Polydisperse systems density distribution

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 strictly 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. One can regard a polydisperse fluid as a mixture of an infinite number of distinct particle species. If we label each species according to the value of its polydisperse attribute, a, the state of a polydisperse system entails specification of a density distribution p(a), rather than a finite number of density variables. It is usual to identify two distinct types of polydispersity variable and fixed. Variable polydispersity pertains to systems such as ionic micelles or oil-water emulsions, where the degree of polydispersity (as measured by the form of p(a)) can change under the influence of external factors. A more common situation is fixed polydispersity, appropriate for the description of systems such as colloidal dispersions, liquid crystals, and polymers. Here the form of p(cr) is determined by the synthesis of the fluid. 

Computationally, polydispersity is best handled within a grand canonical (GCE) or semi-grand canonical ensemble in which the density distribution p(a) is controlled by a conjugate chemical potential distribution p(cr). Use of such an ensemble is attractive because it allows p(a) to fluctuate as a whole, thereby sampling many different realizations of the disorder and hence reducing finite-size effects. Within such a framework, the case of variable polydispersity is considerably easier to tackle than fixed polydispersity The phase behavior is simply obtained as a function of the width of the prescribed p(cr) distribution. Perhaps for this reason, most simulation studies of phase behavior in polydisperse systems have focused on the variable case [90, 101-103]. 

The second method can also be used in the general case, namely, a polydispersed system in which the average density varies with particle size, and requires only three scattering curves. The ratio (p(R)y/y/ p R) can be used as an estimate of the width of the average density distribution. 

PSD, the implementation of different measurement methods varies widely within the industry. Other potential sources of variability in sizing methods include adjustable instrument parameters or material property data required as inputs (density, refractive index), and fundamental differences due to the nature of the technique itself In the latter case, it is commonly acknowledged that different methods may provide different size distribution. A given method may be sensitive to either particle mass, particle number, or projected surface area. As a result, for a polydisperse system, each method produces a distribution with a slightly different weighting. Thus the mean particle diameter values are expected to differ. 

Keeping the spherical symmetry of the interaction potential a long-range repulsive barrier can be added to the short-range attraction, which destabilizes the fluid-fluid separation, and allows the study of fluid states close to gelation even at low densities [25,26], Polydispersity is used to avoid crystallization, either a continuous distribution or binary mixtures of particles. Figure 7.3 presents an interaction potential based on the AO potential, with a repulsive barrier and a polydisperse system. 

Several colloidal systems, that are of practical importance, contain spherically symmetric particles the size of which changes continuously. Polydisperse fluid mixtures can be described by a continuous probability density of one or more particle attributes, such as particle size. Thus, they may be viewed as containing an infinite number of components. It has been several decades since the introduction of polydispersity as a model for molecular mixtures [73], but only recently has it received widespread attention [74-82]. Initially, work was concentrated on nearly monodisperse mixtures and the polydispersity was accounted for by the construction of perturbation expansions with a pure, monodispersive, component as the reference fluid [77,80]. Subsequently, Kofke and Glandt [79] have obtained the equation of state using a theory based on the distinction of particular species in a polydispersive mixture, not by their intermolecular potentials but by a specific form of the distribution of their chemical potentials. Quite recently, Lado [81,82] has generalized the usual OZ equation to the case of a polydispersive mixture. Recently, the latter theory has been also extended to the case of polydisperse quenched-annealed mixtures [83,84]. As this approach has not been reviewed previously, we shall consider it in some detail. 

The different location of polar and amphiphilic molecules within water-containing reversed micelles is depicted in Figure 6. Polar solutes, by increasing the micellar core matter of spherical micelles, induce an increase in the micellar radius, while amphiphilic molecules, being preferentially solubihzed in the water/surfactant interface and consequently increasing the interfacial surface, lead to a decrease in the miceUar radius [49,136,137], These effects can easily be embodied in Eqs. (3) and (4), aUowing a quantitative evaluation of the mean micellar radius and number density of reversed miceUes in the presence of polar and amphiphilic solubilizates. Moreover it must be pointed out that, as a function of the specific distribution law of the solubihzate molecules and on a time scale shorter than that of the material exchange process, the system appears polydisperse and composed of empty and differently occupied reversed miceUes [136],... 

In this chapter, the basic definitions of the equivalent diameter for an individual particle of irregular shape and its corresponding particle sizing techniques are presented. Typical density functions characterizing the particle size distribution for polydispersed particle systems are introduced. Several formulae expressing the particle size averaging methods are given. Basic characteristics of various material properties are illustrated. 

Near the gel point, the system consists of a highly polydisperse distribution of polymers. One of the most important features of gelation is that the number density of polymers near the gel point has a power law... 

The moments of an NDF represent some important physical properties of the underlying population of elements constituting the multiphase system under study. For this reason, they have to satisfy some simple rules. For instance, the positiveness of the density function over its support implies that the moment of order zero must be positive. Additionally, there are other simple, intuitive rules. For example, if the internal coordinate assumes only positive values (or in other words the moment is defined on a positive support) then the moment of order one (as well as all the other moments) must be positive. Another important property of the distribution is its variance (i.e. cr = m2 - m lmQ), which must be zero for a delta-function distribution, while it must be positive for polydisperse distributions. Accordingly, it has to be m2 > m lmo. For higher-order moments the mathematical constraints are less intuitive and cannot be directly related to specific global properties of the multiphase systems. Fortunately, the theory of moments provides some interesting theorems that turn out to be very useful in determining whether a set of moments is invalid. 

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