Particle size distribution polydisperse colloidal system

Colloidal systems are generally of a polydispersed nature - i.e. the molecules or particles in a particular sample vary in size. By virtue of their stepwise build-up, colloidal particle and polymer molecular sizes tend to have skew distributions, as illustrated in Figure 1.2, for which the Poisson distribution often offers a good approximation. Very often, detailed determination of relative molecular mass or particle size distribution is impracticable and less perfect experimental methods, which yield average values, must be accepted. The significance of the word average depends on the relative contributions of the various molecules or particles to the property of the system which is being measured. 

Polydispersity arises in systems composed of particles characterized by a property (e.g., particle diameter) that spans a continuum of values. Small molecules exhibit discrete properties, so they do not form polydisperse mixtures. Only at the level of macromolecules and colloidal aggregates does polydispersity become an issue. Here variations in particle size are known to influence the ordering into a solid phase. Experimentally it has been observed that colloidal systems will not form a solid phase if the size polydispersity (as measured by the standard deviation of the particle-size distribution) is greater than about 5% to 10% of the average diameter [252]. 

The conditions of stability of colloidal particles with respect to further dispersion down to molecular sizes can be found by analyzing the A (d) dependence at d - b. If the value of a does not change with the decrease in d down to the molecular dimensions, and if a can be used to describe the work of dispersing, further dispersion of particles down to molecular sizes is thermodynamically favorable. In a real polydisperse system the dispersed particles of colloidal range with some defined particle size distribution may, however, also fluctuationally form at a = const. 

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. 

Heterodisperse Suspensions. The rate laws given above apply to monodisperse colloids. In polydisperse systems the particle size and the distribution of particle sizes have pronounced effects on the kinetics of agglomeration (O Melia, 1978). For the various transport mechanisms (Brownian diffusion, fluid shear, and differential settling), the rates at which particles come into contact are given in Table 7.2. 

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. 

Particles can be broadly classified as either colloids or as macroparticulate powders. Colloids typically have dimensions smaller than 1000 A and are optically transparent, while dispersed powders are generally larger and form turbid suspensions. Neither colloidal dispersions nor powder suspensions are usually monodisperse, and to the extent that particle size can influence attainable surface charge and area, many such systems will typically reflect a distribution of properties as a function of preparation method. Recent advances in synthetic techniques for providing materials with reduced polydispersity are likely to allow for better characterization of these effects in the near future. 

The bimodal model has also been applied to polydisperse suspensions (Probstein et al. 1994), which in practice generally have particle sizes ranging from the submicrometer to hundreds of micrometers. In order to apply the bimodal model to a suspension with a continuous size distribution, a rational procedure is required for the separation of the distribution into fine and coarse fractions. Such a procedure has not been developed so that an inverse method had to be used wherein the separating size was selected which resulted in the best agreement with the measured viscosity. Again, however, the relatively small fraction of colloidal size particles was identified as the principal agent that acts independently of the rest of the system and characterizes the shear thinning nature of the suspension viscosity. 

What has been said so far applies to dilute systems, but densely packed colloidal particles, such as highly filled nanocomposites, require to take into account the interparticle interference effects. Another assumption that is not always valid is that the particles are homogeneous and monodisperse in size. Particle anisotropy and polydispersity are very common factors that bring about severe deviations of the system from ideality. A distribution of sizes must therefore usually be included in the theoretical models used to reproduce the experimental SAXS patterns. 

In real life, many systems are not monodisperse. For example, polymers prepared by synthetic methods are statistically distributed in molecular weight. Both synthetic and naturally occurring colloidal particles are polydisperse. The same applies to self-assembled systems constituted of surfactant and block copolymers. Owing to both the intrinsic polydispersity of the components and the statistical process of self-assembly, polydispersity in terms of aggregation number and size is evident. 

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