Polydisperse systems: Statistical thermodynamics, with applications to several models including hard and permeable spheres

Salacuse, J. J.; Stell, G.

doi:10.1063/1.444274

Cited by 281 publications

(234 citation statements)

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“…The popular but arbitrary method of discretizing the distribution 6], though e cacious, gives little insight. The in nity of coexistence conditions hampers the formulation of truly polydisperse statistical mechanics (discussed in 5,7,8]), especially in non-mean-eld systems, for which exact phase calculations are consequently scarce 9]. The approach of Gualtieri et al 5] to calculating two-phase coexistence is applicable to a large class of model systems, but gives rise to formidable non-linear equations.…”

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confidence: 99%

Universal Law of Fractionation for Slightly Polydisperse Systems

1998

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By perturbing about a general monodisperse system, we provide a complete description of twophase equilibria in any system which is slightly polydisperse in some property (e.g. particle size, charge, etc.). We derive a universal law of fractionation which is corroborated by comprehensive experiments on a model colloid-polymer mixture. We furthermore predict that phase separation is an e ective method of reducing polydispersity only for systems with a skewed distribution of the polydisperse property.PACS numbers: 64.10.+h, 82.70.Dd Complex uids contain mesoscopic units that are almost inevitably polydisperse, i.e. colloidal or polymeric particles have some characteristic, such as radius, charge, mass or oblateness, which varies quasi-continuously from one to another. A truly polydisperse system contains in nitely many species with a distribution of properties, and could separate into arbitrarily many coexisting phases. The onset of phase separation is at the`cloud curve', the boundary of coexistence with an in nitesimal amount of a second phase on the`shadow curve'. In contrast to simple systems, a complete description of phase equilibria entails determining not just these limiting curves, but also the di erent compositions (described by a distribution) of arbitrary coexisting phases.Experimentally, phase equilibria have been completely determined for polydisperse polymers 1]. In contrast, most experiments on colloidal phase behavior have ignored polydispersity, despite pragmatic interest in using phase separation to fractionate suspensions 2]; limited data on particulate systems derives only from simulations 3,4]. Many calculations of two-phase equilibria have been attempted for speci c polydisperse systems ( 5] and references therein), especially polymers (which admit mean-eld analysis). The popular but arbitrary method of discretizing the distribution 6], though e cacious, gives little insight. The in nity of coexistence conditions hampers the formulation of truly polydisperse statistical mechanics (discussed in 5,7,8]), especially in non-mean-eld systems, for which exact phase calculations are consequently scarce 9]. The approach of Gualtieri et al. 5] to calculating two-phase coexistence is applicable to a large class of model systems, but gives rise to formidable non-linear equations. They calculate cloud/shadow curves for a polydisperse van der Waals model, but give no general result. We present a simpler treatment, applicable to real systems, and use it to solve the two-phase coexistence problem completely in the limit of small polydispersities. A universal law of fractionation is derived. We show signi cant consistency with comprehensive measurements of phase equilibria in a model polydisperse colloid.Following Gualtieri et al. 5], we divide the total free energy, F tot = F id + F ex into two parts: the free energy of a polydisperse ideal gas of the given species distribution, and the excess due to interactions. The ideal part,

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confidence: 99%

Universal Law of Fractionation for Slightly Polydisperse Systems

1998

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“…where m͑s͒ is the chemical potential of species s in the bulk, L͑s͒ is a thermal wavelength [6], V ͑s, r͒ is the external potential acting on species s, and…”

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confidence: 99%

“…More is known about partial structure factors in single-phase fluids [4,5] and liquid-liquid phase equilibrium [6,7] than about crystalline phases [8] or interfacial properties, for example. And, where such inhomogeneous situations have been studied [9] it has often proved necessary to assume that only the mean density, and not the size distribution, can vary in space [10].…”

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confidence: 99%

Local Size Segregation in Polydisperse Hard Sphere Fluids

2000

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The structure of polydisperse hard sphere fluids, in the presence of a wall, is studied by the Rosenfeld density functional theory. Within this approach, the local excess free energy depends on only four combinations of the full set of density fields. The case of continuous polydispersity thereby becomes tractable. We predict, generically, an oscillatory size segregation close to the wall, and connect this, by a perturbation theory for narrow distributions, with the reversible work for changing the size of one particle in a monodisperse reference fluid. PACS numbers: 61.20.Gy, 05.20.Jj, 64.75. + g Understanding the behavior of polydisperse systems is relevant to many materials of practical interest. In particular, colloidal and/or polymeric fluids generally contain particles which have, in effect, a continuous distribution of sizes (and/or other parameters such as charge and chemical composition). This affects their performance in applications ranging from foodstuffs to polymer processing [1]. More fundamentally, colloidal systems also provide the closest experimental approach to the "theorists ideal fluid," namely, that of perfect hard spheres [2]. The fact that all colloids are in practice polydisperse (at least slightly) must then be taken into account in comparing theory with experiment. Only recently has experimental work started to clarify in a systematic way the generic consequences of polydispersity, such as the partitioning of sizes between coexisting phases [3].Despite the continuous interest polydisperse fluids have raised, their theoretical understanding remains far from complete, especially for inhomogeneous cases. More is known about partial structure factors in single-phase fluids [4,5] and liquid-liquid phase equilibrium [6,7] than about crystalline phases [8] or interfacial properties, for example. And, where such inhomogeneous situations have been studied [9] it has often proved necessary to assume that only the mean density, and not the size distribution, can vary in space [10]. This ignores size segregation effects, which (globally) influence the phase diagram [3,7]. A similar tendency to local segregation is implicit in treatments of binary and ternary hard sphere mixtures [11][12][13] and in polydisperse equilibrium structure factors in the homogeneous state [4].In what follows, we treat continuous polydispersity within a density functional theory (DFT) that properly allows for local size segregation. Our work is based on a choice of density functional (that of Rosenfeld [14]) that has previously been used to study finite mixtures of hard spheres. By exploiting the fact that its excess free energy density depends on only a small number of linear combinations of the particle densities (four "moment densities"), we are able to address the case of continuous polydispersity, where the underlying densities are infinite in number. This allows us to study, e.g., the effects of varying the shape of a smooth size distribution. Moreover, by a perturbative analysis of the same functional, we can di...

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“…Within scaled particle theory the pressure of a mixture of hard spheres may be shown readily to be an explicit function of just three diameter moments [55]. The same simplification is evident also in the approximate equation of state obtained from the PercusYevick closure for a system of polydisperse hard spheres [56]; also in the case of the "improved" equation of state obtained by Boublik [57] and Mansoori et al [58] from an interpolation between the Percus-Yevick virial and compressibility equations; and also in A. Santos et al's approaches [59][60][61].…”

Section: Theorymentioning

confidence: 70%

Microstructure and macroscopic properties of polydisperse systems of hard spheres

Ogarko¹

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Polydisperse systems: Statistical thermodynamics, with applications to several models including hard and permeable spheres

Cited by 281 publications

References 22 publications

Universal Law of Fractionation for Slightly Polydisperse Systems

Universal Law of Fractionation for Slightly Polydisperse Systems

Local Size Segregation in Polydisperse Hard Sphere Fluids

Microstructure and macroscopic properties of polydisperse systems of hard spheres

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