Statistical Mechanics of Linear k-mer Lattice Gases: From Theory to Applications

Abstract

The statistical mechanics of structured particles with arbitrary size and shape adsorbed ontodiscrete lattices presents a longstanding theoretical challenge, mainly due to complex spatialcorrelations and entropic effects that emerge at finite densities. Even for simplified systemssuch as hard-core linear k-mers, exact solutions remain limited to low-dimensional or highlyconstrained cases. In this review, we summarize the main theoretical approaches developedby our research group over the past three decades to describe adsorption phenomena involvinglinear k-mers—also known as multisite occupancy adsorption—on regular lattices.We examine modern approximations such as an extension to two dimensions of the exactthermodynamic functions obtained in one dimension, the Fractional Statistical Theory ofAdsorption based on Haldane’s fractional statistics, and the so-called Occupation Balancebased on expansion of the reciprocal of the fugacity, and hybrid approaches such as thesemi-empirical model obtained by combining exact one-dimensional calculations and theGuggenheim–DiMarzio approach. For interacting systems, statistical thermodynamics isexplored within generalized Bragg–Williams and quasi-chemical frameworks. Particularfocus is given to the recently proposed Multiple Exclusion statistics, which capture the correlatedexclusion effects inherent to non-monomeric particles. Applications to monolayerand multilayer adsorption are analyzed, with relevance to hydrocarbon separation technologies.Finally, computational strategies, including advanced Monte Carlo techniques, arereviewed in the context of high-density regimes. This work provides a unified frameworkfor understanding entropic and cooperative effects in lattice-adsorbed polyatomic systemsand highlights promising directions for future theoretical and computational research.