skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Energy variational analysis of ions in water and channels: Field theory for primitive models of complex ionic fluids

Abstract

Ionic solutions are mixtures of interacting anions and cations. They hardly resemble dilute gases of uncharged noninteracting point particles described in elementary textbooks. Biological and electrochemical solutions have many components that interact strongly as they flow in concentrated environments near electrodes, ion channels, or active sites of enzymes. Interactions in concentrated environments help determine the characteristic properties of electrodes, enzymes, and ion channels. Flows are driven by a combination of electrical and chemical potentials that depend on the charges, concentrations, and sizes of all ions, not just the same type of ion. We use a variational method EnVarA (energy variational analysis) that combines Hamilton’s least action and Rayleigh’s dissipation principles to create a variational field theory that includes flow, friction, and complex structure with physical boundary conditions. EnVarA optimizes both the action integral functional of classical mechanics and the dissipation functional. These functionals can include entropy and dissipation as well as potential energy. The stationary point of the action is determined with respect to the trajectory of particles. The stationary point of the dissipation is determined with respect to rate functions (such as velocity). Both variations are written in one Eulerian (laboratory) framework. In variational analysis, an “extra layer” ofmore » mathematics is used to derive partial differential equations. Energies and dissipations of different components are combined in EnVarA and Euler–Lagrange equations are then derived. These partial differential equations are the unique consequence of the contributions of individual components. The form and parameters of the partial differential equations are determined by algebra without additional physical content or assumptions. The partial differential equations of mixtures automatically combine physical properties of individual (unmixed) components. If a new component is added to the energy or dissipation, the Euler–Lagrange equations change form and interaction terms appear without additional adjustable parameters. EnVarA has previously been used to compute properties of liquid crystals, polymer fluids, and electrorheological fluids containing solid balls and charged oil droplets that fission and fuse. Here we apply EnVarA to the primitive model of electrolytes in which ions are spheres in a frictional dielectric. The resulting Euler–Lagrange equations include electrostatics and diffusion and friction. They are a time dependent generalization of the Poisson–Nernst–Planck equations of semiconductors, electrochemistry, and molecular biophysics. They include the finite diameter of ions. The EnVarA treatment is applied to ions next to a charged wall, where layering is observed. Applied to an ion channel, EnVarA calculates a quick transient pile-up of electric charge, transient and steady flow through the channel, stationary “binding” in the channel, and the eventual accumulation of salts in “unstirred layers” near channels. EnVarA treats electrolytes in a unified way as complex rather than simple fluids. Ad hoc descriptions of interactions and flow have been used in many areas of science to deal with the nonideal properties of electrolytes. It seems likely that the variational treatment can simplify, unify, and perhaps derive and improve those descriptions.« less

Authors:
 [1];  [2];  [2]
  1. Department of Molecular Biophysics and Physiology, Rush University, Chicago, Illinois 60612 (United States)
  2. Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota 55455 (United States)
Publication Date:
OSTI Identifier:
22685039
Resource Type:
Journal Article
Journal Name:
Journal of Chemical Physics
Additional Journal Information:
Journal Volume: 133; Journal Issue: 10; Other Information: (c) 2010 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-9606
Country of Publication:
United States
Language:
English
Subject:
37 INORGANIC, ORGANIC, PHYSICAL AND ANALYTICAL CHEMISTRY; DIELECTRIC MATERIALS; ELECTROLYTES; EXPERIMENTAL DATA; FIELD THEORIES; INTERACTIONS; LAYERS; LIQUID CRYSTALS; NANOSTRUCTURES; POTENTIAL ENERGY; TIME DEPENDENCE; VARIATIONAL METHODS

Citation Formats

Eisenberg, Bob, Hyon, YunKyong, Liu, Chun, and Department of Mathematics, Pennsylvania State University, University Park, State College, Pennsylvania 16802. Energy variational analysis of ions in water and channels: Field theory for primitive models of complex ionic fluids. United States: N. p., 2010. Web. doi:10.1063/1.3476262.
Eisenberg, Bob, Hyon, YunKyong, Liu, Chun, & Department of Mathematics, Pennsylvania State University, University Park, State College, Pennsylvania 16802. Energy variational analysis of ions in water and channels: Field theory for primitive models of complex ionic fluids. United States. https://doi.org/10.1063/1.3476262
Eisenberg, Bob, Hyon, YunKyong, Liu, Chun, and Department of Mathematics, Pennsylvania State University, University Park, State College, Pennsylvania 16802. 2010. "Energy variational analysis of ions in water and channels: Field theory for primitive models of complex ionic fluids". United States. https://doi.org/10.1063/1.3476262.
@article{osti_22685039,
title = {Energy variational analysis of ions in water and channels: Field theory for primitive models of complex ionic fluids},
author = {Eisenberg, Bob and Hyon, YunKyong and Liu, Chun and Department of Mathematics, Pennsylvania State University, University Park, State College, Pennsylvania 16802},
abstractNote = {Ionic solutions are mixtures of interacting anions and cations. They hardly resemble dilute gases of uncharged noninteracting point particles described in elementary textbooks. Biological and electrochemical solutions have many components that interact strongly as they flow in concentrated environments near electrodes, ion channels, or active sites of enzymes. Interactions in concentrated environments help determine the characteristic properties of electrodes, enzymes, and ion channels. Flows are driven by a combination of electrical and chemical potentials that depend on the charges, concentrations, and sizes of all ions, not just the same type of ion. We use a variational method EnVarA (energy variational analysis) that combines Hamilton’s least action and Rayleigh’s dissipation principles to create a variational field theory that includes flow, friction, and complex structure with physical boundary conditions. EnVarA optimizes both the action integral functional of classical mechanics and the dissipation functional. These functionals can include entropy and dissipation as well as potential energy. The stationary point of the action is determined with respect to the trajectory of particles. The stationary point of the dissipation is determined with respect to rate functions (such as velocity). Both variations are written in one Eulerian (laboratory) framework. In variational analysis, an “extra layer” of mathematics is used to derive partial differential equations. Energies and dissipations of different components are combined in EnVarA and Euler–Lagrange equations are then derived. These partial differential equations are the unique consequence of the contributions of individual components. The form and parameters of the partial differential equations are determined by algebra without additional physical content or assumptions. The partial differential equations of mixtures automatically combine physical properties of individual (unmixed) components. If a new component is added to the energy or dissipation, the Euler–Lagrange equations change form and interaction terms appear without additional adjustable parameters. EnVarA has previously been used to compute properties of liquid crystals, polymer fluids, and electrorheological fluids containing solid balls and charged oil droplets that fission and fuse. Here we apply EnVarA to the primitive model of electrolytes in which ions are spheres in a frictional dielectric. The resulting Euler–Lagrange equations include electrostatics and diffusion and friction. They are a time dependent generalization of the Poisson–Nernst–Planck equations of semiconductors, electrochemistry, and molecular biophysics. They include the finite diameter of ions. The EnVarA treatment is applied to ions next to a charged wall, where layering is observed. Applied to an ion channel, EnVarA calculates a quick transient pile-up of electric charge, transient and steady flow through the channel, stationary “binding” in the channel, and the eventual accumulation of salts in “unstirred layers” near channels. EnVarA treats electrolytes in a unified way as complex rather than simple fluids. Ad hoc descriptions of interactions and flow have been used in many areas of science to deal with the nonideal properties of electrolytes. It seems likely that the variational treatment can simplify, unify, and perhaps derive and improve those descriptions.},
doi = {10.1063/1.3476262},
url = {https://www.osti.gov/biblio/22685039}, journal = {Journal of Chemical Physics},
issn = {0021-9606},
number = 10,
volume = 133,
place = {United States},
year = {Tue Sep 14 00:00:00 EDT 2010},
month = {Tue Sep 14 00:00:00 EDT 2010}
}