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

Phase Transitions and Electrochemical Properties of Ionic Liquids and Ionic Liquid-Solvent Mixtures

Recent advances in studies of ionic liquids (IL) and ionic liquid-solvent mixtures are reviewed. Selected experimental, simulation, and theoretical results for electrochemical, thermodynamical, and structural properties of IL and IL-solvent mixtures are described. Special attention is paid to phenomena that are not predicted by the classical theories of the electrical double layer or disagree strongly with these theories. We focus on structural properties, especially on distribution of ions near electrodes, on electrical double layer capacitance, on effects of confinement, including decay length of a dissjoining pressure between confinig plates, and on demixing phase transition. In particular, effects of the demixing phase transition on electrochemical properties of ionic liquid-solvent mixtures for different degrees of confinement are presented.

Vapor-liquid phase behavior of a size-asymmetric model of ionic fluids confined in a disordered matrix: The collective-variables-based approach

We develop a theory based on the method of collective variables to study the vapor-liquid equilibrium of asymmetric ionic fluids confined in a disordered porous matrix. The approach allows us to formulate the perturbation theory using an extension of the scaled particle theory for a description of a reference system presented as a two-component hard-sphere fluid confined in a hard-sphere matrix. Treating an ionic fluid as a size- and charge-asymmetric primitive model (PM) we derive an explicit expression for the relevant chemical potential of a confined ionic system which takes into account the third-order correlations between ions. Using this expression, the phase diagrams for a size-asymmetric PM are calculated for different matrix porosities as well as for different sizes of matrix and fluid particles. It is observed that general trends of the coexistence curves with the matrix porosity are similar to those of simple fluids under disordered confinement, i.e., the coexistence region gets narrower with a decrease of porosity and, simultaneously, the reduced critical temperature T_{c}^{*} and the critical density ρ_{i,c}^{*} become lower. At the same time, our results suggest that an increase in size asymmetry of oppositely charged ions considerably affects the vapor-liquid diagrams leading to a faster decrease of T_{c}^{*} and ρ_{i,c}^{*} and even to a disappearance of the phase transition, especially for the case of small matrix particles.

Self-consistent theory for systems with mesoscopic fluctuations

We have developed a theory for inhomogeneous systems that allows for the incorporation of the effects of mesoscopic fluctuations. A hierarchy of equations relating the correlation and direct correlation functions for the local excess [Formula: see text] of the volume fraction of particles ζ has been obtained, and an approximation leading to a closed set of equations for the two-point functions has been introduced for the disordered inhomogeneous phase. We have numerically solved the self-consistent equations for one-dimensional (1D) and three-dimensional (3D) models with short-range attraction and long-range repulsion. Predictions for all of the qualitative properties of the 1D model agree with the exact results, but only semi-quantitative agreement is obtained in the simplest version of the theory. The effects of fluctuations in the two 3D models considered are significantly different, despite the very similar properties of these models in the mean-field approximation. In both cases we obtain the sequence of large-small-large compressibility for increasing ζ. The very small compressibility is accompanied by the oscillatory decay of correlations with correlation lengths that are orders of magnitude larger than the size of particles. In one of the two models considered, the small compressibility becomes very small and the large compressibility becomes very large with decreasing temperature, and eventually van der Waals loops appear. Further studies are necessary in order to determine the nature of the strongly inhomogeneous phase present for intermediate volume fractions in 3D.

The line tension of two-dimensional ionic fluids

Pressure tensor components are very useful in the calculation of the tension associated with a liquid-vapor interface. In this work, we present expressions for the pressure tensor components of two-dimensional ionic fluids, modeled at the level of the primitive model. As an application, we carried out molecular dynamics simulations of liquid-vapor interfaces to calculate the line tension of the 1:1 two-dimensional ionic fluid, whose liquid-vapor coexistence curve had already been obtained in a previous work. The pressure tensor components were validated by simulating states of one phase and reproducing the scalar pressure, previously obtained from bulk simulations and reported in the literature. The effects on the line tension and the coexisting densities, originated by the choice of the Ewald parameters, the cutoff radius, and the interfacial length were also evaluated.

Gas-liquid critical parameters of asymmetric models of ionic fluids

The effects of size and charge asymmetry on the gas-liquid critical parameters of a primitive model (PM) of ionic fluids are studied within the framework of the statistical field theory based on the collective variables method. Recently, this approach has enabled us to obtain the correct trends of the both critical parameters of the equisize charge-asymmetric PM without assuming ionic association. In this paper, we focus on the general case of an asymmetric PM characterized by the two parameters: hard-sphere diameter, lambda=sigma+/sigma-, and charge, z=q+/|q-|, ratios of the two ionic species. We derive an explicit expression for the chemical potential conjugate to the order parameter which includes the effects of correlations up to the third order. Based on this expression we consider the three versions of PM: a monovalent size-asymmetric PM (lambda not equal 1, z=1) , an equisize charge-asymmetric PM (lambda=1, z not equal 1) and a size- and charge-asymmetric PM (lambda not equal 1, z=2) . Similar to simulations, our theory predicts that the critical temperature and the critical density decrease with the increase in size asymmetry. Regarding the effects of charge asymmetry, we obtain the correct trend of the critical temperature with z , while the trend of the critical density obtained in this approximation is inconsistent with simulations, as well as with our previous results found in the higher-order approximation. We expect that the consideration of the higher-order correlations will lead to the correct trend of the critical density with charge asymmetry.

Phase diagrams of charged colloids from thermodynamic integration

We present full phase diagrams (including solid phases) of spherical charged colloids, using Monte Carlo sampling and thermodynamic integration of the Helmholtz free energy. Colloids and their co- and counterions are described by the primitive model for ionic systems that consists of hard-spheres with central point charges, while the solvent is taken into account solely through its dielectric constant. Two systems are considered: (i) a size-asymmetric system of oppositely charged spheres with size ratios q = 0.3 and 0.5 and (ii) a charge- and size-asymmetric system with colloid charge Q = 10 and counterions of charge -1 in the presence of monovalent added salt. In system (i), for both size ratios, the stable solid phase is equivalent to the NaCl crystal where the oppositely charged spheres take the lattice positions of Na and Cl ions. In system (ii), the phase diagram consists of gas-liquid and fluid-solid coexistence regions. We show that added salt stabilizes the fluid phase and shrinks the fluid-solid coexistence region, in agreement with experimental and theoretical results.

Disappearance of the gas-liquid phase transition for highly charged colloids

We calculate the full phase diagram of spherical charged colloidal particles using Monte Carlo free energy calculations. The system is described using the primitive model, consisting of explicit hard-sphere colloids and point counterions in a uniform dielectric continuum. We show that the gas-liquid critical point becomes metastable with respect to a gas-solid phase separation at colloid charges Q > or =20 times the counterion charge. Approximate free energy calculations with only one and four particles in the fluid and solid phases, respectively, are used to determine the critical line for highly charged colloids up to Q=2000. We propose the scaling law T*(c) approximately Q(1/2) for this critical temperature.

Field theory for size- and charge-asymmetric primitive model of ionic systems: mean-field stability analysis and pretransitional effects

The primitive model of ionic systems is investigated within a field-theoretic description for the whole range of diameter-, lambda , and charge, Z ratios of the two ionic species. Two order parameters (OP) are identified. The relation of the OP's to physically relevant quantities is nontrivial. Each OP is a linear combination of the charge density and the number-density waves. Instabilities of the disordered phase associated with the two OP's are determined in the mean-field approximation (MF). In MF a gas-liquid separation occurs for any Z and lambda is not equal to 1 . In addition, an instability with respect to various types of periodic ordering of the two kinds of ions is found. Depending on lambda and Z , one or the other transition is metastable in different thermodynamic states. For sufficiently large size disparity we find a sequence of fluid-crystal-fluid transitions for the increasing volume fraction of ions, in agreement with experimental observations. The instabilities found in MF represent weak ordering of the most probable instantaneous states, and are identified with structural loci associated with pretransitional effects.

Melting line of charged colloids from primitive model simulations

We develop an efficient simulation method to study suspensions of charged spherical colloids using the primitive model. In this model, the colloids and the co- and counterions are represented by charged hard spheres, whereas the solvent is treated as a dielectric continuum. In order to speed up the simulations, we restrict the positions of the particles to a cubic lattice, which allows precalculation of the Coulombic interactions at the beginning of the simulation. Moreover, we use multiparticle cluster moves that make the Monte Carlo sampling more efficient. The simulations are performed in the semigrand canonical ensemble, where the chemical potential of the salt is fixed. Employing our method, we study a system consisting of colloids carrying a charge of 80 elementary charges and monovalent co- and counterions. At the colloid densities of our interest, we show that lattice effects are negligible for sufficiently fine lattices. We determine the fluid-solid melting line in a packing fraction eta-inverse screening length kappa plane and compare it with the melting line of charged colloids predicted by the Yukawa potential of the Derjaguin-Landau-Verwey-Overbeek theory. We find qualitative agreement with the Yukawa results, and we do not find any effects of many-body interactions. We discuss the difficulties involved in the mapping between the primitive model and the Yukawa model at high colloid packing fractions (eta>0.2).

Universality of ionic criticality: size- and charge-asymmetric electrolytes

Grand-canonical simulations designed to resolve critical universality classes are reported for z:1 hard-core electrolyte models with diameter ratios lambda=a+/a- less than or approximately equal 6. For z=1 Ising-type behavior prevails. Unbiased estimates of Tc(lambda) are within 1% of previous (biased) estimates but the critical densities are approximately 5% lower. Ising character is also established for the 2:1 and 3:1 equisized models, along with critical amplitudes and improved Tc estimates. For z=3, however, strong finite-size effects reduce the confidence level although classical and O (n>or=3) criticality are excluded.

Phase coexistence in a polydisperse charged hard-sphere fluid: Polymer mean spherical approximation

We have reconsidered the phase behavior of a polydisperse mixture of charged hard spheres (CHSs) introducing the concept of minimal size neutral clusters. We thus take into account ionic association effects observed in charged systems close to the phase boundary where the properties of the system are dominated by the presence of neutral clusters while the amount of free ions or charged clusters is negligible. With this concept we clearly pass beyond the simple level of the mean spherical approximation (MSA) that we have presented in our recent study of a polydisperse mixture of CHS [Yu. V. Kalyuzhnyi, G. Kahl, and P. T. Cummings, J. Chem. Phys. 120, 10133 (2004)]. Restricting ourselves to a 1:1 and possibly size-asymmetric model we treat the resulting polydisperse mixture of neutral, polar dimers within the framework of the polymer MSA, i.e., a concept that--similar as the MSA--readily can be generalized from the case of a mixture with a finite number of components to the polydisperse case: again, the model belongs to the class of truncatable free-energy models so that we can map the formally infinitely many coexistence equations onto a finite set of coupled, nonlinear equations in the generalized moments of the distribution function that characterizes the system. This allows us to determine the full phase diagram (in terms of binodals as well as cloud and shadow curves), we can study fractionation effects on the level of the distribution functions of the coexisting daughter phases, and we propose estimates on how the location of the critical point might vary in a polydisperse mixture with an increasing size asymmetry and polydispersity.

Thermodynamic properties of lattice hard-sphere models

Thermodynamic properties of several lattice hard-sphere models were obtained from grand canonical histogram- reweighting Monte Carlo simulations. Sphere centers occupy positions on a simple cubic lattice of unit spacing and exclude neighboring sites up to a distance sigma. The nearestneighbor exclusion model, sigma = radical2, was previously found to have a second-order transition. Models with integer values of sigma = 1 or 2 do not have any transitions. Models with sigma = radical3 and sigma = 3 have weak first-order fluid-solid transitions while those with sigma = 2 radical2, 2 radical3, and 3 radical2 have strong fluid-solid transitions. Pressure, chemical potential, and density are reported for all models and compared to the results for the continuum, theoretical predictions, and prior simulations when available.

Critical point of electrolyte mixtures

The critical behavior of electrolyte mixtures was studied using grand canonical Monte Carlo simulations. Mixtures consist of large multivalent macroions and small monovalent co- and counterions. The system can be viewed as a binary mixture of macroions (with their counterions) and salt (co- and counterion pair). The primitive model description was used, in which the ions are point charges with a hard core and the solvent is treated as a uniform dielectric continuum. The grand canonical simulations are based on insertions and removals of neutral molecules: macroion with its counterions or coions and a counterion. We propose a distance biasing method that enables direct grand canonical simulations up to charge asymmetry of 10:1. We calculated the critical loci that connect the salt-free state, which consists of only macroions and counterions, with the pure salt state using mixed-field finite-size scaling with no pressure mixing. The critical parameters are determined for macroion to counterion charge asymmetries of 2:1, 3:1, and 10:1. Our results suggest that binary electrolyte mixtures are type-I mixtures, where the two components mix continuously.

Monte Carlo simulations of oppositely charged macroions in solution

The structure and phase behavior of oppositely charged macroions in solution have been studied with Monte Carlo simulations using the primitive model where the macroions and small ions are described as charged hard spheres. Size and charge symmetric, size asymmetric, and charge asymmetric macroions at different electrostatic coupling strengths are considered, and the properties of the solutions have been examined using cluster size distribution functions, structure factors, and radial distribution functions. At increasing electrostatic coupling, the macroions form clusters and eventually the system displays a phase instability, in analogy to that of simple electrolyte solutions. The relation to the similar cluster formation and phase instability occurring in solutions containing oppositely charged polymers is also discussed.

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.

Theoretical study of catalytic effects in micellar solutions

The catalytic effect of charged micelles as manifested through the increased collision frequency between the counterions of an electrolyte in the presence of such micelles is explored by the Monte Carlo simulation technique and various theoretical approaches. The micelles and ions are pictured as charged hard spheres embedded in a dielectric continuum with the properties of water at 298 K with the charge on micelles varying from zero to z(m) = 50 negative elementary charges. Analytical theories such as (i) the symmetric Poisson-Boltzmann theory, (ii) the modified Poisson-Boltzmann theory, and (iii) the hypernetted-chain integral equation are applied and tested against the Monte Carlo data for micellar ions (m) with up to 50 negative charges in aqueous solution with monovalent counterions (c; z(c) = +1) and co-ions (co; z(co) = -1). The results for the counterion-counterion pair correlation function at contact, g(cc)(sigma(cc)), are calculated in a micellar concentration range from c(m) = 5 x 10(-)(6) to 0.1 mol/dm(3) with an added +1:-1 electrolyte concentration of 0.005 mol/dm(3) (for most cases), and for various model parameters. Our computations indicate that even a small concentration of a highly charged polyelectrolyte added to a +1:-1 electrolyte solution strongly increases the probability of finding two counterions in contact. This result is in agreement with experimental data. For low charge on the micelles (z(m) below -8), all the theories are in qualitative agreement with the new computer simulations. For highly charged micelles, the theories either fail to converge (the hypernetted-chain theory) or, alternatively, yield poor agreement with computer data (the symmetric Poisson-Boltzmann and modified Poisson-Boltzmann theories). The nonlinear Poisson-Boltzmann cell model results yield reasonably good agreement with computer simulations for this system.

Phase coexistence in polydisperse charged hard-sphere fluids: Mean spherical approximation

Taking advantage of the availability of the analytic solution of the mean spherical approximation for a mixture of charged hard spheres with an arbitrary number of components we show that the polydisperse fluid mixture of charged hard spheres belongs to the class of truncatable free energy models, i.e., to those systems where the thermodynamic properties can be represented by a finite number of (generalized) moments of the distribution function that characterizes the mixture. Thus, the formally infinitely many equations that determine the parameters of the two coexisting phases can be mapped onto a system of coupled nonlinear equations in these moments. We present the formalism and demonstrate the power of this approach for two systems; we calculate the full phase diagram in terms of cloud and shadow curves as well as binodals and discuss the distribution functions of the coexisting daughter phases and their charge distributions.

Thermodynamics of electrolytes on anisotropic lattices

The phase behavior of ionic fluids on simple cubic and tetragonal (anisotropic) lattices has been studied by grand canonical Monte Carlo simulations. Systems with both the true lattice Coulombic potential and continuous-space 1/r electrostatic interactions have been investigated. At all degrees of anisotropy, only coexistence between a disordered low-density phase and an ordered high-density phase with the structure similar to ionic crystal was found, in contrast to recent theoretical predictions. Tricritical parameters were determined to be increasing functions of anisotropy parameters which is consistent with theoretical calculations based on the Debye-Hückel approach. At large anisotropies a two-dimensional-like behavior is observed, from which we estimated the dimensionless tricritical temperature and density for the two-dimensional square lattice electrolyte to be T*(tri)=0.14 and rho*(tri)=0.70.