Uniaxial and biaxial structures in the elastic Maier-Saupe model

We perform statistical mechanics calculations to analyze the global phase diagram of a fully connected version of a Maier-Saupe-Zwanzig lattice model with the inclusion of couplings to an elastic strain field. We point out the presence of uniaxial and biaxial nematic structures, depending on temperature $T$ and on the applied stress $\sigma$. Under uniaxial extensive tension, applied stress favors uniaxial orientation, and we obtain a first-order boundary along which there is a coexistence of two uniaxial paranematic phases, and which ends at a simple critical point. Under uniaxial compressive tension, stress favors biaxial orientation; for small values of the coupling parameters, the first-order boundary ends at a tricritical point, beyond which there is a continuous transition between a paranematic and a biaxially ordered structure. For some representative choices of the model parameters, we obtain a number of analytic results, including the location of critical and tricritical points and the line of stability of the biaxial phase.

Figure 1

Sketch of the stress-temperature phase diagram of a uniaxial nematic system for small (a) and large (b) values of the coupling parameters. Solid and dashed lines correspond to first- and second-order transitions. We indicate critical and tricritical points.

Figure 2

Critical line ($E_1=1$, green) of the biaxial solution, and limit of stability of the biaxial solution ($E_2=0$; blue, dot-dashed) for a typical parameter value, $\omega =n_s{\delta }^2=0.2$. The intersection defines the tricritical point.

Figure 3

Free energy as a function of the nematic order parameter $S$ at the tricritical point, for $\omega$ between 0.1 (top curve) and 0.58 (bottom curve). The vertical axis is shifted so that $g\left(S_0\right)=0$.