Peculiarity of the interaction of small particles in smectic liquid crystals

We investigate the peculiarity of the interaction between particles immersed into a smectic liquid crystal with a layered structure. Such a structure of a liquid crystal imposes restrictions on possible deformations of the layer displacement field. Previous studies neglect this fact and give improper results for the interaction potential within one molecular layer. The present paper shows that such restrictions yield an interaction potential substantially different from those of previous studies. Oscillatory behavior, which was not present in the potentials of previous studies, might give rise to superstructures of immersed particles with finite interparticle distance.

Figure 1

Numerical computations of the energy of the interaction between asymmetrical (a) and axially symmetric particles (b) as a function of two variables. $\tilde{U}=U_{p{p}^{\prime }}{\pi }^{2}B/\left(2{\alpha }^2{q}_0^3\right)$ in (a) and $\tilde{U}=U_{p{p}^{\prime }}{\pi }^{2}B/\left(2{\beta }^2{q}_0^5\right)$ in (b). Analytical results are almost identical to the numerical ones and therefore we do not present them here.

Figure 2

$W(q_{\perp },z)$ (line $a$) and $q_{\perp }^3{J}_0\left(q_{\perp }{r}_{\perp }\right)$ (line $b$) as functions of $q_{\perp }/q_0$. Here $r_{\perp }=z=\lambda =1/q_0$, $f\left(q_{\perp }\right)=W(q_{\perp },z)/q_0^3$, and $f\left(q_{\perp }\right)=q_{\perp }^3{J}_0\left(q_{\perp }{r}_{\perp }\right)/q_0^3$, respectively. Though it is not shown here, $W(q_{\perp },z)$ is the “slower” function for all $q_{\perp }/q_0$. Hence the validity of series (16) and, consequently, expression (17) does not depend on values of $r_{\perp }$ and $z$.

Figure 3

Dimensionless energy of the interaction between the particles located in the same smectic layer, $z=0$. Solid lines correspond to the analytical results for “$\alpha$” and “$\beta$” particles, $\tilde{U}=U_{p{p}^{\prime }}{\pi }^{2}B/\left(2{\alpha }^2{q}_0^3\right)$ and $\tilde{U}=U_{p{p}^{\prime }}{\pi }^{2}B/\left(2{\beta }^2{q}_0^5\right)$ in (a) and (b), respectively. Dashed lines depict the results of numerical computations.

Figure 4

Dimensionless energy of the interaction between the particles located along the $z$ axis, $r_{\perp }=0$. Solid lines correspond to the analytical results for “$\alpha$” and “$\beta$” particles, $\tilde{U}=U_{p{p}^{\prime }}{\pi }^{2}B/\left(2{\alpha }^2{q}_0^3\right)$ in (a) and $\tilde{U}=U_{p{p}^{\prime }}{\pi }^{2}B/\left(2{\beta }^2{q}_0^5\right)$ in (b), respectively. Dashed lines depict the results of numerical computations.

Figure 5

Energy of the interaction as a function of $q_0$ for $r_{\perp }=0$, $z=l$ (solid lines) and $r_{\perp }=l$, $z=0$ (dashed lines). $U^q=U_{p{p}^{\prime }}{\pi }^{2}B{l}^3/\left(2{\alpha }^2\right)$ in (a) and $U^q=U_{p{p}^{\prime }}{\pi }^{2}B{l}^5/\left(2{\beta }^2\right)$ in (b).