Polydispersity

Polydispersity of simple bile salt micelles can only be assessed by modem QLS techniques employing the 2nd cumulant analysis of the time decay of the autocorrelation function [146,161]. These studies have shown, in the cases of the 4 taurine conjugates in 10 g/dl concentrations in both 0.15 M and 0.6 M NaCl, that the distribution in the polydispersity index (V) varies from 20% for small n values to 50% for large n values [6,146]. Others [112] have foimd much smaller V values (2-10%) for the unconjugated bile salts in 5% (w/v) solutions. Recently, the significance of QLS-derived polydispersities have been questioned on the basis of the rapid fluctuation in n of micellar assemblies hence V may not actually represent a micellar size distribution [167-169]. This argument is specious, since a micellar size distribution and fast fluctuations in aggregation number are identical quantities on the QLS time scale (jusec-msec) [94]. 

At low concentrations just above the CMC and at low ionic strengths ( 0.2 M NaCl), nearly all simple bile salt micellar solutions contain spherical or nearly spherical micellar particles [5,6,12,146]. Intrinsic viscosity measurements [162,170-172] are in agreement with this and also indicate that the micelles are highly hydrated, cholates2 DC [162,172]. The maximum size of these globular micelles falls in the range Ry, = 10-16 A with h = 10-12 [146]. In the case of NaTC, the water of hydration amounts to about 30 moles H20/mole of monomer in the micelle [162]. By employing the translational mobility of H20, Lindman et al. [173] 

The fractional binding of counterions to ionic micelles is commonly denoted by the parameter /S. It follows that the corresponding micellar charge is l-fi [176]. Work on classic micellar systems such as dodecyl sulfate and octanoate show that this subject is highly complicated and as yet not fully understood [176-179]. In the presence of different counterions (such as the alkali ions Li , Na, K, Rb and Cs ), micellar size and shape, CMC values and CMT values of dodecyl sulfate are altered and do not follow the order of atomic number of the alkali ions in the periodic table nor their hydrated radius (P.J. Missel, G.B. Benedek and M.C. Carey, unpublished observations). Hydrated radius bears an inverse relationship to atomic radius, viz. (in A) Li (2.35), Na (1.85), (1.32), Cs+ (1.27) and Rb (1.26). A 

Much less is known about micellar charge and counterion binding in the case of bile salts. Based on the result of ionic self-diffusion measurements [20,163,173], conductance studies [17,18,187], Na, and Ca activity coefficients [16,19,144,188,189] and NMR studies with Na, Rb and Cs [190], a number of generalities can be made. Below the operational CMC, all bile salts behave as fully dissociated 1 1 electrolytes, yet interionic effects between cations and bile salt anions decrease the equivalent conductance of very dilute solutions [17,18,187]. With the onset of micelle formation, counterions become bound to a small degree values at this concentration are about 0.07-0.13 and are not greatly influenced by the species of monovalent alkali cations [163,190]. At concentrations above the CMC, values remain relatively constant to 100 mM in the case of C and this 

Polydispersity is a measure of the uniformity of a droplet size distribution and typically varies from 0.0 to 1.0 (unit-less) (Constantinides and Yiv, 1995), where values of 0.000 to 0.02 indicate a monodisperse or nearly monodisperse distribution, values of 0.02-0.08 are common for narrowly-distributed droplet sizes, and values higher than 0.08 indicate broad distributions. Despite microemulsions being polydispersed, they are frequently interpreted as being monodispersed to simplify analysis. 

Polysaccharide size polydispersity transcends decades of molecular weights (Fig. 3 in Chapter 5). Such polydispersity is evaluated by a variety of methods (Barth and Sun, 1991). 

Given that fl is a function of mass-to-charge ratio, the number of spots displayed on an electrophoregram is a semiquantitative indicator of the polymolecularity of the parent polysaccharide (Aspinall and Cottrell, 1970). 

HPLC has been adapted to industrial polysaccharides (Barth and Reg-nier, 1981), e.g., guar (Barth and Smith, 1981), starch (Kobayashi et al., 1985), polydextrose (Thomas et al., 1990), pectin (Schols et al., 1989), and other anionic gums (Voragen et al., 1982). Baseline separation is limited to DP =10 (Chester and Innis, 1986), which is more the size of an oligosaccharide than a polysaccharide. Separations to DP = 30 are possible under special conditions (Praznik et al., 1984). 

In normal polysaccharide HPLC, the most polar molecules with identical M are the last to elute in a polar solvent, because of their greater interaction with the aqueous stationary phase. Reverse-phase HPLC is the technique of substituting the LPLC stationary phase with a nonpolar solvent stationary phase and using a less polar mobile phase, with the result that the most polar homologs elute first. Heyraud and Rinaudo (1991) applied reverse-phase HPLC to the separation of low DP anomeric dextrins and Voragen et al. (1982) applied it to analysis of pectin enzyme digests. 

Thin-layer chromatography (TLC) is another liquid-liquid partition technique applicable to polysaccharides, but in two dimensions. In TLC, the M cutoff boundaries between separated molecules are sharpened, because diffusion is minimized or eliminated in favor of capillary transport. The sample capacity of a TLC plate is in microliters. Resolution is enhanced further at high solvent pressure (Rombouts and Thibault, 1986). 

We shall now consider the effects of polydispersity on the interpretation of light scattering measurements on solutions of large molecules. If a solution contains particles having differing weights and shapes, but all scattering independently, then the observed intensity is simply the sum of all the individual intensities  

Usually the only easily measurable quantities are Rftv and the total concentration c = Zct so that as actually applied the relation becomes 

Since the weighting factor, is the same as that in the expression 

1 The Z-average molecular weight for a polydispersed system is defined by the relation 

The weight average(equations 20) for this same distribution is(Z+ i)/Y so that 

Controversial results are reported in the literature regarding the polydispersity of polyesters produced by SSP, associated with the side reactions in the later stages of the reaction. These are not only dependent on the concentrations of the reactive groups but also on their intramolecular distances [11], Additionally, it has been found that cyclization leads to a different polydispersity. According to theoretical considerations, the polydispersity index of an SSP polymer is generally higher than that of prepolymer produced in the melt phase, which should, in an ideal case have a value of 2 [21-24, 59], 

Crystallization is a time-dependent phenomenon [25, 26], Therefore, the increase of crystallinity during SSP reduces the reaction rate due to the hindered 

The crystallization is reduced by an increasing molecular weight of the intrinsic viscosity due to the decreased mobility of the polymer chains. Shorter chains 

It should be pointed out that the crystal size and the surface structure may also affect the desorption of side products. The development of crystallinity 

Here nj is the number fraction, and cj = njMj/ i rikMk is the weight fraction of polymer molecules having molecular weight Mj. The polydispersity of the polymer can be 

The particles in a real eolloidal dispersion are not identical, and any complete statistieal theory must take this into account. The problem of polydispersity seems first to have been recognized by Onsager in his study of orientational order-disorder transitions. By a polydisperse system, we mean one in which there is a continuous distribution f(x) of some characteristic (single-particle) variable x. According to this definition, a binary mixture is not an example of a polydisperse system. The variable x can refer to any characteristic of the particle, but in practice this is almost always the particle size. 

For monodisperse samples, a plot of G(r) against r gives a straight line with a constant slope which is inversely proportional to particle size. For polydisperse samples, the relationship is multi-exponential and a plot of G(r) against r acquires curvature, the degree of which increases with increasing polydispersity [278]. 

The autocorrelation function for a polydisperse system represents the weighted sum of decaying exponential functions, each of which corresponds to a different particle diameter. For such a system  

F F) is the normalized distribution of decay constants of the scatterers in suspension. Given G(r) it is necessary to invert equation (10.42) in order to determine F jT). Unfortunately, the inversion is ill-posed in that there are an infinite number of distributions which satisfy this equation within the experimental error to be found in G( r). A large number of algorithms have been suggested for the inversion and an evaluation of their performance can be found in Stock and Ray [279]. 

The autocorrelation function can also be analyzed by the method of cumulants. In this approach G(r) is fitted to a low order polynomial. Fora third order cumulants fit  

An average particle size is obtained from the average decay rate 7 using equations (10.41-10.43) and an indication of spread (or polydispersity) is given by a.  

Nearly all synthetic polymers and naturally occurring macromolecular substances exist 

Reproduced from F. W. Billmeyer, Textbook of Polymer Science, 2nd edn, Wiley, New York, 1971. 

Polytetrafluoroethylene Polyacrylonitrile Poly(vinyl alcohol) Poly(vinyl acetate) 

In equation (8.2) m, m2, m, . .. are the masses of each species, and m, is obtained hy multiplying the molecular weight of each species hy the number of molecules of that weight that is, m, = Thus the molecular weight 

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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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