Cluster-Based Thermodynamics of Interacting Dice in a Lattice

Abstract

In this paper, a model for two-component systems of six-sided dice in a simple cubic lattice is developed, based on a basic cluster approach previously proposed. The model represents a simplified picture of liquid mixtures of molecules with different interaction sites on their surfaces, where each interaction site can be assigned an individual energetic property to account for cooperative effects. Based on probabilities that characterize the sequential construction of the lattice using clusters, explicit expressions for the Shannon entropy, synonymously used as thermodynamic entropy, and the internal energy of the system are derived. The latter are used to formulate the Helmholtz free energy that is minimized to determine thermodynamic bulk properties of the system in equilibrium. The model is exemplarily applied to mixtures that contain distinct isomeric configurations of molecules, and the results are compared with the Monte-Carlo simulation results as a benchmark. The comparison shows that the model can be applied to distinguish between isomeric configurations, which suggests that it can be further developed towards an excess Gibbs-energy, respectively, activity coefficient model for chemical engineering applications.

Discrete Modeling Approach for Cluster-Based Excess Gibbs-Energy of Molecular Liquids

“…It is equivalent to the Shannon information describing the sequential cluster construction process, illustrated in Figure . The general shape of the entropy function, as defined in eq , is the same as that of the previously published dice model …”

“…It is equivalent to the Shannon information describing the sequential cluster construction process, illustrated in Figure . The general shape of the entropy function, as defined in eq , is the same as that of the previously published dice model s = prefix− R ∑ a ∈ C ∑ b ∈ scriptC ∑ c ∈ C ∑ d ∈ scriptC ∑ i = 1 m ∑ j = 1 m ∑ k = 1 m ∑ l = 1 m p a b c d i j k l nobreak0em0.25em⁡ log ( p a b c d i j k l q b c d j k l ) …”

“…The shape of the cluster is based on a previously established model that uses dice-like molecules in a simple cubic lattice. There, the system is hypothetically constructed by inserting a new molecule into a given neighborhood of already existing molecules from previous insertion steps . One such insertion is illustrated in Figure .…”

“…As a unique feature, this substantially different approach uses Shannon information describing the sequential construction process of the system with clusters as a synonym for thermodynamic entropy. In previous papers, the two-dimensional Vinograd approach was further developed to three-dimensional simple cubic lattices of spherical molecules and dice systems. − …”

“…In previous papers, the two-dimensional Vinograd approach was further developed to three-dimensional simple cubic lattices of spherical molecules and dice systems. 39 − 43 …”