Steric effects in a mean-field model for polar nematic liquid crystals

The existence of uniaxial liquid crystals comprising polar molecules, with all the dipoles aligned in a parallel pattern, is classically ruled out. Generally, there are two different avenues to a mean-field theory for liquid crystals: one is based on short-range, repulsive, steric forces, and the other is based on long-range, globally attractive, dispersion forces. Purely polar steric interactions have been shown to have the potential of inducing unexpected orientationally ordered states. In real molecules, anisotropies both in shape and in polarizability coexist; it has been shown that dispersion forces interaction can be combined with hard-core repulsion in a formal theory, based on a steric tensor. Starting from this, we build an interaction Hamiltonian featuring the average electric dipolar energy exchanged between molecules with the same excluded region. Under the assumption that the molecular shape is spheroidal, we propose a mean-field model for polar nematic liquid crystals which can exhibit both uniaxial and biaxial polar phases. By means of a numerical bifurcation analysis, we discuss the stability of the equilibrium against the choice of two model parameters, one describing the degree of molecular shape biaxiality and the other describing the relative orientation of the electric dipole within each molecule. We find only uniaxial stable phases, which are effectively characterized by a single scalar order parameter.

Figure 1

The van der Waals regions $\mathcal{R}$ and ${\mathcal{R}}^{\prime }$ surrounding the point dipoles $\mathit{p}$ and ${\mathit{p}}^{\prime }$ placed at positions $p_0$ and $p_0^{\prime }$ in space. The unit vector ${\mathit{e}}_r$ is directed from $p_0$ to $p_0^{\prime }$; $\mathit{\nu }$ and ${\mathit{\nu }}^{\prime }$ are the unit outer normals to $\mathcal{R}$ and ${\mathcal{R}}^{\prime }$, respectively. The boundary $\partial {\mathcal{R}}^{\ast }$ of the excluded region ${\mathcal{R}}^{\ast }$ is the set of all points that $p_0^{\prime }$ can reach while $\partial {\mathcal{R}}^{\prime }$ glides without rolling over $\partial \mathcal{R}$. The unit vector ${\mathit{\nu }}^{\ast }$ is the outer normal to $\partial {\mathcal{R}}^{\ast }$.

Figure 2

Cartoon of the ellipsoid describing the molecular shape. The unit vector $\mathbf{\ell }$ describes the molecular electric polarity; $\mathit{m}$ and $\mathit{e}$ are the unit vectors adopted to describe the molecular tensors $\mathit{q}$ and $\mathit{b}$ in Eq. (21).

Figure 3

The order parameter space $\mathcal{P}:\left\{\left(\lambda ,\alpha \right):\lambda ∊\left[-1,1\right],\alpha ∊\left[0,\frac{\pi }{2}\right]\right\}$ is split for later reference in the subregions $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$. The boundary separating $\mathcal{B}$ from $\mathcal{C}$ in $\mathcal{P}$ is described by Eq. (B7).

Figure 4

The microscopic degree of polarization $\Pi$ in the ground state of $H_{\text{dip}}$, as given by Eq. (25), plotted over the parameter space $\mathcal{P}$. $\Pi$ is a smooth function away from the point $\left(\frac{1}{3},\frac{\pi }{2}\right)∊\partial \mathcal{P}$.

Figure 5

Bifurcation diagrams for the scalar order parameters $u$ and $w$ against the reciprocal dimensionless temperature $\beta$. Stable branches are represented by solid lines, while unstable branches are represented by dashed lines. At the critical value $\beta ={\beta }_{\text{c}}$, which agrees with the value delivered by Eq. (C2), a second order transition takes place: the isotropic phase becomes unstable, and stability occurs at a polar uniaxial stable phase with $t=v=0$ and $u/w=\rho$, in complete agreement with Eq. (49). (a) Typical bifurcation diagram in region $\mathcal{A}$, parameters used are $\lambda =-\frac{1}{2}$ and $\alpha =\frac{\pi }{4}$. Here ${\beta }_{\text{c}}=3.5802$, $\rho =0.0554$; the dominant order parameter is $w$. (b) Typical bifurcation diagram in region $\mathcal{B}$, parameters used are $\lambda =0$ and $\alpha =\frac{2}{5}\pi$. Here ${\beta }_{\text{c}}=4.8708$, $\rho =-0.1318$; the dominant order parameter is $w$. (c) Typical bifurcation diagram in region $\mathcal{C}$, parameters used are $\lambda =\frac{4}{9}$ and $\alpha =\frac{7}{16}\pi$. Here ${\beta }_{\text{c}}=80.1608$, $\rho =-1.6879$; the dominant order parameter is $u$.

Figure 6

The graph of the saturated polarization $\mathsf{P}$ defined by Eq. (50), plotted over the parameter space $\mathcal{P}$. $\mathsf{P}$ is a continuous function in $\mathring{\mathcal{P}}$, exhibiting a discontinuity at $\left(\frac{1}{3},\frac{\pi }{2}\right)∊\partial \mathcal{P}$. It fails to be differentiable along the line $\alpha ={\alpha }_{\text{c}}\left(\lambda \right)$, see Eq. (B7), which separates $\mathcal{B}$ from $\mathcal{C}$ in Fig. 3.

Figure 7

The graph of the difference $\Delta ≔\Pi -\mathsf{P}$ between the microscopic degree of polarization $\Pi$ in the ground state of $H_{\text{dip}}$ and the macroscopic saturated polarization defined by Eq. (50). $\Delta$ is appreciably different from zero only along the boundary separating the regions $\mathcal{B}$ and $\mathcal{C}$ in $\mathcal{P}$, see Fig. 3.

Figure 8

The graph of the function $\chi$ defined in Eq. (B8) over the parameter space $\mathcal{P}$; $\chi$ is a smooth function away from the discontinuity for $\lambda =\frac{1}{3}$ and $\alpha =\frac{\pi }{2}$.

Figure 9

Cartoon of two molecules in the ground state of $H_{\text{dip}}$ in Eq. (23), for a choice of parameters $\left(\lambda ,\alpha \right)$ in $\mathcal{P}\backslash \mathcal{A}$, see Fig. 3.

Figure 10

The order parameter space $\mathcal{P}$ enriched with further information obtained from the bifurcation analysis of the mean-field free energy. On three segments on $\partial \mathcal{P}$, namely, $\left\{\left(\lambda ,\alpha \right):\lambda ∊\left[-1,1\right],\alpha =0\right\}$, $\left\{\left(\lambda ,\alpha \right):\lambda =1,\alpha ∊\left[0,\frac{\pi }{2}\right]\right\}$, and $\left\{\left(\lambda ,\alpha \right):\lambda ∊\left[\frac{1}{3},1\right],\alpha =\frac{\pi }{2}\right\}$, represented here as solid lines, the function ${\beta }_{\text{c}}$ in Eq. (C2) diverges to $\infty$. Along the white line, which is in part on $\partial \mathcal{P}$ (for $\alpha =\frac{\pi }{2}$) and in part in $\mathcal{B}$ [for $\alpha ={\alpha }_0\left(\lambda \right)$ as in Eq. (C4)], ${\beta }_{\text{c}}$ is stationary with respect to $\alpha$. Bold points are disseminated along the locus where the function $\rho$ in Eq. (49) vanishes. More specifically, $0<\rho <1$ in $\mathcal{A}$, $-1<\rho <0$ in $\mathcal{B}$, and $\rho <-1$ in $\mathcal{C}$.