Physics of isodesmic chemical equilibria in solution

Abstract

Exactly solvable one-dimensional lattice-gas model mixtures are used to develop a physics of isodesmic chemical equilibria. Potential distribution theory is used to solve directly for the equation of state and to obtain the cluster statistics needed to discuss self-assembly. A mapping of three-dimensional amphiphilic discotic solutions onto one-dimensional models is proposed and is found to explain the remarkable nature of previous computer simulation data. Here, at fixed pressure, the low concentration regime involves an extreme concentration dependence to solute aggregation, associated with a maximum in the equilibrium constant. This behavior is a class of colloidal phenomena, driven by solvent-solvent attractive interactions. In addition, the exact physics of isodesmic chemical equilibria is used to investigate a variety of conceptual issues concerning the phenomenology of self-assembly. One finds that ${\overline{\ensuremath{\mu}}}_{n}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}n{\ensuremath{\mu}}_{A}$ is an exact consequence of statistical mechanics and that it is even possible to give a precise meaning to the chemical potential of an aggregate, that, for example, defines what is meant physically by the identity ${\overline{\ensuremath{\mu}}}_{1}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\mu}}_{A}$. The nonuniqueness of the choice of cluster definition is considered in the context of solvent-excluded clusters; an explicit example appropriate to amphiphilic systems. Finally, the mapping to three-dimensional discotic solutions is extended to inhomogeneous phenomena whereby the disks prefer to adsorb flat onto a surface. This mapping implies that the sticky solvent regime is associated with an overwhelming driving force for chains to attach by one end to a solutelike wall.