Polydispersities

Rowell and co-workers [62-64] have developed an electrophoretic fingerprint to uniquely characterize the properties of charged colloidal particles. They present contour diagrams of the electrophoretic mobility as a function of the suspension pH and specific conductance, pX. These fingerprints illustrate anomalies and specific characteristics of the charged colloidal surface. A more sophisticated electroacoustic measurement provides the particle size distribution and potential in a polydisperse suspension. Not limited to dilute suspensions, in this experiment, one characterizes the sonic waves generated by the motion of particles in an alternating electric field. O Brien and co-workers have an excellent review of this technique [65]. 

After reviewing various earlier explanations for an adsorption maximum, Trogus, Schechter, and Wade [244] proposed perhaps the most satisfactory one so far (see also Ref. 243). Qualitatively, an adsorption maximum can occur if the surfactant consists of at least two species (which can be closely related) what is necessary is that species 2 (say) preferentially forms micelles (has a lower CMC) relative to species 1 and also adsorbs more strongly. The adsorbed state may also consist of aggregates or hemi-micelles, and even for a pure component the situation can be complex (see Section XI-6 for recent AFM evidence of surface micelle formation and [246] for polymeric surface micelles). Similar adsorption maxima found in adsorption of nonionic surfactants can be attributed to polydispersity in the surfactant chain lengths [247], Surface-active impuri-... 

Figure B3.3.9. Phase diagram for polydisperse hard spheres, in the volume fraction ((]))-polydispersity (s) plane. Some tie-lines are shown connecting coexistmg fluid and solid phases. Thanks are due to D A Kofke and P G Bolhuis for this figure. For frirther details see [181. 182].

Boihuis P G and Kofke D A 1996 Monte Carlo study of freezing of polydisperse hard spheres Phys. Rev. E 54... 

Broseta D, Fredriekson G H, Helfand E and Leibler L 1990 Moleeular-weight effeots and polydispersity effeots at polymer-polymer interfaoes Macromolecules 23 132... 

Even when carefully prepared, model colloids are almost never perfectly monodisperse. The spread in particle sizes, or polydispersity, is usually expressed as the relative widtli of tire size distribution,... 

Figure C2.6.1. SEM image of silica spheres of radius a = 15 nm and polydispersity a < 0.01 (courtesy of Professor A van Blaaderen)...

The major class of plate-like colloids is tliat of clay suspensions [21]. Many of tliese swell in water to give a stack of parallel, tliin sheets, stabilized by electrical charges. Natural clays tend to be quite polydisperse. The syntlietic clay laponite is comparatively well defined, consisting of discs of about 1 nm in tliickness and 25 nm in diameter. It has been used in a number of studies (e.g. [22]). 

The fonnation of colloidal crystals requires particles tliat are fairly monodisperse—experimentally, hard sphere crystals are only observed to fonn in samples witli a polydispersity below about 0.08 [69]. Using computer... 

Salgi P and Rajagopalan R 1993 Polydispersity in colloids—implications to static structure and scattering Adv. Colloid Interface Sc/. 43 169-288... 

Figure C2.17.4. Transmission electron micrograph of a field of Zr02 (tetragonal) nanocrystals. Lower-resolution electron microscopy is useful for characterizing tire size distribution of a collection of nanocrystals. This image is an example of a typical particle field used for sizing puriDoses. Here, tire nanocrystalline zirconia has an average diameter of 3.6 nm witli a polydispersity of only 5% 1801.

Ohara P C ef a/1995 Crystallization of opals from polydisperse nanopartioles Phys. Rev. Lett. 75 3466... 

In connection with Eq. (1.4), we noted that the standard deviation measures the spread of a distribution now we see that the ratio M /M also measures this polydispersity. The relationship between these two different measures of polydispersity is easily shown. Equation (1.14) may be written as... 

Sec. 1.8, where polydispersity in ordinary samples was emphasized. Polydis-persity clearly complicates things, especially in the neighborhood of n, where a significant number of molecules are too short to show entanglement effects while an equally significant fraction are entangled. We simply note that any study conducted with the intention of a molecular interpretation should be conducted on a sample with as sharp a distribution as possible. 

For preparative purposes batch fractionation is often employed. Although fractional crystallization may be included in a list of batch fractionation methods, we shall consider only those methods based on the phase separation of polymer solutions fractional precipitation and coacervate extraction. The general principles for these methods were presented in the last section. In this section we shall develop these ideas more fully with the objective of obtaining a more narrow distribution of molecular weights from a polydisperse system. Note that the final product of fractionation still contains a distribution of chain lengths however, the ratio M /M is smaller than for the unfractionated sample. 

Hven fractionated polymer samples are generally polydisperse, which means that the molecular weight determined from intrinsic viscosity experiments is an average value. The average obtained is the viscosity average as defined by Eqs. (1.20) and (2.40) as seen by the following argument ... 

For a polydisperse system containing molecules in different molecular weight categories which we index i, we can write (m,), =, and... 

For polydisperse systems the value of M obtained from the values of s° and D°-or, better yet, the value of the s/D ratio extrapolated to c = 0-is an average value. Different kinds of average are obtained, depending on the method used to define the average location of the boundary. The weight average is the type obtained in the usual analysis. 

Polydispersity obscures the nature of the average obtained, although the possibility of extracting more than one kind of average from the same data under optimum conditions partially offsets this. 

Polydisperse polymers do not yield sharp peaks in the detector output as indicated in Fig. 9.14. Instead, broad bands are produced which reflect the polydispersity of synthetic polymers. Assuming that suitable calibration data are available, we can construct molecular weight distributions from this kind of experimental data. An indication of how this is done is provided in the following example. 

This result uses the already established fact that M = when the molecular weight is determined by light scattering for a polydisperse system. 

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