Volume phase transitions of smectic gels

We present a mean-field theory to describe volume phase transitions of side-chain liquid crystalline (LC) gels, accompanied by isotropic-nematic-smectic-$A$ phase transitions. Three different uniaxial nematic phases (${\text{N}}_1$, ${\text{N}}_2$, and ${\text{N}}_3$) and smectic-$A$ phases (${\text{S}}_1$, ${\text{S}}_2$, and ${\text{S}}_3$) are defined by using orientational order parameter $S_m$ of side-chain liquid crystals (mesogens), $S_b$ of semiflexible backbone chains, and a translational order parameter $\sigma$ for a smectic-$A$ phase. We derive the free energy for smectic-$A$ phases of side-chain LC gels dissolved in an isotropic solvent and examine the swelling curve of the LC gel, the orientational order parameters, and the deformation of the LC gel as a function of temperature. We find that the LC gel discontinuously changes the volume at an isotropic-nematic, an isotropic-smectic-$A$, and a nematic-smectic-$A$ phase transition.

Figure 1

(Color online) Schematic representations of three nematic and smectic-$A$ phases for side-chain liquid crystalline gels. Depending on two orientational order parameters, one is the $S_m$ of nematogenic side chains (mesogens) and the other is the $S_b$ of semiflexible backbone chain, we can define a nematic ${\text{N}}_1$ phase with $S_m>0$ and $S_b<0$, a ${\text{N}}_2$ phase with $S_m<0$ and $S_b>0$, and a ${\text{N}}_3$ phase with $S_m>0$ and $S_b>0$. In the ${\text{N}}_1$ phase, the backbone chain adopts an oblate shape. When these nematic domains are stacked into a smectic-$A$ phase, three different smectic-$A$ phases appear, depending on the value of the orientational order parameters $S_m$ and $S_b$. In the ${\text{S}}_1$ phase, the backbone chain is distributed on the $x\text{−}y$ plane, which is perpendicular to a nematic director. In the ${\text{S}}_2$ and ${\text{S}}_3$ phases, the backbone chain has a prolate shape within a layer of a smectic-$A$ phase.

Figure 2

Layer displacement $u\left(z\right)$ for a smectic-$A$ phase.

Figure 3

Swelling curve of the gel on the $\tau -{\varphi }_g$ plane for $t=10$, $b=1$, $n_m=3$, $c=0.5$, $c_{mb}=-1$, and $c_{bb}=0$. The value of $n_b$ is changed.

Figure 4

(a) Order parameters $S_m$, $S_b$, and $\sigma$ and (b) the deformations ${\kappa }_z/{\kappa }_{iso}$ and ${\kappa }_x/{\kappa }_{iso}$ of the gel, corresponding to $n_b=6$ in Fig. 3, plotted against the reduced temperature $\left(\tau \right)$.

Figure 5

Swelling curve of the gel on the $\tau -{\varphi }_g$ plane for $t=10$, $b=1$, $n_m=3$, $c=0.5$, $c_{mb}=1$, and $c_{bb}=0$. The value of $n_b$ is changed.

Figure 6

(a) Order parameters $S_m$, $S_b$, and $\sigma$ and (b) the deformations ${\kappa }_z/{\kappa }_{iso}$ and ${\kappa }_x/{\kappa }_{iso}$ of the gel, corresponding to $n_b=6$ in Fig. 5, plotted against the reduced temperature $\left(\tau \right)$.

Figure 7

Swelling curve of the gel on the $\tau -{\varphi }_g$ plane for $t=10$, $b=1$, $n_m=4$, $c=0.7$, $c_{mb}=-1$, and $c_{bb}=1$. The value of $n_b$ is changed.

Figure 8

(a) Order parameters $S_m$, $S_b$, and $\sigma$ and (b) the deformations ${\kappa }_z/{\kappa }_{iso}$ and ${\kappa }_x/{\kappa }_{iso}$ of the gel, corresponding to $n_b=6$ in Fig. 7, plotted against the reduced temperature $\left(\tau \right)$.