Molecular-statistical theory of elasticity in nematic liquid crystals composed of polar and nonpolar molecules

A molecular-statistical theory of the orientational elasticity of nematic liquid crystals has been developed employing the orientational deformation tensor which describes the rotation of the director. An explicit expression for the general elasticity tensor of the nematic phase has been obtained and the Frank elastic constants are expressed in terms of the three independent parameters of this tensor. Explicit expressions for the Frank elastic constants have been derived in the molecular field approximation in terms of the orientational order parameters and the corresponding coefficients of expansion of the intermolecular potential in spherical invariants. Frank elastic constants have been calculated numerically for nematic liquid crystals composed of both polar and nonpolar molecules together with the orientational order parameters using the classical Gay-Berne model interaction potential and the two of its popular modifications. The polarity of the uniaxial molecular shape has been directly introduced into the model potential by modifying the distance of closest approach. The elastic constants are presented as functions of temperature for different values of the molecular elongation, the anisotropy of the potential well and the molecular shape polarity. It has been shown that the elastic constants are much more sensitive to the details of the intermolecular interaction potential in comparison with the orientational order parameters. In particular, a relatively weak polarity of the molecular shape may result in an unusual decrease of the splay constant $K_{11}$ which may vanish at some temperature leading to the instability of the homogeneous nematic phase. This may represent a mechanism of the formation of the splay-bend phase.

Figure 1

Colormaps of the dimensionless parameters of the classical G-B pair potential with $\mu =2$ and $\nu =1$ characterizing its expansion in spherical invariants determining the nematic ordering ($v^{202}$ and $v^{404}$) and the elastic constants ($u^{202}$, $u^{404}$, $u^{222}$, and $u^{422}$). The white dotted line of the border between the areas of positive and negative elastic constants and four exemplary points A, B, C, and D are shown in the $v^{202}$ plot.

Figure 2

Colormaps of the dimensionless parameters of the modified G-B pair potential with $\mu =1$ and $\nu =2$ characterizing its expansion in spherical invariants determining the nematic ordering ($v^{202}$ and $v^{404}$) and elastic constants ($u^{202}$, $u^{404}$, $u^{222}$, and $u^{422}$). Four exemplary points A, B, C, and D are marked in the $v^{202}$ plot.

Figure 3

Same as Fig. 2 for the modified G-B pair potential with $\mu =1$ and $\nu =3$.

Figure 4

Thermodynamics of nematic liquid crystal determined by the parameters of classical G-B potential with $\mu =2$, $\nu =1$ (see Fig. 1): (a) Dependencies of the order parameters $S$ and $S_4$ and (b) dimensionless elastic constants $k_{11}$ (solid), $k_{22}$ (dashed), and $k_{33}$ (dotted) on the dimensionless temperature for four sets of parameters corresponding to the points A, B, C, and D,marked in Fig. 1. Same colors correspond to same points.

Figure 5

Same as Fig. 4 for the modified G-B potential with $\mu =1$, $\nu =2$. The four sets of parameters correspond to the points marked as A, B, C, and D in Fig. 2.

Figure 6

Same as Fig. 4 for the modified G-B potential with $\mu =1$, $\nu =3$. The four sets of parameters correspond to the points marked as A, B, C, and D in Fig. 3.

Figure 7

Cuts of the range (71) of polar G-B potential with $\eta =0.03$ (solid) and $\eta =0$ (dashed) by the plane containing the main molecular axes ${\mathbf{a}}_{1,2}$ for their antiparallel and perpendicular orientations. The relative molecular elongation is $\kappa =4$.

Figure 8

Colormaps of dimensionless parameters of the expansion in spherical invariants of the modified polar G-B pair potential with $\mu =1$, $\nu =3$, and $\eta =0.005$. In the $v^{202}$ plot the four exemplary points A, B, C, and D are shown for which we evaluate the elastics constants in Fig. 9 and the white dashed line shows the border of the area with $K_{11}$ singularity. The colormaps of not shown parameters are marginally affected by $\eta$ and remain practically identical to those in Fig. 3.

Figure 9

Same as Fig. 6 for the modified polar G-B potential with $\mu =1$, $\nu =3$, and $\eta =0.005$. The four sets of parameters correspond to the points marked as A, B, C, and D in Fig. 8.

Figure 10

Colourmaps of the contributions from different vector combinations to the dimensionless parameter $u^{222}$ of the classical Gay-Berne pair potential with $\mu =2$ and $\nu =1$.