Modulated phases in a three-dimensional Maier-Saupe model with competing interactions

This work is dedicated to the study of the discrete version of the Maier-Saupe model in the presence of competing interactions. The competition between interactions favoring different orientational ordering produces a rich phase diagram including modulated phases. Using a mean-field approach and Monte Carlo simulations, we show that the proposed model exhibits isotropic and nematic phases and also a series of modulated phases that meet at a multicritical point, a Lifshitz point. Though the Monte Carlo and mean-field phase diagrams show some quantitative disagreements, the Monte Carlo simulations corroborate the general behavior found within the mean-field approximation.

Figure 1

Phase diagram of the Maier-Saupe model with competing interactions, with $J_1>0, J_2<0$, and $p=-J_2/J_1$, obtained by numerically minimizing the free energy with different imposed periodicities of the order parameter. We have identified isotropic, nematic, and modulated (with 4, 6, and 8 periodicity) phases. The nematic and modulated phases meet the isotropic phase at the Lifshitz point located at $p\approx 0.35$ and $t\approx 4.27$.

Figure 2

Bilayer structure of liquid crystal molecules. (a) and (b) are two examples of the bilayer structure: (a) shows a stack of three bilayers with three different molecule orientations; (b) shows an alternating structure of bilayers with two different orientations.

Figure 3

Temperature dependence of energy density at various coupling ratios $p$ for system size $L=8$. As expected, at low temperatures, for system with $p<0.5$ energy approaches to $E=(2p-6)/3$, while for $p>0.5$ it approaches $E=-(9+2p)/6$. The apparent anomalies seen for $p=0.4$ at around $T=1.4$ and $p=0.6$ around $T=1.1$ comes from the second phase transition taking place for this range of $p$.

Figure 4

Temperature dependence of order parameter at various ratios $p$ for system size $L=8$. The inset shows the layer order parameter averaged over the layers for $p\ge 0.5$.

Figure 5

Temperature dependence of the specific heat for $p=0.0$, 0.36, and 0.38. There is only one peak for $p=0$, indicating the isotropic-nematic phase transition, however, a second peak emerges for $p=0.36$ and becomes obvious for $p=0.38$, indicating that dual phase transitions occur at these $p$ values.

Figure 6

Temperature dependence of the specific heat for $p=0.75$, 0.8, and 0.9. There are two peaks for $p=0.75$ and $p=0.8$, while only one peak for $p=0.9$, indicating that the dual phase transition disappears between $p=0.8$ and $p=0.9$.

Figure 7

Temperature dependence of $\chi$ at $p=0.6$ for $L=8$ and $L=12$. The peak near $T\approx 1.44$ indicates the isotropic—single-layer phase transition, while the anomaly at around $T=1.14$ corresponds to the transition from single-layer-structure to bilayer-structure. Comparing the results for the two cases, we can see the finite-size-effect playing some role in the $L=8$ systems, but we believe that the overall effects on the phase boundaries are minor.

Figure 8

Phase diagram of the discrete Maier-Saupe model with competing interactions, obtained on the basis of MC simulations with $8\times 8\times 8$ lattices. The red star point indicates the Lifshitz point (LP) based on our MC results, while the magenta diamond shows the LP based on mean-field results in Fig. 1 adjusted by the factor $9/4$ relating the two versions of the Hamiltonian.