Self-consistent Integral Equation Theory Research Articles

Large effects of tiny structural changes on the cluster formation process in model colloidal fluids: an integral equation study

The formation of aggregates is commonly observed in soft matter such as globular protein solutions and colloidal suspensions. A lively debated issue concerns the possibility to discriminate between a generic intermediate-range order taking place in the fluid, as contrasted with the more specific presence of a clustered state. Recently, we have predicted by Monte Carlo simulations of a standard colloidal model — spherical particles interacting via a short-range attraction followed by a screened electrostatic repulsion at larger distances — the existence of a tiny structural change occurring in the pair structure. This change consists in a reversal of trend affecting a portion of the local density as the attractive strength increases, that is shown to take place precisely at the clustering threshold. Here, we address the same issue by refined thermodynamically self-consistent integral equation theories of the liquid state. We document how such theoretical schemes positively account for the observed phenomenology, highlighting their accuracy to finely describe the aggregation processes in model fluids with microscopic competing interactions.

On the structure, property, and phase behaviour of the symmetric Yukawa mixtures: testing of the consistent integral equation theories

ABSTRACTYukawa mixtures are important microscopic models of liquids. In order to obtain their properties, we must have accurate and consistent theories, without the issues arising from the multi-valued pressures and chemical potentials due to inconsistencies in the theory. This work tests two self-consistent integral equation theories (IETs): the modified hypernetted-chain (MHNC) theory and the zero-separation (ZSEP) based theory. These equations, by construction, automatically satisfy the pressure consistency the Gibbs-Duhem relation, as well as the zero-separation theorem. To verify, new Monte-Carlo (MC) simulations are carried out for temperatures 0.7 < T* < 1.1 and densities 0.4 < ρ* < 0.5. The Yukawa interactions uij are symmetric (u11 = u22). The unlike u12 is made weaker than the like-type by a factor α < 1. The structures, exemplified by the radial distribution functions, are accurately reproduced by both MHNC and ZSEP. The internal energies are predicted to within 1% of the MC data, the pressures to within 4%; and the chemical potentials to less than 3%. The vapour-liquid phase envelope is determined for a typical mixture. In this case the performance of both IETs is also shown to be satisfactory.

Equilibrium theory of the hard sphere fluid and glasses in the metastable regime up to jamming. I. Thermodynamics

We formulate and apply a non-replica equilibrium theory for the fluid-glass transition, glass thermodynamic properties, and jamming of hard spheres in three and all higher spatial dimensions. Numerical predictions for the zero complexity glass transition and jamming packing fractions, and a "densest" equilibrium glass, are made. The equilibrium glass equation of state is regarded as the practical continuation of its fluid analog up to jamming. The analysis provides a possible resolution to the inability of any fluid virial series re-summation based equation of state to capture jamming at a reasonable volume fraction. The numerical results are quantitatively compared with various simulation data for equilibrium hard sphere glasses in 3 to 12 dimensions. Although there are uncertainties in this comparison, the predicted zero complexity or configurational entropy and corresponding jamming packing fractions do agree well with two characteristic packing fractions deduced from the dynamic simulation data. The similarities and differences of our approach compared to the replica approach are discussed. The high dimensional scaling of the equilibrium glass transition and jamming volume fractions are also derived. The developments in this paper serve as input to Paper II [R. Jadrich and K. S. Schweizer, J. Chem. Phys. 139, 054502 (2013)] that constructs a self-consistent integral equation theory of the 3-dimensional hard sphere pair structure, in real and Fourier space, in the metastable regime up to jamming. The latter is employed as input to a microscopic dynamical theory of single particle activated barrier hopping.

Theoretical description of cluster formation in two-Yukawa competing fluids

We investigate the temperature threshold whereupon two-Yukawa systems with microscopic competing interactions transform from normal fluids to ‘cluster fluids’. Specifically, we apply two refined, thermodynamically self-consistent integral equation theories of the liquid state to study a family of two-Yukawa models having the same interaction strength, independent of the choice of potential parameters. We find, over the range of potential parameters and fluid densities investigated, an almost linear scaling between the temperature threshold for the cluster formation and the height of repulsive interactions, with higher barriers stabilizing the cluster fluid at higher temperatures. Cluster–cluster correlation lengths seem instead to be influenced by the long-range features of the interactions.

Temperature study of cluster formation in two-Yukawa fluids

An accurate thermodynamically self-consistent integral equation theory of the liquid state is used to investigate model fluids with competing attractive interaction at short distances and long-range repulsion, focusing on thermodynamic states where the formation of clusters is expected to occur. We find a remarkable accuracy of theoretical predictions, through a detailed assessment against results of Monte Carlo simulations. The behavior of theoretical radial distribution functions and structure factors faithfully follows the onset and growth of cluster aggregates in the homogeneous dense-vapor phase. The thermodynamic properties of the system sensitively depend on the ratio between the repulsive barrier and the attraction strength. We elucidate the role of accurate theoretical tools to investigate the properties of fluids with complex phase behaviors.

Structural and dynamical properties of a core-softened fluid in a supercritical region

We present a theoretical study of the structural, thermodynamic, and transport properties of a supercritical fluid comprising particles interacting via isotropic attractive core-softened potential. The shear viscosity and self-diffusion coefficient are computed on the basis of the mode-coupling theory, with required structural input obtained from the thermodynamically self-consistent integral equation theory. We also consider dilute solutes in a core-softened fluid and use the anisotropic integral equation theory to obtain the solute-solute potential of mean force, which yields the second virial coefficient. We analyze its dependence on the solvent density and solute-solvent interaction strength.

Waterlike dynamic anomalies in a liquid described by a core-softened potential

We present a theoretical study of transport properties of a liquid comprised of particles interacting via isotropic core-softened potential. Shear viscosity and self-diffusion coefficient are computed on the basis of the mode-coupling theory, with required structural input obtained from thermodynamically self-consistent integral equation theory. Both self-diffusion coefficient and viscosity display waterlike anomalous density dependence, with diffusivity increasing and viscosity decreasing with density within a particular density range along several isotherms below a certain temperature. Our theoretical results for both transport coefficients are in good agreement with the simulation data.

Thermodynamics of a long-range triangle-well fluid

The long-range triangle-well fluid has been studied using three different approaches: firstly, by an analytical equation of state obtained by a perturbation theory, secondly via a self-consistent integral equation theory, the so-called self-consistent Ornstein–Zernike approach (SCOZA) which is presently one of the most accurate liquid-state theories, and finally by Monte Carlo simulations. We present vapour–liquid phase diagrams and thermodynamic properties such as the internal energy and the pressure as a function of the density at different temperatures and for several values of the potential range. We assess the accuracy of the theoretical approaches by comparison with Monte Carlo simulations: the SCOZA method accurately predicts the thermodynamics of these systems and the first-order perturbation theory reproduces the overall thermodynamic behaviour for ranges greater than two molecular diameters except that it overestimates the critical point. The simplicity of the equation of state and the fact that it is analytical in the potential range makes it a good candidate to be used for calculating other thermodynamic properties and as an ingredient in more complex theoretical approaches.

Improvement on macroscopic compressibility approximation and beyond

A numerical procedure is proposed to extend the thermodynamic perturbation expansion (TPE) to a higher order. It is shown that the present second order term is superior to that due to a macroscopic compressibility approximation (MCA), a local compressibility approximation, and a superposition approximation by Barker and Henderson [Rev. Mod. Phys. 48, 587 (1976)]. Extensive model calculation and comparison with simulation data available in literature and supplied in the present report indicate that the present third order TPE is superior to a previous second order TPE based on the MCA, two previous perturbation theories, which are respectively based on an analytical mean spherical approximation for an Ornstein-Zernike equation, and an assumed explicit functional form for the Laplace transform of radial distribution function multiplied by radial distance, and a recent generalized van der Waals theory. The present critical temperature for a hard core attractive Yukawa fluid of varying range is in very good agreement with that due to a hierarchical reference theory. The present third order TPE is computationally far more modest than the self-consistent integral equation theory, and therefore is a viable alternative to use of the latter.

Effective intramolecular interactions in weakly charged polyelectrolytes: Relation to structural behavior of solution

A self-consistent integral equation theory in the form of a hybrid Monte Carlo/PRISM computation scheme was used to study a polyelectrolyte solution. The static conformational and structural properties of polyions of different rigidities in a good solvent were studied with explicit allowance for counterions over a wide concentration range. An analysis of the calculated effective potentials and correlation functions confirms the presence of effective attraction between units of the charged polymer in semidilute and concentrated solutions; this attraction leads to the collapse of polyions under certain conditions. It was shown that the cause of effective attraction is the dipole-dipole interaction of ion pairs. For the region of polyelectrolyte transition from the semidilute to the concentrated state of solution, the results qualitatively agree with experimental data and theoretical predictions. Visualized images of conformations in the test range of parameters are given.

Thermodynamically self-consistent integral equation theory for pair-correlation functions of molecular fluids-II

A closure for the pair-correlation functions of molecular fluids is described in which the hypernetted-chain and the Percus–Yevick approximations are “mixed” as a function of interparticle separation. An adjustable parameter α in the mixing function is used to enforce thermodynamic consistency, by which it is meant that identical results are obtained when the equations of state are calculated via the virial and compressibility routes, respectively. The mixed integral equation for the pair-correlation functions has been solved for two model fluids: (i) a fluid of the hard Gaussian overlap model, and (ii) a fluid the molecules of which interact via a modified Gay–Berne model potential. For the modified Gay–Berne fluid we have slightly modified the original Gay–Berne potential to study the effect of attraction on hard core systems. The pair-correlation functions of the isotropic phase which enter in the density-functional theory as input informations have been calculated from the integral equation theories for these model fluids. We have used two different versions of the density-functional theory known as the second order and modified weighted-density-functional theory to locate the isotropic-nematic (I–N) transitions and calculate the values of transition parameters for the hard Gaussian overlap and modified Gay–Berne model fluids. We have compared our results with those of computer simulations wherever they are available. We find that the density-functional theory is good to study the I–N transition in molecular fluids if the values of the pair-correlation functions in the isotropic phase are accurately known.

Monte Carlo simulation and self-consistent integral equation theory for polymers in quenched random media

The conformational properties and static structure of freely jointed hard-sphere chains in matrices composed of stationary hard spheres are studied using Monte Carlo simulations and integral equation theory. The simulations show that the chain size is a nonmonotonic function of the matrix density when the matrix spheres are the same size as the monomers. When the matrix spheres are of the order of the chain size the chain size decreases monotonically with increasing matrix volume fraction. The simulations are used to test the replica-symmetric polymer reference interaction site model (RSP) integral equation theory. When the simulation results for the intramolecular correlation functions are input into the theory, the agreement between theoretical predictions and simulation results for the pair-correlation functions is quantitative only at the highest fluid volume fractions and for small matrix sphere sizes. The RSP theory is also implemented in a self-consistent fashion, i.e., the intramolecular and intermolecular correlation functions are calculated self-consistently by combining a field theory with the integral equations. The theory captures qualitative trends observed in the simulations, such as the nonmonotonic dependence of the chain size on media fraction.

Integral equation theory of random copolymer melts: Self-consistent treatment of intramolecular and intermolecular correlations

A self-consistent integral equation theory is presented for the conformational properties and spinodal lines of random copolymer melts. The theory combines field-theoretic methods with the polymer reference interaction site model (PRISM) theory. The many-chain problem is replaced by a single chain where the sites interact via a bare plus a self-consistently determined medium-induced potential, and the conformational properties are obtained using a variational method. The theoretical prediction for the spinodal line is qualitatively similar to that of non-self-consistent PRISM theory. The theory predicts macroscopic phase separation for all values of the monomer correlation strength, lambda. The inverse spinodal temperature is a nonmonotonic function of lambda with a maximum at lambda(max). For large values of lambda( approximately 1), the values of spinodal temperatures are almost identical to those of non-self-consistent PRISM theory. For low values of lambda, however, the theory predicts higher values for spinodal temperatures than non-self-consistent PRISM theory. The theory predicts significant changes in the mean-square end-to-end distance as the temperature is decreased.

Replica Ornstein-Zernike self-consistent theory for mixtures in random pores.

We present a self-consistent integral equation theory for a binary liquid in equilibrium with a disordered medium, based on the formalism of the replica Ornstein-Zernike (ROZ) equations. Specifically, we derive direct formulas for the chemical potentials and the zero-separation theorems (the latter provide a connection between the chemical potentials and the fluid cavity distribution functions). Next we solve a modified-Verlet closure to ROZ equations, which has built-in parameters that can be adjusted to satisfy the zero-separation theorems. The degree of thermodynamic consistency of the theory is also kept under control. We model the binary fluid in random pores as a symmetrical binary mixture of nonadditive hard spheres in a disordered hard-sphere matrix and consider two different values of the nonadditivity parameter and of the quenched matrix packing fraction, at different mixture concentrations. We compare the theoretical structural properties as obtained through the present approach with Percus-Yevick and Martinov-Sarkisov integral equation theories, and assess both structural and thermodynamic properties by performing canonical standard and biased grand canonical Monte Carlo simulations. Our theory appears superior to the other integral equation schemes here examined and provides reliable estimates of the chemical potentials. This feature should be useful in studying the fluid phase behavior of model adsorbates in random pores in general.

Self-consistent integral equation theory for solutions of finite extensible semiflexible polyelectrolyte chains

We investigate the structural and conformational properties of solutions containing semiflexible polyelectrolyte chains using a self-consistent integral equation theory approach. A one-component system is considered where the polyelectrolyte chains interact with each other via a Debye–Hückel potential. Nonelectrostatic interactions among the polymers are taken into account by a self-consistently determined solvation potential. The conformational properties of the polymer chain are determined from a variational calculation with a semiflexible reference chain. The finite chain extensibility is taken into account by constraints for the bond lengths and bond angles using Lagrangian multipliers. The scaling relation for the size of an isolated semiflexible chain with respect to chain length exhibits a transition from rodlike to excluded volume type for a given Debye screening length. For flexible chains in solution, the theory provides conformational properties which are in excellent agreement with computer simulation results. The bare chain stiffness has a pronounced influence on the conformational and structural properties of the solution. In the semidilute regime a pronounced liquidlike order is obtained for flexible polyelectrolyte chains which diminishes with increasing bare persistence length. This process is accompanied by a shift of the structural peaks to smaller length scales.

Self-consistent integral equation theory for polyolefins: Comparison to molecular dynamics simulations and x-ray scattering

We report on self-consistent polymer reference interaction site model (PRISM) calculations as well as molecular dynamics (MD) simulations for several types of polyolefins. For all polymer types one single set of potential parameters was used. In general we find good semi-quantitative agreement between PRISM and MD results. Further we compare both MD and PRISM results to experimental x-ray scattering data and show that the potentials used give a good to excellent description of these data. From the quality of the PRISM calculations it is clear that PRISM can be used as an efficient tool in model development.

Self-consistent integral equation theory for semiflexible chain polyelectrolyte solutions

A self-consistent integral equation theory is presented for the conformational behavior and static structure of polyelectrolyte solutions composed of semiflexible chain molecules. The free energy of the solution is expressed as the sum of an elastic, a Coulombic, and a medium-induced contribution, and is minimized with respect to the single chain structure factor. In the flexible limit, the theory is accurate for the chain conformations and static structure factor when compared to Monte Carlo simulations of flexible chains. For semiflexible chains near the flexible limit, the chain conformations are sensitive to the bare persistence length in dilute solutions but not in the semidilute regime. In contrast, the effect of the bare persistence length on the static structure factor is not as significant. The theory predicts that charged flexible chains exhibit more pronounced liquidlike order in solution than their rigid-rod counterparts, and the length scale of these liquidlike correlations is also shorter in flexible chains.

Thermochemistry of solvation: A self-consistent three-dimensional reference interaction site model approach

We developed a self-consistent three-dimensional reference interaction site model integral equation theory with the molecular hypernetted chain closure (SC-3D-RISM/HNC) for studying thermochemistry of solvation of ionic solutes in a polar molecular solvent. It is free from the inconsistency in the positions of the ion–solvent site distribution peaks, peculiar to the conventional RISM/HNC approach and improves the predictions for the solvation thermodynamics. The SC-3D-RISM treatment can be readily generalized to the case of finite ionic concentrations, including the consistent dielectric corrections to provide a consistent description of the dielectric properties of ion–molecular solution. The proposed theory is tested for hydration of the Na+ and Cl− ions in ambient water at infinite dilution. An improved agreement of the ion hydration structure and thermodynamics with molecular simulation results is found as compared to the conventional RISM/HNC treatment.

Generalized mean-spherical-approximation description of highly asymmetric hard-sphere mixtures

We use thermodynamically self-consistent integral equation theories to determine the structure of binary hard-sphere mixtures in a regime of moderate to high size asymmetry, and for low concentration of the species with bigger particle size. Calculations are performed by applying the generalized mean-spherical approximation (GMSA) and the Rogers-Young (RY) approximation. The thermodynamic consistency of the GMSA is implemented in terms of adjustable parameters which are used in order to reproduce the Mansoori-Carnahan-Starling-Leland equation of state, and to impose the equality of the osmotic isothermal compressibilities estimated through the virial and fluctuation routes. The structural results obtained for a moderate size asymmetry of the particle species compare rather satisfactorily with the available Monte Carlo (MC) data and their parametrizations, and with previously reported modified hypernetted-chain results. The relative performances of the GMSA and of the RY approximations are also examined for strongly asymmetric mixtures. A regime of semi-dilute concentration for which no simulation data are available is investigated first and a very close agreement emerges between the RY and GMSA radial distribution functions. The case of very high dilution of the component with bigger particle size, for which RY and MC results already exist, is then considered, but it appears impossible to achieve a thermodynamically consistent solution for the GMSA according to the consistency prescriptions adopted. Other possible implementations of the thermodynamic consistency of the GMSA for HSMs and other multicomponent fluids are discussed.

Phase transitions in hard-core Yukawa fluids: toward a theory of phase stability in protein solutions

Thermodynamically self-consistent integral equation theories (TC-IETs) supplemented by a one-phase freezing criterion, and Monte Carlo simulations, are used to investigate the thermodynamic and structural properties, as well as the phase diagram, of the hard-core Yukawa fluid (HCYF). The attention is focused on rapidly decaying Yukawa tails, a potential regime suited to model in an approximate manner the interaction between globular proteins in protein solutions. TC-IETs are found to give a reasonably accurate description of the physical properties of the HCYF in this limit. The position of the sublimation line relative to the liquid-vapour binodal line, known to play a crucial role in the onset of crystallization in protein solutions, seems qualitatively reproducible. We suggest on this basis the possibility of extending the use of TC-IET to more realistic models of protein solutions, so as to take into account the true multicomponent nature of these fluids, a physical situation whose description still challenges the currently available computer simulation capabilities.