Abstract
In a previous paper [G. J. dos Santos, D. H. Linares, and A. J. Ramirez-Pastor, J. Stat. Mech. (2017) 073211] a methodology for the determination of the critical point of the condensation phase transition occurring in monolayers of linear adsorbates ($k$-mers) was presented. The maximum cumulant method was developed from the phenomenological observation that the fourth-order Binder cumulant and the isotherm inflection point are produced at the same value of chemical potential. In the present work, mathematical arguments are presented to show analytically that the previously mentioned relationship is satisfied by evaporation-condensation systems under the conditions that: (i) the surface coverage distribution function is a bimodal distribution composed of a linear combination of two normalized functions ${g}_{1}(\ensuremath{\theta})$ and ${g}_{2}(\ensuremath{\theta})$ with zero overlap and mean values ${\ensuremath{\theta}}_{1}$ and ${\ensuremath{\theta}}_{2}$, respectively; and (ii) ${g}_{1}(\ensuremath{\theta})$ and ${g}_{2}(\ensuremath{\theta})$ are unimodal distributions that are symmetric with respect to the middle point $({\ensuremath{\theta}}_{1}+{\ensuremath{\theta}}_{2})/2$. In addition, numerical results from Monte Carlo simulations of four different adsorption-desorption systems (linear $k$-mers on square and triangular lattices, $S$-shaped $k$-mers on square lattices and ${k}^{2}$-mers on square lattices) are presented to check the theoretical results and to provide evidence of the general validity and robustness of the method.
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