Structure and phase diagram of self-assembled rigid rods: Equilibrium polydispersity and nematic ordering in two dimensions

We investigate the influence of directional or bonding interactions on the structure and phase diagram of complex fluids. Using a generalization of the theory of associating fluids we study the interplay between the self-assembly process, driven by the bonding interactions, and the isotropic-nematic transition, driven by the anisotropic shape of the equilibrium clusters, for a model consisting of particles with two bonding sites and discrete orientational degrees of freedom. The theory is applied over a wide range of temperature and density in two dimensions and the results are compared with Monte Carlo simulations on the square lattice. The specific heat is shown to exhibit pronounced structure at the onset of self-assembly and at the nematic-isotropic transition that occur over a narrow range of temperature, at fixed density. The results reveal that bonding is enhanced by the nematic ordering, although a bonding temperature still occurs in the isotropic phase at low densities. The average rod length is described quantitatively in both phases, while the location of the ordering transition, which was found to be continuous, is predicted semiquantitatively by the theory.

Figure 1

(a) Phase diagram of the model. The nematic phase is stable at low temperatures and high densities. The full line is a line of critical transitions that separates regions of isotropic and nematic stability; (b) equilibrium average rod length on the transition line.

Figure 2

Representative equilibrium configurations for the system with $\rho =0.2$ at four temperatures.

Figure 3

Two signatures of the self-assembly process for a system with density $\rho =0.2$ and length $L=100$: The specific heat per particle as a function of temperature (a) and the total rod length density distribution (b). In the former, simulation data—as measured from the fluctuations in the internal energy—is plotted over the theoretical curve. The total rod length density distributions are shown for three temperatures $T^{*}=0.14$ (triangles), 0.20 (squares), and 1.0 (circles), with the theoretical curves from (35) over-plotted.

Figure 4

Average rod length as a function of temperature from simulation data (points with error bars) and theory (lines) for two densities, $\rho =0.2$ (a) and $\rho =0.4$ (b). For the theoretical curves, the $\Delta =0$ (isotropic) solution is plotted with dashed lines and the $\Delta >0$ (nematic, $T^{*}<T_{c,\mathit{th}}^{*}$) solution with full lines. The temperature of the nematic transition is indicated with two vertical lines, corresponding to the theoretical prediction $T_{c,\mathit{th}}^{*}$ and that obtained from the simulation $T_c^{*}$.

Figure 5

Plot of the orientational order parameter $\Delta$ as a function of temperature for $\rho =0.2$ (a) and $\rho =0.4$ (b) with theoretical curves over-plotted (dashed). The light dashed lines are the best fits to ${(T_c^{*}-T^{*})}^{1∕8}$, where $T_c^{*}[\rho =0.2]=0.144$ and $T_c^{*}[\rho =0.4]=0.2075$.

Figure 6

Fluctuations of the order parameter $\mid \Delta \mid$ as a function of temperature for densities $\rho =0.2$ (a) and $\rho =0.4$ (b). Vertical lines indicate the estimated $T_c^{*}$, as determined by the fit in the insets of Fig. 5.