Computational aspects of the smectization process in liquid crystals: An example study of a perfectly aligned two-dimensional hard-boomerang system

A replica method for calculation of smectic liquid crystal properties within the Onsager theory has been presented and applied to an exemplary case of two-dimensional perfectly aligned needlelike boomerangs. The method allows one to consider the complete influence of the interaction terms in contrast to the Fourier expansion method which uses mostly first or second order terms of expansion. The program based on the replica algorithm is able to calculate a single representative layer as an equivalent set of layers, depending on the size of the considered width of the sample integration interval. It predicts successfully smectic density distributions, energies, and layer thicknesses for different types of layer arrangement—of the antiferroelectric or of the smectic A order type. Specific features of the algorithm performance and influence of the numerical accuracy on the physical properties are presented. Future applications of the replica method to freely rotating molecules are discussed.

Figure 1

Boomerang particles. The arm length is given by $L$ and the apex angle is $\alpha$. Boomerangs are perfectly aligned and two orientations are considered, $R$ and $L$. Different shading is used to distinguish particles with different orientation.

Figure 2

Geometry of the integration points along the $Y$ axis. The size of $S_d$ should be such that the particles placed at the outermost points from the central interval and from the neighboring interval will not lead to an overlap while the particles are being moved along the $X$ axis; i.e., their excluded $X$ interval for such points is already zero. An example of the excluded slice for the boomerangs at $y_i$ and $y_j$ is given as a thick solid (red) line.

Figure 3

Shape of the excluded areas for two different mutual orientations of boomerangs.

Figure 4

Solutions $f_L$ and $f_R$ obtained from the initial configurations due to Eqs. (15) with $A_L=1$ and $A_R=1$: (a) starting profiles of $f_L$ and $f_R$ according to Eqs. (15); (b) an intermediate stage of $f_L$ and $f_R$, when the error is $\text{error}\le \epsilon =50$; (c) an intermediate stage of $f_L$ and $f_R$, when the error is $\text{error}\le \epsilon =10$; and (d) the final solutions of $f_L$ and $f_R$, when the error is $\text{error}\le \epsilon =0.00001$. At each stage the solutions are continuous.

Figure 5

Antiferroelectric phase.

Figure 6

Solutions $f_L$ and $f_R$ obtained from the initial configurations due to Eqs. (15) with $A_L=0.4$ and $A_R=0.7$: (a) starting profiles of $f_L$ and $f_R$ according to Eqs. (15); (b) an intermediate stage of $f_L$ and $f_R$, when the error is $\text{error}\le \epsilon =50$; (c) an intermediate stage of $f_L$ and $f_R$, when the error is $\text{error}\le \epsilon =10$; and (d) the final solutions of $f_L$ and $f_R$, when the error is $\text{error}\le \epsilon =0.00001$. Periodicity of the initial profiles is much different from the final solutions which results in the fact that intermediate profiles are discontinuous, yet final solutions are continuous again.

Figure 7

Free energy versus $S_d$ interval. Two subsequent minimums correspond to the same type of solution. The second minimum corresponds to the solutions covering twice the number of smectic layers as in the first minimum case.

Figure 8

Free energy versus $S_d$ interval. Two subsequent minimums correspond to the same type of solution.

Figure 9

Accuracy of free energy calculations with respect to the number of integration points.

Figure 10

The distribution profiles $f_R$ and $f_L$ for $S_d=0.1775$ and $S_d=0.335$. These are the same functions, but for a longer interval of $S_d$ the number of smectic layers obtained is twice the number as in the case of $S_d=0.1775$.

Figure 11

Comparison of the distributions for the smectic A case and for the antiferroelectric smectic obtained for the same density, $d=800$. In the smectic A case the free energy is $E=10.76$ and in the AF smectic it is $E=7.317$.

Figure 12

Smectic A occurs when the numbers of left and right particles within each layer are on average the same.

Figure 13

Fitting of the probe function of the form $exp(\lambda cos(2\pi y/d\left)\right)$ (dots) to the $f_R$ solution (solid line) for $d=600, L=0.1, S_d=0.16$, and $\alpha =\pi /2$.