Polydispersity coefficient

Analytical ultracentrifugation (AUC) Molecular weight M, molecular weight distribution, g(M) vs. M, polydispersity, sedimentation coefficient, s, and distribution, g(s) vs. s solution conformation and flexibility. Interaction complex formation phenomena. Molecular charge No columns or membranes required [2]... 

The translational diffusion coefficient in Eq. 11 can in principle be measured from boimdary spreading as manifested for example in the width of the g (s) profiles although for monodisperse proteins this works well, for polysaccharides interpretation is seriously complicated by broadening through polydispersity. Instead special cells can be used which allow for the formation of an artificial boundary whose diffusion can be recorded with time at low speed ( 3000 rev/min). This procedure has been successfully employed for example in a recent study on heparin fractions [5]. Dynamic fight scattering has been used as a popular alternative, and a good demonstra-... 

The particle size analysis techniques outlined earlier show promise in the measurement of polydispersed particle suspensions. The asumption of Gaussian instrumental spreading function is valid except when the chromatograms of standard latices are appreciably skewed. Calc ll.ation of diameter averages indicate a fair degree of insensitivity to the value of the extinction coefficient. 

Here, S0 is the signal when G = 0, D is the self-diffusion coefficient, yG is the gyromagnetic ratio and am are roots of the Bessel function equation amaf3 2(ama) — (l/2)J3/2(a ma) = 0. If the system is polydisperse, the signal decay is due to contributions from droplets of different sizes. Then, the signal attenuation is given by the volume average over all sizes as... 

This can be done via Eqs. (29) through (32). From the dissolution data, the coefficient B2 is obtained and through the results from microscopy the moments r2 and p3 can be evaluated. By knowing N, the initial number of particles, and the density of the solid, the average initial volume shape factor for a polydisperse powder can be estimated. 

V is the molar volume of the solvent and pp the density of the polymer. For polydisperse polymers A2 is a more complex average, which shall not be discussed here in detail [7]. For good solvents and high concentrations, the influence of the 3rd virial coefficient A3 cannot be ignored, and n/c versus c sometimes does not lead to a linear plot. In these cases, a linearization can frequently be obtained with the approximation A3 = A (M)n/A by plotting [12,13]... 

In a sedimentation equilibrium run, the stationary radial concentration profile, which is established after a few hours for a 1-mm column, is analyzed according to Equation (72) or, in case of polydisperse samples, Equations (74) or (75). Contrary to the sedimentation velocity experiment, the diffusion coefficient D is not required. 

Strictly speaking, monodisperse samples would be required for the determination of the Mark-Houwink coefficients. Since, however, the poly-dispersities of the nine individual fractions are only moderate (Mw/Mn 2) and since both Mw and [tj] are measured as weight averages with the same statistical weights, the error introduced by the incorrect treatment of the polydispersity could be neglected. 

If one does not use the short gradient pulse (SGP) approximation, the term A has to be substituted with (A 8/3). In the case of a mono-disperse system, the plot of ln(E) versus y2g282A is a straight line having the absolute value of the slope equal to the self-diffusion coefficient. For polydisperse sample, the signal intensity decay can be interpreted in terms of a distribution of diffusing species ... 

Figures 3 and 4 show the variation of the average attachment coefficient with CMD. It can be seen that for particles of CMD less than 0.06 ym and Og = 2 the kinetic theory predicts an attachment coefficient similar to the hybrid theory, whereas for CMD greater than about 1 ym the diffusion theory and the hybrid theory give approximately the same results. For a more polydisperse aerosol (Og = 3) the kinetic theory deviates from the hybrid theory even at a CMD = 0.01 ym. The diffusion theory is accurate for a CMD greater than about 0.6 ym. These results are easily explained since as the aerosol becomes more polydisperse, there are more large diameter particles (CMD >0.3 ym) which attach according to the diffusion theory. In contrast, the kinetic theory becomes more inaccurate as the aerosol becomes more polydisperse.

Figure 5 shows the variation of the hybrid theory with CMD for various Og. It is obvious that assuming an aerosol to be mono-disperse when it is in fact polydisperse leads to an underestimation of the attachment coefficient, leading in turn to large errors in calculation of theoretical unattached fraction. 

Another uncertainty arises from the influence of polydispersity. Intrinsic viscosity data were mostly obtained from fractions but the second virial coefficient data were chosen from unfractionated samples. The resulting error is probably not large since A2 depends only slightly on the width of the distribution [183, 184]. 

Partition coefficients of NPE surfactants were determined for hexadecane-water mixtures (Table 1). Similar results were obtained by Crook et al. (2) for octyl analogs in an isooctane-water system. However, Log K is not a linear function of the mol% ethylene oxide in the surfactant as predicted by Eq. 2. Nonlinearity in the octyl analogs was due to polydispersity in the polymer chain length (2), and similar effects presumably operate here. 

Once the and coefficients are experimentally determined, they can be used to characterize an unknown monodisperse or polydisperse sample from the experimental determination of X under given experimental conditions (see Equation 12.48). 

Among the many industrial applications, one can recall the analyses on carbon black, where FIFFF and SdFFF were used in synergy, and on carbon nanotube, for which a frit inlet AsFlFFF channel was used. Water-soluble polydisperse polymers were fractionated, with a very high selectivity, according to differences in the diffusion coefficient, yielding a diffusion coefficient spectrum which was then converted into a molecular weight (M) distribution curve based on the relationship between D and molecular weight [36]. 

Volume-normalized extinction is plotted in Fig. 11.2 as a function of photon energy for several polydispersions of MgO spheres both scales are logarithmic. For comparison of bulk and small-particle properties the bulk absorption coefficient a = Airk/X is included. Some single-particle features, such as ripple structure, are effaced by the distribution of radii. The information contained in these curves is not assimilated at a glance they require careful study. 

Here, /u ° and ju are, respectively, the chemical potentials of pure solvent and solvent at a certain concentration of biopolymer V is the molar volume of the solvent Mn=2 y/M/ is the number-averaged molar mass of the biopolymer (sum of products of mole fractions, x, and molar masses, M, over all the polymer constituent chains (/) as determined by the polymer polydispersity) (Tanford, 1961) A2, A3 and A4 are the second, third and fourth virial coefficients, respectively (in weight-scale units of cm mol g ), characterizing the two-body, three-body and four-body interactions amongst the biopolymer molecules/particles, respectively and C is the weight concentration (g ml-1) of the biopolymer. 

It is important for us to keep in mind that biopolymers are generally not monodisperse components. Proteins are typically paucidisperse — mixtures of monomers, dimers and multimers. And polysaccharides are polydisperse their chain lengths and molar masses can be represented as a continuous distribution. For this reason the virial coefficients appearing in equations (5.16) and (5.17) should be interpreted as averages. So the inverse of the number-averaged molar mass of component / is given by... 

The averaged second virial coefficients for a polydisperse system are given by (Schaink and Smit, 2007) ... 

It is important to note that the values of the second virial coefficients determined by light scattering are weight-average quantities. The general expression for the second virial coefficient of a polydisperse polymer is as follows (Casassa, 1962) ... 

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