Singular perturbation solutions of steady-state Poisson-Nernst-Planck systems

Polyelectrolyte Hydrogel-Functionalized Photothermal Sponge Enables Simultaneously Continuous Solar Desalination and Electricity Generation Without Salt Accumulation

Technologies that can simultaneously generate electricity and desalinate seawater are highly attractive and required to meet the increasing global demand for power and clean water. Here, a bifunctional solar evaporator that features continuous electric generation in seawater without salt accumulation is developed by rational design of polyelectrolyte hydrogel-functionalized photothermal sponge. This evaporator not only exhibits an unprecedentedly high water evaporation rate of 3.53 kg m-2 h-1along with 98.6% solar energy conversion efficiency but can also uninterruptedly deliver a voltage output of 0.972 V and a current density of 172.38 µA cm-2 in high-concentration brine over a prolonged period under one sun irradiation. Many common electronic devices can be driven by simply connecting evaporator units in series or in parallel without any other auxiliaries. Different from the previously proposed power generation mechanism, this study reveals that the water-enabled proton concentration fields in intermediate water region can also induce an additional ion electric field in free water region containing solute, to further enhance electricity output. Given the low-cost materials, simple self-regeneration design, scalable fabrication processes, and stable performance, this work offers a promising strategy for addressing the shortages of clean water and sustainable electricity.

Finite ion size effects on I-V relations via Poisson-Nernst-Planck systems with two cations: A case study

We consider a quasi-one-dimensional Poisson-Nernst-Planck model with two cations having the same valances and one anion. Bikerman's local hard-sphere potential is included to account for ion size effects. Under some further restrictions on the boundary conditions of the two cations, we obtain approximations of the I-V (current-voltage) relations by treating the ion sizes as small parameters. Critical potentials are identified, which play critical roles in characterizing finite ion size effects on ionic flows. Nonlinear interplays between system parameters, such as boundary concentrations and diffusion coefficients, are analyzed. To provide more intuitive illustrations of our analytical results and better understanding of the dynamics of ionic flows through membrane channels, numerical simulations are performed.

Unipolar Solution Flow in Calcium-Organic Frameworks for Seawater-Evaporation-Induced Electricity Generation

Seawater-flow- and -evaporation-induced electricity generation holds significant promise in advancing next-generation sustainable energy technologies. This method relies on the electrokinetic effect but faces substantial limitations when operating in a highly ion-concentrated environment, for example, natural seawater. We present herein a novel solution using calcium-based metal-organic frameworks (MOFs, C12H6Ca2O19·2H2O) for seawater-evaporation-induced electricity generation. Remarkably, Ca-MOFs show an open-circuit voltage of 0.4 V and a short-circuit current of 14 μA when immersed in seawater under natural conditions. Our experiments and simulations revealed that sodium (Na) ions selectively transport within sub-nanochannels of these synthetic superhydrophilic MOFs. This selective ion transport engenders a unipolar solution flow, which drives the electricity generation behavior in seawater. This work not only showcases an effective Ca-MOF for electricity generation through seawater flow/evaporation but also contributes significantly to our understanding of water-driven energy harvesting technologies and their potential applications beyond this specific context.

Bounds for systems of coupled advection-diffusion equations with application to the Poisson-Nernst-Plank equations

We consider coupled systems of advection-diffusion equations with initial and boundary conditions and determine conditions on the advection terms that allow us to obtain solutions that can be explicitly bounded above and below using the initial and boundary conditions. Given the advection terms, using our methodology one can easily check if such bounds can be obtained and then one can construct the necessary nonlinear transformation to allow the bounds to be determined. We apply this technique to determine bounding quantities for a number of examples. In particular, we show that the three-ion electroneutral Poisson-Nernst-Planck system of equations can be transformed into a system, which allows for the use of our techniques and we determine the bounding quantities. In addition, we determine the general form of advection terms that allow these techniques to be applied and show that our method can be applied to a very wide class of advection-diffusion equations.

Competition between Cations via Classical Poisson-Nernst-Planck Models with Nonzero but Small Permanent Charges

We study a one-dimensional Poisson–Nernst–Planck system for ionic flow through a membrane channel. Nonzero but small permanent charge, the major structural quantity of an ion channel, is included in the model. Two cations with the same valences and one anion are included in the model, which provides more rich and complicated correlations/interactions between ions. The cross-section area of the channel is included in the system, and it provides certain information of the geometry of the three-dimensional channel, which is critical for our analysis. Geometric singular perturbation analysis is employed to establish the existence and local uniqueness of solutions to the system for small permanent charges. Treating the permanent charge as a small parameter, through regular perturbation analysis, we are able to derive approximations of the individual fluxes explicitly, and this allows us to study the competition between two cations, which is related to the selectivity phenomena of ion channels. Numerical simulations are performed to provide a more intuitive illustration of our analytical results, and they are consistent.

Hysteresis and linear stability analysis on multiple steady-state solutions to the Poisson-Nernst-Planck equations with steric interactions

In this work, we numerically study linear stability of multiple steady-state solutions to a type of steric Poisson-Nernst-Planck equations with Dirichlet boundary conditions, which are applicable to ion channels. With numerically found multiple steady-state solutions, we obtain S-shaped current-voltage and current-concentration curves, showing hysteretic response of ion conductance to voltages and boundary concentrations with memory effects. Boundary value problems are proposed to locate bifurcation points and predict the local bifurcation diagram near bifurcation points on the S-shaped curves. Numerical approaches for linear stability analysis are developed to understand the stability of the steady-state solutions that are only numerically available. Finite difference schemes are proposed to solve a derived eigenvalue problem involving differential operators. The linear stability analysis reveals that the S-shaped curves have two linearly stable branches of different conductance levels and one linearly unstable intermediate branch, exhibiting classical bistable hysteresis. As predicted in the linear stability analysis transition dynamics, from a steady-state solution on the unstable branch to one on the stable branches, are led by perturbations associated to the mode of the dominant eigenvalue. Further numerical tests demonstrate that the finite difference schemes proposed in the linear stability analysis are second-order accurate. Numerical approaches developed in this work can be applied to study linear stability of a class of time-dependent problems around their steady-state solutions that are computed numerically.

Dynamics of ionic flows via Poisson-Nernst-Planck systems with local hard-sphere potentials: Competition between cations

We study a quasi-one-dimensional steady-state Poisson-Nernst-Planck type model for ionic flows through a membrane channel with three ion species, two positively charged with the same valence and one negatively charged. Bikerman's local hard-sphere potential is included in the model to account for ion sizes. The problem is treated as a boundary value problem of a singularly perturbed differential system. Under the framework of a geometric singular perturbation theory, together with specific structures of this concrete model, the existence and uniqueness of solutions to the boundary value problem for small ion sizes is established. Furthermore, treating the ion sizes as small parameters, we derive an approximation of individual fluxes, from which one can further study the qualitative properties of ionic flows and extract concrete information directly related to biological measurements. Of particular interest is the competition between two cations due to the nonlinear interplay between finite ion sizes, diffusion coefficients and boundary conditions, which is closely related to selectivity phenomena of open ion channels with given protein structures. Furthermore, we are able to characterize the distinct effects of the nonlinear interplays between these physical parameters. Numerical simulations are performed to identify some critical potentials which play critical roles in examining properties of ionic flows in our analysis.

Boundary Layer Effects on Ionic Flows Via Classical Poisson-Nernst-Planck Systems

A quasi-one-dimensional steady-state Poisson-Nernst-Planck model of two oppositely charged ion species through a membrane channel is analyzed. The model problem is treated as a boundary value problem of a singularly perturbed differential system. Our analysis is based on the geometric singular perturbation theory but, most importantly, on specific structures of this concrete model. The existence and (local ) uniqueness of solutions to the boundary value problem is established. In particular, an approximation of both the individual flux and the I-V (current-voltage) relation are derived explicitly from the zeroth order approximation (in ") solutions, from which the boundary layer effects on ionic flows are studied in great details.

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).

Electroneutral models for dynamic Poisson-Nernst-Planck systems

The Poisson-Nernst-Planck (PNP) system is a standard model for describing ion transport. In many applications, e.g., ions in biological tissues, the presence of thin boundary layers poses both modeling and computational challenges. In this paper, we derive simplified electroneutral (EN) models where the thin boundary layers are replaced by effective boundary conditions. There are two major advantages of EN models. First, it is much cheaper to solve them numerically. Second, EN models are easier to deal with compared to the original PNP system; therefore, it would also be easier to derive macroscopic models for cellular structures using EN models. Even though the approach used here is applicable to higher-dimensional cases, this paper mainly focuses on the one-dimensional system, including the general multi-ion case. Using systematic asymptotic analysis, we derive a variety of effective boundary conditions directly applicable to the EN system for the bulk region. This EN system can be solved directly and efficiently without computing the solution in the boundary layer. The derivation is based on matched asymptotics, and the key idea is to bring back higher-order contributions into the effective boundary conditions. For Dirichlet boundary conditions, the higher-order terms can be neglected and the classical results (continuity of electrochemical potential) are recovered. For flux boundary conditions, higher-order terms account for the accumulation of ions in boundary layer and neglecting them leads to physically incorrect solutions. To validate the EN model, numerical computations are carried out for several examples. Our results show that solving the EN model is much more efficient than the original PNP system. Implemented with the Hodgkin-Huxley model, the computational time for solving the EN model is significantly reduced without sacrificing the accuracy of the solution due to the fact that it allows for relatively large mesh and time-step sizes.