Elasticity of smectic liquid crystals with in-plane orientational order and dispiration asymmetry

The Nelson-Peliti formulation of the elasticity theory of isolated fluid membranes with orientational order emphasizes the interplay between geometry, topology, and thermal fluctuations. Fluid layers of lamellar liquid crystals such as smectic-$C$, hexatic smectics, and smectic-$C^{*}$ are endowed with in-plane orientational order. We extend the Nelson-Peliti formulation so as to bring these smectics within its ambit. Using the elasticity theory of smectics-$C^{*}$, we show that positive and negative dispirations (topological defects in Smectic-$C^{*}$ liquid crystals) with strengths of equal magnitude have disparate energies—a result that is amenable to experimental tests.

Figure 1

Schematic of smectics: ${\mathit{n}}_{\mathrm{F}}\equiv -{\mathit{n}}_{\mathrm{F}}$ is the unit, apolar Frank director that specifies the average orientation of molecules; $\mathit{N}$ is the unit layer normal. The equilibrium layer spacing is $d$. In $\text{Sm}A$ ${\mathit{n}}_{\mathrm{F}}\parallel \mathit{N}$; $\text{Sm}A$ does not have in-plane orientational order, the molecular orientation is not tilted. In $\mathrm{Sm}C$, the projection of ${\mathit{n}}_{\mathrm{F}}$ onto the layer plane, the polar vector $\mathit{c}$, spontaneously breaks the continuous azimuthal symmetry. The plane spanned by ${\mathit{n}}_{\mathrm{F}}$ and $\mathit{N}$ (the $xz$ plane) is a mirror plane with a point of inversion. $\mathrm{Sm}{C}^{*}$ has a chiral structure, in which the mirror symmetry of $\mathrm{Sm}C$ is lost; $\mathit{c}=c(cos\left(q^{*}z\right),sin\left(q^{*}z\right),0)$ in the ground state, i.e., ${\mathit{n}}_{\mathrm{F}}$ lies on a cone with its tip on a helix with pitch $P^{*}=2\pi /q^{*}$.

Figure 2

The planar surface $L_i$ labels the $i\mathrm{th}$ smectic layer in the reference lattice of $\mathrm{Sm}A$, with interlayer spacing $d$. Making a vertical cut $\mathbb{C}$ (shaded rectangle) through the layers, identifying the left lip of the cut on $L_{i-1}$ to the right lip of the cut on $L_i$, and letting the system relax leads to the right-handed, half-helicoidal surface shown. This is the Volterra construction of a screw dislocation. The $z$ axis is the singular dislocation line, and $\mathit{b}=d\widehat{\mathit{z}}$ is the Burgers vector.

Figure 3

Elliptic $\left(+1\right)$ and hyperbolic $\left(-1\right)$ disclinations in the $xy$ model. Upon traversing an anticlockwise, closed circuit, the vector field rotates in anticlockwise sense through $2\pi$ for a $+1$ disclination and in clockwise sense through $2\pi$ for a $-1$ disclination.

Figure 4

Volterra construction for a dispiration in $\mathrm{Sm}{C}^{*}$: $L_i$ (light planar surfaces) and ${\mathit{c}}_i=c_0(cos{\psi }_i,sin{\psi }_i)$ (thick arrows) represent the $i\mathrm{th}$ smectic layer and the corresponding $\mathit{c}$ field with pitch $P^{*}$. In the laboratory frame, ${\mathit{c}}_0\parallel \widehat{\mathit{x}}$. Thin arrows on $L_i$ represent ${\mathit{c}}_{i-1}$. Making a vertical cut $\mathbb{C}$ (shaded rectangle) through the layers and identifying the left lip of the cut on $L_{i-1}$ to the right lip of the cut on $L_i$ leads to a mismatch in the $\mathit{c}$ field across $\mathbb{C}$. To eliminate the mismatch, wedges of angle $\omega =d/|P^{*}|$ in the $\mathit{c}$ field need to be inserted at the central singular line—one wedge of angle $\omega$ per layer. Each wedge is a negative, partial disclination, since it does not correspond to a symmetry operation of the ground state of $\mathrm{Sm}{C}^{*}$. Post-relaxation, the construction described above leads to a wedge-screw dispiration—a screw dislocation associated with partial disclinations in each smectic layer.

Figure 5

A dispiration with ${\mathtt{s}}_d>0$. As in Fig. 4, the screw dislocation is right-handed; $\mathit{b}=d\widehat{\mathit{z}}$. However, the chirality of $\mathrm{Sm}{C}^{*}$ is opposite of that shown in Fig. 4; $\widehat{\mathbf{\lambda }}=-\widehat{\mathit{z}}$. Note that eliminating the mismatch in the $\widehat{\mathit{c}}$ field requires removal of wedges of angle $\omega$.