Beyond the single-file fluid limit using transfer matrix method: Exact results for confined parallel hard squares

Abstract

We extend the transfer matrix method of one-dimensional hard core fluids placed between confining walls for that case where the particles can pass each other and at most two layers can form. We derive an eigenvalue equation for a quasi-one-dimensional system of hard squares confined between two parallel walls, where the pore width is between σ and 3σ (σ is the side length of the square). The exact equation of state and the nearest neighbor distribution functions show three different structures: a fluid phase with one layer, a fluid phase with two layers and a solid-like structure where the fluid layers are strongly correlated.The structural transition between differently ordered fluids develops continuously with increasing density, i.e. no thermodynamic phase transition occurs. The high density structure of the system consists of clusters with two layers which are broken with particles staying in the middle of the pore.

Ordering of hard rectangles in strong confinement

“…the traditional transfer operator method cannot be applied for this model. However, on the basis of our previous study for parallel hard squares [38], it is feasible to work out a dimer-approach, where the two neighboring particles are considered as a dimer particle. If N is an even number, i.e.…”

“…1). We now extend the transfer operator method for rotating rectangles, which was developed for parallel hard squares [38]. To do this, we start with the configurational part of the isobaric partition function of second neighbor interacting systems, which can be written as…”

“…[38]. The probability distribution of the inner coordinates of a dimer is related to the eigenfunction of λ eigenvalue through…”

Highly confined mixtures of parallel hard squares: A density-functional-theory study

“…In essence, the technique calculates, apart from the partition function, the probability density and pair correlations between particles. The method was successfully applied to the study of hard disks, squares, rhombuses and rectangles under strong confinement [33][34][35][36][37][38][39][40]. The results can be summarized as follows: (i) Phase transitions between different spatial structures are ruled out, a confirmation of the general result that fluids composed of particles interacting via hard-core potentials do not exhibit phase transitions in 1 + ǫ dimensions.…”

“…A comment on the real nature of phase transitions obtained here for the confined system is in order. Exact calculations using the TMM for one-component hard disks, squares, rectangles or rhombuses confined in a slit geometry, with at most two layers of particles, show that these 1 + ǫ-dimensional systems do not exhibit true phase transitions [34,35,37,38]. However their structural properties can dramatically change as pressure is increased.…”