationally disordered than the high aerosil density regime, which is more translationally disordered. An interpretation of these results based on a temperature dependent disorder strength is discussed. Finally, a detailed thermal hysteresis study has found that crystallization of a well homogenized sample perturbs and increases the disorder for low aerosil density samples but does not influence high density samples.
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