Unrestricted Primitive Model Research Articles

Computing excess functions of ionic solutions: the smaller-ion shell model versus the primitive model. 2. Ion-size parameters.

A recent Monte Carlo (MC) simulation study of the primitive model (PM) of ionic solutions ( Abbas, Z. et al. J. Phys. Chem. B 2009 , 113 , 5905 ) has resulted in an extensive "mapping" of real aqueous solutions of 1-1, 2-1, and 3-1 binary electrolytes and a list of "recommended ionic radii" for many ions. For the smaller cations, the model-experiment fitting process gave much larger radii than the respective crystallographic radii, and those cations were therefore claimed to be hydrated. In Part 1 (DOI 10.1021/ct5006938 ) of the present work, the above study for the unrestricted PM - dubbed MC-UPM - has been confronted with the Smaller-ion Shell (SiS) treatment ( Fraenkel, D. Mol. Phys. 2010 , 108 , 1435 ), or "DH-SiS", by comparing the range and quality of model-experiment fits of the mean ionic activity coefficient as a function of ionic concentration. Here I compare the ion-size parameters (ISPs) of "best fit" of the two models and argue that since ISPs derived from DH-SiS are identical with (or close to) crystallographic or thermochemical ionic diameters for both cations and anions, and they do not depend on the counterion - they are more reliable, as physicochemical entities, than the PM-derived "recommended ionic radii".

Computing excess functions of ionic solutions: the smaller-ion shell model versus the primitive model. 1. Activity coefficients.

The present study compares the Monte Carlo (MC) simulation of the primitive model (PM) ( Abbas, Z. et al. J. Phys. Chem. B 2009 , 113 , 5905 ), by which activity coefficients of many binary ionic solutions have been computed through adjusting ion-size parameters (ISPs) for achieving best fit with experiment, with a parallel fit and ISP adjustment, employing the Smaller-ion Shell (SiS) treatment ( Fraenkel, D. Mol. Phys. 2010 , 108 , 1435 ), a Debye-Hückel type theory ("DH-SiS") considering counterions of unequal size. DH-SiS is analogous to the unrestricted PM (UPM), so the comparison is with the MC simulation of the UPM, "MC-UPM". Among the representative electrolytes NaCl, KCl, NaClO4, CaCl2, Ca(ClO4)2, and LaCl3, in water at 25 °C, the 1-1 electrolytes exhibit a far better fit quality for DH-SiS than for MC-UPM, and the fit extends to higher concentration. Moreover, theoretical single-ion activity coefficients derived from DH-SiS agree with experimental estimation for solutions of NaCl, CaCl2, and other electrolytes ( Fraenkel, D. J. Phys. Chem. B 2012 , 116 , 3603 ), whereas parallel MC-UPM predictions are at odds with experiment. Additional advantages of DH-SiS over MC-UPM are in (a) employing co-ion ISPs that are usually equal to the crystallographic ion diameters and (b) easily applying ISP nonadditivity in adjusting counterion ISPs.

Repulsion between oppositely charged planar macroions.

The repulsive interaction between oppositely charged macroions is investigated using Grand Canonical Monte Carlo simulations of an unrestricted primitive model, including the effect of inhomogeneous surface charge and its density, the depth of surface charge, the cation size, and the dielectric permittivity of solvent and macroions, and their contrast. The origin of the repulsion is a combination of osmotic pressure and ionic screening resulting from excess salt between the macroions. The excess charge over-reduces the electrostatic attraction between macroions and raises the entropic repulsion. The magnitude of the repulsion increases when the dielectric constant of the solvent is lowered (below that of water) and/or the surface charge density is increased, in good agreement with experiment. Smaller size of surface charge and the cation, their discreteness and mobility are other factors that enhance the repulsion and charge inversion phenomenons.

Looking deeper into the structure of mixed electric double layers near the point of zero charge

Molecular simulations have been carried out using the Metropolis Monte Carlo approach to investigate the structure of planar electric double layers containing counterion mixture within the framework of the unrestricted primitive model. The results reveal that near the point of zero charge, the rise of monovalent salt drastically elevates the collapse of ions regardless of their polarity. In particular, we fail to observe the formation of a strongly correlated liquid in the first counterion layer due to favorable entropic effects, in contrast to the early data from molecular dynamics simulations [corrected] for a spherical electric double layer [R. Messina, E. González-Tovar, M. Lozada-Cassou, and C. Holm, Europhys. Lett. 60, 383 (2002)]. Moreover, the large size of coions is found to be a pivotal factor in determining the reversal of electrophoretic mobility. On the other hand, the repulsive image charge forces thoroughly annihilate this peculiar reversal of mobility within the investigated scope of concentrations, but exert no effect on the emergence of charge reversal. These findings highlight potential applications of coion's characteristics to control gene delivery and colloidal stability as well as to design viral packing and polyelectrolyte self-assembly.

The thermodynamic properties prediction of aqueous ionic liquid ([Emim][X]) solutions by Monte Carlo simulation

Unrestricted Primitive Model (UPM) has been applied to predict various properties of room-temperature ionic liquids (RTILs) systems. Five RTILs with different characteristics are selected for this study: 1-etyl-3-methylimidazolium ([Emim][X], X=iodide, chloride, tri-fluoromethanesulfonate, trifluoroacetate, and ethyl sulfate) aqueous solutions. Monte Carlo (MC) simulations are carried out in the NVT ensemble to calculate the thermodynamic properties. Binary mixtures i.e. [Emim][Cl]+water are studied in three temperatures (298.15K, 308.15K and 318.15K) to calculate internal energy and osmotic coefficient. Moreover, activity of solvent, osmotic coefficient and activity coefficient for [Emim][Otf]+water and [Emim][Tfa]+water are reported at 298.15K. Available experimental values of osmotic coefficients for [Emim][EtSO4] are compared with simulation results. In agreement with previous experimental studies, our results support the validity of the UPM to investigate aqueous solutions of RTILs.

Insights from Monte Carlo simulations on charge inversion of planar electric double layers in mixtures of asymmetric electrolytes

Monte Carlo simulations of a planar negatively charged dielectric interface in contact with a mixture of 1:1 and 3:1 electrolytes are carried out using the unrestricted primitive model under more realistic hydrated ion sizes. Two typical surface charge densities are chosen to represent the systems from the weak to strong coupling regimes. Our goal is to determine the dependence of the degree of charge inversion on increasing concentration of both mono- and trivalent salts and to provide a systematic study on this peculiar effect between short-range and electrostatic correlations. The numerical results show that addition of monovalent salt diminishes the condensation of trivalent counterions due to either the favorable solvation energy or the available space constraints. As the concentration of trivalent salt increases, on the other hand, the inclusion of the ionic size and size asymmetry results in a damped oscillatory charge inversion at low enough surface charge and another counterintuitive surface charge amplification. It is proposed that both of the anomalous events in the weak coupling regime are thought to be entropic in origin which is completely different from the electrostatic driven charge inversion in the strong coupling regime. In addition, the electrostatic images arising from the dielectric mismatch lead to a decaying depletion effect on the structure of double layer with growing salt concentration in the case of low charged interface but have no effect at high surface charge values. The microscopic information obtained here points to the need for a more quantitative theoretical treatment in describing the charge inversion phenomenon of real colloidal systems.

Using Monte Carlo simulation to compute osmotic coefficients of aqueous solutions of ionic liquids

We perform, for the first time to our knowledge, Monte Carlo simulation to compute osmotic coefficient of ionic liquid aqueous solutions. The ionic liquids chosen are 1-ethyl-3-methylimidazolium bromide [Emim][Br], 1-methyl-3-methylimidazolium chloride [Mmim][Cl], 1-methyl-3-methylimidazolium bromide [Mmim][Br], 1-methyl-3-methylimidazolium iodide [Mmim][I] and 1-methyl-3-methylimidazolium hexafluorophosphate [Mmim][PF 6]. Simulations are carried out in the NVT ensemble at 298.15 K. The Unrestricted Primitive Model (UPM) of electrolyte is used as microscopic model in simulation process. Accuracy of simulation to predict osmotic coefficients is verified by a direct comparison of simulation results with experimental data. Computed osmotic coefficients are in good agreement with available experimental values.

Monte Carlo Simulations of Salt Solutions: Exploring the Validity of Primitive Models

An extensive series of Monte Carlo (MC) simulations were performed in order to explore the validity of simple primitive models of electrolyte solutions and in particular the effect of ion size asymmetry on the bulk thermodynamic properties of real salt solutions. Ionic activity and osmotic coefficients were calculated for 1:1, 2:1, and 3:1 electrolytes by using the unrestricted primitive model (UPM); i.e., ions are considered as charged hard spheres of different sizes dissolved in a dielectric continuum. Mean ionic activity and osmotic coefficients calculated by the MC simulations were fitted simultaneously to the experimental data by adjusting only the cation radius while keeping the anion radius fixed at its crystallographic value. Ionic radii were further optimized by systematically varying the cation and anion radii at a fixed sum of ionic radii. The success of this approach is found to be highly salt specific. For example, experimental data (mean ionic activity and osmotic coefficients) of salts which are usually considered as dissociated such as HCl, HBr, LiCl, LiBr, LiClO(4), and KOH were successfully fitted up to 1.9, 2.5, 1.9, 3, 2.5, and 4.5 M concentrations, respectively. In the case of partially dissociated salts such as NaCl, the successful fits were only obtained in a more restricted concentration range. Consistent sets of the best fitted cation radii were obtained for acids, alkali, and alkaline earth halides. A list of recommended ionic radii is also provided. The reliability of the optimized ionic radii was further tested in simulations of the osmotic coefficients of LiCl-NaCl-KCl salt mixtures. A very good agreement between the simulated and experimental data was obtained up to ionic strength of 4.5 M.

From restricted towards realistic models of salt solutions: Corrected Debye–Hückel theory and Monte Carlo simulations

The properties of bulk salt solutions over wide concentration ranges are explored by a combination of simple physical theory and Monte Carlo (MC) simulations. The corrected Debye–Hückel (CDH) theory which incorporates ion size effects in a linear response approximation is extended to yield free energy and other thermodynamic properties by integration of the chemical potential over concentration. Charging integration which is usually used to obtain an electrostatic contribution of total free energy of electrolytes is avoided in this new direct approach. MC simulations are performed with a modified Widom particle insertion method, which also provides directly the ionic activity coefficients. The validity of the CDH theory is tested by comparison with the MC simulation data for 1:1, 2:1, 2:2 and 3:1 restricted primitive model (RPM) electrolytes over a wide concentration range and at various ion sizes. Mean ionic activity and osmotic coefficients calculated by the CDH theory in RPM approximation of electrolyte are fitted to experimental data by adjusting only a mean ionic diameter. Good fits up to 1 molal (m) concentration are obtained for a large number of salt solutions. MC simulations data for unrestricted primitive model (UPM) of 1:1 and 2:1 electrolytes are also fitted to the experimental data by varying the cation radius while keeping the anion radius fixed at a crystallographic value. The success of this approach is found to be salt specific. For example good fits up to 2 and 3.5 m concentrations were obtained for LiCl and LiBr, respectively. However in the case of less dissociated salts such as NaCl and KI the experimental data could only be fitted up to one molal concentration. Possibility of extending the applicability range of the CDH theory to concentrations >2 m is explored by including a concentration dependent dielectric constant as measured in experiments. Mean ionic activity coefficients for a number of salts could successfully be fitted up to 3 m concentration by adjusting only a mean ionic diameter. Difficulties encountered in simultaneously fitting the mean ionic activity and osmotic coefficients at salt concentrations > 2 m are discussed.

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.

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.

Charge and density fluctuations in electrolytes: The Lebowitz and other correlation lengths

Puzzling features in the observed critical behavior of electrolyte solutions have stimulated the development of a generalized Debye-Hückel (GDH) theory which yields approximate but thermodynamically consistent correlation functions. Since understanding charge and density fluctuations proves crucial, we have employed Meeron's (1958) analysis supplemented by the HNC graphical resummation (1959) for the unrestricted primitive model (with distinct diameters, e.g., a ++, a +−, and a −−) to compute exact low-density expansions, with leading correction terms, for the zeroth, second, and fourth moment charge-charge and density-density correlation lengths, ξ Z,1 ( T, ϱ), ξ N,1 ( T, ϱ), etc., and for the Lebowitz length, ξ L ( T, ϱ). The latter quantifies the area law of charge fluctuations, 〈 Q Λ 2〉, in a large domain Λ. All results confirm the predictions of the GDH approximation in leading orders which, thus, currently seems the best basis for studying criticality in ionic systems.

Modeling Self-diffusion in Multicomponent Aqueous Electrolyte Systems in Wide Concentration Ranges

A comprehensive model has been developed for calculating self-diffusion coefficients in multicomponent aqueous electrolyte systems. The model combines contributions of long-range (Coulombic) and short-range interactions. The long-range interaction contribution, which manifests itself in the relaxation effect, is obtained from the dielectric continuum-based mean-spherical approximation (MSA) theory for the unrestricted primitive model. The short-range interactions are represented using the hard-sphere model. In the combined model, aqueous species are characterized by effective radii, which depend on the ionic environment. For multicomponent systems, a mixing rule has been developed on the basis of phenomenological equations of nonequilibrium thermodynamics. The effects of complexation are taken into account by combining the diffusivity model with thermodynamic speciation calculations. The model accurately reproduces self-diffusivities of ions and neutral species in aqueous solutions ranging from infinitely...

Computation of Electrical Conductivity of Multicomponent Aqueous Systems in Wide Concentration and Temperature Ranges

A comprehensive model for calculating the electrical conductivity of multicomponent aqueous systems has been developed. In the infinite-dilution limit, the temperature dependence of ionic conductivities is calculated on the basis of the concept of structure-breaking and structure-making ions. At finite concentrations, the concentration dependence of conductivity is calculated from the dielectric continuum-based mean-spherical-approximation (MSA) theory for the unrestricted primitive model. The MSA theory has been extended to concentrated solutions by using effective ionic radii. A mixing rule has been developed to predict the conductivity of multicomponent systems from those of constituent binary cation−anion subsystems. The effects of complexation are taken into account through a comprehensive speciation model coupled with a technique for predicting the limiting conductivities of complex species from those of simple ions. The model reproduces the conductivity of aqueous systems ranging from dilute to concentrated solutions (up to 30 mol/kg) at temperatures up to 573 K with an accuracy that is sufficient for modeling industrially important systems. In particular, the conductivity of multicomponent systems can be accurately predicted using data for single-solute systems.

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

Widom's 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.

Cluster theory applied to aqueous (2:2) electrolytes over a wide concentration range. Osmotic coefficients, association and redissociation

The cluster theory recently derived by Tani and Henderson has been extended to the unrestricted primitive model limiting the number of ions in the clusters to a pair. In this way it was possible to describe the osmotic coefficients of MSO4 aqueous solutions at 298 K up to 3 mol dm–3 employing a single freely adjustable parameter, the cation diameter σ+.

The structure of size-asymmetric electrolytes at charged surfaces: The unrestricted primitive model in the HNC/MSA approximation

Results are presented of ion density distribution functions, potential of zero charge and differential capacitance calculated for a primitive model electrolyte with hard core ions of unequal size new an electrode surface. The hypernetted chain/mean spherical approximation (HNC/MSA) is used. Comparing with the modified Gouy-Chapman (MGC) theory we emphasize the effects of size asymmetry.