Molecular origin of the heterogeneity in the nematic and smectic liquid crystals: Elastic constants, gradients of order parameters, and visualization of small objects

The formation of heterogeneous nematic and smectic liquid crystals in the general case of an arbitrary geometry is investigated in the framework of molecular-statistical approach [Emelyanenko and Khokhlov, J. Chem. Phys. 142, 204905 (2015)]. The molecular aspects of the orientational and translational orderings at the curved surfaces of small solid objects dispersed in liquid crystal are considered, and the differential equations for gradients of the order parameters in vicinities of the small objects are presented in the general form. The five elastic constants are obtained within the same approach, from which we were able to predict that a significant space variations of the order parameters can be observed within the $0.5–0.8\mu \text{m}$ area around any small object, almost independently of its own dimension. Therefore, the liquid crystals can be a simple tool for the optical visualization of nano-objects. It is also demonstrated that the kind of molecular self-organization (smectic, nematic or conventionally isotropic) at the surfaces of small solid objects can be different from that in the bulk of mesogenic material. Totally we predict eight various combinations of simple states at the surfaces and in the bulk depending on the solid objects' size and temperature. It is also shown that the surfaces of $10\mu \text{m}$-size solid objects and larger act almost as flat surfaces, while the surfaces of $1\mu \text{m}$-size solid objects and smaller act almost as point defects.

Figure 1

Pair of neighboring molecules in the inhomogeneous LC (nematic or smectic). The plane perpendicular to director ${\mathbf{n}}_1$ is shown by dotted line. The projection of intermolecular vector ${\mathbf{r}}_{12}$ on this plane is shown by dash arrow. Angle $\beta$ is the angle between ${\mathbf{r}}_{12}$ and its projection on the plane mentioned above.

Figure 2

Model of a liquid crystal molecule (solid ellipsoid) and its nearest surrounding: side view (a) and top view (b). The surrounding molecules are shown by empty ellipsoids. The contours of the whole nearest surrounding are shown by dash lines.

Figure 3

The temperature dependencies of the elastic constants in the homogeneous LC at ${\sigma }_0{J}_{202}^{\left(0\right)}/k_B=1100\text{K}$, ${\sigma }_0{J}_{220}^{\left(0\right)}/k_B={\sigma }_0{J}_{022}^{\left(0\right)}/k_B=1023\text{K}$, ${\sigma }_0{J}_{222}^{\left(0\right)}/k_B=-1122\text{K}$, ${\sigma }_0{J}_{242}^{\left(0\right)}/k_B=-3080\text{K}$, ${\sigma }_0{J}_{404}^{\left(0\right)}/k_B=-55\text{K}$, ${\sigma }_0{J}_{202}^{\left(2\right)}/k_B=51\text{K}\mu {\text{m}}^2$, ${\sigma }_0{J}_{222}^{\left(2\right)}/k_B=39\text{K}\mu m^2$, ${\sigma }_0{J}_{422}^{\left(2\right)}/k_B={\sigma }_0{J}_{224}^{\left(2\right)}/k_B=35\text{K}\mu {\text{m}}^2$, ${\sigma }_0{J}_{220}^{\left(2\right)}/k_B={\sigma }_0{J}_{022}^{\left(2\right)}/k_B=9\text{K}\mu {\text{m}}^2$ (solid lines) or ${\sigma }_0{J}_{220}^{\left(2\right)}/k_B={\sigma }_0{J}_{022}^{\left(2\right)}/k_B=0\text{K}\mu {\text{m}}^2$ (dash lines), and all other constants participating in approximation Eq. (17) equal to zero. Here $V_0\approx 1.1\times {10}^{-24}{\text{m}}^3$ (see discussion at the end of Sec. 2b).

Figure 4

Liquid crystal media with solid particles dispersed inside; spherical LC droplet with radial director distribution (shown by dotted lines) and solid spherical particle in the center; an illustration of the orientational order parameter variation with distance from a small particle. Here the radius of each particle is $r_0$; LC exhibits homeotropic anchoring at each sphere.

Figure 5

The temperature dependencies of the prime orientational order parameter (a) and of the variance of ${sin}^2\beta$ (b) at ${\sigma }_0{J}_{202}^{\left(0\right)}/k_B=1100\text{K}$, ${\sigma }_0{J}_{220}^{\left(0\right)}/k_B={\sigma }_0{J}_{022}^{\left(0\right)}/k_B=1023\text{K}$, ${\sigma }_0{J}_{222}^{\left(0\right)}/k_B=-1122\text{K}$, ${\sigma }_0{J}_{242}^{\left(0\right)}/k_B=-3080\text{K}$, ${\sigma }_0{J}_{404}^{\left(0\right)}/k_B=-55\text{K}$, $J_2^{\mathrm{surf}}/k_B=0.55\text{K}$, $J_4^{\mathrm{surf}}/k_B=5.5\text{K}$, ${\sigma }_0{J}_{202}^{\left(2\right)}/k_B=51\text{K}\mu {\text{m}}^2$, ${\sigma }_0{J}_{222}^{\left(2\right)}/k_B=39\text{K}\mu {\text{m}}^2$, ${\sigma }_0{J}_{422}^{\left(2\right)}/k_B={\sigma }_0{J}_{224}^{\left(2\right)}/k_B=35\text{K}\mu {\text{m}}^2$, and all other constants participating in approximation Eqs. (17) and (51) equal to zero. The black thick line corresponds to the homogeneous LC (far from any particles), while the thin lines with different numbers correspond to the LC at the surfaces of spherical particles of various radii: (1) $r_0=\infty$ [flat surface]; (2) 10; (3) 3; (4) 2.5; (5) 1.9; (6) 1.55; (7) $1.25\mu \text{m}$.

Figure 6

The $r_0\text{−}T$ phase diagram at the same molecular parameters [constants participating in approximation Eqs. (17) and (51)] as in Fig. 5. The blue thick lines correspond to the structure transitions in the bulk of LC, while the red thin lines correspond to the structure transitions at the surfaces of particles of diameter $r_0$. The dash lines correspond to the structure transitions at a flat surface. The triangles correspond to the points in the diagram, at which the $r$-dependence of the $S_2$ and $\mathrm{Var}\left({sin}^2\beta \right)$ parameters is presented in Figs. 7 and 8. The color of each triangle corresponds to the color of particular lines in Figs. 7 and 8.

Figure 7

The dependencies of the prime orientational order parameter (a) and of the variance of ${sin}^2\beta$ (b) on the distance $r-r_0$ from the particle's surface just below (at $T=342.0\text{K}$, solid triangles in Fig. 6) and just above (at $T=342.2\text{K}$, hollow triangles in Fig. 6) the smectic-nematic transition temperature in the bulk of LC at various sizes of the particles: (1) $r_0=\infty$ [flat surface]; (2) 10; (3) 3; (4) 2.5; (5) 1.9; (6) 1.55; (7) $1.25\mu$m. Molecular parameters [constants participating in approximation Eqs. (17) and (51)] are the same as in Figs. 5 and 6. The color of each curve corresponds to the color of particular triangle in Fig. 6.

Figure 8

The dependencies of the prime orientational order parameter (a) and of the variance of ${sin}^2\beta$ (b) on the distance $r\text{–}{r}_0$ from the particle's surface just below (at $T=345.3\text{K}$, half up triangles in Fig. 6) and just above (at $T=343.4\text{K}$, half down triangles in Fig. 6) the nematic-isotropic transition temperature in the bulk of LC at various sizes of the particles: (1) $r_0=\infty$ [flat surface]; (2) 10; (3) 3; (4) 2.5; (5) 1.9; (6) 1.55; (7) $1.25\mu$m. Molecular parameters [constants participating in approximation Eqs. (17) and (51)] are the same as in Figs. 5 and 6. The color of each curve corresponds to the color of particular triangle in Fig. 6. The $4/45$ level of $\mathrm{Var}\left({sin}^2\beta \right)$, corresponding to the uniform distribution of the intermolecular vectors, is shown by the dash line.