Modeling smectic layers in confined geometries: Order parameter and defects

We identify problems with the standard complex order parameter formalism for smectic-A (SmA) liquid crystals and discuss possible alternative descriptions of smectic order. In particular, we suggest an approach based on the real smectic density variation rather than a complex order parameter. This approach gives reasonable numerical results for the smectic layer configuration and director field in sample geometries and can be used to model smectic liquid crystals under nanoscale confinement for technological applications.

Figure 1

Disclination in a two-dimensional smectic phase. Bright and dark regions correspond to higher and lower density, respectively, and the red dotted line is the branch cut in $\psi$ and $\widehat{\mathbf{n}}$.

Figure 2

Simulation of smectic layers using a fixed director field (shown by short lines) with two half-charged disclinations. (a) Results using the complex order parameter approach of Eq. (2), with parameters $\alpha =-1,\beta =100,b_1=-3,b_2=5,e_1=-b_1-8b_2{(\pi /10)}^2$. Note that the line defect between the defect cores is unphysical. (b) Results using the real order parameter approach of Eq. (9), with parameters $a=-0.1,b=0,c=10,q=2\pi /10$, and $B=0.1/q^4$.

Figure 3

Simulations of a circular domain with boundary conditions requiring a single half-charged disclination. (a) Results using the complex order parameter approach of Eq. (2), with parameters $\alpha =-1,\beta =100,b_1=-3,b_2=5,e_1=-b_1-8b_2{(\pi /10)}^2,K=0.0025$. The line defect from the defect core to the boundary is unphysical. (b) Results using the real order parameter approach of Eq. (9), with parameters $a=-0.1,b=0,c=10,q=2\pi /10,B=0.1/q^4$, and $K=0.008$.

Figure 4

Visualization of the phase variation $\Phi (x,y)$ around a single edge dislocation.

Figure 5

Behavior as the dislocation is displaced with respect to the layer structure. Plots show the smectic density variation, the free energy density for the complex order parameter formalism, and the free energy density for the real density formalism (calculated for the cubic parameter $b=0$ and $b\ne 0$). Red represents highest values of density or free energy; blue represents lowest values.

Figure 6

Simulations of a circular domain with boundary conditions requiring tangential alignment (parts a and b) or radial alignment (c and d) of the director. The density functional free energy (9) is used. In a and c, parameters are $a=-5,b=0,c=5,B={10}^{-5},q=40$, and $K=0.3$. In b and d, the Frank constant is reduced to $K=0.05$.

Figure 7

Integration of the free energy density over a region containing an integer number of smectic layers.

Figure 8

Integrated free energy in the real density formalism as the dislocation is displaced with respect to the layer structure, as given by the parameter $\Delta \Phi$. (a) Parameters $a=-10,c=10,q=1,B=0.1/q^4$, and $b=0$, so that there is a symmetry between density minima and maxima. (b) Parameter $b=1$, breaking the symmetry between density minima and maxima.

Figure 9

Simulations of smectic layer configurations around disclinations of total topological charge $+2$ (parts a and c) and $-2$ (parts b and d). In a and b, the nematic director field inside the domain is specified, and the smectic layers respond to it. In c and d, the boundary conditions require a total topological charge of $\pm 2$, and the director field and the smectic layers inside the domain are both free to relax.