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The critical behavior of long straight rigid rods of length $k$ ($k$-mers) onsquare and triangular lattices at intermediate density has been studied. Anematic phase, characterized by a big domain of parallel $k$-mers, was found.This ordered phase is separated from the isotropic state by a continuoustransition occurring at a intermediate density $\theta_c$. Two analyticaltechniques were combined with Monte Carlo simulations to predict the dependenceof $\theta_c$ on $k$, being $\theta_c(k) \propto k^{-1}$. The first involvessimple geometrical arguments, while the second is based on entropyconsiderations. Our analysis allowed us also to determine the minimum value of$k$ ($k_{min}=7$), which allows the formation of a nematic phase on atriangular lattice.

Critical behavior of long straight rigid rods on two-dimensional lattices: Theory and Monte Carlo simulations