Finite Ion Size Research Articles

Nonlinear electrophoresis of ideally polarizable particles

We focus in this paper on the nonlinear electrophoresis of ideally polarizable particles. At high applied voltages, significant ionic exchange occurs between the electric double layer, which surrounds the particle, and the bulk solution. In addition, steric effects due to the finite size of ions drastically modify the electric potential distribution in the electric double layer. In this situation, the velocity field, the electric potential, and the ionic concentration in the immediate vicinity of the particle are described by a complicated set of coupled nonlinear partial differential equations. In the general case, these equations must be solved numerically. In this study, we rely on a numerical approach to determine the electric potential, the ionic concentration, and the velocity field in the bulk solution surrounding the particle. The numerical simulations rely on a pseudo-spectral method which was used successfully by Chu and Bazant [J. Colloid Interface Sci. 315(1), 319–329 (2007)] to determine the electric potential and the ionic concentration around an ideally polarizable metallic sphere. Our numerical simulations also incorporate the steric model developed by Kilic et al. [Phys. Rev. E 75, 021502 (2007)] to account for crowding effects in the electric double layer, advective transport, and for the presence of a body force in the bulk electrolyte. The simulations demonstrate that surface conduction significantly decreases the electrophoretic mobility of polarizable particles at high zeta potential and at high applied electric field. Advective transport in the electric double layer and in the bulk solution is also shown to significantly impact surface conduction.

Measurement of bow tie profiles in CT scanners using a real-time dosimeter.

Several areas of computed tomography (CT) research require knowledge about the intensity profile of the x-ray fan beam that is introduced by a bow tie filter. This information is considered proprietary by CT manufacturers, so noninvasive measurement methods are required. One method using real-time dosimeters has been proposed in the literature. A commercially available dosimeter was used to apply that method, and analysis techniques were developed to extract fan beam profiles from measurements. A real-time ion chamber was placed near the periphery of an empty CT gantry and the dose rate versus time waveform was recorded as the x-ray source rotated about the isocenter. In contrast to previously proposed analysis methods that assumed a pointlike detector, the finite-size ion chamber received varying amounts of coverage by the collimated x-ray beam during rotation, precluding a simple relationship between the source intensity as a function of fan beam angle and measured intensity. A two-parameter model for measurement intensity was developed that included both effective collimation width and source-to-detector distance, which then was iteratively solved to minimize the error between duplicate measurements at corresponding fan beam angles, allowing determination of the fan beam profile from measured dose-rate waveforms. Measurements were performed on five different scanner systems while varying parameters such as collimation, kVp, and bow tie filters. On one system, direct measurements of the bow tie profile were collected for comparison with the real-time dosimeter technique. The data analysis method for a finite-size detector was found to produce a fan beam profile estimate with a relative error between duplicate measurement intensities of <5%. It was robust over a wide range of collimation widths (e.g., 1-40 mm), producing fan beam profiles that agreed with a relative error of 1%-5%. Comparison with a direct measurement technique on one system produced agreement with a relative error of 2%-6%. Fan beam profiles were found to differ for different filter types on a given system and between different vendors. A commercially available real-time dosimeter probe was found to be a convenient and accurate instrument for measuring fan beam profiles. An analysis method was developed that could handle a wide range of collimation widths by explicitly considering the finite width of the ion chamber. Relative errors in the profiles were found to be less than 5%. Measurements of five different clinical scanners demonstrate the variation in bow tie designs, indicating that generic bow tie models will not be adequate for CT system research.

Capacitance of graphene in aqueous electrolytes: The effects of dielectric saturation of water and finite size of ions

We present a theoretical model for electrolytically top-gated graphene, in which we analyze the effects of dielectric saturation of water due to possibly strong electric fields near the surface of a highly charged graphene, as well as the steric effects due to the finite size of salt ions in an aqueous electrolyte. By combining two well-established analytical models for those two effects, we show that the total capacitance of the solution-gated graphene is dominated by its quantum capacitance for gating potentials $\ensuremath{\lesssim}1\phantom{\rule{0.28em}{0ex}}\mathrm{V}$, which is the range of primary interest for most sensor applications of graphene. On the other hand, at the potentials $\ensuremath{\gtrsim}1\phantom{\rule{0.28em}{0ex}}\mathrm{V}$ the total capacitance is dominated by a universal capacitance of the electric double layer in the electrolyte, which exhibits a dramatic decrease of capacitance with increasing gating potential due to the interplay of a fully saturated dielectric constant of water and ion crowding near graphene.

Invited: Continuum Modeling of Interfacial and Transport Phenomena in Electrochemical Capacitors

Electrochemical capacitors are promising electrical energy storage devices bridging the gap between batteries and conventional dielectric capacitors. They can be divided in two categories: electric double layer capacitors (EDLCs) and pseudocapacitors. EDLCs store electric charge physically in the electric double layer forming at the interfaces between the electrolyte and nanoporous electrodes. During charging or discharging, ions are transported in the electrolyte under the combined influence of diffusion and electrostatic forces as well as steric effects. On the other hand, pseudocapacitors store energy not only via the electric double layer but also via fast surface redox reactions and possibly ion intercalation within the redox-active electrode.This talk will present physical modeling of interfacial phenomena, transport phenomena, and redox reactions in electrochemical capacitors under charging and discharging conditions. The model accounts for finite ion size, asymmetric electrolytes, and three-dimensional electrode morphology. It also considers faradaic reactions and ion intercalation when simulating pseudocapacitors. Typical cyclic voltammetry measurements will be reproduced numerically for various electrolyte and electrode materials and different scan rates. The results will be used to provide detailed physical interpretation of experimental measurements. Special effort will be made to systematically compare model predictions with experimental data.

Role of non-ideality for the ion transport in porous media: Derivation of the macroscopic equations using upscaling

This paper is devoted to the homogenization (or upscaling) of a system of partial differential equations describing the non-ideal transport of a N-component electrolyte in a dilute Newtonian solvent through a rigid porous medium. Realistic non-ideal effects are taken into account by an approach based on the mean spherical approximation (MSA) model which takes into account finite size ions and screening effects. We first consider equilibrium solutions in the absence of external forces. In such a case, the velocity and diffusive fluxes vanish and the equilibrium electrostatic potential is the solution of a variant of the Poisson–Boltzmann equation coupled with algebraic equations. Contrary to the ideal case, this nonlinear equation has no monotone structure. However, based on invariant region estimates for the Poisson–Boltzmann equation and for small characteristic value of the solute packing fraction, we prove existence of at least one solution. To our knowledge this existence result is new at this level of generality. When the motion is governed by a small static electric field and a small hydrodynamic force, we generalize O’Brien’s argument to deduce a linearized model. Our second main result is the rigorous homogenization of these linearized equations and the proof that the effective tensor satisfies Onsager properties, namely is symmetric positive definite. We eventually make numerical comparisons with the ideal case. Our numerical results show that the MSA model confirms qualitatively the conclusions obtained using the ideal model but there are quantitative differences arising that can be important at high charge or high concentrations.

TH‐E‐BRE‐03: A Novel Method to Account for Ion Chamber Volume Averaging Effect in a Commercial Treatment Planning System Through Convolution

Purpose:Fourier‐based deconvolution approaches used to eliminate ion chamber volume averaging effect (VAE) suffer from measurement noise. This work aims to investigate a novel method to account for ion chamber VAE through convolution in a commercial treatment planning system (TPS).Methods:Beam profiles of various field sizes and depths of an Elekta Synergy were collected with a finite size ion chamber (CC13) to derive a clinically acceptable beam model for a commercial TPS (Pinnacle3), following the vendor‐recommended modeling process. The TPS‐calculated profiles were then externally convolved with a Gaussian function representing the chamber (σ = chamber radius). The agreement between the convolved profiles and measured profiles was evaluated with a one dimensional Gamma analysis (1%/1mm) as an objective function for optimization. TPS beam model parameters for focal and extra‐focal sources were optimized and loaded back into the TPS for new calculation. This process was repeated until the objective function converged using a Simplex optimization method. Planar dose of 30 IMRT beams were calculated with both the clinical and the re‐optimized beam models and compared with MapCHEC™ measurements to evaluate the new beam model.Results:After re‐optimization, the two orthogonal source sizes for the focal source reduced from 0.20/0.16 cm to 0.01/0.01 cm, which were the minimal allowed values in Pinnacle. No significant change in the parameters for the extra‐focal source was observed. With the re‐optimized beam model, average Gamma passing rate for the 30 IMRT beams increased from 92.1% to 99.5% with a 3%/3mm criterion and from 82.6% to 97.2% with a 2%/2mm criterion.Conclusion:We proposed a novel method to account for ion chamber VAE in a commercial TPS through convolution. The reoptimized beam model, with VAE accounted for through a reliable and easy‐to‐implement convolution and optimization approach, outperforms the original beam model in standard IMRT QA verification.

Influence of the finite size and effective permittivity of ions on the equilibrium double layer around colloidal particles in aqueous electrolyte solution

The equilibrium properties of the electrical double layer surrounding a charged spherical colloidal particle immersed in an aqueous electrolyte solution are examined taking into account the finite ion size. This study includes the representation of the steric interactions among ions using both the Bikerman and the Carnahan–Starling models, an account of all the effects related to the representation of hydrated ions as dielectric spheres (dependence of the electrolyte solution permittivity, on the local ion concentration, and appearance of the Born and the dielectrophoretic forces acting on the ions), and solution of the problem for both high and low surface charge values. We find that the Carnahan–Starling model together with effective ion permittivity related effects appears to be able to provide an interpretation to the electrokinetic potential vs. surface charge dependence in the case of colloidal particles suspended in aqueous electrolyte solutions. On the contrary, for electrode–electrolyte systems, both the Bikerman and the Carnahan–Starling models might be able to explain this dependence.

Electrochemical Double Layers in Electrochemical Energy Storage and Conversion: A Multiscale Simulation Study

The nanometric liquid/solid electrochemical interface is the pivotal scale behind the operation principles of low-temperature batteries and fuel cells. It involves ionic transport by diffusion and electro-migration in the electrolyte, surface multi-step REDOX reactions, among other mechanisms. Developing an understanding of its functional role is thus crucial to help on determining the limiting factors of batteries and fuel cells performance and stability [1-3].A very significant amount of work has been carried out since more than 100 years to understand the structure of such interfaces in electrochemical systems (also known as electrochemical double layers -EDL-) but under ideal conditions (equilibrium conditions, absence of REDOX reactions): this is not representative of the non-equilibrium conditions typically found in electrochemical power generators. In this work a new non-equilibrium EDL multiscale model is presented, able to capture the following features of relevance on designing high performant and stable electrolytes in batteries and fuel cells [4]:- the impact of solvent composition, additives and lithium salt concentrations onto the charge-discharge curves in lithium ion and lithium air batteries (LIBs and LABs), and thus discriminating between thermodynamic and kinetic effects;- the impact of the platinum-based catalysts oxidation state onto the structure of the EDL, the ionomer morphology and the effectiveness of the Oxygen Reduction Reaction in Polymer Electrolyte Membrane Fuel Cells (PEMFCs).The model is supported on an extension of the so-called "Poisson-Nernst-Planck approach" and accounts for solvent orientational polarization effects, finite ion size effects and extended Transition State Theory. REDOX reactions occuring on the surface of the active particles (LIBs and LABs) and catalysts (PEMFCs) are described through an elementary kinetic approach parametrized with DFT-calculated activation energies [5]. The influence of the coverage onto the effective activation barriers is evaluated through on lattice Kinetic Monte Carlo simulations within the Variable Step Size Method (VSSM) describing surface diffusion, reactions between adspecies and adsorption/desorption.Some application cases of this model are discussed: for example, the model reproduces nicely well complex phenomena such as the irreversible Nafion reduction/oxidation peaks observed in cyclic voltammetry experiments in [6] (PEMFC), and it predicts how solvent composition impacts polarization and lithium ion transport properties (LIB/LAB). Finally, a cell device multiscale model integrating these EDL models is presented [7], and the EDL impact onto the overall cell performance dynamics (as function of the applied potentiodynamic and galvanodynamic conditions in PEMFCs, and cycling rate for batteries) is studied. Acknowledgements. Prof. Dominiquer Larcher and Dr. Abdelouahab El Kharbachi (LRCS) are gratefully acknowledged for their experiments devoted to our models validation.

Effect of finite ion sizes in an electrostatic potential distribution for a charged soft surface in contact with an electrolyte solution

We provide a theory to analyze the impact of finite ion sizes (or steric effect) in electrostatic potential distribution for a charged soft surface in contact with an electrolyte solution. The theory is based on a free energy model that appropriately accounts for the contribution of finite ion sizes as well as the structural characteristics of a soft interface, represented by a combination of a rigid surface and a fixed charge layer (FCL), with the FCL being in contact with an electrolyte solution forming an electric double layer (EDL). This FCL contains a particular kind of ion which is impermeable to the electrolyte solution, and this impermeability is quantified in terms of the corresponding Donnan potential of the "membrane" represented by the FCL-electrolyte interface. We find that consideration of the finite ion size increases the magnitude of this Donnan potential, with the extent of increase being dictated by three length scales, namely, the thickness of the FCL, the thickness of the electrolyte EDL, and the thickness of an equivalent EDL within the FCL. Such regulation of the Donnan potential strongly affects the distribution of the permeable electrolyte ions within the FCL, which in turn will have significant implications in several processes involving "soft" biological membranes.

Effects of Finite-size Ions and Relative Permittivity in a Nanopore Model of a Polymer Electrolyte Membrane

In this contribution, modified Poisson-Nernst-Planck equations are applied to study the distribution and flow of protons within cylindrical, water-filled nanopores of polymer electrolyte membranes (PEM). The model incorporates finite-size effects and an explicit expression for the relative permittivity. Due to the geometry, the problem decouples into a modified Poisson-Boltzmann equation in the radial direction and the Stokes equation in the axial direction. Ion saturation near pore walls, radial permittivity profiles, conductivity and water drag are properties of particular interest. Shortcomings and the usefulness of continuum models are elucidated and how these models might be employed to simulate macroscopic membrane properties.

A Stabilized Finite Element Method for Modified Poisson-Nernst-Planck Equations to Determine Ion Flow Through a Nanopore.

The conventional Poisson-Nernst-Planck equations do not account for the finite size of ions explicitly. This leads to solutions featuring unrealistically high ionic concentrations in the regions subject to external potentials, in particular, near highly charged surfaces. A modified form of the Poisson-Nernst-Planck equations accounts for steric effects and results in solutions with finite ion concentrations. Here, we evaluate numerical methods for solving the modified Poisson-Nernst-Planck equations by modeling electric field-driven transport of ions through a nanopore. We describe a novel, robust finite element solver that combines the applications of the Newton's method to the nonlinear Galerkin form of the equations, augmented with stabilization terms to appropriately handle the drift-diffusion processes. To make direct comparison with particle-based simulations possible, our method is specifically designed to produce solutions under periodic boundary conditions and to conserve the number of ions in the solution domain. We test our finite element solver on a set of challenging numerical experiments that include calculations of the ion distribution in a volume confined between two charged plates, calculations of the ionic current though a nanopore subject to an external electric field, and modeling the effect of a DNA molecule on the ion concentration and nanopore current.

Correlated Ions in a Calcium Channel Model: A Poisson–Fermi Theory

We derive a continuum model, called the Poisson-Fermi equation, of biological calcium channels (of cardiac muscle, for example) designed to deal with crowded systems in which ionic species and side chains nearly fill space. The model is evaluated in three dimensions. It includes steric and correlation effects and is derived from classical hard-sphere lattice models of configurational entropy of finite size ions and solvent molecules. The maximum allowable close packing (saturation) condition is satisfied by all ionic species with different sizes and valences in a channel system, as shown theoretically and numerically. Unphysical overcrowding ("divergence") predicted by the Gouy-Chapman diffuse model (produced by a Boltzmann distribution of point charges with large potentials) does not occur with the Fermi-like distribution. Using probability theory, we also provide an analytical description of the implicit dielectric model of ionic solutions that gives rise to a global and a local formula for the chemical potential. In this primitive model, ions are treated as hard spheres and solvent molecules are described as a dielectric medium. The Poisson-Fermi equation is a local formula dealing with different correlations at different places. The correlation effects are apparent in our numerical results. Our results show variations of dielectric permittivity from bath to channel pore described by a new dielectric function derived as an output from the Poisson-Fermi analysis. The results are consistent with existing theoretical and experimental results. The binding curve of Poisson-Fermi is shown to match Monte Carlo data and illustrates the anomalous mole fraction effect of calcium channels, an effective blockage of permeation of sodium ions by a tiny concentration (or number) of calcium ions.

Simulations of Cyclic Voltammetry for Electric Double Layers in Asymmetric Electrolytes: A Generalized Modified Poisson–Nernst–Planck Model

This paper presents a generalized modified Poisson–Nernst–Planck (MPNP) model derived from first principles based on excess chemical potential and Langmuir activity coefficient to simulate electric double-layer dynamics in asymmetric electrolytes. The model accounts simultaneously for (1) asymmetric electrolytes with (2) multiple ion species, (3) finite ion sizes, and (4) Stern and diffuse layers along with Ohmic potential drop in the electrode. It was used to simulate cyclic voltammetry (CV) measurements for binary asymmetric electrolytes. The results demonstrated that the current density increased significantly with decreasing ion diameter and/or increasing valency |zi| of either ion species. By contrast, the ion diffusion coefficients affected the CV curves and capacitance only at large scan rates. Dimensional analysis was also performed, and 11 dimensionless numbers were identified to govern the CV measurements of the electric double layer in binary asymmetric electrolytes between two identical planar electrodes of finite thickness. A self-similar behavior was identified for the electric double-layer integral capacitance estimated from CV measurement simulations. Two regimes were identified by comparing the half cycle period τCV and the “RC time scale” τRC corresponding to the characteristic time of ions’ electrodiffusion. For τRC ≪ τCV, quasi-equilibrium conditions prevailed and the capacitance was diffusion-independent while for τRC ≫ τCV, the capacitance was diffusion-limited. The effect of the electrode was captured by the dimensionless electrode electrical conductivity representing the ratio of characteristic times associated with charge transport in the electrolyte and that in the electrode. The model developed here will be useful for simulating and designing various practical electrochemical, colloidal, and biological systems for a wide range of applications.

A two-scale non-local model of swelling porous media incorporating ion size correlation effects

A new two-scale model is proposed for derivation of the macroscopic modified effective stress principle for swelling porous media saturated by an electrolyte solution containing finite size ions. A non-local pore-scale model is developed within the framework of Statistical Mechanics in conjunction with the thermodynamic approach based on Density Functional Theory leading to a nonlinear integral Fredholm equation of second kind for the ion/nanopore correlation function coupled with Poisson problem for the electric double layer potential. When combined with the fluid equilibrium condition such non-local electrochemical problem gives rise to a constitutive law for the fluid stress tensor in terms of the disjoining pressure which is decomposed into several components of different nature. The homogenization procedure based on formal asymptotic expansions is applied to up-scale the model to the macroscale leading to a two-scale constitutive law for the swelling pressure appearing in the modified effective stress principle with improved accuracy incorporating the deviations from the Gouy–Chapman Poisson–Boltzmann-based theory due to the finite size short-range ion–ion correlation effects. The integro-differential problem posed in a periodic cell is discretized by collocation schemes. Numerical results are obtained for a stratified arrangement of parallel macromolecules showing that the effects of ion–ion correlation forces give rise to anomalous attraction patterns between the particles for divalent ions.

Lattice-gas Poisson-Boltzmann approach for sterically asymmetric electrolytes

We derive a modification of the Poisson-Boltzmann (PB) equation brought about by the finite and different sizes of mobile ions. The equation we obtain contains explicitly the size asymmetry of the electrolyte, enabling one to apply the PB approach to the cases where this effect may be of importance. We show that the modification we propose predicts the asymmetry of the double-layer capacitance. Our approach thus provides a transparent analytical framework for a previously proposed heuristic modification of the PB equation, which also predicts the same effect. In addition, a detailed exposition of the lattice-based calculation of the entropy, accounting for steric, spatial correlations, enables us to clearly pinpoint some of the drawbacks of such an approach, which are not entirely obvious in previous attempts to include the steric effects in the PB equation.

Effect of Ion–Ion Correlations on Polyelectrolyte Gel Collapse and Reentrant Swelling

Polyelectrolyte gels are known to exhibit strong response to changes in salt concentration, an effect attributed to osmotic effects. Traditional theories are insufficient to describe all the properties observed in experiments; namely, reentrant swelling effects at high salt concentrations and complete polyelectrolyte collapse at high salt valencies. We incorporate liquid-state integral equation theories to account for the many-body correlations between finite-sized ions in a strongly charged polyelectrolyte gel. This represents a theory for such systems that goes well beyond well-known Debye–Huckel (DH) and Poisson–Boltzmann (PB) theories. We demonstrate that charge correlations give rise to enhanced gel deswelling, a first-order collapse to a highly correlated state at large valencies, and counterion size-dependent reentrant swelling. Such ion-specific effects provide routes for practical control of stimuli-responsive polymeric materials based on polyelectrolyte gels, and deeper understanding of biologic...

Influence of the dielectrophoretic force in mixed electrical double layers

The equilibrium properties of a charged plane immersed in an aqueous electrolyte solution are examined using a generalized Poisson–Boltzmann equation that takes into account the finite ion size by modeling the solution as a suspension of polarizable insulating spheres in water. This formalism is applied to a general solution composed of two or more counterion species with different valences, sizes, and effective permittivity values. It is shown that, due to the dependence of the dielectrophoretic force on the ion size and effective permittivity value, the concentration of the smaller counterion strongly increases while that of the larger one decreases in the immediate vicinity of the charged surface. As a result the surface potential value strongly increases as compared to the usual modified Poisson–Boltzmann theory that only includes steric interactions among ions. This effect is particularly important in the case of mixtures of univalent and divalent counterions, being significant even for relatively low surface charge values.

Predictions of the maximum energy extracted from salinity exchange inside porous electrodes

Capacitive energy extraction based on double layer expansion (CDLE) is the name of a new method devised for extracting energy from the exchange of fresh and salty water in porous electrodes. It is based on the change of the capacitance of electrical double layers (EDLs) at the electrode/solution interface when the concentration of the bulk electrolyte solution is modified. The use of porous electrodes provides huge amounts of surface area, but given the typically small pore size, the curvature of the interface and EDL overlap should affect the final result. This is the first aspect dealt with in this contribution: we envisage the electrode as a swarm of spherical particles, and from the knowledge of their EDL structure, we evaluate the stored charge, the differential capacitance and the extracted energy per CDLE cycle. In all cases, different pore radii and particle sizes and possible EDL overlap are taken into account. The second aspect is the consideration of finite ion size instead of the usual point-like ion model: given the size of the pores and the relatively high potentials that can be applied to the electrode, excluded volume effects can have a significant role. We find an extremely strong effect: the double layer capacitance is maximum for a certain value of the surface potential. This is a consequence of the limited ionic concentration at the particle-solution interface imposed by the finite size of ions, and leads to the presence of two potential ranges: for low electric potentials the capacitance increases with the ionic strength, while for large potentials we find the opposite trend. The consequences of these facts on the possibility of net energy extraction from porous electrodes, upon changing the solution in contact with them, are evaluated.

Numerical methods for the Poisson–Fermi equation in electrolytes

The Poisson–Fermi equation proposed by Bazant, Storey, and Kornyshev [Phys. Rev. Lett. 106 (2011) 046102] for ionic liquids is applied to and numerically studied for electrolytes and biological ion channels in three-dimensional space. This is a fourth-order nonlinear PDE that deals with both steric and correlation effects of all ions and solvent molecules involved in a model system. The Fermi distribution follows from classical lattice models of configurational entropy of finite size ions and solvent molecules and hence prevents the long and outstanding problem of unphysical divergence predicted by the Gouy–Chapman model at large potentials due to the Boltzmann distribution of point charges. The equation reduces to Poisson–Boltzmann if the correlation length vanishes. A simplified matched interface and boundary method exhibiting optimal convergence is first developed for this equation by using a gramicidin A channel model that illustrates challenging issues associated with the geometric singularities of molecular surfaces of channel proteins in realistic 3D simulations. Various numerical methods then follow to tackle a range of numerical problems concerning the fourth-order term, nonlinearity, stability, efficiency, and effectiveness. The most significant feature of the Poisson–Fermi equation, namely, its inclusion of steric and correlation effects, is demonstrated by showing good agreement with Monte Carlo simulation data for a charged wall model and an L type calcium channel model.

Energy loss straggling in collisions of fast finite-size ions with atoms

The influence of ion size on straggling of energy losses by fast partially stripped ions is studied using the nonperturbative approach based on the eikonal approximation. It is shown that such a consideration of collisions of ions with complex atoms can lead to considerable corrections in calculating root-mean-square straggling of energy losses by fast ions compared to the results obtained for point ions. The root-mean-square straggling of energy losses are calculated for bromide and iodine ions in collisions with copper, silver, and aluminum atoms. It is shown that allowance for the size of the electron “coat” of an ion noticeably improves the agreement with experimental data.