Theory and simulation of two-dimensional nematic and tetratic phases

Recent experiments and simulations have shown that two-dimensional systems can form tetratic phases with fourfold rotational symmetry, even if they are composed of particles with only twofold symmetry. To understand this effect, we propose a model for the statistical mechanics of particles with almost fourfold symmetry, which is weakly broken down to twofold. We introduce a coefficient $\kappa$ to characterize the symmetry breaking, and find that the tetratic phase can still exist even up to a substantial value of $\kappa$. Through a Landau expansion of the free energy, we calculate the mean-field phase diagram, which is similar to the result of a previous hard-particle excluded-volume model. To verify our mean-field calculation, we develop a Monte Carlo simulation of spins on a triangular lattice. The results of the simulation agree very well with the Landau theory.

Figure 1

Schematic illustration of an interacting particle system in the tetratic phase. The shape of the particles indicates that the rotational symmetry of the interaction is broken down from fourfold to twofold.

Figure 2

Numerical mean-field calculation of the order parameters $C_2$ and $C_4$ as functions of $\gamma$ (inverse temperature), for several values of $\kappa$ (twofold distortion in the interaction): (a) $\kappa =0.4$; (b) $\kappa =0.75$; (c) $\kappa =1.5$; (d) $\kappa =2.25$.

Figure 3

Phase diagram of the model in terms of $\gamma$ (inverse temperature) and $\kappa$ (twofold distortion in the interaction). The gray solid lines represent second-order transitions, and the dark solid lines are first-order transitions. The dashed lines indicate the extrapolated second-order transitions, which give the cooling limits of the metastable phases. Point B (0.79,2) is the triple point, and A (0.61,2.2) and D (2,1) are the two tricritical points. Point C (1,2) is the intersection of the extrapolated second-order transitions.

Figure 4

A snapshot of the spins on a triangular lattice in the tetratic phase. The shape of the rectangles is just a schematic illustration of the symmetry of their interaction.

Figure 5

Simulation results for the order parameters $C_2$ and $C_4$ as functions of $\gamma$ (inverse temperature), for several values of $\kappa$ (twofold distortion in the interaction): (a) $\kappa =0.3$; (b) $\kappa =0.5$; (c) $\kappa =1$; (d) $\kappa =3$.

Figure 6

Simulation results for the phase diagram in terms of $\gamma$ (inverse temperature) and $\kappa$ (twofold distortion in the interaction). The triple point is at approximately $\gamma =3.2$ and $\kappa =0.60$.