Universal Law of Fractionation for Slightly Polydisperse Systems

Abstract

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,

Local Size Segregation in Polydisperse Hard Sphere Fluids

“…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].…”

“…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.…”

“…Moreover, by a perturbative analysis of the same functional, we can distinguish certain generic features that do not depend on the shape of the parent, if it remains narrow. These two aspects of our work build on recent nonperturbative [15] and perturbative [3,16] progress in understanding polydisperse phase equilibria.The state of a polydisperse fluid is specified by the local number density of each species, r͑s, r͒, with s the particle diameter (let us say). The spatial average of this quantity must recover the (known) global size distribution, r͑s͒, which we call the "parent."…”

“…Moreover, by a perturbative analysis of the same functional, we can distinguish certain generic features that do not depend on the shape of the parent, if it remains narrow. These two aspects of our work build on recent nonperturbative [15] and perturbative [3,16] progress in understanding polydisperse phase equilibria.…”

Microstructure and macroscopic properties of polydisperse systems of hard spheres

“…In many experimental or industrially relevant circumstances, the particular components (spheres) of a system are not uniform in size but rather display some distribution of sizes or "polydispersity" [164]. Interesting phenomena arise in the presence of a wide polydispersity [25,34,207], but due to the wide size distribution across many scales, it is difficult to provide simple accurate statistical models [208,209] for such systems. In many fluidbased theories, for example, Percus-Yevick integral equation theory [210,211], scaled-particle theory [55], Boublík, Mansoori, Carnahan, Starling, and Leland (BMCSL) equation of state (EOS) [57,58], Rosenfeld's fundamental measure theory [212], A. Santos' approaches [59][60][61], and others [49,52,213], it is assumed that the dependence of the polydisperse pressure of N hard spheres on the N degrees of freedom (volume fraction plus N − 1 size ratios) can be encapsulated into the dependence of only three parameters, i.e., the volume fraction and the second and third scaled moments (divided by the first moment, with appropriate power).…”