Poisson-Fermi Model of Single Ion Activities

A Poisson-Fermi model is proposed for calculating activity coefficients of single ions in strong electrolyte solutions based on the experimental Born radii and hydration shells of ions in aqueous solutions. The steric effect of water molecules and interstitial voids in the first and second hydration shells play an important role in our model. The screening and polarization effects of water are also included in the model that can thus describe spatial variations of dielectric permittivity, water density, void volume, and ionic concentration. The activity coefficients obtained by the Poisson-Fermi model with only one adjustable parameter are shown to agree with experimental data, which vary nonmonotonically with salt concentrations.Comment: 12 pages, 6 figure

Molecular Mean-Field Theory of Ionic Solutions: a Poisson-Nernst-Planck-Bikerman Model

We have developed a molecular mean-field theory -- fourth-order Poisson-Nernst-Planck-Bikerman theory -- for modeling ionic and water flows in biological ion channels by treating ions and water molecules of any volume and shape with interstitial voids, polarization of water, and ion-ion and ion-water correlations. The theory can also be used to study thermodynamic and electrokinetic properties of electrolyte solutions in batteries, fuel cells, nanopores, porous media including cement, geothermal brines, the oceanic system, etc. The theory can compute electric and steric energies from all atoms in a protein and all ions and water molecules in a channel pore while keeping electrolyte solutions in the extra- and intracellular baths as a continuum dielectric medium with complex properties that mimic experimental data. The theory has been verified with experiments and molecular dynamics data from the gramicidin A channel, L-type calcium channel, potassium channel, and sodium/calcium exchanger with real structures from the Protein Data Bank. It was also verified with the experimental or Monte Carlo data of electric double-layer differential capacitance and ion activities in aqueous electrolyte solutions. We give an in-depth review of the literature about the most novel properties of the theory, namely, Fermi distributions of water and ions as classical particles with excluded volumes and dynamic correlations that depend on salt concentration, composition, temperature, pressure, far-field boundary conditions etc. in a complex and complicated way as reported in a wide range of experiments. The dynamic correlations are self-consistent output functions from a fourth-order differential operator that describes ion-ion and ion-water correlations, the dielectric response (permittivity) of ionic solutions, and the polarization of water molecules with a single correlation length parameter.Comment: 18 figure

Poisson-Nernst-Planck-Fermi Theory for Ion Channels

A Poisson-Nernst-Planck-Fermi (PNPF) theory is developed for studying ionic transport through biological ion channels. Our goal is to deal with the finite size of particle using a Fermi like distribution without calculating the forces between the particles, because they are both expensive and tricky to compute. We include the steric effect of ions and water molecules with nonuniform sizes and interstitial voids, the correlation effect of crowded ions with different valences, and the screening effect of water molecules in an inhomogeneous aqueous electrolyte. Including the finite volume of water and the voids between particles is an important new part of the theory presented here. Fermi like distributions of all particle species are derived from the volume exclusion of classical particles. The classical Gibbs entropy is extended to a new entropy form --- called Gibbs-Fermi entropy --- that describes mixing configurations of all finite size particles and voids in a thermodynamic system where microstates do not have equal probabilities. The PNPF model describes the dynamic flow of ions, water molecules, as well as voids with electric fields and protein charges. The PNPF results are in good accord with experimental currents recorded in a 10^8-fold range of Ca++ concentrations. The results illustrate the anomalous mole fraction effect, a signature of L-type calcium channels. Moreover, numerical results concerning water density, dielectric permittivity, void volume, and steric energy provide useful details to study a variety of physical mechanisms ranging from binding, to permeation, blocking, flexibility, and charge/space competition of the channel.Comment: 23 pages, 12 figures. arXiv admin note: text overlap with arXiv:1506.0595

A GPU Poisson-Fermi Solver for Ion Channel Simulations

The Poisson-Fermi model is an extension of the classical Poisson-Boltzmann model to include the steric and correlation effects of ions and water treated as nonuniform spheres in aqueous solutions. Poisson-Boltzmann electrostatic calculations are essential but computationally very demanding for molecular dynamics or continuum simulations of complex systems in molecular biophysics and electrochemistry. The graphic processing unit (GPU) with enormous arithmetic capability and streaming memory bandwidth is now a powerful engine for scientific as well as industrial computing. We propose two parallel GPU algorithms, one for linear solver and the other for nonlinear solver, for solving the Poisson-Fermi equation approximated by the standard finite difference method in 3D to study biological ion channels with crystallized structures from the Protein Data Bank, for example. Numerical methods for both linear and nonlinear solvers in the parallel algorithms are given in detail to illustrate the salient features of the CUDA (compute unified device architecture) software platform of GPU in implementation. It is shown that the parallel algorithms on GPU over the sequential algorithms on CPU (central processing unit) can achieve 22.8x and 16.9x speedups for the linear solver time and total runtime, respectively.Comment: 21 pages, 5 figure

Theory of phase separation and polarization for dissociated ionic liquids

Room temperature ionic liquids are attractive to numerous applications and particularly, to renewable energy devices. As solvent free electrolytes, they demonstrate a paramount connection between the material morphology and Coulombic interactions: unlike dilute electrolytes, the electrode/RTIL interface is a product of both electrode polarization and spatiotemporal bulk properties. Yet, theoretical studies have dealt almost exclusively with independent models of morphology and electrokinetics. In this work, we develop a novel Cahn-Hilliard-Poisson type mean-field framework that couples morphological evolution with electrokinetic phenomena. Linear analysis of the model shows that spatially periodic patterns form via a finite wavenumber instability, a property that cannot arise in the currently used Fermi-Poisson-Nernst-Planck equations. Numerical simulations in above one-space dimension, demonstrate that while labyrinthine type patterns develop in the bulk, stripe patterns emerge near charged surfaces. The results qualitatively agree with empirical observations and thus, provide a physically consistent methodology to incorporate phase separation properties into an electrochemical framework.Comment: 5 pages, 3 figure

A local approximation of fundamental measure theory incorporated into three dimensional Poisson-Nernst-Planck equations to account for hard sphere repulsion among ions

The hard sphere repulsion among ions can be considered in the Poisson-Nernst-Planck (PNP) equations by combining the fundamental measure theory (FMT). To reduce the nonlocal computational complexity in 3D simulation of biological systems, a local approximation of FMT is derived, which forms a local hard sphere PNP (LHSPNP) model. It is interestingly found that the essential part of free energy term of the previous size modified model has a very similar form to one term of the LHS model, but LHSPNP has more additional terms accounting for size effects. Equation of state for one component homogeneous fluid is studied for the local hard sphere approximation of FMT and is proved to be exact for the first two virial coefficients, while the previous size modified model only presents the first virial coefficient accurately. To investigate the effects of LHS model and the competitions among different counterion species, numerical experiments are performed for the traditional PNP model, the LHSPNP model, the previous size modified PNP (SMPNP) model and the Monte Carlo simulation. It's observed that in steady state the LHSPNP results are quite different from the PNP results, but are close to the SMPNP results under a wide range of boundary conditions. Besides, in both LHSPNP and SMPNP models the stratification of one counterion species can be observed under certain bulk concentrations.Comment: This paper has been withdrawn by the author due to the misquotation of Ref. 2

A Nonlocal Poisson-Fermi Model for Ionic Solvent

We propose a nonlocal Poisson-Fermi model for ionic solvent that includes ion size effects and polarization correlations among water molecules in the calculation of electrostatic potential. It includes the previous Poisson-Fermi models as special cases, and its solution is the convolution of a solution of the corresponding nonlocal Poisson dielectric model with a Yukawa-type kernel function. Moreover, the Fermi distribution is shown to be a set of optimal ionic concentration functions in the sense of minimizing an electrostatic potential free energy. Finally, numerical results are reported to show the difference between a Poisson-Fermi solution and a corresponding Poisson solution.Comment: 12 pages, 3 figure

Poisson-Fermi Modeling of Ion Activities in Aqueous Single and Mixed Electrolyte Solutions at Variable Temperature

The combinatorial explosion of empirical parameters in tens of thousands presents a tremendous challenge for extended Debye-H\"uckel models to calculate activity coefficients of aqueous mixtures of most important salts in chemistry. The explosion of parameters originates from the phenomenological extension of the Debye-H\"uckel theory that does not take steric and correlation effects of ions and water into account. In contrast, the Poisson-Fermi theory developed in recent years treats ions and water molecules as nonuniform hard spheres of any size with interstitial voids and includes ion-water and ion-ion correlations. We present a Poisson-Fermi model and numerical methods for calculating the individual or mean activity coefficient of electrolyte solutions with any arbitrary number of ionic species in a large range of salt concentrations and temperatures. For each activity-concentration curve, we show that the Poisson-Fermi model requires only three unchanging parameters at most to well fit the corresponding experimental data. The three parameters are associated with the Born radius of the solvation energy of an ion in electrolyte solution that changes with salt concentrations in a highly nonlinear manner.Comment: 21 pages, 7 figure

Do Bi-Stable Poisson-Nernst-Planck Models Describe Single Channel Gating?

Experiments measuring currents through single protein channels show unstable currents, a phenomena called the gating of a single channel. Channels switch between an 'open' state with a well defined single amplitude of current and 'closed' states with nearly zero current. The existing mean-field theory of ion channels focuses almost solely on the open state. The physical modeling of the dynamical features of ion channels is still in its infancy, and does not describe the transitions between open and closed states, nor the distribution of the duration times of open states. One hypothesis is that gating corresponds to noise-induced fast transitions between multiple steady (equilibrium) states of the underlying system. In this work, we aim to test this hypothesis. Particularly, our study focuses on the (high order) steric Poisson-Nernst-Planck-Cahn-Hilliard model since it has been successful in predicting permeability and selectivity of ionic channels in their open state, and since it gives rise to multiple steady states. We show that this system gives rise to a gating-like behavior, but that important features of this switching behavior are different from the defining features of gating in biological systems. Furthermore, we show that noise prohibits switching in the system of study. The above phenomena are expected to occur in other PNP-type models, strongly suggesting that one has to go beyond over-damped (gradient flow) Nernst-Planck type dynamics to explain the spontaneous gating of single channels

Analysis of Generalized Debye-H\"uckel Equation from Poisson-Fermi Theory

The Debye-H\"uckel equation is a fundamental physical model in chemical thermodynamics that describes the free energy (chemical potential, activity) of an ion in electrolyte solutions at variable salt concentration, temperature, and pressure. It is based on the linear Poisson-Boltzmann equation that ignores the steric (finite size), correlation, and polarization effects of ions and water (or solvent molecules). The Poisson-Fermi theory developed in recent years takes these effects into account. A generalized Debye-H\"uckel equation is derived from the Poisson-Fermi theory and is shown to consistently reduce to the classical equation when these effects vanish in limiting cases. As a result, a linear fourth-order Poisson-Fermi equation is presented for which unique solutions are shown to exist for spherically symmetric systems. Moreover, a generalized Debye length is proposed to include the size effects of ions and water.Comment: 22 pages, 1 figures. arXiv admin note: text overlap with arXiv:1801.0347