Peter Sloth

Monte Carlo simulations of single-ion chemical potentials. Preliminary results for the restricted primitive model

Abstract Widoms formula has been used to calculate activity coefficients for the restricted primitive model of electrolyte systems in Monte Carlo simulations based on the Metropolis sampling procedure. The agreement with literature data based on other approaches is found to be very satisfactory. The method can be used to evaluate single-ion chemical potentials.

Monte Carlo simulations of single ion chemical potentials. Results for the unrestricted primitive model

Abstract Widoms formula has been used to calculate single-ion activity coefficients for the primitive model of electrolyte systems, with unequal ionic radii, in Monte Carlo simulations based on the Metropolis sampling procedure. The results are found to be in reasonable agreement with values obtained by the MSA theory.

Monte Carlo calculations of chemical potentials in ionic fluids by application of Widom's formula: Correction for finite-system effects

Abstract A method for correcting (single) ion chemical potentials obtained by Monte Carlo calculations using Widoms test particle formula is proposed. The method is shown to remove most of the considerable system-size dependence, which is found for these quantities.

Ion exchange and membrane potentials in cellulose acetate membranes separating solutions of mixed electrolytes

The properties of (almost) symmetrical cellulose acetate membranes are studied by measuring the electromotive force (EMF) of concentration cells with the membrane as separator. One or two 1:1 salts with a common anion (Cl−) were used in different concentrations on each side of the membrane and AgAgCl were used as electrodes. The concentration(s) on one side of the membrane is fixed, whereas one concentration is varied on the other side. A theory is developed based upon ideal Nernst—Planck transport equations in the membrane, Donnan equilibrium at the membrane interfaces and Nernst equilibrium at electrodes. Furthermore, Hendersons continuous mixture hypothesis is used in the case of two different electrolytes. Comparing measurements and theory for pure salts (NaCl and KCl), the fixed charge of the membrane divided by the intrinsic binding constant for the salt can be evaluated. At low variable concentrations, however, considerable deviations from the one-electrolyte theory are observed. The reason is the competition of the H+ ion from the autoprotolysis of water with the Na+ or K+ ion. The measured values of EMF are therefore highly sensitive to pH at low variable concentrations. The variations with pH (pH between 5 and 8) are well described by the theory assuming unaltered fixed charge and a binding constant for H+ to the membrane which is around 50 times as great as the binding constant for Na+ or K+. The H+ ion is distributed almost evenly between aqueous phase and membrane phase, whereas K+ and Na+ are sterically excluded from the membrane. Comparison between theory and experiment further shows that half of the negative fixed charge is titrated at pH 4 and almost all has been titrated at pH 3. This indicates that the fixed charge in cellulose acetate membranes is caused by carboxylic acid groups (glucuronic acids) as has been reported also in earlier work. A certain asymmetry in the EMF measurements with pure NaCl was observed at high variable concentrations (up to 10 mV), when the membrane sides were changed. The asymmetry is ascribed to the evaporation asymmetry during preparation of the membrane. No asymmetry is observed at low or intermediate variable concentrations.

Experimental activity coefficients in aqueous mixed solutions of KCl and KF at 25 °C compared to Monte Carlo simulations and mean spherical approximation calculations

A system of three different ions in solution has been studied experimentally and theoretically. Mean molar ionic activity coefficients have been measured in pure and mixed, aqueous solutions of KF and KCl at 25 °C and 1 atm by means of valinomycin, LaF3 and Ag / AgCl electrodes. More than 200 independent electrometric measurements were considered. The ionic strength varied from 0.0005 to 4 mol dm–3. The activity coefficients were close to unity for pure KF than for KF in equimolar mixture with KCl at the same ionic strength for ionic strengths higher than 1 mol dm–3. The activity coefficients for KCl in pure and mixed solutions could not be statistically separated up to 4 mol dm–3. The Harned coefficients are estimated to be 0 ± 0.0025 dm3 mol–1 for KCl and 0.055 ± 0.025 dm3 mol–1 for KF. The Debye–Huckel limiting law is obeyed within 1.5% in the region from 0.0005 to 0.01 mol dm–3, indicating that the ions involved are small.Comparison with calculations for the primitive electrolyte model using the Kirkwood–Buff equations and the generalized DHX theory has shown, that the experimental data are approximately fitted using diameters of 2.9, 2.9, and 3.4A for the K+, Cl–, and F– ions, respectively. The same values fit approximately the data in the mean spherical approximation (MSA). The MSA calculations demonstrate the validity of Harneds rule. From the latter theory we also obtain single-ion activity coefficients. Monte Carlo (MC) simulations of single-ion activity coefficients have been performed for the primitive, electrolyte model for the above-mentioned ionic diameters and in addition for a diameter of the F– ion equal to 3.7A. Widoms test-particle method is used in conjunction with the extrapolation procedure suggested by Sloth and Sorensen. The MSA as well as the MC calculations support earlier suggestions, that the single-ion activity coefficient of F– in KF–KCl mixed solutions is almost independent of the salt ratio. The same is approximately true for the Cl– ion. Increasing the diameter of the F– ion from 3.4 to 3.7A does not alter this conclusion, but the separation between single-ion activity coefficients becomes larger.

Ion and potential distributions in charged and non-charged primitive spherical pores in equilibrium with primitive electrolyte solution calculated by grand canonical ensemble Monte Carlo simulation. Comparison with generalized Debye–Hückel and Donnan theory

Grand canonical ensemble Monte Carlo simulations (GCEMC) have been performed for dilute to moderately concentrated restricted primitive model electrolytes (1 : 1) in equilibrium with spherical micropores with radii from 1.5 to 10 times the ionic diameter. The pores are primitive : hard walls, with the same relative permittivity as the pore fluid and a smeared out fixed charge on the walls. The fixed charge is set to zero, one or five elementary charges. The constraining chemical potentials in the bulk solution are found by canonical Monte Carlo simulations by HNC calculations.The following topics are emphasized : 1, The need to move ions independently without regard to electroneutrality. 2, The deviation from electroneutrality in isolated small pores. 3, Electroneutrality may be artificially induced by the application of an overall ‘Donnan potential’. 4, Electroneutrality is a collective phenomenon of a ensemble of many pores. 5, Average mean activity coefficients in the pores depend slightly on the total applied electric potential for the dilute solutions corresponding to electrosorption. The dependence is even more pronounced for the average single ion activity coefficients. 6, Except in dilute solutions, the mean ionic singlet distribution functions G±(r) are unaffected by the presence of the wall charge. 7, G± near the wall is a compromise between the hard sphere ‘piling up’ at high concentrations (an ionic diameter effect) and the tendency of the ions at lower concentrations to avoid the zone, where symmetric ionic clouds cannot be formed (a Debye length effect). The latter effect points to the need for a generalisation of the simple Onsager–Samaras theory of surface adsorption and surface tension due to image charges. 8, The electric potential may be effectively found and smoothed by a Poisson equation integration of the GCEMC data for G+ to G–. 9, At low concentrations, a simple analytic generalisation of the Debye–Huckel theory explains well the observed potential distribution. 10, At higher concentrations, the potential distribution may still be well fitted by one or two eigenfunctions of the Laplace operator. 11, There is evidence of a quasi-crystalline structure induced by the wall charge in the middle of large pores at higher ionic concentrations. 12, The present simulation method may be used also for simulation of bulk properties of electrolytes without introduction of periodic boundary conditions, since bulk densities are obtained in a great volume fraction in a pore with radius only ca. 10 times the ionic diameter.We discuss some implications of the present model calculations and future generalisations for the theory of real desalination membranes with an alveolar structure of the skin layer and for the theory of ion-exchange membranes.

Hard‐sphere fluids inside spherical, hard pores. Grand canonical ensemble Monte Carlo calculations and integral equation approximations

Density profiles and partition coefficients are obtained for hard‐sphere fluids inside hard, spherical pores of different sizes by grand canonical ensemble Monte Carlo calculations. The Monte Carlo results are compared to the results obtained by application of different kinds of integral equation approximations. Also, some exact, analytical results for the partition coefficients are given, which are valid in the case of (very) small pores or at low density, respectively.

Hard, charged spheres in spherical pores. Grand canonical ensemble Monte Carlo calculations

A model consisting of hard charged spheres inside hard spherical pores is investigated by grand canonical ensemble Monte Carlo calculations. It is found that the mean ionic density profiles in the pores are almost the same when the wall of the pore is moderately charged as when it is uncharged. Also, a bulklike phase is found to be present at the center of the pores in surprisingly small systems. Finally, the Poisson–Boltzman approximation is discussed in the light of our Monte Carlo results.

Monte Carlo calculations of thermodynamic properties of the restricted, primitive model of electrolytes at extreme dilution using 32, 44, 64, 100, 216 and 512 ions and ca. 106 configurations per simulation

Monte Carlo (MC) simulations have been performed for the restricted primitive electrolyte model at a Bjerrum parameter B= 1.546 and three extremely small concentrations. The precision of the obtained thermodynamic properties (the excess energy, the heat capacity and the osmotic coefficient) is unprecedented in the literature. The objective is to study the first deviations from the Debye-Huckel limiting law (DHLL) without any assumptions. The values of the mentioned thermodynamic properties are significantly different from the values obtained from the extended Debye-Huckel expression (DH). MC values are situated between DH and DHLL values and are close to the values of the DHX theory (which uses a DH screened potential as the effective potential connected with the radial distribution functions). However, the MC values are also significantly different from the DHX values. The MC values of the radial distribution functions at contact are statistically identical to the values given by the DHX theory, but the contact values are quite uncertain because of the rare close encounters at extreme dilution. Nevertheless, the osmotic coefficients are well determined, since the contact term is vanishingly small in comparison to the third of the excess energy. The excess Helmholtz free energy has been calculated in the same Metropolis sampling by means of the Salsburg–Chesnut method. For an infinite Metropolis Markov chain, this method is shown to lead to the correct result for the excess electrostatic free energy. However, the method has not quite converged even with 106 configurations, and the electrostatic free energies are found to be ca. 5% higher than the DHLL values. The DH theory predicts up to 5% lower values. Because of the present systematic deviation, the electrostatic entropies calculated by the difference between the excess energy and the electrostatic free energy are 15–20% lower than the DHLL values.

On the calculation of single-ion activity coefficients by the Kirkwood-Buff theory

Abstract An expression which applies for single-ion activity coefficients is obtained from the theory of Kirkwood and Buff by introduction of a single plausible assumption. When this expression is used in connection with the extended mean spherical approximation for the primitive model of electrolyte solutions, the limiting law of Debye and Huckel is recovered in the limit of low ionic concentration.