Ion Size Effects Research Articles

A parallel finite element simulator for ion transport through three‐dimensional ion channel systems

A parallel finite element simulator, ichannel, is developed for ion transport through three-dimensional ion channel systems that consist of protein and membrane. The coordinates of heavy atoms of the protein are taken from the Protein Data Bank and the membrane is represented as a slab. The simulator contains two components: a parallel adaptive finite element solver for a set of Poisson-Nernst-Planck (PNP) equations that describe the electrodiffusion process of ion transport, and a mesh generation tool chain for ion channel systems, which is an essential component for the finite element computations. The finite element method has advantages in modeling irregular geometries and complex boundary conditions. We have built a tool chain to get the surface and volume mesh for ion channel systems, which consists of a set of mesh generation tools. The adaptive finite element solver in our simulator is implemented using the parallel adaptive finite element package Parallel Hierarchical Grid (PHG) developed by one of the authors, which provides the capability of doing large scale parallel computations with high parallel efficiency and the flexibility of choosing high order elements to achieve high order accuracy. The simulator is applied to a real transmembrane protein, the gramicidin A (gA) channel protein, to calculate the electrostatic potential, ion concentrations and I - V curve, with which both primitive and transformed PNP equations are studied and their numerical performances are compared. To further validate the method, we also apply the simulator to two other ion channel systems, the voltage dependent anion channel (VDAC) and α-Hemolysin (α-HL). The simulation results agree well with Brownian dynamics (BD) simulation results and experimental results. Moreover, because ionic finite size effects can be included in PNP model now, we also perform simulations using a size-modified PNP (SMPNP) model on VDAC and α-HL. It is shown that the size effects in SMPNP can effectively lead to reduced current in the channel, and the results are closer to BD simulation results.

Ion Correlations in Nanofluidic Channels: Effects of Ion Size, Valence, and Concentration on Voltage- and Pressure-Driven Currents

The effects of ion-ion and ion-wall correlations in nanochannels are explored, specifically how they influence voltage- and pressure-driven currents and pressure-to-voltage energy conversion. Cations of different diameters (0.15, 0.3, and 0.9 nm) and different valences (+1, +2, and +3) at concentrations ranging from 10(-6) M to 1 M are considered in 50-nm- and 100-nm-wide nanoslits with wall surface charges ranging from 0 C/m(2) to -0.3 C/m(2). These parameters are typical of nanofluidic devices. Ion correlations have significant effects on device properties over large parts of this parameter space. These effects are the result of ion layering (oscillatory concentration profiles) for large monovalent cations and charge inversion (more cations in the first layer near the wall than necessary to neutralize the surface charge) for the multivalent cations. The ions were modeled as charged, hard spheres using density functional theory of fluids, and current was computed with the Navier-Stokes equations with two different no-slip conditions.

Poisson--Nernst--Planck Systems for Ion Flow with a Local Hard-Sphere Potential for Ion Size Effects

In this work, we analyze a one-dimensional steady-state Poisson--Nernst--Planck-type model for ionic flow through a membrane channel with fixed boundary ion concentrations (charges) and electric potentials. We consider two ion species, one positively charged and one negatively charged, and assume zero permanent charge. A local hard-sphere potential that depends pointwise on ion concentrations is included in the model to account for ion size effects on the ionic flow. 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 of solutions to the boundary value problem for small ion sizes is established and, treating the ion sizes as small parameters, we also derive an approximation of the I-V (current-voltage) relation and identify two critical potentials or voltages for ion size effects. Under electroneutrality (zero net charge) boundary conditions, each of these two critical potentials separates the potential into two regions over which the ion size effects are qualitatively opposite to each other. On the other hand, without electroneutrality boundary conditions, the qualitative effects of ion sizes will depend not only on the critical potentials but also on boundary concentrations. Important scaling laws of I-V relations and critical potentials in boundary concentrations are obtained. Similar results about ion size effects on the flow of matter are also discussed. Under electroneutrality boundary conditions, the results on the first order approximation in ion diameters of solutions, I-V relations, and critical potentials agree with those with a nonlocal hard-sphere potential examined by Ji and Liu [J. Dynam. Differential Equations, 24 (2012), pp. 955--983].

Efficient Algorithms for a Nonlocal Dielectric Model for Protein in Ionic Solvent

The nonlocal dielectric approach can significantly enhance the classical Poisson dielectric model by including polarization correlations among water molecules. In this paper, a nonlocal dielectric model for protein in ionic solvent is proposed and analyzed, alongside a new efficient numerical algorithm and program package for solving the model. In particular, by using solution splitting and reformulation techniques, it is shown that the solution of the nonlocal dielectric model is unique, and can be found from solving two well-defined partial differential systems and one Poisson-like boundary value problem. Consequently, the singular and computational difficulties caused by Dirac delta distributions and convolution terms are overcome. Furthermore, a nonlocal linearized Poisson--Boltzmann equation with uniform ionic size effect is proposed and numerically tested on three protein molecules with up to 6062 atoms by using a fast finite element solver from the FEniCS project. A nonlocal point charge Born model...

Effects of rare-earth ion size on the stability of the coherent Jahn-Teller distortions in undoped perovskite manganites

We present a theoretical study on the relation between the size of the rare earth ions, often known as chemical pressure, and the stability of the coherent Jahn-Teller distortions in undoped perovskite manganites. Using a Keating model expressed in terms of atomic scale symmetry modes, we show that there exists a coupling between the uniform shear distortion and the staggered buckling distortion within the Jahn-Teller energy term. It is found that this coupling provides a mechanism by which the coherent Jahn-Teller distortion is more stabilized by smaller rare earth ions. We analyze the appearance of the uniform shear distortion below the Jahn-Teller ordering temperature, estimate the Jahn-Teller ordering temperature and its variation between NdMnO3 and LaMnO3, and obtain the relations between distortions. We find good agreement between theoretical results and experimental data.

Steric-effect-induced alteration of thermal transport phenomenon for mixed electroosmotic and pressure driven flows through narrow confinements

The present paper addresses, for the first time, the thermal transport process for mixed electroosmotic and pressure-drive flows of electrolyte solutions, through nanoscopic confinements with step-change in the wall temperature, by going beyond the prevalent simplifying assumption of non-interacting, point charge behaviour of the ions in the electrolyte solution. An attempt is made here to delineate the alterations in the heat transfer characteristics induced by the inclusion of the finite ionic size effect or the ‘steric effect’, in the electrokinetic framework, which becomes very significant for electro-hydrodynamic flows through narrow-confinements with high surface charge density. Under such situations the finite size of the ions cannot be trivially precluded from the analysis, thereby rendering the point charge assumption erroneous. The distinctive influence of the ‘steric’ effect of ions on the heat transfer characteristics is numerically investigated by delineating the variations in the local liquid temperature, local Nusselt number and the thermal entrance length for the thermally developing regime. The observed significant influences of the bulk volume fraction of ions, as represented by the steric factor, on the thermal transport phenomenon are physically explained by simultaneously highlighting their intrinsic differences with the effects of other existing parameters, like Joule heating and viscous dissipation. The utilitarian scope of the work lies in the fact that it proposes a more comprehensive methodology for analyzing the heat transfer characteristics in state-of-the-art nanoscale electromechanical devices.

Poisson–Nernst–Planck Systems for Ion Flow with Density Functional Theory for Hard-Sphere Potential: I–V Relations and Critical Potentials. Part II: Numerics

We consider a one-dimensional steady-state Poisson–Nernst–Planck type model for ionic flow through membrane channels. Improving the classical Poisson–Nernst–Planck models where ion species are treated as point charges, this model includes ionic interaction due to finite sizes of ion species modeled by hard sphere potential from the Density Functional Theory. The resulting problem is a singularly perturbed boundary value problem of an integro-differential system. We examine the problem and investigate the ion size effect on the current–voltage (I–V) relations numerically, focusing on the case where two oppositely charged ion species are involved and only the hard sphere components of the excess chemical potentials are included. Two numerical tasks are conducted. The first one is a numerical approach of solving the boundary value problem and obtaining I–V curves. This is accomplished through a numerical implementation of the analytical strategy introduced by Ji and Liu in [Poisson–Nernst–Planck systems for ion flow with density functional theory for hard-sphere potential: I–V relations and critical potentials. Part I: Analysis, J. Dyn. Differ. Equ. (to appear)]. The second task is to numerically detect two critical potential values V c and V c .The existence of these two critical values is first realized for a relatively simple setting and analytical approximations of V c and V c are obtained in the above mentioned reference. We propose an algorithm for numerical detection of V c and V c without using any analytical formulas but based on the defining properties and numerical I–V curves directly. For the setting in the above mentioned reference, our numerical values for V c and V c agree well with the analytical predictions. For a setting including a nonzero permanent charge in which case no analytic formula for the I–V relation is available now, our algorithms can still be applied to find V c and V c numerically.

Thermal degradation behaviour of some metal chelate polymer compounds with bis(bidentate) ligand by TG/DTG/DTA

Seven novel divalent transitional metal chelate polymers compounds (commonly known as chelate compounds or metal coordination complexes or polymer complexes) have been characterized by thermogravimetry (TG), differential thermal gravimetry (DTG) and differential thermal analysis (DTA) methods. Thermal decomposition behaviour of Mn(II), Co(II), Ni(II), Cu(II), Zn(II), Cd(II) and Hg(II) polymers with terphthaoyl-bis(p-methoxyphenylcarbamide) has been investigated by thermogravimetric analysis (TGA) at heating rate 10 °C min−1 under nitrogen atmosphere. TG/DTA of chelate compounds were shown to be a stable compound against thermal decomposition which was measured on the basis of final decomposing temperature, but it is observed in some curves that decomposition takes place at low temperature due to the lattice water, which is always placed at outer coordination sphere of the central metal ion. The presence of both lattice and coordinated water were noteworthy investigated in Co(II), Ni(II) and Cu(II) chelate polymer compounds, whereas lattice water found in Zn(II), Cd(II) and Hg(II). However, Mn(II) showed only coordinated water. Thermal stabilities for release of lattice water, coordinated water and organic moiety that occur in sequential decomposition of chelate compounds are explained on the basis of ionic size effect and electronegativity. The processes of thermal degradation taking place in seven chelate polymers were studied comparatively by TG/DTG/DTA curves which indicating the difference in the thermal decomposition. Coats–Redfern integral method is used to determine the kinetic parameters for the successive steps in the decomposition sequence of TG curves. Scanning electron microscope images of some chelate polymers were shown in previous publication revealed that particle sizes of chelate polymers were found to be of nanomaterial level therefore, resulting chelate compounds might be called as nanomaterial.

Ion size effects on the electrokinetics of salt-free concentrated suspensions in ac fields

We analyze the influence of finite ion size effects in the response of a salt-free concentrated suspension of spherical particles to an oscillating electric field. Salt-free suspensions are just composed of charged colloidal particles and the added counterions released by the particles to the solution that counterbalance their surface charge. In the frequency domain, we study the dynamic electrophoretic mobility of the particles and the dielectric response of the suspension. We find that the Maxwell–Wagner–O’Konski process associated with the counterions condensation layer is enhanced for moderate to high particle charges, yielding an increment of the mobility for such frequencies. We also find that the increment of the mobility grows with ion size and particle charge. All these facts show the importance of including ion size effects in any extension attempting to improve standard electrokinetic models.

Ion size effects on electric double layers and ionic transport through ion-exchange membrane systems

The density function theory is used to study the density profiles and the transport properties of an ion-exchange membrane system submitted to an electric potential drop. As the ionic density increases, hard sphere interaction between ions becomes dominant and the ion size must be taken into account. The results show that the density distribution and the transport properties depend on the bulk electrolyte density. At equilibrium the charge inside the electric double layer (EDL) adjacent to the membrane decreases and the membrane electric potential increases as the bulk density increases. For high bulk density of unsymmetric electrolyte, secondary charge layers are observed inside the EDL. In the membrane the anion-density-to-bulk-density ratio increases when the bulk density increases from small to moderate values owing to the membrane potential increase. But it decreases abruptly at high bulk density values owing to the increase of the non-ideal electrostatic interaction. At a given electric potential drop, the current/voltage curves follow the variation of this ratio with respect to the bulk density at equilibrium. As the current density approaches the limiting one, the amplitude of the secondary charge layers decreases and the EDL thickness increases.

Equilibrium properties of charged spherical colloidal particles suspended in aqueous electrolytes: Finite ion size and effective ion permittivity effects

The equilibrium properties of a charged spherical colloidal particle immersed in an aqueous electrolyte solution are examined using an extension of the Standard Electrokinetic Model that takes into account the finite ion size by modeling the aqueous electrolyte solution as a suspension of polarizable insulating spheres in water. We find that this model greatly amplifies the steric effects predicted by the usual modified Poisson–Boltzmann equation, which only imposes a restriction on the ability of ions to approach one another. This suggests that a solution of the presented model under nonequilibrium conditions could have important consequences in the interpretation of dielectric and electrokinetic data in colloidal suspensions.

Competitive adsorption and ordered packing of counterions near highly charged surfaces: From mean-field theory to Monte Carlo simulations

Competitive adsorption of counterions of multiple species to charged surfaces is studied by a size-effect-included mean-field theory and Monte Carlo (MC) simulations. The mean-field electrostatic free-energy functional of ionic concentrations, constrained by Poisson's equation, is numerically minimized by an augmented Lagrangian multiplier method. Unrestricted primitive models and canonical ensemble MC simulations with the Metropolis criterion are used to predict the ionic distributions around a charged surface. It is found that, for a low surface charge density, the adsorption of ions with a higher valence is preferable, agreeing with existing studies. For a highly charged surface, both the mean-field theory and the MC simulations demonstrate that the counterions bind tightly around the charged surface, resulting in a stratification of counterions of different species. The competition between mixed entropy and electrostatic energetics leads to a compromise that the ionic species with a higher valence-to-volume ratio has a larger probability to form the first layer of stratification. In particular, the MC simulations confirm the crucial role of ionic valence-to-volume ratios in the competitive adsorption to charged surfaces that had been previously predicted by the mean-field theory. The charge inversion for ionic systems with salt is predicted by the MC simulations but not by the mean-field theory. This work provides a better understanding of competitive adsorption of counterions to charged surfaces and calls for further studies on the ionic size effect with application to large-scale biomolecular modeling.

A systematic study on the effect of histidine position and fragment ion size on the formation of bn ions

In this study, we utilized gas-phase hydrogen/deuterium (H/D) exchange reactions and ion mobility-mass spectrometry (IM-MS) to examine the effect of: (i) histidine (His) amino acid position and (ii) fragment ion size on the structures of bn (n=4–7) ions generated from nozzle-skimmer fragmentation. Both H/D exchange patterns and semi-log temporal plots of histidine-containing bn fragment ions generated from seven isobaric model heptapeptides {i.e., (His)AAAAAA-NH2, A(His)AAAAA-NH2, AA(His)AAAA-NH2, AAA(His)AAA-NH2, AAAA(His)AA-NH2, AAAAA(His)A-NH2, and AAAAAA(His)-NH2} suggested the presence of at least two structures for all b fragment ions: “fast” and “slow” H/D-exchanging ion populations. The observed H/D exchange rate constants (for disappearance of isolated 12Call or D0) for b4+ and b5+ fragments were higher than those observed for b7+ fragment ions, suggesting more compact and/or stable structures for b7+ fragment ions. Among the studied histidine-containing bn+ fragments, b4+ and b5+ showed the most variation in H/D exchange reactivity as a function of histidine position in the original peptide sequence. Ion mobility arrival time distributions (ATDs) for histidine-containing b5+ fragments from AA(His)AAAA-NH2, AAA(His)AAA-NH2, and AAAA(His)AA-NH2 showed two ion populations. H/D exchange and ion mobility results both imply the potential presence of a mixture of compact and open structures for b5+. ATDs for b4+ fragments generated from A(His)AAAAA-NH2 and AA(His)AAAA-NH2 (compared to ATDs of the other model heptapeptides) suggest the appearance or increase in the percentage of a more compact ion population. ATDs of histidine-containing b6+ and b7+ fragments varied significantly as a function of histidine position in the original heptapeptides.

Formulation of a new and simple nonuniform size‐modified poisson–boltzmann description

The nonlinear Poisson-Boltzmann equation (PBE) governing biomolecular electrostatics neglects ion size and ion correlation effects, and recent research activity has focused on accounting for these effects to achieve better physical modeling realism. Here, attention is focused on the comparatively simpler challenge of addressing ion size effects within a continuum-based solvent modeling framework. Prior works by Borukhov et al. (Phys. Rev. Lett. 1997, 79, 435; Electrochim. Acta 2000, 46, 221) have examined the case of uniform ion size in considerable detail. Generalizations to accommodate different species ion sizes have been performed by Li (Nonlinearity 2009, 22, 811; SIAM J. Math. Anal. 2009, 40, 2536) and Zhou et al. (Phys. Rev. E 2011, 84, 021901) using a variational principle, Chu et al. (Biophys. J. 2007, 93, 3202) using a lattice gas model, and Tresset (Phys. Rev. E 2008, 78, 061506) using a generalized Poisson-Fermi distribution. The current work provides an alternative derivation using simple statistical mechanics principles that place the ion size effects and energy distributions on a consistent statistical footing. The resulting expressions differ from the prior nonuniform ion-size developments. However, all treatments reduce to the same form in the cases of uniform ion-size and zero ion size (the PBE). Because of their importance to molecular modeling and salt-dependent behavior, expressions for the salt sensitivities and ionic forces are also derived using the nonuniform ion size description. Emphasis in this article is on formulation and numerically robust evaluation; results are presented for a simple sphere and a previously considered DNA structure for comparison and validation. More extensive application to biomolecular systems is deferred to a subsequent article.

Splitting of the pairing transition and coherence transition in R1−xCexBa2Cu3O7−δ (R = Y, Er, Ho and Gd)

Systematic conductivity measurements in granular R1−xCexBa2Cu3O7−δ (R = Y, Er, Ho and Gd) superconductors are presented. The focus is the role of the cerium ion on the superconductor–normal transition. The results showed that the resistive transition proceeds in two stages, intragranular and intergranular, as seen by the temperature derivative of the resistivity near TC. Therefore, from the critical exponent analysis, we have also observed two close and sharp genuine superconducting transitions in all samples investigated. Two critical temperatures were identified, but only one is strongly affected by Ce doping. The splitting of the pairing transition was associated with Ce and related to the occurrence of a phase separation. R1−xCexBa2Cu3O7−δ can be regarded as a mixed crystalline solid solution consisting of two crystal unit cells: R-123 (R cell) and Ce-123 (Ce cell), where a two-dimensional (2D) site-percolation structure is built along the ab plane. The ionic size effect of rare earth element R on the superconductor–normal transition has been investigated and it was observed that the temperature splitting Δ varies between approximately 4.2 K for Ho to 12.1 K for Gd.

Voids and necks in liquid ammonia and their roles in diffusion of ions of varying size

Voids in a medium are defined as the regions that are located outside an appropriately defined occupied space associated with molecules. Dynamical properties like diffusion can be related to the structure and distribution of voids present in the medium. This work deals with an analysis of voids and diffusion in liquid ammonia. The analysis of voids is done by the construction of Voronoi polyhedra and Delaunay tessellation. We have performed a series of molecular dynamics simulations of monovalent cations and anions of varying size in liquid ammonia at two different temperatures of 210 and 240 K to investigate the effects of ion size on the diffusion of ions and roles of voids in determining the observed diffusion behavior. It is found that with the increase of ion size, the diffusion coefficients first increase and then pass through a maximum similar to the behavior observed earlier for diffusion in water. The observed results are explained in terms of passage through voids and necks that are present in liquid ammonia.

Electric-double-layer potential distribution in multiple-layer immiscible electrolytes: Effect of finite ion sizes

In a recent study [S. Das and S. Hardt, Phys. Rev. E 84, 022502 (2011).], we provided analytical results for the electric-double-layer (EDL) electrostatic potential distribution in a system of immiscible electrolyte layers confined between plates with gap dimensions comparable to the EDL thickness. We demonstrated that the intrinsic jumps in the ion-solvent interactions across the interface of the immiscible electrolytes lead to nontrivial electrostatic potential distributions that may completely defy or substantially augment the effect of the boundary wall potential. In this Brief Report, I extend this calculation to obtain analytical and numerical results for the case with finite ion sizes (or a finite ionic steric effect). It is found that the finite steric effect substantially enhances the contributions of jump in the ion-solvent interactions in the overall electrostatic potential distribution. More importantly, I demonstrate that such jumps in the ion-solvent interactions, owing to the immiscibility of the electrolytes, ensure that even for very weak wall zeta potentials, the steric effect can significantly affect the electrostatic potential distribution.

TmDOTA-tetraglycinate encapsulated liposomes as pH-sensitive LipoCEST agents.

Lanthanide DOTA-tetraglycinate (LnDOTA-(gly)4 −) complexes contain four magnetically equivalent amide protons that exchange with protons of bulk water. The rate of this base catalyzed exchange process has been measured using chemical exchange saturation transfer (CEST) NMR techniques as a function of solution pH for various paramagnetic LnDOTA-(gly)4 − complexes to evaluate the effects of lanthanide ion size on this process. Complexes with Tb(III), Dy(III), Tm(III) and Yb(III) were chosen because these ions induce large hyperfine shifts in all ligand protons, including the exchanging amide protons. The magnitude of the amide proton CEST exchange signal differed for the four paramagnetic complexes in order, Yb>Tm>Tb>Dy. Although the Dy(III) complex showed the largest hyperfine shift as expected, the combination of favorable chemical shift and amide proton CEST linewidth in the Tm(III) complex was deemed most favorable for future in vivo applications where tissue magnetization effects can interfere. TmDOTA-(gly)4 − at various concentrations was encapsulated in the core interior of liposomes to yield lipoCEST particles for molecular imaging. The resulting nanoparticles showed less than 1% leakage of the agent from the interior over a range of temperatures and pH. The pH versus amide proton CEST curves differed for the free versus encapsulated agents over the acidic pH regions, consistent with a lower proton permeability across the liposomal bilayer for the encapsulated agent. Nevertheless, the resulting lipoCEST nanoparticles amplify the CEST sensitivity by a factor of ∼104 compared to the free, un-encapsulated agent. Such pH sensitive nano-probes could prove useful for pH mapping of liposomes targeted to tumors.

Poisson–Boltzmann Description of the Electrical Double Layer Including Ion Size Effects

The electrical double layer is examined using a generalized Poisson-Boltzmann equation that takes into account the finite ion size by modeling the aqueous electrolyte solution as a suspension of polarizable insulating spheres in water. We find that this model greatly amplifies the steric effects predicted by the usual modified Poisson-Boltzmann equation, which imposes only a restriction on the ability of ions to approach one another. This amplification should allow for an interpretation of the experimental results using reasonable effective ionic radii (close to their well-known hydrated values).

Potential of mean force between identical charged nanoparticles immersed in a size-asymmetric monovalent electrolyte

In a previous theoretical and simulation study [G. I. Guerrero-García, E. González-Tovar, and M. Olvera de la Cruz, Soft Matter 6, 2056 (2010)], it has been shown that an asymmetric charge neutralization and electrostatic screening depending on the charge polarity of a single nanoparticle occurs in the presence of a size-asymmetric monovalent electrolyte. This effect should also impact the effective potential between two macroions suspended in such a solution. Thus, in this work we study the mean force and the potential of mean force between two identical charged nanoparticles immersed in a size-asymmetric monovalent electrolyte, showing that these results go beyond the standard description provided by the well-known Derjaguin-Landau-Verwey-Overbeek theory. To include consistently the ion-size effects, molecular dynamics (MD) simulations and liquid theory calculations are performed at the McMillan-Mayer level of description in which the solvent is taken into account implicitly as a background continuum with the suitable dielectric constant. Long-range electrostatic interactions are handled properly in the simulations via the well established Ewald sums method and the pre-averaged Ewald sums approach, originally proposed for homogeneous ionic fluids. An asymmetric behavior with respect to the colloidal charge polarity is found for the effective interactions between two identical nanoparticles. In particular, short-range attractions are observed between two equally charged nanoparticles, even though our model does not include specific interactions; these attractions are greatly enhanced for anionic nanoparticles immersed in standard electrolytes where cations are smaller than anions. Practical implications of some of the presented results are also briefly discussed. A good accord between the standard Ewald method and the pre-averaged Ewald approach is attained, despite the fact that the ionic system studied here is certainly inhomogeneous. In general, good agreement between the liquid theory approach and MD simulations is also found.