Small polydisperse systems

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. 

Microelectrophoresis is the most common technique for electrokinetic measurements in colloidal systems. Here individual particles can be observed, in their normal environment, under the microscope. Very dilute dispersions can be studied and very small particles, down to about 0.1 pm diameter, can be observed using the dark-field microscope (ultramicroscope). High magnifications allow minimization of observation times, and in polydisperse systems a given size range of particles can be studied to the exclusion of others. 

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. 

It is concluded on the basis of equation [4.S] that the intensity of the particle loss due to the thermal coagulation is directly proportional to square of the particle concentration, while the coagulation efficiency increases with decreasing particle radius. This means that the coagulation of small particles at a high concentration is a very rapid process. Equation [4.S] is valid only for monodisperse aerosols, i.e. aerosols composed of particles of uniform size. However, the same qualitative conclusion can also be drawn in the case of polydisperse systems. 

To obtain the q(r) / Pmm curve from sedimentation curve, c (R, A/=const.), one can plot the relative concentration, c/c0, as a function of particle radius obtained from particle displacement, AR, that occurred over the time, At, using eq. (V.53). If the diffusion rate is negligibly small, the c = c (AR / AO curves match each other at all times, At. The latter allows one to separate sedimentation and diffusion in polydisperse systems as well. To... 

Both LaMer s and the falling film aerosol generators yield only small quantities of products. Much larger amounts of aerosols can be produced by dispersing liquids with the help of various mechanical devices, e.g. rotating disks or ultrasonic nozzles [6]. These techniques, however, usually yield aerosols with broad distribution of droplet sizes and thus lead to polydisperse systems. The dispersion aerosol generators can, consequently, be used... 

Figures 2.15-2.18 show a small difference between the results of analytical (Ramazanov et al., 1983a) and simple geometrical smoothing and the results of direct computation of the functions for the model distribution in a wide range of polydispersity. The subscript p denotes the gcimma distribution p oo and p = 0 are the limiting cases of a monodisperse system and an utmost polydisperse system (see Equations 106 and 107).

For very small particles, there is no angular dependence of the scattering intensity, and not even an gyration radius can be extracted. However, the absolute scattering intensity contains information about the particle size, but in most cases only an average particle size can be extracted. However, if the system investigated allows a systematic variation of the optical contrast, one has a powerful method at hand, from which even small polydispersities can be obtained reliably. Such situations are not that uncommon - they readily occur for water-in-oil microemulsions, but are also conceivable for other types of layered nanoparticles. 

The technique works best for monodisperse suspensions, and can be used to make size measurements with an accuracy which is better than a few percent. However, the method is not too sensitive to small polydispersi-ties. Moderately polydisperse systems can be analysed as well, and different methods are available for this purpose. A popular approach is the so-called cumulant analysis , in this one expands the correlation function as follows ... 

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