Polydisperse systems properties

Another very important physical parameter one must consider is the size distribution of the colloids. A system consisting of particles of the same size is called a monodis-perse. A system with different sizes is called polydisperse. It is also obvious that systems with monodisperse will exhibit different properties from those of polydispersed systems. In many industrial application (such as coating on tapes used for recording music and coatings on CDs or DVDs), latter kind of quality of coatings is needed. 

In the past few decades, a specific kind of colloidal system based on monodis-perse size has been developed for various industrial applications. A variety of metal oxides and hydroxides and polymer lattices have been produced. Monodisperse systems are obviously preferred since their properties can be easily predicted. On the other hand, polydisperse systems will exhibit varying characteristics, depending on the degree of polydispersity. 

As polydispersity is an inherent property of the sample being processed, it is not a system property and can be ignored when optimizing an instrument. 

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. 

Cell elongation in the direction of expansion, which is responsible for the anisotropy of the properties of plastic foams may be quantitatively evaluated using the cell aspect ratio, i.e. the ratio of the largest to the smallest dimension of the cell. Like any other quantitative shape or size characteristic of polydisperse systems such as foamed polymers, these ratios are averaged values. The smaller the size, the more difficult it becomes to evaluate the cell aspect ratio... 

The synthesis of fluorene oligomers is intensive in terms of number of steps, but the unique properties and the new insight into structure-property relationships are well worth the effort. A number of approaches have been developed from the initial brute force fractionation of polydisperse systems to the more elegant deterministic synthesis of monodisperse systems. 

Number-average data of a polydisperse system are of little value. It is important to use other methods, such as small-angle X-ray scattering, to determine the extent of polydispersity of the system before making conclusions from colligative-property measurements. 

The rest of this chapter is organized as follows. First, in Section 6.1, we consider the collision term for monodisperse hard-sphere collisions both for elastic and for inelastic particles. We introduce the kinetic closures due to Boltzmann (1872) and Enksog (1921) for the pair correlation function, and then derive the exact source terms for the velocity moments of arbitrary order and then for integer moments. Second, in Section 6.2, we consider the exact source terms for polydisperse hard-sphere collisions, deriving exact expressions for arbitrary and integer-order moments. Next, in Section 6.3, we consider simplified kinetic models for monodisperse and polydisperse systems that are derived from the exact collision source terms, and discuss their properties vis-d-vis the hard-sphere collision models. In Section 6.4, we discuss properties of the moment-transport equations derived from Eq. (6.1) with the hard-sphere collision models. Finally, in Section 6.5 we briefly describe how quadrature-based moment methods are applied to close the collision source terms for the velocity moments. 

So far, little progress has been made in understanding the rheological properties of polydisperse systems, especially at high concentrations. 

The semigrand ensemble method can be implemented in different forms for calculation of equilibrium properties, and phase equilibria for inert or reacting mixtures. Recently, it has been applied to simulate phase coexistence for binary polymer blends [85], where advantage was taken of the fact that identity exchanges are employed in lieu of insertions or deletions of full molecules. The semigrand ensemble also provides a convenient framework to treat polydisperse systems (see Section III.F). 

The above equations can be used to deduce the properties of the suspension from observations of the front speeds, typically the one separating the clarified layer from the suspension. For example, knowing the fall speed (Eq. 5.4.6), we can determine the effective particle size if the particle density has been found independently. The extension of the results to infinitely dilute systems containing particles of two or more sizes (polydisperse systems) is straightforward and will not be discussed further here. It may only be mentioned that with different fall speeds there will be as many distinct downward-moving fronts as there are particle sizes. From measurements of these front speeds the particle sizes can be determined as for the monodisperse system. 

Hwang W F, Wiff D R, Benner C L and Helminiak T E (1983) Composites on a molecular level phase relationships, processing and properties, J Macromol Sci Part B Phys 22 231-257. Aharoni S M (1980) Rigid backbone polymers. 17. Solution viscosity of polydispersed systems, Polymer 21 1413-1422. 

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. 

The thermodynamic properties of silicate melts however, are complicated by the presence of a large number of different silicate anions. While silicate minerals are usually monor disperse, containing only a single type of anion (e.g. SiojJ in olivine), molten silicates contain a distribution of different polymeric silicate anions of different molecular weights and are thus polydisperse systems. The presence of a distribution of silicate anions in the melt can be inferred from the mixing properties of silicate melts (Richardson, 1956 Masson, I965). However, it has recently become possible to separate some of the... 

Modern colloid chemistry makes an important distinction between monodisperse and polydisperse systems. In monodisperse solutions all particles of the dissolved substance are of the same size and same shape, and have the same properties in polydisperse solutions we find the most diverse particle sizes and varying charges, in short, different properties. Polydisperse systems do not behave like monodisperse solutions whose particles have the size of colloidal particles. 

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