Monte Carlo study of the ordering in a strongly frustrated liquid crystal

We have performed Monte Carlo simulations to investigate the temperature dependence of the ordering of the side chains of the $\mathsf{X}$-shaped liquid crystal molecules which are arranged in a hexagonal array. Each hexagon contains six side chains, one from each side of the hexagon. Each liquid crystal molecule has two, dissimilar, side chains, one that contains silicon and one that contains fluorine. Like chains attract each other more strongly than unlike chains and this drives an order-disorder transition. The system is frustrated because it is not possible to find a configuration in which all the hexagons are occupied by either all silicon or all fluorine chains. There are two phase transitions. If only pairwise interactions are included it is found that there is an interesting fluctuating phase between the disordered phase and the fully ordered ground state. This did not agree with the experiments where an intermediate phase was seen that had long range order on one of the three sublattices. Agreement was found when the calculations were modified to include attractive three-body interactions between the silicon chains.

Figure 1

(a) The structure of the molecules A1 and A2. (b) Schematic drawing of the liquid crystalline $\mathsf{X}$-shaped molecule. (c) The state where the side chains are as fully ordered as allowed by the frustration. (d) The fully ordered state in terms of hexagons. (e) The experimentally observed partially ordered state, as confirmed by x-ray and neutron diffraction and AFM. (f) The fully disordered phase.

Figure 2

The configurations (i) and (ii) of the chain that is flipped between hexagons a and b.

Figure 3

Temperature dependence of the parameters of the pairwise model obtained from Monte Carlo simulations; (a) the sublattice magnetizations, $m_A,m_B,m_C$, as defined by $m_A=\frac{2n_A-6}{6}$ where the hexagons on sublattice $A$ contain $n_A$ Si chains, etc.; (b) the sublattice susceptibilities; (c) the specific heat and the entropy; these plots were obtained from the average over 50 heating runs; (d) the sublattice magnetizations in a single cooling run; (e) the sublattice magnetizations in a single heating run.

Figure 4

Indicates partial order (by the use of pale colors) in (a) the sites labeled 1, 2, 3 which are mixed (green), and then fluctuate to blue; this destabilizes the site (0) and the green and blue sublattices start to interchange.

Figure 5

The temperature dependence of the parameters of the model that included three-body interactions obtained from simulations: (a) sublattice magnetizations, $m_A,m_B,m_C$, as defined by $m_A=\frac{2n_A-6}{6}$ where the hexagons on sublattice $A$ contain $n_A$ Si chains, etc.; (b) heat capacity and the entropy; (c) susceptibility for sublattice $A$; and (d) susceptibility for sublattices $B$ and C.

Figure 6

(a) The unit cells for the disordered phase and the partially and fully ordered phases and (b) the x-ray scattering from sample A1 with $T_c=80{}^{\circ }\mathrm{C}$ showing the emergence of the superlattice peaks at low temperature.

Figure 7

(a) Intensity of the (10) diffraction peak as a function of temperature, for compounds A1 and A2, together with the fitted curves. The curves are plotted both as $I_{\left(10\right)}$ vs $T$ and $\mathrm{lo}{\mathrm{g}}_{10}\left(I_{\left(10\right)}\right)$ vs $\mathrm{lo}{\mathrm{g}}_{10}(T_c-T)$ for clarity. The heating rate used is $1{}^{\circ }\mathrm{C}/\min$. $\mathbf{A}\mathbf{1},T_c=56.3{}^{\circ }\mathrm{C},2\beta =0.36$, and $\beta =0.18;\mathbf{A}\mathbf{2},T_c=81.1{}^{\circ }\mathrm{C}, 2\beta =0.33$, and $\beta =0.17$; (b) the log-log plots of the order parameter red (dots) obtained from the simulation below $T_H$ fitted to $\beta =0.12$ using blue (dotted) line.