Statistical thermodynamics of multi-body effects in associating fluids: A cluster size distribution theory
Chapman, Walter G.
Doctor of Philosophy
Associating fluids are central to all natural and engineered systems. Notable examples are water, the matrix of life, and solvents typically encountered in industrial processes. In contrast to simple liquids, in associating fluids attractive interactions play a decisive role in the structure and thermodynamics of the fluid. Patchy colloids, particles with engineered directional interactions, are characterized by short range directional interactions and are archetypes of associating fluids. For these molecules with short range directional interactions, by varying the shape, number, and position of the patches, different self assembled geometries leading to complex structures can be obtained. Thus these systems have drawn intense scrutiny in designing materials from the nanoscale level. But despite the simplicity in describing and engineering the inter-molecular interactions in such systems, a general theory to predict the phase behavior is not yet available. Statistical associating fluid theory (SAFT) has proven useful in modeling associating fluids of both scientific and industrial importance. In SAFT, the fluid properties are obtained by incorporating the role of association over the reference system properties. The properties of the reference are typically based on modeling two-particle or three-particle distribution functions. As the complexity of the interaction increases in the physical system, such as may result from multiple bonding and size asymmetries, multi-body interactions become more important and information about two-body or three-body correlations in the reference no longer suffices. The difficulty in determining these interactions arises due to the limited knowledge in describing multi-body correlation functions particularly for three-body and higher correlations, also at separations other than contact value. In this work, we develop a new type of perturbation theory that allows us to incorporate multi-body effects given the properties of the reference. Drawing upon SAFT and quasichemical theory, theoretical models for the molecular level interactions are developed to describe the thermodynamic properties and structure of these self-assembling mixtures. We first study mixtures of spherically symmetric solute in patchy colloid solvents, a model for solvated ions and supramolecular star molecules. We present a ``complete reference'' perturbation theory that can describe multiple bonding at a single patch by representing higher order correlation functions in the hard sphere reference fluid in terms of average $n$-ordered clusters. Studies for symmetric and asymmetric colloidal mixtures show excellent agreement with the computer experiments for these systems. We also perform a quasichemical theory based analysis to develop a physically transparent, statistical mechanical model to describe the occupancy probabilities in the hard sphere fluids for different packing fractions. This model corrects multi-body effects obtained for isolated clusters by incorporating the role of the cluster-bulk interface and the bulk medium effects. Next we model associating fluids entirely within the quasichemical organization of the potential distribution theorem and explore a range of bonding configurations from a solute that can bond only once, to a solute that can bond multiple solvents but only on one-hemisphere of its surface, i.e.\ a Janus particle, and to a solute with a sticky patch that covers its entire surface. Quasichemical theory leads to the identification of the occupancy of a patch conditional on the total occupancy of the spherical observation volume, all in the reference fluid, as an important quantity within the theory. Based on this study we extend Wertheim's theory beyond second order thermodynamic perturbation to study different multiple bonding geometries of a single site on the solute molecules. Again excellent agreement with computer simulations is obtained for different patch geometries. Our theoretical phase equilibrium study for these mixture suggests empty fluid regime for these mixtures. Lastly, the application of these multi-body effects to accurately model ion-solvent interactions in electrolyte systems is indicated with a SAFT model for electrolyte systems. We also present various future directions of our approach.
Statistical mechanics; patchy colloids; multi-body effects; cluster distributions; soft-matter