Influence of latent heat and thermal diffusion on the growth of nematic liquid crystal nuclei

The growth of nematic liquid crystal nuclei from an isotropic melt follows a power law behavior with exponent $n$ found experimentally to vary between $\frac{1}{2}$ for low quench depths, up to 1 for high quench depths. This behavior has been attributed to the competition between curvature and free energy. We show that curvature cannot account for the low quench depth behavior of the nucleus growth, and attribute this behavior to the diffusion of latent heat. We use a multiscale approach to solve the Landau-Ginzburg order parameter evolution equation coupled to a diffusive heat equation, and discuss this behavior for material parameters experimentally measured for the liquid crystal 8CB.

Figure 1

Free-energy density $f$ from Eq. (6) as a function of the order parameter (OP) $m$ in order of increasing temperature. The dotted line shows the free energy at $u=0$, where both the disordered phase at $m=0$ and the ordered phase at $m=1$ are equally stable. The black solid line shows the free energy at the spinodal temperature $u_{s,1}$, the thin solid gray line shows the free energy used in Ref. 7 at its spinodal temperature $u_{s,2}$. The dashed and dashed-dotted line denote the highest and lowest temperature we use in this paper. Note that the lowest temperature is very close to the spinodal and beyond the reach of the free energy used in Ref. 7.

Figure 2

(Color online) Radius $R-R_0$ of a spherical nucleus as a function of time $t$, for increasing quench depth $\Delta T=u_{\text{quench}}{L}_c∕c_p$: $\Delta T=-0.084\mathrm{K}$, $-0.2415\mathrm{K}$, $-0.42\mathrm{K}$, $-0.59\mathrm{K}$, $-0.75\mathrm{K}$, $-0.92\mathrm{K}$, $-1.1\mathrm{K}$, and $-1.26\mathrm{K}$. In (a) the data is displayed on a log-log scale plot to reveal the growth exponent, whereas (b) shows the same data linearly on a time scale where the two growth regimes diverge. Parameters are taken from Table . The dashed line denotes the temperature $u_{\text{quench}}=-1$, $(\Delta T=-L_c∕c_p)$. Lines below the dashed line (blue in online version) denote shallow quenches where ${\lim }_{t\to \infty }R\propto t^{1∕2}$, lines above the dashed line (red in online version) denote deep quenches where ${\lim }_{t\to \infty }R\propto t^1$.

Figure 3

(Color online) Thin lines: Temperature field $\Delta T\left(r\right)$ at a given time during the growth of a three-dimensional nucleus for two temperatures. The profiles are shown from left to right at exponentially increasing intervals of time, $t_n=t_0{5}^n$. The thick line is the temperature $\Delta T\mathbf{\left(}R\left(t\right)\mathbf{\right)}$ at the interface position $R\left(t\right)$. In (a) the isotropic phase is quenched to $\Delta T=-0.92\mathrm{K}<-L_c∕c_p$, at which a nucleus will grow as $R\propto t^1$ asymptotically. In (b) the isotropic phase is quenched to $\Delta T=-0.084\mathrm{K}>-L_c∕c_p$, at which $R\propto t^{1∕2}$ asymptotically. In (a) the first profile is taken at $t=0.2\mathrm{s}$ and the last at $t=2\times {10}^6\mathrm{s}$. In (b) the first profile is taken at $t=10\mathrm{s}$ and the last at $t=2\times {10}^9\mathrm{s}$. Parameters are taken from Table .

Figure 4

(Color online) Exponent $n$ found by least-squares fitting of the data of Fig. 2 to $(R-R_0)\propto t^n$, as a function of quench depth $-\Delta T$, where the negative sign is introduced to be compatible with Ref. 1. The dotted (blue, online) line is $n$ fitted for radii ${10}^{-5}\mathrm{m}<R\left(t\right)<{10}^{-3}\mathrm{m}$, the dashed (red, online) line is a fit for ${10}^{-4}\mathrm{m}<R\left(t\right)<{10}^{-2}\mathrm{m}$, and the solid (black, online) line for ${10}^{-3}\mathrm{m}<R\left(t\right)<{10}^{-1}\mathrm{m}$. The thin solid line denotes the asymptotic value of $n$: (a) one dimension, (b) two dimensions, (c) three dimensions.