(553h) Discrete Modeling Approach Using Dice-like Molecules for Excess Gibbs-Energy Models

Thermodynamic models for fluid phase equilibria calculations, such as equations of state and activity coefficients, are being challenged by the need to describe complex molecules. Especially systems with strong interactions which show large deviations from ideal mixture behavior are increasingly difficult to describe using conventional modeling approaches. In this context, previous papers proposed ‘discrete modeling’ as a novel approach to incorporate a more detailed molecular picture into thermodynamics from scratch. The approach is characterized by the rigorous use of Shannon information equivalently to thermodynamic entropy. As a proof of concept, the thermal and caloric equations of state, heat capacity and Maxwell-Boltzmann distribution for ideal gas were derived on the basis of discrete states of individual molecules [1-2]. To further extend this approach to strongly interacting condensed-phase systems [3], a previous application of discrete Markov-chains to thermodynamic modeling of two-dimensional lattices describing solid solutions [4] was modified and extended from a flat lattice towards a three-dimensional, Ising-type lattice model. The initial step of this model was the description of spherical molecules which are characterized by a uniform energetic interaction across the surface [5]. In this paper, the molecules are modeled with a dice-like geometry, allowing there to be up to six different interaction sites per molecule.

The lattice system is constructed by sequentially inserting one molecule at a time. Such an insertion process is described using conditional probabilities which account for the probability of inserting a molecule of a certain type into the already existing configuration of nearest-neighbors. This sequential construction process using conditional probabilities is based on the theory of discrete Markov-chains. From the whole lattice system, a small, three-dimensional subgroup respective cluster of sites is picked out as a representative part of the system and the basis for thermodynamic modeling. Its stepwise formation occurs in the same manner as described for the entire lattice. Such clusters can be treated as statistically independent subsystems, yet account sufficiently for cooperative effects due to molecular interactions inside the cluster. Through the sequential construction, a correlation between probabilities of pairwise interactions and the conditional insertion probabilities is formed. The internal energy and Shannon entropy are formulated on the basis of these insertion probabilities. Constrained minimization of the Helmholtz free energy yields the equilibrium distribution for the probabilities of pairwise interactions. To create a link to real molecules, the energetic interactions between molecules are determined using a sampling algorithm similar to the PAC-MAC approach [6].

The model results are compared to data from Monte-Carlo simulations for uniform and dice-like molecules. A key aspect of this model is the retention of geometrical information about the cluster. This enables the model to intrinsically distinguish between isomers that only differ in relative charge positions on a molecule. Also the boundary case of a systematic ordering of molecules for strongly interacting systems under consideration of geometric constraints is observable. Due to its better representation of uniform molecules and its ability to distinct between isomers, the proposed modeling approach is a promising basis for further developing the method towards an activity coefficient model for liquids.

References

[1] Pfleger, M., Wallek, T., and Pfennig, A., “Constraints of Compound Systems: Prerequisites for Thermodynamic Modeling Based on Shannon Entropy.” Entropy 2014, 16, 2990-3008.

[2] Pfleger, M., Wallek, T., and Pfennig, A., “Discrete Modeling: Thermodynamics Based on Shannon Entropy and Discrete States of Molecules.” Ind. Eng. Chem. Res. 2015, 54, 4643-4654.

[3] Wallek, T., Pfleger, M., and Pfennig, A., “Discrete Modeling of Lattice Systems: The Concept of Shannon Entropy Applied to Strongly Interacting Systems.” Ind. Eng. Chem. Res. 2016, 55, 2483–2492.

[4] Vinograd, V.L., Perchuk, L.L., “Informational Models for the Configurational Entropy of Regular Solid Solutions: Flat Lattices.” J. Phys. Chem. 1996, 100, 15972-15985.

[5] Wallek, T., Mayer, C., Pfennig, A., “Discrete Modeling Approach as a Basis of Excess Gibbs-Energy Models for Chemical Engineering Applications”, Ind. Eng. Chem. Res. 2018, 57, 1294–1306.

[6] Sweere, A. J. M.; Fraaije, J. G. E. M., “Force-Field Based Quasichemical Method for Rapid Evaluation of Binary Phase Diagrams.” J. Phys. Chem. B 2015, 119, 14200−14209.