Enhanced stability of the 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 $i\text{th}$ 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.

Figure 1

(Color online) Configuration of hard rectangles of aspect ratio $\kappa =7$ at packing fraction $\eta =0.655$, as obtained from MC simulation. Clusters, defined by the pair connectedness criterion explained in the text, have been colored according to their size.

Figure 2

Size distribution function $x_i$ as obtained from MC simulation, in natural logarithmic scale, for a fluid of HRs with different values of aspect ratio and packing fraction: filled circles, $\kappa =3$ and $\eta =0.635$; open squares, $\kappa =7$ and $\eta =0.600$; and gray squares, $\kappa =5$ and $\eta =0.657$. Straight lines are linear fits. Inset: area distribution function $i{x}_i$ for the case $\kappa =5$ and $\eta =0.692$.

Figure 3

Monomer $h_m\left(\varphi \right)$ (solid line) and cluster $h_c\left(\varphi \right)$ (dashed line) orientational distribution functions for the case $\kappa =7$ and $\eta =0.655$, as obtained from MC simulation.

Figure 4

(Color online) Multicomponent isotropic mixture composed of different species, where species, depicted in different colors, correspond to clusters of a particular size. Clusters are defined as aggregates of rectangular monomers in a side-by-side configuration. In this instance the monomer aspect ratio is $\kappa =5$.

Figure 5

(Color online) (a) Schematic representation of the excluded area between clusters $i$ and $j$. Their characteristic lengths and relative orientation are shown. (b) Excluded area $A_{ij}\left(\varphi \right)$ for the particular case $L∕\sigma =2$, $i=3$, and $j=4$.

Figure 6

Functions $\Delta \left(\kappa \right)$ in terms of inverse aspect ratio ${\kappa }^{-1}$. Symbols indicate the nematic phase which is stable in the region in question. The region of tetratic stability is shaded. Open squares indicate values of polydispersity, for values of aspect ratio $\kappa =3$, 5, and 7, at the $I\text{−}{N}_t$ transition, as obtained from MC simulation; error bars correspond to uncertainty in $\Delta$ originating from uncertainty in location of spinodal.

Figure 7

Free energy per particle $\Phi ∕\rho$ vs inverse packing fraction ${\eta }^{-1}$ from SPT results for the multicomponent HR fluid of monomer aspect ratio $\kappa =3$ and with different values of polydispersity: $q=\left(\mathrm{a}\right)$ 0.25, (b) 0.35, (c) 0.50, and (d) 0.65. Continuous curves, tetratic phase; dashed curves, uniaxial nematic phase; dotted curves, isotropic phase. In (c) and (d) symbols indicate bifurcation points from the isotropic to the tetratic (filled circles) and uniaxial nematic (open circles) phases. In (a) and (b) a straight line in ${\eta }^{-1}$ has been subtracted to better visualize the curves.

Figure 8

Cluster (a) and monomer (b) orientational distribution functions for $\kappa =5$ and $q=0.65$. The continuous line corresponds to a stable $N_u$ phase with $\eta =0.85$, whereas the discontinuous line is for the metastable $N_t$ phase at the same value of $\eta$.

Figure 9

Packing fractions $\eta$ of the $I\text{−}{N}_t$ (continuous line) and $I\text{−}{N}_u$ (dashed line) transitions versus aspect ratio $\kappa$.

Figure 10

Area distribution functions for tetratic $N_t$ and uniaxial $N_u$ nematic phases. $\kappa =\left(\mathrm{a}\right)$ 10 and (b) 100.