Zimm plots

When the light-scattering data are available only for one concentration, but the concentration is sufficiently low, we can regard that the line connecting the measured data as the c = 0 line in the Zinun plot. The intercept of the line gives and 

A variant of this scheme is also used. A mirror is on a rotation stage. The mirror is rotated so that the beam hits the two parts of the detector with the same intensity. The angle of rotation gives the refractive index difference An. 

As an alternative, we can use a regular Abbe refractometer that reads the refractive index of a liquid for a typical light source of a sodium lamp (D line A = 589.3 nm) at a given temperature. Unlike the differential refractometer, the fluid is exposed to the atmosphere therefore, this method is not suitable for a solution dissolved in a volatile solvent. 

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One of the most important fiinctions in the application of light scattering is the ability to estimate the object dimensions. As we have discussed earlier for dilute solutions containing large molecules, equation (B 1.9.38) can be used to calculate tire radius of gyration , R, which is defined as the mean square distance from the centre of gravity [12]. The combined use of equation (B 1.9.3 8) equation (B 1.9.39) and equation (B 1.9.40) (tlie Zimm plot) will yield infonnation on R, A2 and molecular weight. 

There is really nothing in this method that we have not already considered, one aspect at a time, in this chapter. The Zimm plot simply brings it all together in a single analysis. 

We shall presently construct a Zimm plot in detail in an example. In anticipation of this, we label each of the paragraphs describing Zimm s procedure for ease of cross-referencing in the example. 

Prepare a Zimm plot using the data in Table 10.2 and evaluate M, B,and for this solution of polystyrene in benzene. The effective wavelength in the medium is Xq/u = 546/1.501 = 364 nm for this experiment. 

The abscissa values for the Zimm plot are given by sin (9/2) + kCj- For these data, k = 100 cm g" gives a good array of points. Table 10.3 shows the values of the abscissa in the same format used in Table 10,2. 

The objective of the Zimm plot is to conduct all extrapolations on a single graph. The three-stage development of Fig. 10.14 is not typical, but it is intended to clarify the discussion. 

Not all Zimm plots show the same grid of essentially parallel straight lines found in Fig. 10.14. In some cases there is considerable curvature, and quite a bit of interpretation is required to extract the molecular parameters from the data. In this connection we note that the reciprocal of Eq. (10.83)... 

In addition, the intercept obtained by extrapolating this asymptote back to sin (0/2) = 0 equals (2M )". Note that both Mand are number averages when this asymptotic limit is used. This is illustrated schematically in Fig. 10.15 and indicates that even more information pertaining to polymer characterization can be extracted from an analysis of the curvature in Zimm plots. 

Zimm plots at 546 nm were prepared for a particular polystyrene at two ... 

For polystyrene in butanone at 67 C the following values of KC2/R0 X 10 were measuredf at the indicated concentrations and angles. Construct a Zimm plot from the data below using k = 100 cm g" for the graphing constant. Evaluate M, B, and from the results. In this experiment... 

For c = Cp the value of the term sin (0/2) + 100 is calculated for each value of 0, the factor being an arbitrary choice. The values obtained for this term are plotted against A. The line that results is then extrapolated to 6 = 0, Le. to lOOcp This is the uppermost line in the Zimm plot illustrated in Figure 6.2. 

These calculations are repeated for C2, C3, and respectively. This results in a series of parallel lines inclined from the horizontal in the Zimm plot each line is extrapolated to 100c to give a result for 0=0,... 

Both extrapolated lines meet on the A axis at the same point, and this corresponds to 1/M. Other solution properties of the polymer may also be determined once the Zimm plot has been prepared. Along the line of 0 = 0, A = 1/M (1 + 2T2 c +. ..). Hence the slope of this line is 2T2/M, from which r of the Flory equation may be evaluated. 

Figure 2. Zimm plot for poly(dichlorophosphazene) Sample IL-22...

Figure 7 Schematic Zimm plot The determined Kc/(it values ( ), the extrapolated values for fixed concentrations to q — 0 ( ), and the extrapolated values for fixed scattering vector to infinite dilution (°) c — 0.

Figure 8 Zimm plot for xanthan in an aqueous solution of 10 mM NaCl. Reproduced with permission from Berth et al. [38]. Copyright 1996 American Chemical Society.

The weight-average Molecular weightin the inverse of the intercept at c = 0 and q = 0 in the Zimm plot. The second virial coefficient can be calculated from the slope of the lines at constant angle by eq. 8.34, and to a first approximation is independent of angle. The radius of gyration is... 

The extrapolation may be done separately or simultaneously as in case of Zimm plot method. In Zimm... 

For large particles, Eq. (40) may be treated via different methods the most popular of which is the Zimm plot. Essentially, all the data are displayed on a grid which yields on appropriate extrapolations, two lines representing (a) the dependence of (Kc/Ra)e = 0 on concentration and (b) the dependence of (Kc/Rg)c=o on angle. 

Fig. 49. Zimm plot for a styrene/isoprene block copolymer in methyl isobutyl ketone153)...

Fig. SO. Zimm plot for same sample as in Fig. 49, but with chlorobenzene as solvent1 3)...

Zimm plot analychem A graphical determination of the root-square-mean end-to-end distances of coillike polymer molecules during scattered-light photometric analyses. zim,plat ... 

Note Several modifications of the Zimm plot are in frequent use the most common one uses the excess scattering instead of the excess Rayleigh ratio. 

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