Poisson–Fermi model of single ion activities in aqueous solutions

Biocompatible Conductive Hydrogels: Applications in the Field of Biomedicine

The impact of COVID-19 has rendered medical technology an important factor to maintain social stability and economic increase, where biomedicine has experienced rapid development and played a crucial part in fighting off the pandemic. Conductive hydrogels (CHs) are three-dimensional (3D) structured gels with excellent electrical conductivity and biocompatibility, which are very suitable for biomedical applications. CHs can mimic innate tissue's physical, chemical, and biological properties, which allows them to provide environmental conditions and structural stability for cell growth and serve as efficient delivery substrates for bioactive molecules. The customizability of CHs also allows additional functionality to be designed for different requirements in biomedical applications. This review introduces the basic functional characteristics and materials for preparing CHs and elaborates on their synthetic techniques. The development and applications of CHs in the field of biomedicine are highlighted, including regenerative medicine, artificial organs, biosensors, drug delivery systems, and some other application scenarios. Finally, this review discusses the future applications of CHs in the field of biomedicine. In summary, the current design and development of CHs extend their prospects for functioning as an intelligent and complex system in diverse biomedical applications.

Review and Modification of Entropy Modeling for Steric Effects in the Poisson-Boltzmann Equation

The classical Poisson-Boltzmann model can only work when ion concentrations are very dilute, which often does not match the experimental conditions. Researchers have been working on the modification of the model to include the steric effect of ions, which is non-negligible when the ion concentrations are not dilute. Generally the steric effect was modeled to correct the Helmholtz free energy either through its internal energy or entropy, and an overview is given here. The Bikerman model, based on adding solvent entropy to the free energy through the concept of volume exclusion, is a rather popular steric-effect model nowadays. However, ion sizes are treated as identical in the Bikerman model, making an extension of the Bikerman model to include specific ion sizes desirable. Directly replacing the ions of non-specific size by specific ones in the model seems natural and has been accepted by many researchers in this field. However, this straightforward modification does not have a free energy formula to support it. Here modifications of the Bikerman model to include specific ion sizes have been developed iteratively, and such a model is achieved with a guarantee that: (1) it can approach Boltzmann distribution at diluteness; (2) it can reach saturation limit as the reciprocal of specific ion size under extreme electrostatic conditions; (3) its entropy can be derived by mean-field lattice gas model.

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.

A Flux Ratio and a Universal Property of Permanent Charges Effects on Fluxes

In this work, we consider ionic flow through ion channels for an ionic mixture of a cation species (positively charged ions) and an anion species (negatively charged ions), and examine effects of a positive permanent charge on fluxes of the cation species and the anion species. For an ion species, and for any given boundary conditions and channel geometry,we introduce a ratio _(Q) = J(Q)/J(0) between the flux J(Q) of the ion species associated with a permanent charge Q and the flux J(0) associated with zero permanent charge. The flux ratio _(Q) is a suitable quantity for measuring an effect of the permanent charge Q: if _(Q) > 1, then the flux is enhanced by Q; if _ < 1, then the flux is reduced by Q. Based on analysis of Poisson-Nernst-Planck models for ionic flows, a universal property of permanent charge effects is obtained: for a positive permanent charge Q, if _1(Q) is the flux ratio for the cation species and _2(Q) is the flux ratio for the anion species, then _1(Q) < _2(Q), independent of boundary conditions and channel geometry. The statement is sharp in the sense that, at least for a given small positive Q, depending on boundary conditions and channel geometry, each of the followings indeed occurs: (i) _1(Q) < 1 < _2(Q); (ii) 1 < _1(Q) < _2(Q); (iii) _1(Q) < _2(Q) < 1. Analogous statements hold true for negative permanent charges with the inequalities reversed. It is also shown that the quantity _(Q) = |J(Q) − J(0)| may not be suitable for comparing the effects of permanent charges on cation flux and on anion flux. More precisely, for some positive permanent charge Q, if _1(Q) is associated with the cation species and _2(Q) is associated with the anion species, then, depending on boundary conditions and channel geometry, each of the followings is possible: (a) _1(Q) > _2(Q); (b) _1(Q) < _2(Q).

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-Hückel models to calculate activity coefficients of aqueous mixtures of the most important salts in chemistry. The explosion of parameters originates from the phenomenological extension of the Debye-Hückel theory that does not take steric and correlation effects of ions and water into account. By 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.

Atomistic simulations of cation hydration in sodium and calcium montmorillonite nanopores

During the last four decades, numerous studies have been directed to the swelling smectite-rich clays in the context of high-level radioactive waste applications and waste-liners for contaminated sites. The swelling properties of clay mineral particles arise due to hydration of the interlayer cations and the diffuse double layers formed near the negatively charged montmorillonite (MMT) surfaces. To accurately study the cation hydration in the interlayer nanopores of MMT, solvent-solute and solvent-clay surface interactions (i.e., the solvation effects and the shape effects) on the atomic level should be taken into account, in contrast to many recent electric double layer based methodologies using continuum models. Therefore, in this research we employed fully atomistic simulations using classical molecular dynamics (MD) simulations, the software package GROMACS along with the CLAYFF forcefield and the SPC/E water model. We present the ion distributions and the deformation of the hydrated coordination structures, i.e., the hydration shells of Na⁺ and Ca²⁺ in the interlayer, respectively, for MMT in the first-layer, the second-layer, the third-layer, the fourth-layer, and the fifth-layer (1W, 2W, 3W, 4W, and 5W) hydrate states. Our MD simulations show that Na⁺ in Na-MMT nanopores have an affinity to the ditrigonal cavities of the clay layers and form transient inner-sphere complexes at about 3.8 Å from clay midplane at water contents less than the 5W hydration state. However, these phenomena are not observed in Ca-MMT regardless of swelling states. For Na-MMT, each Na⁺ is coordinated to four water molecules and one oxygen atom of the clay basal-plane in the first hydration shell at the 1W hydration state, and with five to six water molecules in the first hydration shell within a radius of 3.1 Å at all higher water contents. In Ca-MMT, however each Ca²⁺ is coordinated to approximately seven water molecules in the first hydration shell at the 1W hydration state and about eight water molecules in the first hydration shell within a radius of 3.3 Å at all higher hydration states. Moreover, the MD results show that the complete hydration shells are nearly spherical with an orthogonal coordination sphere. They could only be formed when the basal spacing d₀₀₁ ≥ 18.7 Å, i.e., approximately, the interlayer separation h ≥ 10 Å. Comparison between DFT and MD simulations shows that DFT failed to reproduce the outer-sphere complexes in the Stern-layer (within ∼5.0 Å from the clay basal-plane), observed in the MD simulations.

Mass Action and Conservation of Current

The law of mass action does not force a series of chemical reactions to have the same current flow everywhere. Interruption of far-away current does not stop current everywhere in a series of chemical reactions (analyzed according to the law of mass action), and so does not obey Maxwell’s equations. An additional constraint and equation is needed to enforce global continuity of current. The additional constraint is introduced in this paper in the special case that the chemical reaction describes spatial movement through narrow channels. In that case, a fully consistent treatment is possible using different models of charge movement. The general case must be dealt with by variational methods that enforce consistency of all the physical laws involved. Violations of current continuity arise away from equilibrium, when current flows, and the law of mass action is applied to a non-equilibrium situation, different from the systems considered when the law was originally derived. Device design in the chemical world is difficult because simple laws are not obeyed in that way. Rate constants of the law of mass action are found experimentally to change from one set of conditions to another. The law of mass action is not robust in most cases and cannot serve the same role that circuit models do in our electrical technology. Robust models and device designs in the chemical world will not be possible until continuity of current is embedded in a generalization of the law of mass action using a consistent variational model of energy and dissipation.

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-like kernel function. The Fermi distribution is shown to be a set of optimal ionic concentration functions in the sense of minimizing an electrostatic potential free energy. Numerical results are reported to show the difference between a Poisson-Fermi solution and a corresponding Poisson solution.