Enhanced stability of tetratic phase due to clustering

We show that the relative stability of the nematic tetratic phase with respect to the usual uniaxial nematic phase can be greatly enhanced by clustering effects. Two--dimensional rectangles of aspect ratio $\kappa$ interacting via hard interactions are considered, and the stability of the two nematic phases (uniaxial and tetratic) is examined using an extended scaled--particle theory applied to a polydispersed fluid mixture of $n$ species. Here the $i$--th species is associated with clusters of $i$ rectangles, with clusters defined as stacks of rectangles containing approximately parallel rectangles, with frozen internal degrees of freedom. The theory assumes an exponential cluster size distribution (an assumption fully supported by Monte Carlo simulations and by a simple chemical--reaction model), with fixed value of the second moment. The corresponding area distribution presents a shoulder, and sometimes even a well-defined peak, at cluster sizes approximately corresponding to square shape (i.e. $i\simeq\kappa$), meaning that square clusters have a dominant contribution to the free energy of the hard--rectangle fluid. The theory predicts an enhanced region of stability of the tetratic phase with respect to the standard scaled--particle theory, much closer to simulation and to experimental results, demonstrating the importance of clustering in this fluid.