Local Composition for a Binary Mixture of Particles on a Three-Dimensional Ising Lattice

Abstract

Results from Monte Carlo (MC) simulations are reported for a binary mixture of particles A and B placed at the nodes of a three-dimensional cubic Ising lattice, with pairwise nearest neighbor interactions between species i and j, E ij . The local composition is studied as a function of composition and of the value of the interchange energy (δ = 2E AB – E AA – E BB). The MC results are used to assess the accuracy of analytical theories proposed in the literature as a function of the interchange energy. The local composition is shown to obey an exact limiting law at high dilution of one component, thus providing a simple expression for the limiting activity coefficient. The activity coefficients of the species and the entropy per particle are computed from the MC data.

Local Composition in a Binary Mixture on a One-Dimensional Ising Lattice

“…The deviation becomes very large for Δ values outside the range of −0.5 to 0.5. 22 The present study in 1D provides insight into a model 18,19 proposed for the approximate determination of the local composition. It was developed as an extension of the 1D Ono− Kondo equation for adsorption to various types of systems in higher dimensions.…”

“…It has been found that the result from the QC theory for the first-neighbor composition, x 1 , is exact in 1D, as expected. 27 This is not true in 2D (results not shown) and in 3D, 22 in which case the QC formula overestimates the value of x 1 for negative values of Δ, and vice versa for Δ > 0. The deviation becomes very large for Δ values outside the range of −0.5 to 0.5.…”

“…The composition of a site adjacent to a particle B can be deduced from the value of x 1 . 22 We first calculate the internal energy of the binary mixture and then derive therefrom an expression for x 1 . The internal energy per particle of the mixture, u, may be calculated from the relation…”

“…Then it is likely that the Helmholtz energy variation given by eq 49 may be used with sufficient accuracy, as in the original model, 18 for Δ typically in the range of −0.5 to +0.5. 22 On the other hand, the scheme seems more uncertain in case ii. This is so because it is not obvious that the entropy variation will be small relative to the other contributions (eq 56) in this case.…”

“…A look at the literature shows that local composition models have seldom been examined in light of the Ising problem. In a recent work, we have assessed the accuracy of a few approximate treatments in the 3D case by using Monte Carlo numerical simulation . In this work, the possibility of solving the problem in 1D is used in order to obtain exact information on the local composition.…”