Orientational energy of anisometric particles in liquid-crystalline suspensions

We obtain a general expression for the orientational energy of an individual anisometric particle suspended in uniform nematic liquid crystals when the main axis of the particle rotates with respect to the nematic director. We show that there is a qualitative and quantitative analogy between the internal and external problems for cylindrical volumes of nematic liquid crystals, and on this basis we obtain an estimate of the orientational energy of a particle of cylindrical (rodlike, needlelike, or ellipsoidal) shape. For an ensemble of such particles we propose a modified form of their orientational energy in the nematic matrix. This orientational energy has the usual second-order term, and additional fourth-order term in the scalar product of the nematic director and the vector which characterizes an average direction of the main axes of the particles. As an example we obtain the expression for the free energy density of ferronematics, i.e., colloidal suspensions of needlelike magnetic particles in nematic liquid crystals. Unlike previous models, the free energy density includes the proposed modified form of the particle orientational energy, and also a contribution describing the surface saddle-splay deformations of the liquid crystal matrix.

Figure 1

Profiles of the function $\left(F_{S,4}/\left|{\tilde{A}}_2\right|\right)$ which corresponds to the angular part of bistable anchoring energy and angular dependence of orientational energy of ferroparticles (4), with ${\tilde{A}}_2>0$ (a) and ${\tilde{A}}_2<0$ (b) for different values of $\tilde{\zeta }$.

Figure 2

Qualitative analogy between internal and external problems for cylindrical volumes of NLC: (a) ER structure in cylindrical capillary; (b) director field near a particle with rigid homeotropic anchoring for $\mathit{u}\parallel {\mathit{n}}_0$, obtained from the ER structure with the use of transformation (11); (c) distribution of the director on going from rigid to soft anchoring at a particle surface, obtained with use of transformation $R\to R_{\mathrm{ef}}=p_{\parallel }R$ [see Eq. (14)].

Figure 3

Qualitative analogy between internal and external problems for cylindrical volumes of NLC: (a) PP configuration in cylindrical capillary; (b) director field near a particle with rigid homeotropic anchoring for $\mathit{u}\perp {\mathit{n}}_0$, obtained from the PP structure with use the of transformation (11); (c) distribution of the director on going from rigid to soft anchoring at a particle surface, obtained with the use of transformation $R\to R_{\mathrm{ef}}=p_{\perp }R$ [see Eq. (14)].

Figure 4

Normalized coefficient ${\overline{A}}_2$ as a function of the dimensionless parameter $\overline{w}$. Solid curves are the results of exact calculation of ${\overline{A}}_2$ with the use of Eqs. (16), (17), and (21) for ${\overline{k}}_{24}=0$ (curve 1), 1 (curve 2), and 2 (curve 3); the dashed line is the approximate value of ${\overline{A}}_2$ from Eq. (19). The plot in the inset shows the behavior of curves 2 and 3 at large values of $\overline{w}$.