Thermodynamic Perturbation Theory for Associating Fluids: Beyond First Order
Marshall, Bennett Davis
Cox, Kenneth R.
Doctor of Philosophy
Association interactions such as the hydrogen bond are a key component in many physical and biological systems. For this reason accurate theories are needed to describe both the thermodynamics and self – assembly of associating species. Applications of these theories range from those of industrial importance, such as equations of state for process simulations, to the realm of materials science where these theories can be used to predict how molecular structure determines the self – assembly of associating species into advanced supramolecular materials. Semi – empirical equations of state based on chemical or lattice theories do not contain the molecular level detail to make predictions on how molecular structure affects self – assembly of associating species. For this, one needs a theory whose starting point is the interaction potential between two associating species which includes this molecular level detail. Development of accurate molecular theories for associating fluids is hampered by the strength, directionality and limited valence of the association interaction. This has proven particularly true in the extension of Mayer’s cluster theory to these associating fluids. This problem was largely solved by Wertheim in the 1980’s who developed an exact cluster expansion using a multi – density formalism. Wertheim’s cluster theory incorporates the geometry of the association interaction at an early point in the derivation. This allowed Wertheim to develop the theory in such a way that accurate and simple approximation methods could be applied such as thermodynamic perturbation theory (TPT). When treated at first order in perturbation (TPT1), Wertheim’s theory gives a simple and general equation of state which forms the basis of the statistical associating fluid theory (SAFT) free energy model. SAFT has been become a standard in both academia and industry as an equation of state for associating (hydrogen bonding) fluids. While simple in form and widely applied, the development of TPT1 rest on a number of, sometimes severe, simplifying assumptions: no interaction between associated clusters beyond that of the reference fluid, association sites are singly bondable, no double bonding of molecules, no cycles of association bonds, no steric hindrance between association sites, association is independent of bond angle and there is no bond cooperativity. The purpose of this dissertation is to relax these assumptions. Chapters 2 – 4 extend TPT to allow for multiple bonds per association site. Chapter 2 focuses on the case of associating spheres with a doubly bondable association site as a model for patchy colloids with a multiply bondable patch. Chapters 3 – 4 extend TPT to associating mixtures of spheres where the first component has directional association sites and the second component has spherically symmetric association sites. This theory is applicable as a model for mixtures of patchy and spherically symmetric colloids and ion – water association. Chapters 5 – 6 extend TPT to account for the effect of relative association site location. In chapter 5 the case of associating hard spheres with two association sites is considered. For this case the angle between the centers of the association sites (bond angle) is treated as a independent variable. This is the first application of TPT which has included the effect of bond angle. It is shown that as bond angle is decreased the effects of steric hindrance, ring (cycle) formation and eventually double bonding of molecules must be accounted. The developed theory accounts for each of these higher order interactions as a function of bond angle. The resulting theory is shown to be accurate over the full bond angle range for both the distribution of cluster types (chains, rings, double bonded) as well as the equation of state. In chapter 6 this theory is extended to the case there are more than two association sites. In chapter 7 TPT is extended to account for hydrogen bond cooperativity for the case of molecules with two association sites. The derived theory is shown to be highly accurate for molecules which exhibit positive or negative hydrogen bond cooperativity. Finally, in chapter 8 the case of associating fluids in spatially uniform orienting external fields is considered. An example system for this case would be associating dipolar molecules in a uniform electric field. By employing classical density functional theory in the canonical ensemble exact results are obtained for the orientational distribution function. These exact results contain the monomer fraction which must be approximated in TPT1. The resulting theory is in good agreement with simulation data for the prediction of the effect of a linear orienting field on the chain length and orientation of associated chains of spheres with two association sites.