Zimm s plot

Figure 18.6 Zimm s plot of an alternating copolymer of ethylene and tetrafluoroethylene (PETFE) in diisobutyl adipate at 240 °C. Source Reprinted with permission from Chu B, Wu C. Macromolecules 1987 20 93-98. Copyright 1987 American Chemical Society.

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. 

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 ... 

Hence, if Ro can be found. My, can be calculated without prior knowledge of the molecular shape. Obviously, Ro cannot be found directly, but can be extrapolated from values at sufficiently low angles. In Zimm s method, data obtained at various concentrations (C) and angles (8) are simultaneously extrapolated to zero values of C and 8 by plotting KC/Re against... 

In order to estimate (5 ). and from experimental data of Rq versus c for solutions at various scattering angles 0 at constant temperature, it is convenient to use a graphical method, proposed by Zimm [28]. In this method, Kc/Rq is plotted against (sin (0/2) + kc), where k is an arbitrary constant chosen to produce a reasonable spread of data points. In Zimm s method, (S ) and A2 l are estimated from the relations of lime Kc/Rq versus sin (0/2) and lime >o c// e versus c, respectively. is obtained as the inverse of the value of the intercept of a line of lime Kc/Rq at 0 = 0° or that of a line of lime >o atc = 0. Note that the intercepts of the two lines coincide with each other. Zimm s procedure is a double-extrapolation method. 

Fig. 3.12 Ratio of the hydrodynamic radius Rh to (in our notation), plotted versus Here the hydrodynamic radius is not defined as a purely geometric quantity, but rather as the Stokes radius of a sphere which would have the same sedimentation velocity as the polymer. The latter is obtained via static dynamics , also taking rotational diffusion into account (Zimm s approach. ). Data are shown for star polymers on the cubic lattice and/= 1,3,4,6 (from top to bottom) for different effective monomeric Stokes radii a = 1/4 (full symbols) and 0=1/2 (open symbols) (from Ref. 106).

From Fig. 35, where the normalized coherent scattering laws S(Q, t)/S(Q,0) are plotted as a function of 2 (Q)t for Zimm as well as for Rouse relaxation, one sees that hydrodynamic interaction results in a much faster decay of the dynamic structure factor. 

The Zimm-plot technique, like q.l.s., also involves a difficult extrapolation to zero angle. Another difficulty is that its qiplication assumes that the particles are Rayleigh-Cans-Debye scatterers (namely, that there is no change of phase or other distortions of the incident radiation by the particle). For the larger mucins in particular, this may not be the case. 

A similar expansion can be obtained for scattering intensity S q,C) oc 1/(1 + q R +...). For very low q, in the Guinier regime qR<, this expansion is the basis for the so-called Zimm plots that are commonly used to determine the radius of a chain and the second virial coefficient of a solution. 

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