Domenico Gazzillo

Fluid–fluid phase separation of nonadditive hard‐sphere mixtures as predicted by integral‐equation theories

We investigate the existence of a fluid–fluid phase separation in binary mixtures of equal‐size hard spheres with positively nonadditive diameters [i.e., d11=d22≡d, d12=(1+Δ)d with Δ>0]. An integral‐equation approach is used to evaluate both thermodynamics and structure of many symmetric (equal to equimolar) mixtures (with Δ=0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 1) and some asymmetric cases. We present the results obtained via the Percus–Yevick, the Martynov–Sarkisov, and the Ballone–Pastore–Galli–Gazzillo closures; the thermodynamic consistency of these approximations is discussed and some possible ways to get further improvements are proposed too. The integral‐equation results are then compared with the available ‘‘exact’’ simulation data, a first‐order perturbation approach, and a scaled particle theory. Our study predicts that there exists a demixing for each considered value of the nonadditivity parameter Δ.

Analytic solutions for Baxter's model of sticky hard sphere fluids within closures different from the Percus: Yevick approximation

We discuss structural and thermodynamical properties of Baxters adhesive hard sphere model within a class of closures which includes the Percus-Yevick (PY) one. The common feature of all these closures is to have a direct correlation function vanishing beyond a certain range, each closure being identified by a different approximation within the original square-well region. This allows a common analytical solution of the Ornstein-Zernike integral equation, with the cavity function playing a privileged role. A careful analytical treatment of the equation of state is reported. Numerical comparison with Monte Carlo simulations shows that the PY approximation lies between simpler closures, which may yield less accurate predictions but are easily extensible to multicomponent fluids, and more sophisticate closures which give more precise predictions but can hardly be extended to mixtures. In regimes typical for colloidal and protein solutions, however, it is found that the perturbative closures, even when limited to first order, produce satisfactory results.

Structure factors for the simplest solvable model of polydisperse colloidal fluids with surface adhesion

Closed analytical expressions for scattering intensity and other global structure factors are derived for a new solvable model of polydisperse sticky hard spheres. The starting point is the exact solution of the “mean spherical approximation” for hard core plus Yukawa potentials, in the limit of infinite amplitude and vanishing range of the attractive tail, with their product remaining constant. The choice of factorizable coupling (stickiness) parameters in the Yukawa term yields a simpler “dyadic structure” in the Fourier transform of the Baxter factor correlation function qij(r), with a remarkable simplification in all structure functions with respect to previous works. The effect of size and stickiness polydispersity is analyzed and numerical results are presented for two particular versions of the model: (i) when all polydisperse particles have a single, size-independent, stickiness parameter, and (ii) when the stickiness parameters are proportional to the diameters. The existence of two different reg...

Stability of fluids with more than two components

A complete theoretical basis is presented for integral equation investigations on the phase stability of simple fluid mixtures with any number of components. Thermodynamic criteria for local stability in terms of the Helmholtz free energy are reviewed accurately and compared with those expressed through the Gibbs free energy. The equivalence of several forms of stability condition is proved by means of new thermodynamic identities. Each form of stability criterion consists of a set of inequalities, and the reducibility of each set to only one necessary condition, namely to the most restrictive inequality, is emphasized. As a main result of the paper, Kirkwood-Buff theory is used and thermodynamic stability criteria for an M component fluid are related to microscopic structural properties, namely to correlation functions which can be calculated from integral equation theories. To complete such an extension of known formulas for binary mixtures, a generalization S (M) CC(k) is proposed of the concentration-...

Interaction of Proteins in Solution from Small-Angle Scattering: A Perturbative Approach

In this work an improved methodology for studying interactions of proteins in solution by small-angle scattering is presented. Unlike the most common approach, where the protein-protein correlation functions g(ij)(r) are approximated by their zero-density limit (i.e., the Boltzmann factor), we propose a more accurate representation of g(ij)(r) that takes into account terms up to the first order in the density expansion of the mean-force potential. This improvement is expected to be particularly effective in the case of strong protein-protein interactions at intermediate concentrations. The method is applied to analyze small-angle x-ray scattering data obtained as a function of the ionic strength (from 7 to 507 mM) from acidic solutions of beta-lactoglobulin at the fixed concentration of 10 gl(-1). The results are compared with those obtained using the zero-density approximation and show significant improvement, particularly in the more demanding case of low ionic strength.

Scattering functions for multicomponent mixtures of charged hard spheres, including the polydisperse limit: Analytic expressions in the mean spherical approximation

We present a closed analytical formula for the scattering intensity from charged hard sphere fluids with any arbitrary number of components. Our result is an extension to ionic systems of Vrij’s analogous expression for uncharged hard sphere mixtures. Use is made of Baxter’s factor correlation functions within the mean spherical approximation (MSA). The polydisperse case of an infinite number of species with a continuous distribution of hard sphere diameters and charges is also considered. As an important by-product of our investigation, we present some properties of a particular kind of matrices (sum of the identity matrix with a dyadic matrix) appearing in the solution of the MSA integral equations for both uncharged and charged hard sphere mixtures. This analysis provides a general framework to deal with a wide class of MSA solutions having dyadic structure and allows an easy extension of our formula for the scattering intensity to different potential models. Finally, the relevance of our results for t...

Multicomponent adhesive hard sphere models and short-ranged attractive interactions in colloidal or micellar solutions.

We investigate the dependence of the stickiness parameters tij=1/(12tauij)--where the tauij are the conventional Baxter parameters--on the solute diameters sigmai and sigmaj in multicomponent sticky hard sphere (SHS) models for fluid mixtures of mesoscopic neutral particles. A variety of simple but realistic interaction potentials, utilized in the literature to model short-ranged attractions present in real solutions of colloids or reverse micelles, is reviewed. We consider: (i) van der Waals attractions, (ii) hard-sphere-depletion forces, (iii) polymer-coated colloids, and (iv) solvation effects (in particular hydrophobic bonding and attractions between reverse micelles of water-in-oil microemulsions). We map each of these potentials onto an equivalent SHS model by requiring the equality of the second virial coefficients. The main finding is that, for most of the potentials considered, the size-dependence of tij(T,sigmai,sigmaj) can be approximated by essentially the same expression, i.e., a simple polynomial in the variable sigmaisigmaj/sigmaij2, with coefficients depending on the temperature T, or--for depletion interactions--on the packing fraction eta0 of the depletant particles.

Thermodynamic instabilities of a binary mixture of sticky hard spheres

The thermodynamic instabilities of a binary mixture of sticky hard spheres (SHS) in the modified mean spherical approximation (mMSA) and the Percus-Yevick (PY) approximation are investigated using an approach devised by Chen and Forstmann [corrected] [J. Chem. Phys. [corrected] 97, 3696 (1992)]. This scheme hinges on a diagonalization of the matrix of second functional derivatives of the grand canonical potential with respect to the particle density fluctuations. The zeroes of the smallest eigenvalue and the direction of the relative eigenvector characterize the instability uniquely. We explicitly compute three different classes of examples. For a symmetrical binary mixture, analytical calculations, both for mMSA and for PY, predict that when the strength of adhesiveness between like particles is smaller than the one between unlike particles, only a pure condensation spinodal exists; in the opposite regime, a pure demixing spinodal appears at high densities. We then compare the mMSA and PY results for a mixture where like particles interact as hard spheres (HS) and unlike particles as SHS, and for a mixture of HS in a SHS fluid. In these cases, even though the mMSA and PY spinodals are quantitatively and qualitatively very different from each other, we prove that they have the same kind of instabilities. Finally, we study the mMSA solution for five different mixtures obtained by setting the stickiness parameters equal to five different functions of the hard sphere diameters. We find that four of the five mixtures exhibit very different type of instabilities. Our results are expected to provide a further step toward a more thoughtful application of SHS models to colloidal fluids.

Symmetric mixtures of hard spheres with positively nonadditive diameters: An approximate analytic solution of the Percus–Yevick integral equation

We study the properties of symmetric binary mixtures of hard spheres with positive nonadditive diameters Rij, namely two‐component systems with R11=R22=R and R12>R, at equimolar concentration. The functional form of the direct correlation functions cij(r) is investigated, within the Percus–Yevick approximation, by using Hiroike and Fukui’s version of the Ornstein–Zernike integral equation for multicomponent fluids. It is shown that, by introducing simple polynomial expressions for the cross term C12(r)=rc12(r), the problem of finding an approximate analytic solution of the abovementioned integral equations can be reduced to an algebraic one, i.e., to solving a closed set of a few nonlinear algebraic equations for some unknown parameters. Results corresponding to three different approximations are presented for the radial distribution functions at contact, the virial pressure and the so called bulk modulus. Comparison is made with our exact numerical solutions of the Percus–Yevick integral equation and a v...

A scaling approximation for structure factors in the integral equation theory of polydisperse nonionic colloidal fluids

The integral equation theory of pure liquids, combined with a new “scaling approximation” based on a corresponding states treatment of pair correlation functions, is used to evaluate approximate structure factors for colloidal fluids constituted of uncharged particles with polydispersity in size and energy parameters. Both hard sphere and Lennard-Jones interactions are considered. For polydisperse hard spheres, the scaling approximation is compared to theories utilized by small angle scattering experimentalists (decoupling approximation and local monodisperse approximation) and to the van der Waals one-fluid theory. The results are tested against predictions from analytical expressions, exact within the Percus–Yevick approximation. For polydisperse Lennard-Jones particles, the scaling approximation, combined with a “modified hypernetted chain” integral equation, is tested against molecular dynamics data generated for the present work. Despite its simplicity, the scaling approximation exhibits a satisfacto...