Primitive Model Electrolyte Research Articles

Effective Interaction Potentials for Alkali and Alkaline Earth Metal Ions in SPC/E Water and Polarization Model of Hydrated Ions

In the first paper (J. Phys. Chem. B, 2006, 110, 10878), effective ion-ion potentials in SPC/E water were obtained for Me-Me, Me-Cl-, and Cl(-)-Cl- pairs, where Me is Li+, Na+, K+, Mg2+, Ca2+, Sr2+, and Ba2+ cations. In this second part of the study of effective interionic potentials, ion-ion distribution functions obtained from implicit-water Monte Carlo simulations of electrolyte solution with these potentials have been explored. This analysis verifies the range of applicability of the primitive model of electrolyte. It is shown that this approximation can be applied to monovalent electrolyte solutions in a wide range of concentrations, whereas the nature of ion-ion interactions is notably different for 2:1 electrolytes. An improved model of ions is discussed. The model includes approximations of the ion hydration shell polarization and specific short-range ion-ion interaction. It allows approximation of the potential of mean force acting on ions in strong electric fields of highly charged macromolecules and bilayers.

Screened Coulomb potential and the renormalized charges of ions and molecules inelectrolyte solutions

We consider solutions that consist of solute and solvent molecules of arbitrarysizes, shapes and internal charge distributions. The free energy of interaction(the potential of mean force) between the molecules is analysed in terms of ascreened Coulomb potential and renormalized charge distributions of the molecules.The emphasis in this work is to bring out a simple physical interpretation ofthe theory, but the treatment is based on exact statistical mechanical theorywithout any approximations. The results thereby are the true properties of thesystem for given pair interaction potentials between the constituent particles. Ingeneral, the electric potential from any molecule in the solution can be exactlyobtained for all distances by using a (generalized) screened Coulomb potential,provided the source charges constitute a renormalized charge distribution of themolecule. When charge renormalization is done consistently, macroions, small ionsand solvent molecules are treated in fundamentally the same manner and allparticles acquire renormalized charge distributions that generally are different fromtheir actual (bare) distributions. The electrostatic free energy of interaction isgiven by the interaction between the renormalized charge distributions of themolecules as mediated by the screened Coulomb potential. The exact formalism isalso used for the primitive model of electrolytes. The concepts in the generaltheory are illustrated by expressing the Poisson–Boltzmann and hypernettedchain (HNC) approximations in this alternative framework. Conditions are givenunder which the exact theory predicts the existence of attractive electrostaticinteraction between two identical particles at large distances from each other.

Singular coexistence-curve diameters: Experiments and simulations

Precise calculations of the coexistence-curve diameters of a hard-core square-well (HCSW) fluid and the restricted primitive model (RPM) electrolyte exhibit marked deviations from rectilinear behavior. The HCSW diameter displays a | t| 1− α singularity that sets in sharply for | t| ≡ | T − T c|/ T c ≲ 10 −3; this compares favorably with extensive data for SF 6, also reflected in C 2H 6, N 2, etc. By contrast, the curvature of the RPM diameter varies slowly over a wide range | t| ≲ 0.1; this behavior mirrors observations for liquid alkali metals, specifically Rb and Cs. Amplitudes for the leading singular terms can be estimated numerically but their values cannot be taken literally.

The electrical double layer for a fully asymmetric electrolyte around a spherical colloid: An integral equation study

The hypernetted chain/mean spherical approximation (HNC/MSA) integral equation for a totally asymmetric primitive model electrolyte around a spherical macroparticle is obtained and solved numerically in the case of size-asymmetric systems. The ensuing radial distribution functions show a very good agreement when compared to our Monte Carlo and molecular-dynamics simulations for spherical geometry and with respect to previous anisotropic reference HNC calculations in the planar limit. We report an analysis of the potential versus charge relationship, radial distribution functions, mean electrostatic potential, and cumulative reduced charge for representative examples of 1:1 and 2:2 salts with a size-asymmetry ratio of 2. Our results are collated with those of the modified Gouy-Chapman (MGC) and unequal radius modified Gouy-Chapman (URMGC) theories and with those of HNC/MSA in the restricted primitive model (RPM) to assess the importance of size-asymmetry effects. One of the most striking characteristics found is that, contrary to the general belief, away from the point of zero charge the properties of an asymmetric electrical double layer (EDL) are not those corresponding to a symmetric electrolyte with the size and charge of the counterion, i.e., counterions do not always dominate. This behavior suggests the existence of a new phenomenology in the EDL that genuinely belongs to a more realistic size-asymmetric model where steric correlations are taken into account consistently. Such novel features cannot be described by traditional mean-field theories such as MGC, URMGC, or even by enhanced formalisms, such as HNC/MSA, if they are based on the RPM.

Yang-Yang anomalies and coexistence diameters: Simulation of asymmetric fluids

A general method for estimating the Yang-Yang ratio, R(mu) , of a model fluid via Monte Carlo simulations is presented on the basis of data for a hard-core square-well (HCSW) fluid and the restricted primitive model (RPM) electrolyte. The isothermal minima of Q(L)(triple bond)<m(2)>2L/<m(4)>L scaling are evaluated at T(c) in an LxLxL box where m=rho-<rho>L is the density fluctuation. The "complete" finite-size scaling theory for the Q(+/-)(min) (T(c);L) incorporates pressure mixing in the scaling fields, thereby allowing for a Yang-Yang anomaly. It yields a dominant term in the asymmetry, Q(+)(min)-Q(-)(min) , varying as L(-beta/nu) with an amplitude proportional to the crucial pressure-mixing coefficient, j(2) . The reliably known critical order-parameter distribution for (d=3) Ising systems then enables one to estimate j(2) , thereby yielding R(mu) , from the Q minima together with information on the nonuniversal amplitudes for the order parameter and the susceptibility. The detailed analysis needed to estimate j(2) for an HCSW fluid and the RPM is presented. Furthermore, the Q-minima below T(c) can also provide the coexistence-curve diameters, rho(diam) (T) (triple bond)1/2 (rho(+) + rho(-)) , very close to T(c) for both models: here rho +/-(T) are the densities of the coexisting liquid and gas phases. The recently developed recursive scaling algorithm for Deltarho(infinity) (T) (triple bond)rho(+)-rho(-) is adapted to investigate the corresponding universal scaling functions. The two extremal forms of these scaling functions are computed with the aid of the exactly soluble decorated lattice-gas model. The critical densities for the RPM and HCSW fluid found via this route are consistent with previous estimates obtained from the data above T(c) ; the magnitudes of the |T- T(c)|(2beta) and |T- T(c)|(1-alpha) corrections to rho(diam)(T) are estimated.

Fluid coexistence close to criticality: scaling algorithms for precise simulation

A novel algorithm is presented that yields precise estimates of coexisting liquid and gas densities, ρ ± ( T ) , from grand canonical Monte Carlo simulations of model fluids near criticality. The algorithm utilizes data for the isothermal minima of the moment ratio Q L ( T ; 〈 ρ 〉 L ) ≡ 〈 m 2 〉 L 2 / 〈 m 4 〉 L in L × ⋯ × L boxes, where m = ρ − 〈 ρ 〉 L . When L → ∞ the minima, Q m ± ( T ; L ) , tend to zero while their locations, ρ m ± ( T ; L ) , approach ρ + ( T ) and ρ − ( T ) . Finite-size scaling relates the ratio Y = ( ρ m + − ρ m − ) / Δ ρ ∞ ( T ) universally to 1 2 ( Q m + + Q m − ) , where Δ ρ ∞ = ρ + ( T ) − ρ − ( T ) is the desired width of the coexistence curve. Utilizing the exact limiting ( L → ∞ ) form, the corresponding scaling function can be generated in recursive steps by fitting overlapping data for three or more box sizes, L 1 , L 2 , … , L n . Starting at a T 0 sufficiently far below T c and suitably choosing intervals Δ T j = T j + 1 − T j > 0 yields Δ ρ ∞ ( T j ) and precisely locates T c . The algorithm has been applied to simulation data for a hard-core square-well fluid and the restricted primitive model electrolyte for sizes up to L / a = 8 – 12 (where a is the hard-core diameter): the coexistence curves can be computed to a precision of ± 1 – 2 % of ρ c up to | T − T c | / T c = 10 −4 and 10 −3, respectively. Universality of the scaling functions and the exponent β is verified and the ( T c , ρ c ) estimates confirm previous values based on data from above T c . The algorithm extends directly to calculating the diameter, ρ diam ( T ) ≡ 1 2 ( ρ + + ρ − ) , and can lead to estimates of the Yang-Yang ratio. Furthermore, a new, explicit approximant for the basic scaling function Y permits straightforward estimates of Δ ρ ∞ ( T ) from limited Q-data when Ising-type criticality may be assumed.

Convergence of Fine-Lattice Discretization for Near-Critical Fluids

In simulating continuum model fluids that undergo phase separation and criticality, significant gains in computational efficiency may be had by confining the particles to the sites of a lattice of sufficiently fine spacing, a(0) (relative to the particle size, say a). But a cardinal question, investigated here, then arises; namely, How does the choice of the lattice discretization parameter, zeta identical with a/a(0), affect the values of interesting parameters, specifically, critical temperature and density, T(c) and rho(c)? Indeed, for small zeta ( less, similar 4-8) the underlying lattice can strongly influence the thermodynamic properties. A heuristic argument, essentially exact in d = 1 and d = 2 dimensions, indicates that, for models with hard-core potentials, both T(c)(zeta) and rho(c)(zeta) should converge to their continuum limits as 1/zeta((d)(+1)/2) for d </= 3 when zeta --> infinity; but the behavior of the error is highly erratic for d >/= 2. For smoother interaction potentials, the convergence is faster. Exact results for d = 1 models of van der Waals character confirm this; however, an optimal choice of zeta can improve the rate of convergence by a factor 1/zeta. For d >/= 2 models, the convergence of the second virial coefficients to their continuum limits likewise exhibits erratic behavior, which is seen to transfer similarly to T(c) and rho(c); but this can be used in various ways to enhance convergence and improve extrapolation to zeta = infinity as is illustrated using data for the restricted primitive model electrolyte.

Calculation of ionic surface excess concentrations in the diffuse double layer at low field strengths for a restricted primitive model electrolyte

Analytical expressions have been derived for the ionic surface excesses of a restricted primitive-model z–z electrolyte near a charged wall, by combination of a thermodynamic result with a recently derived equation for the surface potential as a function of salt concentration and surface charge density within the context of the Fawcett–Henderson method. This approximate result accounts for the finite size of ions and applies in the limit of low field strengths at the surface, corresponding to absolute surface charge densities not exceeding ca. 10 μC cm −2 and concentrations of at least 0.1 M. The theoretical expression has been tested via comparison with predictions based on Gouy–Chapman theory and data obtained with Monte Carlo simulations, and has been found to show a superior performance at high concentrations and large ion sizes in the case of 1–1 electrolytes and, to a lesser extent, 2–2 electrolytes.

Self-diffusion coefficients of ions in primitive model electrolyte solutions by smart Brownian dynamics simulation

The structural properties and transport properties of electrolyte solutions are investigated by smart Brownian dynamics simulation based on the primitive model. To test the simulation program, the radial distribution functions of ions g+−(r) and g++(r) for KCl solution are calculated and compared with the results of previous simulations. And then, the effects of ions diameter and concentration on the structural properties and transport properties of electrolyte solution at 298K are extensively simulated. The results indicate that both the radial distribution function and the self-diffusion coefficient are more sensitive to the ion size than salt concentration. As the radius ratio of cation to anion increases, self-diffusion coefficients of both cation and anion become larger.

New mean-field theories for the liquid–vapour transition of charged hard spheres

The phase behaviour of the primitive model of electrolytes is studied in the framework of the various mean-field approximations obtained recently by means of methods pertaining to statistical field theory (J.-M. Caillol, J. statist. Phys., 115, 1461 (2004); extended version: arXiv:cond-mat/0305465). The role of the regularization of the Coulomb potential at short distances is discussed in detail and the link to more traditional approximations of the theory of liquids is discussed. The values computed for the critical temperatures, chemical potentials, and densities are compared with available Monte Carlo data and other theoretical predictions.

Phase behavior and critical properties of size-asymmetric, primitive-model electrolytes.

The theory of J. Jiang et al. [J. Chem. Phys. 116, 7977 (2002)] for size-symmetric electrolytes is extended to size-asymmetric electrolytes. When compared to molecular-simulation results, this extension gives the correct trend of critical properties with size asymmetry.

Symmetric Poisson–Boltzmann structural properties near the gas–liquid point of a de-ionized colloidal suspension

The symmetrical Poisson–Boltzmann structural properties of a colloid–counterion system are analyzed in the neighbourhood of the gas–liquid critical point. The de-ionized colloidal suspension is modelled by a primitive model electrolyte having point counterions. A small counterion–counterion attraction is predicted and a shallow minimum seen in the colloid–counterion radial distribution functions.

Monte Carlo Simulations of Primitive Model (PM) Electrolytes in Non-Euclidean Geometries

Monte Carlo canonical ensemble (NVT) simulations of unrestricted primitive model electrolytes (PM) on the 3D “surface” of a 4D hypersphere are reported here. The effects of concentration, ionic charge and/or ionic radii on the mean internal energy (⟨U⟩), individual and mean ionic activities (⟨γ +⟩, ⟨γ -⟩ or ⟨γ ±⟩) and the mean net charge density distributions were studied. Our results showed that either higher charge, higher concentration or smaller ionic radii favoured ion–ion aggregation. The individual ionic activity coefficients of ions with different radii and/or charge differed significantly at higher concentrations. We also compared mean ionic activity coefficients for aqueous KCl and NaBr from simulations with reported experimental values in the literature and the agreement within 0.7–10%, depending upon concentration were found.

Structure of peptide solutions: a light scattering and numerical study.

We investigated the interactions between protein molecules in solution, in particular for low salt concentrations and thus strong electrostatic interactions where a treatment based on the second virial coefficient is not sufficient. Static and dynamic light scattering experiments on solutions containing the peptide human calcitonin (hCT) were combined with calculations based on the Ornstein-Zernike equation with the hypernetted chain (HNC) closure and computer simulations within the primitive electrolyte model. The simulation illustrates the distribution of proteins in solution and the formation of (transient) protein aggregates. It furthermore allows us to predict the physical stability of hCT solutions in dependence of ionic strength, pH and hCT concentration.

The Dot and Line Method: A Long Range Correction to Coulomb Interaction in a Cylindrical Pore

The charge line (CL) method had been used in the past to represent the periodic charges in Monte Carlo simulations of ions in a cylindrical pore. In this method, there exists a possible singularity when the edge of the image line overlaps with an ion in the central cylinder. This singularity is more problematic for molecular dynamics when the force is evaluated. Molecular dynamics simulations with the CL method have not been reported in the literature. By replacing the first section of the image charge line with an image point, we show that the CL method can be improved and be applicable in the molecular dynamics simulation of electrolytes in a cylindrical geometry. The modified method is demonstrated to be effective by simulations of a high packing primitive model electrolyte, representing the state of a molten salt.

Temperature Effects in Adsorption of a Primitive Model Electrolyte in Disordered Quenched Media: Predictions of the Replica OZ/HNC Approximation

A theoretical study of a quenched−annealed system where both components were modeled as charge symmetric +1:−1 primitive model electrolytes is presented. The adsorbed model electrolyte, mimicking lithium chloride solution, was assumed to be in thermodynamic equilibrium with an external reservoir of the same electrolyte. This partly quenched system was studied by applying the replica Ornstein−Zernike (ROZ) integral equation in the hypernetted chain (HNC) approximation and the grand canonical Monte Carlo technique. The effects of the concentration of matrix ions, pre-quenching conditions, and the electrolyte and solvent conditions on the adsorption of electrolyte were examined. The results indicate that the mean activity coefficient of the adsorbed electrolyte may differ substantially from the value of the corresponding quantity in the equilibrium bulk solution. The concentration of the annealed electrolyte in the matrix can be higher (sorption) or lower (rejection) than the corresponding equilibrium bulk c...

Molecular Thermodynamics of Size Asymmetrical Charged Hard-dumbbell Fluids

A molecular thermodynamic model of size asymmetrical charged hard dumbbell fluids is proposed, which consists of two parts: one is the reference electrolyte solution contribution and the other is the positive and negative ions association contribution. For the former we use MSA result of primitive model electrolyte and the latter is a function of the cavity correlation function of the positive and the negative ions. To test the theory, Monte Carlo simulations are performed to calculate the thermodynamic properties at high temperature. We find that if the original MSA solution is used, the variations of thermodynamic properties with the asymmetry of the ions size is qualitatively wrong. If we take the size asymmetrical ions mixtures as symmetrical ones with effective diameter sgr 12 (sgr 12=(sgr 1+sgr 2)/2 is the mean diameter of the positive and negative ions.) not only the formula is greatly simplified but also the agreement between theory and simulation is greatly improved.

Recent Advances in the Use of MSA Approach to Electrolyte and Charged Colloid Solutions

In this paper, a comprehensive review of recent advances in the use of mean spherical approximation (MSA) to the primitive electrolyte model for ionic fluids, the non-primitive electrolyte model for ion–dipole mixtures, the one-Yukawa EOS for neutral fluids, and the two-Yukawa EOS for charged colloid systems is introduced. A reasonable mathematical derivation and theoretical explanation are proposed why the dielectric constant of the vacuum cannot be used in the long-range electrostatic term in some successful EOSs for real electrolyte solutions, although the dipolar solvent is treated as explicit molecule in these theories. All the studies show that the MSA approach is of significance both in theoretical research and practical use.

Critical parameters of unrestricted primitive model electrolytes with charge asymmetries up to 10:1

The phase behavior of charge- and size-asymmetric primitive model electrolytes has been investigated using reservoir grand canonical Monte Carlo simulations. The simulations rely on the insertion and removal of neutral ion clusters from a reservoir of possible configurations. We first validated our approach by investigating the effect of Rc, the maximum allowable distance between the central cation and its associated anions, on the critical parameters of 2:1 and 3:1 electrolytes. We have shown that the effect of Rc is weak and does not change the qualitative dependence of the critical parameters on size and charge asymmetry. The critical temperature for 2:1 and 3:1 electrolytes shows a maximum at Rc≈3, while the critical volume fraction decreases more or less monotonically, consistent with previous results for 1:1 electrolytes by Romero-Enrique et al. [Phys. Rev. E 66, 041204 (2002)]. We have used the reservoir method to obtain the critical parameters for 5:1 and 10:1 electrolytes. The critical temperature decreases with increasing charge asymmetry and shows a maximum as a function of δ, the size asymmetry parameter. The critical volume fraction however, defined as the volume occupied by ions divided by the total volume of the simulation box, increases with increasing charge asymmetry and exhibits a minimum as a function of δ. This trend is contrary to what is generally predicted by theories, although more recent approaches based on the Debye–Hückel theory reproduce this observed trend. Our results deviate somewhat from the predictions of Linse [Philos. Trans. R. Soc. London, Ser. A 359, 853 (2001)] for the scaling of the critical temperature for a system of macroions with point counterions.

Ion pairing in model electrolytes: A study via three-particle correlation functions

A novel integral equations approach is applied for studying ion pairing in the restricted primitive model electrolyte, i.e., the three-point extension (TPE) to the Ornstein–Zernike integral equations. In the TPE approach, the three-particle correlation functions g[3](r1,r2,r3) are obtained. The TPE results are compared to molecular dynamics (MD) simulations and other theories. Good agreement between TPE and MD is observed for a wide range of parameters, particularly where standard integral equations theories fail, i.e., low salt concentration and high ionic valence. Our results support the formation of ion pairs and aligned ion complexes.