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|عنوان فارسی مقاله:||مطالعات شبیه سازی مجموعه گرند کانونیکال سیالات پلی دیسپرس|
|عنوان انگلیسی مقاله:||Grand canonical ensemble simulation studies of polydisperse fluids|
|رشته های مرتبط:||شیمی، شیمی کاربردی، شیمی پلیمر، شیمی تجزیه|
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We describe a Monte Carlo scheme for simulating polydisperse fluids within the grand canonical ensemble. Given some polydisperse attribute s, the state of the system is described by a density distribution r(s) whose form is controlled by the imposed chemical potential distribution m(s). We detail how histogram extrapolation techniques can be employed to tune m(s) such as to traverse some particular desired path in the space of r(s). The method is applied in simulations of size-disperse hard spheres with densities distributed according to Schulz and log-normal forms. In each case, the equation of state is obtained along the dilution line, i.e., the path along which the scale of r(s) changes but not its shape. The results are compared with the moment-based expressions of Monsoori et al. @J. Chem. Phys. 54, 1523 ~1971!# and Salacuse and Stell @J. Chem. Phys. 77, 3714 ~1982!#. It is found that for high degrees of polydispersity, both expressions fail to give a quantitatively accurate description of the equation of state when the overall volume fraction is large.
I. INTRODUCTION Statistical mechanics was originally formulated to describe the properties of systems of identical particles such as atoms or small molecules. However, many materials of industrial and commercial importance do not fit neatly into this framework. For example, the particles in a colloidal suspension are never precisely identical to one another, but have a range of radii ~and possibly surface charges, shapes, etc!. This dependence of the particle properties on one or more continuous parameters is known as polydispersity. To process a polydisperse colloidal material, one needs to know its phase behavior, i.e., the conditions of temperature and pressure under which a given structure is thermodynamically stable. The main obstacles to gaining this information arise from the effectively infinite number of particle species present in a polydisperse system. Labeling these by the continuous polydispersity attribute s, the state of the system must be described by a density distribution r(s), rather than a finite number of density variables. The phase diagram is therefore infinite dimensional, a feature that poses serious problems to experiment and theory alike. The chief difficulty faced in experimental studies of polydisperse systems is that the infinite dimensionality of the phase diagram precludes a complete mapping of the phase behavior. Instead one is forced to focus attention on particular low dimensional manifolds ~slices! of the full diagram. Typically this involves determining the system properties along some desired trajectory through the space of r(s). Such a strategy is often pursued in experiments on colloidal suspensions,1,2 where the phase behavior is studied along a so-called dilution line. The experimental procedure for tracking this line involves adding a prescribed quantity of colloid of some known degree of polydispersity to a vessel of fixed volume V, the remaining volume being occupied by a solvent. The number distribution of colloidal particles N(s) determines the density distribution of the suspension, r(s) [N(s)/V. Since in a given substance, the relative proportions of the number of particles of each s are fixed, changing the amount of colloid added simply alters the scale of r(s), not its shape. Thus, by varying N(s) at fixed V ~or vice versa! one traces out a locus in the phase diagram in which only the overall scale of r(s) changes. As regards theoretical studies of phase behavior, these typically endeavor to calculate the system free energy as a function of a set of density variables. The difficulty in achieving this for a polydisperse system is that the free energy f@r(s)# is a functional of r(s), and therefore itself occupies an infinite dimensional space. This renders intractable the task of identifying phase boundaries, and obliges one to resort to approximation schemes. Of these, perhaps the most simple is a generalization of the van der Waals approximation to polydisperse systems.3 A more sophisticated approach involves approximating the full free energy by a so-called ‘‘moment free energy’’ containing the full ideal gas contribution plus an excess part that depends only on certain principal moments of the full excess free energy.4 Doing so reduces the problem to a finite number of density variables and allows calculation of phase coexistence properties within a systematically refinable approximation scheme. Additionally the theory delivers ~for the given free energy! exact results for the location of spinodals, critical points, and the cloud and shadow curves. Use of this approach promises to enhance significantly our understanding of phase behavior in polydisperse systems.