Polydispersion corrections

For reasons of simplicity, theoreticians like to assume that, in polymer solutions, all the chains have the same number of links. It is also quite clear that experimentalists try to use polymer samples that are as monodisperse as possible. Nevertheless, it is true that at the present time (1988), the best samples are 

Here S1/2 defines the size of the average polymer in the absence of interaction. In the same way, in the presence of interaction, the size eS1/2 = X of the average polymer in an infinitely dilute solution can be considered as a proper reference length. 

each time we want to take polydispersion into account, we use the Schulz-Zimm law and we choose the size of the isolated average polymer as the reference length. Consequently, all formulae are expressed in the polydisperse case in the same way as in the non-disperse case, the only difference being that now a new dimensionless parameter p defining the polydispersion of the system has to be introduced. 

In Chapter 5, we defined the osmotic pressure of a polymer solution, we indicated how it can be measured, and we described various effects concerning the compressibility and the preferential adsorption. When the polymers are very long and when the volume fraction occupied by the polymers in the solution is small, the complex reality can be represented by a simple model which is the standard continuous model, studied in Chapter 10 in the context of perturbation theory. This model is especially useful because it allows us to perform effective calculations. In particular, it can be used in the limit of long polymers to determine universal quantities because, then, the general properties of long polymers become independent of the chemical microstructure. Calculations are 

The osmotic pressure II of a very dilute polymer solution obeys Van t Hoff s law 

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We can reproduce the result by calculation, assuming that the chain is Brownian and performing the adequate polydispersion corrections (with the dispersion law of Schulz and Zimm). In this way it is possible to obtain a function °Hz(q) compatible with the experiment, and this leads to acceptable values of °Ro,z and of p ... 

No polydispersion correction is necessary to interpret the experiment. However,... 

We wish to compare the experimental points with values derived from the universal function G(x) (15.4.21). For this, it is necessary to make polydispersion corrections. The authors claim that... 

In order to make polydispersion corrections, we shall take as a reference system a monodisperse sample with concentration C and molecular mass M, which we shall define as being equal to Mn. For this monodisperse system... 

For the real system, the second virial coefficient is given by an average quantity. Such an average is difficult to compute. Nevertheless, it can easily by shown that the simple-tree approximation (see Chapter 10, Section 6) does not lead to any polydispersion correction. This brings up the hypothesis that, for d = 3, polydispersion corrections are negligible with respect to osmotic pressure, and we shall therefore assume to a first approximation that... 

The previously discussed theories were developed for monodisperse diblock copolymers, which are not TPEs. However, Leibler s mean-field theory has been extended to include polydispersity (Leibler and Benoit, 1981) and to include triblock, star, and graft copolymers (Olvera de la Cruz and Sanchez, 1986 Mayes and Olvera de la Cruz, 1989). In the former case, polydispersity corrections tend to lower x N corresponding to the ODT. As would be expected from the analogy between blends and diblocks, triblocks will phase separate at higher xN values than the corresponding diblocks. This theory predicts a monotonic increase in the critical value of x A as the symmetry of the triblock increases, to a maximum of about 18 for the symmetric triblock. Surprisingly, the minimum xN value that separates the order and disordered regions in triblocks does not necessarily correspond to the critical point. 

Polydispersity correction factor for number average molar masses... 

Since the SEC/LALLS technique always yields a weight-average molecular weight (l )y for the slightly polydisperse fraction at V, a small overestimation of the sample Rn is expected (, 1 ). As noted previously (Results) a 1% to decrease in the narrow MWD polystyrene Mp values (Table I) accompanied application of the band-spreading correction ... 

Table II shows that for SRM 706 good agreementis obtained between SEC/LALLS and conventional SEC sample My, and Rp values when the band-spreading correction was used. However, the NBS 706 polydispersity index (Ry/Rp) given by the supplier (ca. 2.1) does not agree with that 1.°) found here using the SEC/LALLS and conventional SEC techniques. Insensitivity of the LALLS detector to a small amount of low molecular weight material may account for a larger sample R however, this is not supported by the conventional SEC data. The reason for the discrepancy remains unclear.

It is fairly clear that as re approaches rd the role of Rouse relaxation is significant enough to remove the dip altogether in the shear stress-shear rate curve. As the relaxation process broadens, this process is likely to disappear, particularly for polymers with polydisperse molecular weight distributions. The success of the DE model is that it correctly represents trends such as stress overshoot. The result of such a calculation is shown in Figure 6.23. 

Fig. 12. The rheological functions G ((o) and G"(co) for an H-shaped PI of arm molecular weigh 20 kg mol and backbone 110 kg mol" [46]. The high-frequency arm-retraction modes can be seen as the shoulder from co 10 to co 10 together with a low-frequency peak due to the cross-bar dynamics at co 10. The smooth curves are the predictions of a model which takes Eq. (33) as the basis for the arm-retraction times and a Doi-Edwards reptation spectrum with fluctuations for the backbone. The reptation time is correctly predicted, as is the spectrum from the arm modes, though the low frequency form is more polydisperse than the simple theory predicts...

It is also possible to observe upward curvature in a plot of In c versus r, as in curve C of Figure 21.3. This curvature occurs when the macromolecules are polydisperse, that is, when they possess a range of molecular weights. Common sense tells us, in this case correctly, that the heavier species in the class i will congregate toward the bottom of the cell. As the slope depends on Mj, curve C will become steeper as we move toward the bottom of the cell, where r is greater. 

Instrumental band broadening or axial dispersion can cause calibration errors when employing polydisperse standards. Correction of the polydisperse standard calibration data for instrumental band broadening will minimize the effect on molecular weight analyses of polymer samples. However, as previously demonstrated in this report, when low dispersion SEC columns are employed instrumental band broadening is minimized and the effect on use of linear calibration methodology is negligible. 

The most important consideration of software, however, is that is must provide the correct answers. This requires that the appropriate molecular weight be associated with each data point, which relates to the techniques and algorithms used in constructing the calibration curve. Calibration curves are generated from chromatographic data obtained on standards of known molecular weight both monodisperse and polydisperse standards have been used. A discussion of the relative merits of each technique is beyond the scope of this paper suffice it to say that the model used by the software should reflect the true calibration as closely as possible. 

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