Order-disorder molecular model of the smectic-$A$–smectic-$C$ phase transition in materials with conventional and anomalously weak layer contraction

We develop a molecular-statistical theory of the smectic-$A$–smectic-$C$ transition which is described as a transition of the order-disorder type. The theory is based on a general expansion of the effective interaction potential and employs a complete set of orientational order parameters. All the order parameters of the smectic-$C$ phase including the tilt angle are calculated numerically as functions of temperature for a number of systems which correspond to different transition scenario. The effective interaction potential and the parameters of the transition are also calculated for specific molecular models based on electrostatic and induction interaction between molecular dipoles. The theory successfully reproduces the main properties of both conventional and so-called “de Vries–type” smectic liquid crystals, clarifies the origin of the anomalously weak layer contraction and describes the tricritical behavior at the smectic-$A$–smectic-$C$ transition. The “de Vries behavior,” i.e., anomalously weak layer contraction is also obtained for a particular molecular model based on interaction between longitudinal molecular dipoles. A simple phenomenological model is presented enabling one to obtain explicit expressions for the layer spacing and the tilt angle which are used to fit the experimental data for a number of materials.

Figure 1

Typical variation of smectic layer spacing versus nematic order parameter according to phenomenological formulas (3, 5). Conventional contraction (solid line) is shown for $c^{\prime }=2$, $f_1=2$, and $f_2=3$. De Vries–like behavior (dashed line) corresponds to $c^{\prime }=0$, $f_1=2$, and $f_2=1.5$.

Figure 2

Experimental data on temperature dependence of layer thickness (a) and tilt angle (b) for four different compounds fitted by the phenomenological relations (3, 4, 5). The parameters used are listed in Table .

Figure 3

Schematic of molecular orientation.

Figure 4

(Color online) $\mathrm{Sm}A$-$\mathrm{Sm}C$ phase transition obtained for the parameters $w_1=-0.03$, $w_2=-1$, $w_3=-0.85$ (a) or $w_3=-0.78$ (b), $w_4=-0.78$.

Figure 5

(Color online) $\mathrm{Sm}A$-$\mathrm{Sm}C$ phase transition perturbing the nematic order. Thin dashed line shows the unperturbed $S$. The interaction constants are $w_1=-0.3$, $w_2=-1$, $w_3=-1.4$, $w_4=-1.2$.

Figure 6

Temperature dependence of the average projection of long molecular axis on the layer normal. The interaction constants are taken as in Fig. 4a (solid line) and Fig. 4b (dashed line).

Figure 7

The effect of the perturbation of nematic order on the layer spacing. The solid line shows the real behavior with constants as in Fig. 5, while the dashed line is calculated according to the classical formula (42).

Figure 8

Discontinuity of the heat capacity due to a conventional second order $A$-$C$ transition. The interaction constants are the same as in Figs. 4a (solid line) 4b (dashed line).

Figure 9

Anomaly of heat capacity due to a perturbation of the nematic order by the tilting transition. The interaction constants are the same as in Figs. 5, 7.

Figure 10

(Color online) Schematics of model pair molecular interactions: hardcore repulsion attraction (a), electrostatic dipole-dipole coupling (b), and dipolar induction interaction (c).

Figure 11

(Color online) The ratios $w_i∕\mid w_2\mid$ as a function of well depth anisotropy calculated for the Gay-Berne potential with the molecular elongation $\kappa =4$. The indices $i$ are indicated.

Figure 12

(Color online) Normalized coefficients $w_i∕\mid w_2\mid$, $i=1,2,3,4$ arising from the dipole-dipole electrostatic potential as a function of the dipole location $\nu$ calculated with the hard rod cutoff $\kappa =4$.

Figure 13

(Color online) Example of the $\mathrm{Sm}A$-$\mathrm{Sm}C$ phase transition occurring when the dipole-dipole interaction is added to the Gay-Berne potential. The parameters were chosen to provide the dimensionless ratio $\frac{d^2\nu }{r_0^4{\epsilon }_0}=0.11$.

Figure 14

(Color online) Normalized coefficients $w_i∕\mid w_2\mid$ arising from the dipole-dipole induction interaction potential as a function of the dipole location $\nu$ calculated for the isotropic molecular polarizability and the hard rod cutoff $\kappa =4$.