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Channel-based reaction-diffusion equations are solved analytically for two electron transfer (ET) models, where the fast inner-sphere coordinate leads to an ET reaction treated by Fermi's golden rule, and the slow solvent coordinate moves via diffusion. The analytic solution has let us derive an ET rate constant that modifies the Marcus-Jortner formula by adding a constant alpha which we call a dynamic correction factor. The dynamic correction factor measures the effect of solvent friction. When the relaxation of solvent dynamics is fast, the dynamic correction can be neglected and the ET rate constant reduces to the traditional Marcus-Jortner formula. If the solvent dynamic relaxation is slow, the dynamic correction can be very large and the ET rate can be reduced by orders of magnitude. Using a generalized Zusman-Sumi-Marcus model as a starting point, we introduce two variants, GZSM-A and GZSM-B, where in model A, only one quantum mode is considered for inner-sphere motion and in model B, a classical mode for inner-sphere motion is added. By comparing the two models with experimental data, it is shown that model B is better than model A. For the solvents that have a relaxation time ranging between 0 and 5 ps, our result agrees fairly well with experimental data; for the solvents that have a relaxation time ranging between 5 and 40 ps, our result deviates from the experimental values. After introducing an adjustable scaling index in the effective time correlation function of the reaction coordinate, good agreement is achieved between the experiment and the theory for model B for all of the solvents studied in this paper.

We report a number of new exact expressions, within the realm of classical statistical mechanics, for the solvation contribution ΔA to the free energy change of a solute that undergoes a transformation between two states that differ in the way they interact with the solvent. Our basic result is a coupling-parameter expression that is equivalent to the usual cumulant expansion of ΔA. In our formula, which is derived from a linear coupling-parameter scheme, all the cumulant terms beyond the first one are collected into an integral over the coupling parameter. From this formula, with the help of the mean-value theorems of calculus, we derive exact one- and two-reference-point mean-energy expressions for the solvation free energy change, neither of which is limited to the linear response regime. Both the one- and the two-reference point formulae are given in terms of one distinguished value of the coupling parameter ( and ξ*, respectively) that is not known a priori (except in the linear response regime where...

We study the effects of cooperative forces on protein folding kinetics using statistical energy landscape theory. We find that the mean first passage time for folding changes the shape from V to U-like curve dependence on the temperature when the effects of the cooperative forces are important, which is consistent with kinetic experiments. This supports the necessity for cooperative force. In addition, we find that cooperative forces tend to increase the free energy barrier so the kinetics is in general slower. Furthermore, we find that cooperative forces tend to suppress the fluctuations in kinetics more than non-cooperative forces. The cooperativity causes to the protein folding kinetics to be more two state-like.

Potential distribution and coupling parameter theories are combined to interrelate previous solvation thermodynamic results and derive several new expressions for the solvent reorganization energy at both constant volume and constant pressure. We further demonstrate that the usual decomposition of the chemical potential into noncompensating energetic and entropic contributions may be extended to obtain a Gaussian fluctuation approximation for the chemical potential plus an exact cumulant expansion for the remainder. These exact expressions are further related to approximate first-order thermodynamic perturbation theory predictions and used to obtain a coupling-parameter integral expression for the sum of all higher-order terms in the perturbation series. The results are compared with the experimental global solvation thermodynamic functions for xenon dissolved in n-hexane and water (under ambient conditions). These comparisons imply that the constant-volume solvent reorganization energy has a magnitude of at most approximately kT in both experimental solutions. The results are used to extract numerical values of the solute-solvent mean interaction energy and associated fluctuation entropy directly from experimental solvation thermodynamic measurements.

A new theory of the thermodynamics and structure of polymer-chain fluids is developed. It is based on relating, in a novel way, polymer-chain structure with random-walk structure in the context of Ornstein−Zernike equations. It incorporates much of the interaction-site description of correlation used in reference-system interaction site model (RISM) theory but introduces several simplifying approximations in connecting the RISM equation to random-walk theory that departs in a significant way from the extension of RISM used in polymer RISM (PRISM) theory. Both continuum-space and lattice-gas versions of the theory are summarized, and the extension of the theory to a self-consistent Ornstein−Zernike approach (SCOZA) for polymers is given.

A simple version of the self-consistent Ornstein–Zernike approach (SCOZA) is investigated quantitatively. The simplification consists of retaining thermodynamic self-consistency between the compressibility and energy routes to thermodynamics from the pair distribution function g(r). However, we do not require that g(r) be identically zero within the hard repulsive core of the models we consider here. The resulting theory lends itself to analysis of Hamiltonian models to which the earlier SCOZA formulation cannot be readily applied as well as providing a convenient means of analyzing simpler models. We choose the hard-core Yukawa fluid as an illustrative example to calibrate the accuracy of the new version. The results are compared with our earlier SCOZA and simulation results, and a suggestion is given to systematically improve the accuracy of the theory in the applications that warrant so doing.

A new variant of fluid thermodynamic perturbation theory is constructed by building on approximations developed by Andersen, Chandler, and Weeks (WCA), directly using a hard-sphere (HS) reference system rather than indirectly via a soft-repulsive reference system. The resulting HS-WCA theory is more forgiving with respect to the choice of a reference HS diameter than the original WCA theory. For example, a simple analytical Boltzmann factor criteria, which relates the reference HS diameter to the distance at which the repulsive pair-potential is equal to kT, is shown to produce global Lennard-Jones fluid thermodynamic predictions that are essentially identical to those of the original WCA theory (or its self-consistent refinement by Lado).

The thermodynamic properties of fluids with very soft repulsive-core potentials, resembling those of some liquid metals, are predicted with unprecedented accuracy using a new first-order thermodynamic perturbation theory. This theory is an extension of Mansoori-Canfield/Rasaiah-Stell (MCRS) perturbation theory, obtained by including a configuration integral correction recently identified by Mon, who evaluated it by computer simulation. In this work we derive an analytic expression for Mon's correction in terms of the radial distribution function of the soft-core fluid, g(0)(r), approximated using Lado's self-consistent extension of Weeks-Chandler-Andersen (WCA) theory. Comparisons with WCA and MCRS predictions show that our new extended-MCRS theory outperforms other first-order theories when applied to fluids with very soft inverse-power potentials (nor =6), and predicts free energies that are within 0.3 kT of simulation results up to the fluid freezing point.

We appreciably broaden the scope of existing thermodynamic perturbation theory for pure fluids in several ways. Analytical approximation strategies are combined to express the thermodynamic properties of fluids in analytical and/or algebraic form and applied to a generalized class of pair potentials. The results are illustrated by comparing previously reported simulation measurements with our predictions obtained using a new class of generalized Lennard-Jones potential functions with independently adjustable repulsive and attractive length scales. Results are also obtained for various purely repulsive fluids, including inverse-power potentials with exponents ranging from 12 to 72 and a Lennard-Jones repulsive reference fluid. Applications to argon (Lennard-Jones) and C60 (Girifalco potential) are used to probe the limits of applicability of first-order perturbation theory to systems with very short range attractive potentials. Furthermore, we discuss the role of these new results in a number of additional...

The product±reactant Ornstein±Zernike approach, represented by the polymer mean-spherical approximation (PMSA), is utilized to describe the structure and thermodynamic properties of theuid of Yukawa hard sphere chain molecules. An analytical solution of the PMSA for the most general case of the multicomponent freely jointed hard sphere multi-Yukawa chainuid is presented. As in the case of the regular MSA for the hard sphere Yukawauid, the problem is reduced to the solution of a set of nonlinear algebraic equations in the general case, and to a single equation in the case of the factorizable Yukawa potential coeA cients. Closed form analytical expressions are presented for the contact values of the monomer±monomer radial distribution function, structure factors, internal energy, Helmholtz free energy, chemical potentials and pressure in terms of the quantities, which follows directly from the PMSA solution. By way of illustration, several diA erent versions of the hard sphere Yukawa chain model are considered, represented by one-Yukawa chains of length m, where m ˆ 2;4;8;16. To validate the accuracy of the present theory, Monte Carlo simulations were carried out and the results are compared systematically with the theoretical results for the structure and thermodynamic properties of the system at hand. In general it is found that the theory per- forms very well, thus providing an analytical route to the equilibrium properties of a well de®ned model for chainuids.

We report three new closures to the reference interaction site method (RISM) integral equation. The formulation of the closures is motivated by an analogy with an alternative form of the Ornstein−Zernike hypernetted-chain closure for a simple ionic mixture. Unlike other RISM integral equation closures proposed to date, the closures reported here predict nontrivial results for the dielectric constant e0 of model polar fluids. Interestingly, we find that, for models of polar fluids comprising molecules with only two nonequivalent interaction sites (such as dipolar dumbbells or the popular three-site representations of water), all of the new closures give simple analytical formulas for e0. These simple analytical results indicate that only one of the closures is based on a relation that can account for the dependence of e0 upon the thermodynamic state parameters of the liquid and the features of the molecular model (size and shape) separately, rather than in the ubiquitous dipolar strength parameter combinat...

We study a model for an argon-like fluid parameterised in terms of a hard-core repulsion and a two-Yukawa potential. The liquid-gas phase behaviour of the model is obtained from the thermodynamically self-consistent Ornstein-Zernike approximation (SCOZA) of Hoye and Stell, the solution of which lends itself particularly well to a pair potential of this form. The predictions for the critical point and the coexistence curve are compared to new high resolution simulation data and to other liquid-state theories, including the hierarchical reference theory (HRT) of Parola and Reatto. Both SCOZA and HRT deliver results that are considerably more accurate than standard integral-equation approaches. Among the versions of SCOZA considered, the one yielding the best agreement with simulation successfully predicts the critical point parameters to within 1%., Comment: 10 pages 6 figures

We continue our long-term study of the application of the product-reactant Ornstein-Zernike approach (PROZA) to polymere-fluid models. The PROZA is a theory that yields monomer-monomer pair correlation functions, from which the thermodynamics of a model fluid of polymer molecules can be obtained. Here we extend our earlier work on hard-sphere chains by adding a van der Waals attraction, modelled by a Yukawa term, to the hard-sphere interaction between monomers.

A restricted primitive model (RPM) for ionic systems in which the Coulomb and hard-core interactions are supplemented with short-range (SR) interactions between all the components, including solvent particles, is introduced and studied within a mean-field approximation. Continuum-space as well as simple-cubic lattice systems are considered. A continuous and a first-order phase transition, separated by a tricritical point (tcp), are found between uniform and charge-ordered phases in all the systems considered. The position of the tcp as well as the slope of the line of the continuous transition depend on both the model and the SR interactions. For weak or vanishing SR interactions, at temperatures lower than the transition temperature, two oppositely charged sublattices are found on the simple-cubic lattice, whereas in the continuum case a lamellar structure consisting of charged layers of alternating sign occurs. For strong SR interactions the structure becomes incommensurate with the lattice in the lattice model. Both on the lattice and in the continuum a transition between uniform ion-poor and ion-rich phases occurs for sufficiently strong SR interactions. This critical point (scp) is not to be confused with the liquid–gas type critical point (cp) that already occurs in the continuum-space version of the RPM in the absence of SR interactions. The density at the scp is significantly higher than the density at the cp. The way this latter critical point is perturbed by the presence of SR interactions is not addressed in the analysis here. The SR interactions influence the charge ordering in such a way that the tcp can be located in the same range of densities as the stable critical point. For strong SR the tcp is located close to the scp, whereas for weak SR it is close to the cp.

We show that a rich variety of crystalline structures, and a corresponding diversity of the associated phase diagrams, result from the presence in the pair potential of a soft repulsion in addition to a hard core. We use different forms for the soft repulsion, and show that the results are sensitive to the details of the potentials (in particular, their convexity) even if the range of the soft repulsion is limited to a small fraction of the hard-core diameter. Our demonstration combines exact ground-state analysis with first-order perturbation theory at finite temperatures. The relevance of our work to certain features found in real systems is also discussed.

A Landau-Ginzburg functional of two order parameters (charge-density $\phi$ and mass-density deviation $\eta$) is developed in order to yield a field theory relevant to ionic lattice gases as well as a family of off-lattice models of ionic fluids that go beyond the restricted primitive model (RPM). In a mean-field (MF) approximation an instability of a uniform phase with respect to charge fluctuations with a wave-number $k\ne 0$ is found. This second-order transition to a charge-ordered phase terminates at a tricritical point (tcp). Beyond MF, a singularity of a mass correlation function for $k\to 0$ occurs at ion concentration lower than that of the MF tcp. An effective functional depending only on $\eta$ is constructed. For low ion concentration the usual Landau form of the simple-fluid (Ising) functional is obtained; hence in this theory the critical point is in the Ising universality class., Comment: 23 pages and 4 figures. Figure 4 has parts a and b, which are in separate figs. (i. e., the figures take 5 pages)

The analytical solution of the associative mean spherical approximation (AMSA) for a Yukawa dimerizing multicomponent hard-sphere fluid is derived. The general multi-Yukawa case is discussed. The simpler one-Yukawa case with factorizable coefficients is explicitly solved. As in the previously discussed electrolyte case the solution of the AMSA reduces to the solution of only one nonlinear algebraic equation for the scaling parameter ΓB. The analytical results for the AMSA closure is illustrated by numerical examples and computer simulation for the one-component one-Yukawa dimerizing fluid. Good agreement between theoretical and computer simulation results was found for both the thermodynamic properties and the structure of the system.

In this paper, we employ the product-reactant Ornstein–Zernike approach (PROZA) developed by the authors to investigate the structural and thermodynamic properties of freely-jointed hard-sphere ring fluids. Using an M×m component sticky two-point (S2P) model and specifying an appropriate association rule between various species, the associating monomers will form M rings with each ring composed of m beads in the complete-association limit. Applying the PROZA to such a Hamiltonian and considering the limit of complete association, we are able to derive analytical expressions for the average monomer–monomer radial distribution function (RDF) as well as its intermolecular and intramolecular contributions and a closed form of the compressibility pressure. To test the theory, we also perform Monte Carlo simulations for the freely-jointed hard-sphere ring model over a wide range of densities and ring sizes. Compared to the simulation results, we find that the predictions of the PROZA for the compressibility fac...

Using the analytical solution of a multidensity integral equation solved in our previous papers [J. Chem. Phys. 108, 6513, 6525 (1998)], we derive two compressibility and two virial equations of state (EOS) for freely jointed hard-sphere chain fluids on the basis of the approximations defined by the polymer Percus–Yevick (PPY) closure and of the PPY ideal-chain closure for the integral equations. We also extend a version of first-order thermodynamic perturbation theory to polymers, using a dimer fluid as the reference system, to treat mixtures of heteronuclear chain fluids and polymer solutions; the structural information of the dimer fluid is obtained from the PPY ideal-chain approximation in the complete-association limit. The attractive forces between monomers of chain molecules are treated using simple perturbation theory. We find that the compressibility EOS derived on the basis of the PPY approximation subject to the chain-connectivity condition reduces to the compressibility EOS based upon the PPY ...

We continue our series of studies in which the equations of state (EOS) are derived based on the product-reactant Ornstein–Zernike approach (PROZA) and first-order thermodynamic perturbation theory (TPT1). These include two compressibility EOS, two virial EOS, and one TPT1 EOS (TPT1-D) that uses the structural information of the dimer fluid as input. In this study, we carry out the numerical implementation for these five EOS and compare their numerical results as well as those obtained from Attard’s EOS and GF-D (generalized Flory-dimer) EOS with computer simulation results for the corresponding chain models over a wide range of densities and chain length. The comparison shows that our compressibility EOS, GF-D, and TPT1-D are in quantitative agreement with simulation results, and TPT1-D is the best among various EOS according to its average absolute deviation (AAD). On the basis of a comparison of limited data, our virial EOS appears to be superior to the predictions of Attard’s approximate virial EOS an...

We discuss several interrelated single-ion thermodynamic properties required to calculate the liquid junction potential Ψ between two solutions of the same binary electrolyte. According to a previously reported molecular theory of nonuniform electrolyte solutions in nonequilibrium, Ψ is determined by the transport numbers of the ions, and by the second moment coefficients Hα(2) of the charge densities around the ions. The latter may be viewed as the single-ion contributors to the second moment condition of Stillinger and Lovett. For a solution of a single binary electrolyte, we relate the Hα(2) (R) to the derivatives of the single-ion activity coefficients γα with respect to the ionic strength. In the light of these results, we examine, in some detail, the role played by the specific short-range interionic interactions in determining Ψ. We investigate this matter by means of integral equation calculations for realistic models of LiCl and NaCl aqueous solutions in the 0–1 mol-dm−3 range. In addition to the hypernetted-chain (HNC) relation, we perform calculations under a new integral equation closure that is a hybrid between the HNC and Percus–Yevick closures. Like the HNC approximation, the new closure satisfies the Stillinger and Lovett condition. However, for the models considered in this study, the two closures predict different dependence of the Hα(2) and of Ψ on the specific part of the interionic interactions.

A thermodynamically self-consistent Ornstein-Zernike approximation (SCOZA) is applied to a fluid of spherical particles with a pair potential given by a hard-core repulsion and a Yukawa attractive tail $w(r)=-\exp [-z(r-1)]/r$. This potential allows one to take advantage of the known analytical properties of the solution to the Ornstein-Zernike equation for the case in which the direct correlation function outside the repulsive core is given by a linear combination of two Yukawa tails and the radial distribution function $g(r)$ satisfies the exact core condition $g(r)=0$ for $r, Prepared for the John Barker festschrift issue of Molecular Physics. 22 pages Latex, 6 ps figures

A multidensity integral-equation theory for polymerization into freely jointed hard-sphere homonuclear chain fluids proposed earlier [J. Chem. Phys. 106, 1940 (1997)] is extended to the case of multicomponent heteronuclear chain polymerization. The theory is based on the analytical solution of the polymer Percus–Yevick (PPY) approximation for the totally flexible sticky two-point (S2P) model of associating fluids. The model consists of an n-component mixture of hard spheres of different sizes with species 2,…,n−1 bearing two sticky sites A and B, randomly distributed on its surface, and species 1 and n with only one B and A site per particle, respectively. Due to some specific restrictions imposed on the possibility of forming bonds between particles of various species, the present version of the S2P model represents an associating fluid that is able to polymerize into a mixture of heteronuclear chain macromolecules. The structural properties of such a model are studied in the complete-association limit a...

We continue here our series of studies in which integral-equation theory is developed and used for the monomer-monomer correlation functions in a fluid of multicomponent freely jointed hard-sphere polymers. In this study our approach is based on Wertheim’s polymer Percus–Yevick (PPY) theory supplemented by the ideal-chain approximation; it can be regarded as a simplified version of Wertheim’s four-density PPY approximation for associating fluids considered in the complete-association limit. The numerical procedure of this simplified theory is much easier than that of the original Wertheim’s four-density PPY approximation, but the degree of accuracy is reduced. The theory can also be regarded as an extension of the PPY theory for the homonuclear polymer system proposed by Chang and Sandler [J. Chem. Phys. 102, 437 (1995)]. Their work is based upon a description of a system of hard-sphere monomers that associate into a polydisperse system of chains of prescribed mean length. Our theory instead directly desc...

A charge-dipole term in the ion-solvent contribution to the density-density correlation matrix of an ionic solution is identified, and its effect on the thermodynamics of the solution near its critical point is investigated. It is pointed out that such a term will appear in the description of an ionic melt in the absence of a molecular solvent if the melt is regarded as a mixture of free ions in a solvent of ions that have ionically associated to some extent into dipolar pairs. It is concluded that the presence of such a charge-dipole term might significantly perturb the Ising-like thermodynamic behavior that one expects in a pure dipolar solvent in its critical region, but in a way that is too system-dependent to permit universal statements to be made about its quantitative effect.

The structural properties of the totally flexible sticky two-point (S2P) model for polymerization into chain molecules of fixed length are studied. The model is represented by an n-component mixture of hard spheres of the same size with species 2,…,n−1 bearing two attractive sticky sites A and B, randomly distributed on the surface. The hard spheres of species 1 and n have only one site per particle, site B for species 1 and site A for species n. Due to the specific choice for the attractive interaction, which is present only between site B of the particles of species a and site A of the particles of species a+1, this version of the S2P model represents an associating fluid that polymerizes into freely jointed tangent hard-sphere chain molecules. The correlation functions of this model are studied at all degrees of association using a recently obtained general solution of the polymer Percus–Yevick (PPY) approximation [Yu. Kalyuzhnyi and P. Cummings, J. Chem. Phys. 103, 3265 (1995)]. Comparison of the results of the present theory in the complete association limit with corresponding computer-simulation results and results of other theories is presented and discussed. The complete-association results constitute a quantitatively successful theory of the mean monomer–monomer distribution functions for n⩽16 but for n=50 these functions are no longer quantitatively accurate.

We use the methods of continuum percolation theory to develop a consistent, essentially analytic theory for the properties of the restricted primitive model (RPM) of electrolytes. Contributions to the thermodynamic properties of this system are divided into two types; those from pairs of ions in the same cluster, and those from pairs in different clusters (we call these IN and OUT contributions, respectively, for brevity). We give exact expressions for the IN contributions as weighted integrals over the ionic pair connectedness functions. We give an exact analytic solution for these functions in the generalized mean-spherical approximation. The OUT contributions are calculated by replacing the system of ionic clusters by a system of charged hard spheres having the same statistics, and using the analytic results available for the latter system. Because the method requires no input from simulations, it can be readily adapted to treat many different electrolyte systems. Our method closely models simulation data for the thermodynamic quantities of the RPM. An earlier note [J. Chem. Phys. 96, 9233 (1992)] sketched our theory and compared our results to electrolyte data. Here we present in detail the analytic basis for our method. In future papers we expect to present detailed numerical results.

The Ebeling-Grigo (EG) improvement of the mean-spherical approximation (MSA) via the Bjerrum theory of ionic association in the restricted primitive model (RPM) is reviewed and extended. We find that, in terms of reduced density and temperature, the critical point ({rho}{sub c}{sup *},T{sub c}{sup *}) of the EG approximation (EGA) is located at (0.026, 0.0789), rather than the (0.018, 0.0837) quoted by EG. The corrected result is closer to the most recent stimulation estimates of the RPM critical point. A further improvement is obtained by incorporating the activity coefficient of the associated ionic pair to yield a dipole-ion (DI) contribution. The new approximation, EGA+DI, has a critical point at (0.027, 0.0716), which is in still better agreement with the simulation estimates. A critical discussion comparing these results with recent simulation results and other recent theoretical work is given. 18 refs., 2 figs., 2 tabs.

Exact results for the thermodynamic properties of isolated freely jointed sticky trimers and tetramers are obtained by taking appropriate limits on the exact results for a single square-well chain. Two new methods for approximating the square-well potential by a sticky potential are suggested. These new methods extend greatly the range of square-well widths over which the thermodynamic properties of a single square-well chain can be estimated accurately in terms of the corresponding properties of a single sticky chain.

We find that first‐order thermodynamic perturbation theory (TPT1) which incorporates the reference monomer fluid used in the generalized Flory–AB (GF–AB) theory yields an equation of state for fused hard‐sphere (FHS) chain fluids that has accuracy comparable to the GF–AB and GF–dimer–AC theories. The new TPT1 equation of state is significantly more accurate than other extensions of the TPT1 theory to FHS chain fluids. The TPT1 is also extended to two‐dimensional fused hard‐disk chain fluids. For the fused hard‐disk dimer fluid, the extended TPT1 equation of state is found to be more accurate than the Boublik hard‐disk dimer equation of state.

An analytical solution of the polymer MSA (mean spherical approximation) for the polymerizing primitive model of electrolytes is obtained. The approximation is an extension of the associative MSA proposed recently [M. Holovko and Yu. Kalyuzhnyi, Mol. Phys. 73 (1991) 1145] to study the effects of association in ionic systems. The model is defined by adding charges to the totally flexible sticky two-point model of associating monomers introduced by Wertheim. In the limiting case of an uncharged system, our solution reduces to the solution of the Wertheim polymer PY approximation solved recently [J. Chang and S.I. Sandler, J. Chem. Phys. 102 (1995) 437].

A multiplicative interpolation scheme (MIS) for the cavity function of associating hard‐core particles is proposed and applied to the case of our shielded sticky‐point model in which the particles associate into fused hard‐sphere diatomic molecules (hard dumbbells). The results indicate that the MIS is a significant improvement over the simple interpolation scheme (SIS) of our earlier paper when the bonding length is much less than the hard‐core diameter. The MIS is also applied to the case of a weak electrolyte modelled by shielded sticky‐point particles that are charged, which associate into hard dipolar dumbbells. Using a near‐exact hard‐sphere cavity function along with the mean spherical approximation (MSA) for the electrostatic part of the activity coefficient, we find that the resulting version of the MIS is already in reasonable agreement with the experimental data for monochloroacetic acid, despite the equal‐diameter restriction of the simple version of the model that we are using.

The pairing mean spherical approximation (PMSA) developed in the previous paper of this series is applied to a binary mixture of hard spheres and ions of the restricted primitive model. The resulting equation of state is used to investigate the phase equilibrium for the binary mixture. It is found that the model exhibits type‐III phase behavior. This work serves as the basis for more realistic models.

Integral equation theory based on a recently developed multidensity formalism [Mol. Phys. 78, 1247 (1993)] is proposed to study highly asymmetric electrolyte (polyelectrolyte) solutions. The system studied consists of large and highly charged polyions and small counterions having one or two elementary charges. The potential energy of interaction between counterions and polyions is separated into two parts, a strongly attractive part responsible for the association and a nonassociative part. Due to the strong asymmetry in size we can treat each counterion as bondable to a limited number of polyions n, while each polyion can bond arbitrary number of counterions. In our cluster expansion appropriate to the problem the diagrams appearing in the activity expansion of the one‐point counterion density are classified in terms of the number of associating bonds incident upon the labeled white counterion circle. The corresponding diagrams for the one‐point polyion density are classified in the usual way. A generali...

A new method, which improves upon the mean spherical approximation (MSA), is developed by including the ionic‐pairing contribution using a recent theory of association. The association constant of the new approximation is obtained through the second ionic virial coefficient. In the simplest version of our theory, which we call the pairing MSA 1 (PMSA1), we neglect the activity coefficient of the fully associated ionic‐pairs, which are regarded as a separate dipolar species, and obtain the critical point (ρc*, Tc*) at (0.025, 0.075). In the second PMSA ( or PMSA2), we include the activity coefficient of these dipolar particles at the MSA level. The new critical point is located at (0.023, 0.073). In the third PMSA ( or PMSA3), we further include the effect of the presence of the dipolar‐particle cores. The final critical point is located at (0.0245, 0.0745). These critical points are considerably closer than the MSA result (0.014, 0.079) to the most recent Monte Carlo estimates of ρc* from 0.025 to 0.04 an...

An analytical solution of the multidensity Ornstein-Zernike equation together with the Percus-Yevick-like approximation for the model of polymerizing fluid is obtained in closed form. The solution is illustrated by its application to the sticky shielded shell model of Stell; it can be also easily utilized for a number of other simple models of polymerizing fluids.

The contribution of charge–dipole interactions to the thermodynamics and structure of ionic solutions is considered from the standpoint of two models—a quasicontinuum cavity model of charged spheres of dielectric constant e0 in a continuum of dielectric constant e as well as a molecular model consisting of charged‐sphere solute particles and dipolar‐sphere solvent particles. With suitable scaling of the interaction potentials one expects to obtain the cavity‐model description from the molecular model in the limit σd/σi→0, where σd and σi are the ion and dipolar‐particle diameters, respectively. We recover for the first time the full r−4 ‘‘cavity‐term’’ of the ion–ion cavity‐model pair potential in such a limit as well as a corresponding screened r−4 term in the solvent‐averaged direct correlation function, which is screened by the factor e−2λr and multiplied by the prefactor (1+λr)2, where λ is the ion–ion inverse screening length. Here, r is the distance between a pair of ions. We also show that if one m...

The structure and thermodynamic properties of a model of associating particles that dimerize into fused‐sphere dumbbells are investigated by MC simulation and by integral‐equation theory. The model particles, introduced by Cummings and Stell, associate as a result of shielded attractive shells. The integral equation theories are of two types. The first is an extension of Wertheim’s associative Percus–Yevick (APY) equation to the case of the shielded sticky shell model, which is the limiting case of the shielded attractive shell model that can be handled analytically. The second is the extended mean spherical approximation (EMSA) of Zhou and Stell applied to the shielded sticky shell model. In the case of partially associated systems, the EMSA requires as input the equilibrium association constant, which is obtained here using an exact relation between monomer density and a cavity correlation function, together with an equation of state due to Boublik. The structure obtained from the EMSA is in good agreem...

Using the replica method, we derive the thermodynamic relations for a fluid in equilibrium with a quenched porous matrix. In particular, the appropriate Gibbs–Duhem equation is obtained as well as the equivalence between grand canonical and canonical ensembles. The exact compressiblity and virial equations are derived. Whereas the compressibility equation remains a direct and practical way to obtain the adsorption isotherm, the virial equation involves terms which do not relate easily to the properties of the fluid/matrix system. This explains the inconsistency between previous theoretical predictions and computer simulation results.

It is well known that the numerical solution of the hypernetted chain (HNC) equations yields satisfactory results for the pair correlation function of the primitive model of electrolytes and similar models of ionic particles over a considerable range of thermodynamic states. Despite this, it has become apparent that for low densities (or low ionic concentration in electrolytes) the numerical solution breaks down for temperatures well above the expected coexistence region between gas and liquid phases. Here we study the situation by analytic means, comparing it to a similar problem for sticky hard spheres in the Percus-Yevick (PY) approximation. On the basis of our analysis we conclude that the failure of the HNC is of the same nature and is connected to the existence of two possible solutions for low densities. When the temperature is lowered these solutions will merge into one at a particular temperature, below which a real solution is no longer possible. By extending our analysis to systems like the mon...

The exact asymptotic behavior of the particle–particle direct correlation function for dissociative dipolar dumbbells is discussed. It reveals the sense in which the complete association limit is like a critical point and suggests several approximations as well as the conditions under which they can be expected to be useful. The simplest of these is an extended mean spherical approximation (EMSA) that can be solved analytically for a model liquid of symmetric dissociative dipolar dumbbells with two centers (each bearing a point charge of opposite sign) a distance L apart, when σ/3≤L≤σ/2, where σ is the diameter of the spheres that consistute the dumbbells. The analytical expressions for the Born solvation free energy of a symmetric dipolar dumbbell in a symmetric dipolar dumbbell solvent and in a dipolar hard‐sphere solvent are also obtained. Such expressions can be expected to be useful in investigating intramolecular electron‐transfer reactions. Results for σ/2≤L≤σ that have a somewhat different conceptual status are obtained as well. They suggest a new interpretation of the Percus–Yevick solution to the sticky‐sphere model considered by Baxter.

A lattice version of the microemulsion model introduced by A. Ciach, J. Ho/ye, and G. Stell [J. Phys. A 21, L111 (1989)] (CHS) is studied within a mean‐field approximation. In the absence of (orientational) surfactant–surfactant interactions, an exact integration of the amphiphiles’ orientational degrees of freedom in the CHS model yields an effective spin‐one Hamiltonian with multibody, temperature‐dependent interactions between particles, closely resembling the model introduced by M. Schick and W. H. Shih [Phys. Rev. Lett. 59, 1205 (1987)] and subsequently studied by Gompper and Schick. The phase diagram for the CHS effective Hamiltonian on a two‐dimensional lattice is calculated at a mean‐field level. Comparisons with selected results from Schick’s model are then discussed. The calculated structure functions are in qualitative agreement with experimental results, showing a structural evolution from water‐in‐oil, to bicontinuous, to oil‐in‐water microemulsions as the water‐to‐oil concentration ratio is ...

In a reformulation and extension of our earlier published work, exact equations which relate the cavity function to excess solvation free energies and equilibrium association constants are rederived by using a thermodynamic cycle. A zeroth-order approximation, derived previously by us as a simple interpolation scheme, is found to be very accurate if the associative bonding occurs on or near the surface of the repulsive core of the interaction potential. If the monomeric bonding radius is substantially less than the core radius, our zeroth-order approximation overestimates the association degree and association constant, and an alternative approach is found advantageous. In the case of the association A+B ⇌ AB, it is based on a calculation of the thermodynamics of a mixture of A, B, and AB species, with the species concentrations regarded as independent. From this, near-exact analytic results for binary hard-sphere association into RISM molecules are obtained. Solvent effects on binary hard-sphere association and ionic association are also investigated, and a theory of particle-particle correlation is given to supplement the above purely thermodynamic study. It reveals the sense in which the complete-association limit is like a critical point.

We use the methods of continuum percolation theory to develop a consistent, essentially analytic theory for the properties of the restricted primitive model (RPM) of electrolytes. Our preliminary results are in good agreement with simulation data for the free energy and osmotic coefficient of the RPM.

New theoretical results for the three-particle equilibrium probability distribution function g 3 (123) for classical hard-sphere fluids are given on the basis of the Percus—Yevick approximation applied to be the inhomogeneous pair distribution function in the presence of a third particle, g 2 ( 12 3 ). In particular, comprehensive results are given for three spheres in rolling contact and partial results are given for two spheres in contact with a third particle in arbitrary position. When compared with available simulation data, our results are found to be accurate, and the inhomogeneous Percus—Yevick approach appears to be the first quantitatively successful one for the three-particle configurations considered here.

A lattice counterpart of a bicontinuous phase is found in a three‐dimensional lattice model of ternary surfactant mixtures. The structure of this phase is determined and discussed. The region of stability of the new phase in the parameter space is found at zero temperature. Also determined are the boundary of stability of the disordered phase, the water–water structure factor in this phase, and the disorder line.

The microscopic model for microemulsions, introduced earlier by Ciach, Ho/ye, and Stell [J. Chem. Phys. 90, 1214 (1989)] is here specialized to a one‐dimensional lattice and solved exactly by the transfer matrix method. The microemulsion phase is identified by the formation of thermally distributed surfactant‐bounded domains of oil. For this phase we find scattering functions and characteristic lengths that have some of the same features found in experimental data for microemulsions. Mean‐field interactions beyond nearest‐neighbor sites are introduced in order to study the phase diagram for the nonperiodic phases we encounter.

The molecular theory of the frequency-dependent and wavevector-dependent longitudinal dielectric function ϵL(k, ω) is derived for fluids comprising interaction-site model molecules in which point charges are located on the interaction sites. We find that ϵL(k, ω) is a simple functional of a particular charge susceptibility χμϕ0 (k, ω), which in turn is related to a collective charge-charge equilibrium time correlation function. The electrostatic part FBorn (t) of the time-dependent free energy of solvation of a solute that instantaneously changes its charge state is, on the other hand, determined by a charge susceptibility χμϕ0(1) (k, kt, ω) of the solvent in the presence of the solute molecule in its initial charge state. Using an approximate relation between χμϕ0(1) and χμϕ0 we express F Born(t) in terms of χμϕ0. The resulting theory is applied to calculate the solvation time correlation function of the solute immersed in a dipolar hard sphere (DS) and in dipolar dumbbell (DD) model solvent; the mean spherical approximation and an extended mean spherical approximation are used to compute the structure of the DS and DD solvent models, respectively. With parameters chosen so that the two models have the same molecular volume and the same electric dipole moment, it is found that they have very nearly the same ϵL(k, ω) except at large wavevector, but significantly different solvation time correlation functions.

Analytical expressions for the Born solvation free energy (BSFE) of a pair of ions with arbitrary charges, unequal sizes, and fixed separation are obtained via an interaction-site approach under the mean spherical approximation (MSA). The results for ions in a dipolar dumbbell solvent, in a dipolar hard-sphere solvent, and in an ionic solution in a dielectric continuum solvent are compared. A general equation for the reorganization free energy in a fluid of nonpolarizable molecules indicates that a state-dependent force constant and a non-parabolic free-energy surface will be obtained if a nonlinear theory instead of the MSA is used. It is found that the Marcus cross relation is very accurate within the MSA and the dielectric Green function theory. Results for the Fe +2 -Fe +3 electron transfer reaction are qualitatively compared with recent simulation results. An analytical expression (under the MSA) for the average relaxation time that characterizes the response of the solvent to charge transfer between the ions is also obtained.