The statistical thermodynamics of straight rigid rods of length $k$ on triangular lattices was developed on a generalization in the spirit of the lattice-gas model and the classical Guggenheim-DiMarzio approximation. In this scheme, the Helmholtz fre
e energy and its derivatives were written in terms of the order parameter $delta$, which characterizes the nematic phase occurring in the system at intermediate densities. Then, using the principle of minimum free energy with $delta$ as a parameter, the main adsorption properties were calculated. Comparisons with Monte Carlo simulations and experimental data were performed in order to evaluate the reaches and limitations of the theoretical model.
Numerical simulations and finite-size scaling analysis have been carried out to study the percolation behavior of straight rigid rods of length $k$ ($k$-mers) on two-dimensional square lattices. The $k$-mers, containing $k$ identical units (each one
occupying a lattice site), were adsorbed at equilibrium on the lattice. The process was monitored by following the probability $R_{L,k}(theta)$ that a lattice composed of $L times L$ sites percolates at a concentration $theta$ of sites occupied by particles of size $k$. A nonmonotonic size dependence was observed for the percolation threshold, which decreases for small particles sizes, goes through a minimum, and finally asymptotically converges towards a definite value for large segments. This striking behavior has been interpreted as a consequence of the isotropic-nematic phase transition occurring in the system for large values of $k$. Finally, the universality class of the model was found to be the same as for the random percolation model.
The critical behavior of self-assembled rigid rods on a square lattice was recently reinvestigated by Almarza et al. [Phys. Rev. E 82, 061117 (2010)]. Based on the Binder cumulants and the value of the critical exponent of the correlation length, the
authors found that the isotropic-nematic phase transition occurring in the system is in the two-dimensional Ising universality class. This conclusion contrasts with that of a previous study [Lopez et al., Phys. Rev. E 80, 040105 (R) (2009)] which indicates that the transition at intermediate density belongs to the q = 1 Potts universality class. Almarza et al. attributed the discrepancy to the use of the density as the control parameter by Lopez et al. The present work shows that this suggestion is not sufficient, and that the discrepancy arises solely from the use of different statistical ensembles. Finally, the necessity of making corrections to the scaling functions in the canonical ensemble is discussed.
Monte Carlo simulations and finite-size scaling analysis have been carried out to study the critical behavior in a two-dimensional system of particles with two bonding sites that, by decreasing temperature or increasing density, polymerize reversibly
into chains with discrete orientational degrees of freedom and, at the same time, undergo a continuous isotropic-nematic (IN) transition. A complete phase diagram was obtained as a function of temperature and density. The numerical results were compared with mean field (MF) and real space renormalization group (RSRG) analytical predictions about the IN transformation. While the RSRG approach supports the continuous nature of the transition, the MF solution predicts a first-order transition line and a tricritical point, at variance with the simulation results.
In the Edwards-Anderson model of spin glasses with a bimodal distribution of bonds, the degeneracy of the ground state allows one to define a structure called backbone, which can be characterized by the rigid lattice (RL), consisting of the bonds tha
t retain their frustration (or lack of it) in all ground states. In this work we have performed a detailed numerical study of the properties of the RL, both in two-dimensional (2D) and three-dimensional (3D) lattices. Whereas in 3D we find strong evidence for percolation in the thermodynamic limit, in 2D our results indicate that the most probable scenario is that the RL does not percolate. On the other hand, both in 2D and 3D we find that frustration is very unevenly distributed. Frustration is much lower in the RL than in its complement. Using equilibrium simulations we observe that this property can be found even above the critical temperature. This leads us to propose that the RL should share many properties of ferromagnetic models, an idea that recently has also been proposed in other contexts. We also suggest a preliminary generalization of the definition of backbone for systems with continuous distributions of bonds, and we argue that the study of this structure could be useful for a better understanding of the low temperature phase of those frustrated models.
Using Monte Carlo simulations and finite-size scaling analysis, the critical behavior of attractive rigid rods of length k (k-mers) on square lattices at intermediate density has been studied. A nematic phase, characterized by a big domain of paralle
l k-mers, was found. This ordered phase is separated from the isotropic state by a continuous transition occurring at a intermediate density theta_c, which increases linearly with the magnitude of the lateral interactions.
