Ionic interactions in biological and physical systems: a variational treatment

Abstract

Chemistry is about chemical reactions. Chemistry is about electrons changing their configurations as atoms and molecules react. Chemistry has for more than a century studied reactions as if they occurred in ideal conditions of infinitely dilute solutions. But most reactions occur in salt solutions that are not ideal. In those solutions everything (charged) interacts with everything else (charged) through the electric field, which is short and long range extending to the boundaries of the system. Mathematics has recently been developed to deal with interacting systems of this sort. The variational theory of complex fluids has spawned the theory of liquid crystals (or vice versa). In my view, ionic solutions should be viewed as complex fluids, particularly in the biological and engineering context. In both biology and electrochemistry ionic solutions are mixtures highly concentrated (∼10 M) where they are most important, near electrodes, nucleic acids, proteins, active sites of enzymes, and ionic channels. Ca²⁺ is always involved in biological solutions because the concentration (really free energy per mole) of Ca²⁺ in a particular location is the signal that controls many biological functions. Such interacting systems are not simple fluids, and it is no wonder that analysis of interactions, such as the Hofmeister series, rooted in that tradition has not succeeded as one would hope. Here, we present a variational treatment of hard spheres in a frictional dielectric with the hope that such a treatment of an electrolyte as a complex fluid will be productive. The theory automatically extends to spatially nonuniform boundary conditions and the nonequilibrium systems and flows they produce. The theory is unavoidably self-consistent since differential equations are derived (not assumed) from models of (Helmholtz free) energy and dissipation of the electrolyte. The origin of the Hofmeister series is (in my view) an inverse problem that becomes well posed when enough data from disjoint experimental traditions are interpreted with a self-consistent theory.