Molecular theory of a ferromagnetic nematic liquid crystal

We employ a version of classical density functional theory to study the phase behavior of a simple model liquid crystal in an external field. The uniaxially symmetric molecules have a spherically symmetric core with superimposed orientation-dependent attractions. The interaction between the cores consists of a hard-sphere repulsion plus an isotropic square-well attraction. The anisotropic part of the interaction potential allows for the formation of a uniaxially symmetric nematic phase. The orientation of the molecules couples to an external polar field. The external field is capable of rotating the nematic director $\widehat{\mathit{n}}$ in the $x-z$ plane. The field is also capable of changing the topology of the phase diagram in that it suppresses the phase coexistence between an isotropic liquid and a nematic phase observed in the absence of the field. We study the transition from an unpolar to a polar nematic phase in terms of the orientation-distribution function (odf), nematic and polar order parameters, and components of $\widehat{\mathit{n}}$. If represented suitably the odf allows us to study orientational changes during the switching process between nonpolar and polar nematic phases. We also give a simple argument that explains why nematic order is lost whereas polar order persists up to the gas-liquid critical point along the coexistence curve. We also discuss the relevance of our theory for future experimental studies.

Figure 1

The phase diagram of the liquid crystal in temperature ($T$)–density ($\rho$) representation for ${\varepsilon }^{\prime }=0.40$ [see Eq. (2.7b)] and $H_{\mathrm{n}}=5.0\times {10}^{-2}$ [see Eq. (2.16)]; ($\blacksquare$) $N$, ($▲$), $L$, ($⚫$) $G$ phases (see text). The inset shows an enhancement of a part of the phase diagram.

Figure 2

Plots of the nematic order parameter $S$ as a function of temperature $T$ along the various phase boundaries shown in Fig. 1; ($⚫$) $G$, ($▲$) $L$, and ($\blacksquare$) $N$ phases. The vertical lines demarcate $T=T_{\mathrm{tr}}$ ($—$) and $T=T_{\mathrm{c}}$ ($┄$), respectively. The horizontal line ($···$) represents the threshold value $S_{\mathrm{LdG}}=\frac{1}{3}$ obtained from the LdG theory (see text). In the curve corresponding to the $L$ phase, the upper branch refers to phase coexistence with the $N$ phase whereas the lower branch is computed at coexistence with $G$ phase (see Fig. 1). Note the different scales used on the ordinate.

Figure 3

The orientation distribution function $\alpha \left(\omega \right)$ in the $N$ phase as a function of the transformed polar and azimuthal angles $\tilde{\theta }$ and $\tilde{\varphi }$, respectively (see Sec. 5). To make the plots look more symmetric, the azimuthal angle has been shifted by $\pi$ so that $\tilde{\phi }\in \left[-\pi ,\pi \right]$. The value of $\alpha \left(\omega \right)$ can be read off the attached color bars. The plots have been obtained for $H_{\mathrm{n}}^{\prime }=5.0\times {10}^{-2}$ and $H_{\mathrm{p}}^{\prime }=0.0$; (a) $T=1.20$ and (b) $T=0.75$ at phase coexistence above and below $T_{\mathrm{tr}}\simeq 0.96$, respectively (see Fig. 1).

Figure 4

As Fig. 1, but in the presence of the polar field. The phase diagram has been obtained for $H_{\mathrm{p}}=0.75$; ($\blacksquare$) $L$ phase, ($\blacksquare$) $G$ phase.

Figure 5

Plots of the nematic order parameter $S$ [see Eq. (5.3)] and the polarization $P$ [see Eq. (5.8)] as functions of the temperature $T$; ($⚪$) $S$ and ($⚫$) $P$ in the $N$ and $L$ phases, ($⚪$) $S$ and ($⚫$) $P$ in the $G$ phase. The curves have been obtained for $H_{\mathrm{p}}^{\prime }=0.75$ (see also Fig. 4). The dotted horizontal line represents $S_{\mathrm{LdG}}=\frac{1}{3}$ (see text).

Figure 6

As Fig. 3, but for $H_{\mathrm{p}}^{\prime }=0.75$. We emphasize the different scales used for the color bars attached to the right of each part of the figure; (a) $T=0.75$, (b) $T=1.00$, (c) $T=1.20$, and the plots refer to the $L$ phase (see Fig. 5).

Figure 7

Plots of components $n_{\alpha }$ of the nematic director $\widehat{\mathit{n}}$ and of components $P_{\alpha }$ of the unit polarization vector $\widehat{\mathit{P}}$ as functions of the coupling constant $H_{\mathrm{p}}^{\prime }$; ($\blacksquare$) $n_{\mathrm{x}}$, ($\blacksquare$) $n_{\mathrm{z}}$; ($⚫$) $P_{\mathrm{x}}$, ($⚫$) $P_{\mathrm{z}}$. The curves have been obtained for $H_{\mathrm{n}}^{\prime }=1.25\times {10}^{-1}$ at a temperature $T=0.85$.

Figure 8

Plot of the angle $\vartheta$ as a function of the coupling strength $H_{\mathrm{p}}^{\prime }$ of the polar field ${\mathit{H}}_{\mathrm{p}}$. Notice also the broken ordinate.

Figure 9

As Fig. 7, but for the nematic order parameter $S$ ($\blacksquare$) (right ordinate) and the polarization $P$ ($⚫$) (left ordinate) (cf. Fig. 7).

Figure 10

As Fig. 3, but for $H_{\mathrm{n}}^{\prime }=1.25\times {10}^{-1}$ and $T=0.85$. (a) $H_{\mathrm{p}}^{\prime }=0.700$, (b) $H_{\mathrm{p}}^{\prime }=0.701$, (c) $H_{\mathrm{p}}^{\prime }=0.731$.