Scattering data extrapolation

A graphical method, proposed by Zimm (thus tenned the Zinnn plot), can be used to perfomi this double extrapolation to detemiine the molecular weight, the radius of gyration and the second virial coefficient. An example of a Zinnn plot is shown in figure Bl.9.6 where the light scattering data from a solution of poly... 

Applied to scattering data we encounter the same numerical problems as in the isotropic case we have to extrapolate inward into the center as well as outward towards infinity. We can avoid the outward extrapolation, if at the outer border of the... 

This equation implies a double dependence of scattering intensities on concentration and observation angle [9]. By extrapolating the scattering data for each concentration to zero angle, the second virial coefficient, which is related to thermodynamic properties, may be measured [9,10,15-18],... 

By extrapolating scattering data for each angle to zero concentration, the mean-square radius of gyration may be measured [9,10,15-18]... 

Figure 3. Computer plot obtained by isochronous interpolation of the experimental light-scattering data (O) of HEC during endocellulase attack. The Langrangian interpolation functions are given by the horizontal curves ana the isochronous interpolated Kc/Re values by A. The quadratic least squares extrapolations to zero angle ( ) are given by the...

Figure 4. Changes in Mw as a function of the enzymic hydrolysis times, calculated from the light-scattering data by isochronous interpolation and by subsequent extrapolation to zero concentration using Equation 17.

Figure 2.6 A Zimm plot, in which light-scattering data are simultaneously extrapolated to zeroscattering angle and zero concentration. Reprinted with permission from J. E. Mark, Physical Chemistry of Polymers, ACS Audio Course C-89, American Chemical Society, Washington, DC, 1986. Copyright 1986, American Chemical Society.

In conclusion it can be stated that the results from scattering data and atom experiments are still contradictory and therefore need further investigation. From the side of the atom experiments it should be clarified whether the shift and the width values of pionic hydrogen and deuterium are true strong interaction effects and are not spoiled by the interaction of the pionic atom with the surrounding molecules. In other words the shift and the width measurements for pionic hydrogen and deuterium should be extrapolated to zero pressure. In a second step state of the art electromagnetic corrections should be applied. 

This formula is the basis of the Zimm Plot which consists in plotting the inverse of the scattering intensity, S "1 (Q), vs Q2 which shows a linear variation at low Q and in dilute solutions. Extrapolated values (Q - 0, Cp - 0) of the intercept and the slope yield the degree of polymerization N and the excluded volume v (or second virial coefficient A2) respectively. Zimm s formula can describe scattering data accurately well into the semidilute concentration region. This region is defined for concentrations above an overlap concentration C which is defined in either of the two following ways ... 

Fig. Schematic illustration of Zimm plot for analysing light-scattering data. The solid points ( ) represent experimental measuements the open circles (O) are extrapolated points.

Fig. 3-4. Ziinm plot for simultaneous extrapolation of light scattering data to zero angle (H) and zero concentration (c). The symbols are defined in the text, o, experimental points extrapolated points x, double extrapolation.

Based on arbitrary extrapolation.. Scattered data point, around IEP. IEP from [1662] was cited in [1] and in a few other paper. as IEP of synthetic goethite. Particles were aged for 7-10 days, probably in glass at a very low solid-to-liquid ratio. The low IEP can be then explained by silicate adsorption. 

Fig. 8.5.1. Zimm plot for the double extrapolation of light-scattering data to zero concentration and zero q. The intercept is (Af) 1 and the initial slope of the c = 0 line is Rg2 / 3.

For the determination of molecular weights and associated MWDs, both A and dn/dc are measured in advance. Although the cellulose specimens used in this analysis have characteristic broad MWDs (M= 1.8-4.0), a single determination of A from a representative CTC preparation was considered appropriate for all subsequent determinations of M and the MWD. This was achieved by using the LALLS photometer off-line and by extrapolating scattering data from a dilution series at 9 = 4 -5 j u ing eq. C3). The value of A determined was 3.5 x 10 ml-mole/g Likewise, dn/dc was determined for the same CTC preparation in THF at 632.8 nm on a Chromatix KMX-16 laser differential refrac-tometer, and resulted in a value of 0.163 ml/g. 

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