Phase behavior of polydisperse sticky hard spheres: analytical solutions and perturbation theory

We discuss phase coexistence of polydisperse colloidal suspensions in the presence of adhesion forces. The combined effect of polydispersity and Baxter's sticky-hard-sphere (SHS) potential, describing hard spheres interacting via strong and very short-ranged attractive forces, give rise, within the Percus-Yevick (PY) approximation, to a system of coupled quadratic equations which, in general, cannot be solved either analytically or numerically. We review and compare two recent alternative proposals, which we have attempted to by-pass this difficulty. In the first one, truncating the density expansion of the direct correlation functions, we have considered approximations simpler than the PY one. These $C_{n}$ approximations can be systematically improved. We have been able to provide a complete analytical description of polydisperse SHS fluids by using the simplest two orders $C_{0}$ and $C_{1}$, respectively. Such a simplification comes at the price of a lower accuracy in the phase diagram, but has the advantage of providing an analytical description of various new phenomena associated with the onset of polydispersity in phase equilibria (e.g. fractionation). The second approach is based on a perturbative expansion of the polydisperse PY solution around its monodisperse counterpart. This approach provides a sound approximation to the real phase behavior, at the cost of considering only weak polydispersity. Although a final seattlement on the soundness of the latter method would require numerical simulations for the polydisperse Baxter model, we argue that this approach is expected to keep correctly into account the effects of polydispersity, at least qualitatively.Comment: 12 pages, 4 figures, to appear in Molec. Phys. special issue Liblice 200

On the compressibility equation of state for multicomponent adhesive hard sphere fluids

The compressibility equation of state for a multicomponent fluid of particles interacting via an infinitely narrow and deep potential, is considered within the mean spherical approximation (MSA). It is shown that for a class of models leading to a particular form of the Baxter functions $q_{ij}(r)$ containing density-independent stickiness coefficient, the compressibility EOS does not exist, unlike the one-component case. The reason for this is that a direct integration of the compressibility at fixed composition, cannot be carried out due to the lack of a reciprocity relation on the second order partial derivatives of the pressure with respect to two different densities. This is, in turn, related to the inadequacy of the MSA. A way out to this drawback is presented in a particular example, leading to a consistent compressibility pressure, and a possible generalization of this result is discussed.Comment: 13 pages, no figures, accepted for publication Molec. Physics (2002

Polydisperse fluid mixtures of adhesive colloidal particles

We investigate polydispersity effects on the average structure factor of colloidal suspensions of neutral particles with surface adhesion. A sticky hard sphere model alternative to Baxter's one is considered. The choice of factorizable stickiness parameters in the potential allows a simple analytic solution, within the ``mean spherical approximation'', for any number of components and arbitrary stickiness distribution. Two particular cases are discussed: i) all particles have different sizes but equal stickiness (Model I), and ii) each particle has a stickiness proportional to its size (Model II). The interplay between attraction and polydispersity yields a markedly different behaviour for the two Models in regimes of strong coupling (i.e. strong adhesive forces and low temperature) and large polydispersity. These results are then exploited to reanalyze experimental scattering data on sterically stabilized silica particles.Comment: 9 pages, 2 figures (included), Physica A (2001) to appea

Effect of Polydispersity and Anisotropy in Colloidal and Protein Solutions: an Integral Equation Approach

Application of integral equation theory to complex fluids is reviewed, with particular emphasis to the effects of polydispersity and anisotropy on their structural and thermodynamic properties. Both analytical and numerical solutions of integral equations are discussed within the context of a set of minimal potential models that have been widely used in the literature. While other popular theoretical tools, such as numerical simulations and density functional theory, are superior for quantitative and accurate predictions, we argue that integral equation theory still provides, as in simple fluids, an invaluable technique that is able to capture the main essential features of a complex system, at a much lower computational cost. In addition, it can provide a detailed description of the angular dependence in arbitrary frame, unlike numerical simulations where this information is frequently hampered by insufficient statistics. Applications to colloidal mixtures, globular proteins and patchy colloids are discussed, within a unified framework.Comment: 17 pages, 7 figures, to appear in Interdiscip. Sci. Comput. Life Sci. (2011), special issue dedicated to Prof. Lesser Blu

The Psychodynamic Diagnostic Manual – 2nd edition (PDM-2)

For decades many clinicians, especially psychodynamic and humanistic therapists, have resisted thinking about their patients in terms of categorical diagnoses. In the current era, they find themselves having to choose between reluctantly “accepting” the DSM diagnostic labels, “denying” them, or developing alternatives more consistent with the dimensional, inferential, contextual, biopsychosocial diagnostic formulations characteristic of psychoanalytic and humanistic approaches. The Psychodynamic Diagnostic Manual (PDM) reflects an effort to articulate a psychodynamically oriented diagnosis that bridges the gap between clinical complexity and the need for empirical and methodological validity. In this paper the authors (the steering committee of the PDM-2) describe the process of construction of the PDM-1 and discuss changes proposed for implementation in PDM-2

Stability boundaries, percolation threshold, and two phase coexistence for polydisperse fluids of adhesive colloidal particles

We study the polydisperse Baxter model of sticky hard spheres (SHS) in the modified Mean Spherical Approximation (mMSA). This closure is known to be the zero-order approximation (C0) of the Percus-Yevick (PY) closure in a density expansion. The simplicity of the closure allows a full analytical study of the model. In particular we study stability boundaries, the percolation threshold, and the gas-liquid coexistence curves. Various possible sub-cases of the model are treated in details. Although the detailed behavior depends upon the particularly chosen case, we find that, in general, polydispersity inhibits instabilities, increases the extent of the non percolating phase, and diminishes the size of the gas-liquid coexistence region. We also consider the first-order improvement of the mMSA (C0) closure (C1) and compare the percolation and gas-liquid boundaries for the one-component system with recent Monte Carlo simulations. Our results provide a qualitative understanding of the effect of polydispersity on SHS models and are expected to shed new light on the applicability of SHS models for colloidal mixtures.Comment: 37 pages, 7 figures, 1 tabl

Probing the existence of phase transitions in one-dimensional fluids of penetrable particles

Phase transitions in one-dimensional classical fluids are usually ruled out by making appeal to van Hove's theorem. A way to circumvent the conclusions of the theorem is to consider an interparticle potential that is everywhere bounded. Such is the case of, {\it e.g.}, the generalized exponential model of index 4 (GEM-4 potential), which in three dimensions gives a reasonable description of the effective repulsion between flexible dendrimers in a solution. An extensive Monte Carlo simulation of the one-dimensional GEM-4 model [S. Prestipino, {\em Phys. Rev. E} {\bf 90}, 042306 (2014)] has recently provided evidence of an infinite sequence of low-temperature cluster phases, however also suggesting that upon pushing the simulation forward what seemed a true transition may eventually prove to be only a sharp crossover. We hereby investigate this problem theoretically, by three different and increasingly sophisticated approaches ({\it i.e.}, a mean-field theory, the transfer matrix of a lattice model of clusters, and the exact treatment of a system of point clusters in the continuum), to conclude that the alleged transitions of the one-dimensional GEM4 system are likely just crossovers.Comment: 18 pages, 9 figure

Small Angle Scattering data analysis for dense polydisperse systems: the FLAC program

FLAC is a program to calculate the small-angle neutron scattering intensity of highly packed polydisperse systems of neutral or charged hard spheres within the Percus-Yevick and the Mean Spherical Approximation closures, respectively. The polydisperse system is defined by a size distribution function and the macro-particles have hard sphere radii which may differ from the size of their scattering cores. With FLAC, one can either simulate scattering intensities or fit experimental small angle neutron scattering data. In output scattering intensities, structure factors and pair correlation functions are provided. Smearing effects due to instrumental resolution, vertical slit, primary beam width and multiple scattering effects are also included on the basis of the existing theories. Possible form factors are those of filled or two-shell spheres.Comment: 18 pages, 1 figure, uses elsart.st

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 $q_{ij}(r)$, 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 regimes for the average structure factor, respectively above and below a generalized Boyle temperature which depends on size polydispersity, is recognized and discussed. Because of its analycity and simplicity, the model may be useful in the interpretation of small-angle scattering experimental data for polydisperse colloidal fluids of neutral particles with surface adhesion.Comment: 32 pages, 7 figures, RevTex style, to appear in J. Chem. Phys. 1 December 200

Pathologies in the sticky limit of hard-sphere-Yukawa models for colloidal fluids. A possible correction

A known `sticky-hard-sphere' model, defined starting from a hard-sphere-Yukawa potential and taking the limit of infinite amplitude and vanishing range with their product remaining constant, is shown to be ill-defined. This is because its Hamiltonian (which we call SHS2) leads to an {\it exact}second virial coefficient which {\it diverges}, unlike that of Baxter's original model (SHS1). This deficiency has never been observed so far, since the linearization implicit in the `mean spherical approximation' (MSA), within which the model is analytically solvable, partly {\it masks} such a pathology. To overcome this drawback and retain some useful features of SHS2, we propose both a new model (SHS3) and a new closure (`modified MSA'), whose combination yields an analytic solution formally identical with the SHS2-MSA one. This mapping allows to recover many results derived from SHS2, after a re-interpretation within a correct framework. Possible developments are finally indicated.Comment: 21 pages, 1 figure, accepted in Molecular Physics (2003