## On liquid crystal phase transitions between isotropic and nematic phase states

#### Changyou WangPurdue University

In this talk, I will consider asymptotic behavior of minimizers $Q_\epsilon$, as the parameter $\epsilon\rightarrow 0$, of the Landau- de Gennes energy
$$E_{LdG}(Q)=\int_\Omega (f_e(\nabla Q)+\frac{1}{\epsilon^2} f_b(Q))\,dx, \ \Omega\subset\mathbb R^3,$$
for uniaxial nematic liquid crystals $Q$, where the bulk energy density function $f_b(Q)$
has two minimal wells at isotropic phase
$Q=0$ and nematic phase $Q=s_+({\bf n}\otimes {\bf n}-1/3 I)$. We will discuss the limit value of
$E_{LdG}(Q_\epsilon)$ in terms of the area of the sharp interface $S$ between the two phases, which is an area minimizing surface, and the 1d-minimal connecting energy
$c_0$ between the two phases. We will also discuss the limiting behavior of $Q_\epsilon$ in term of minimizing configurations of the Oseen-Frank energy functional,
including a surface energy from $S$. This is a joint work with Yuanzhen Shao from Purdue University.

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