Bueche theory

Figure 2.13 Model of several orders of coupling through entanglements according to Bueche theory.

Bueche was able to incorporate these ideas into a quantitative theory, the mathematical details of which need not concern us. The result is complex, but simplifies when applied to polymers of very large molecular weight. In this limit the Bueche theory predicts... 

Equation (2.61) predicts a 3.5-power dependence of viscosity on molecular weight, amazingly close to the observed 3.4-power dependence. In this respect the model is a success. Unfortunately, there are other mechanical properties of highly entangled molecules in which the agreement between the Bueche theory and experiment are less satisfactory. Since we have not established the basis for these other criteria, we shall not go into specific details. It is informative to recognize that Eq. (2.61) contains many of the same factors as Eq. (2.56), the Debye expression for viscosity, which we symbolize t . If we factor the Bueche expression so as to separate the Debye terms, we obtain... 

Next let us consider the differences in molecular architecture between polymers which exclusively display viscous flow and those which display a purely elastic response. To attribute the entire effect to molecular structure we assume the polymers are compared at the same temperature. Crosslinking between different chains is the structural feature responsible for elastic response in polymer samples. If the crosslinking is totally effective, we can regard the entire sample as one giant molecule, since the entire volume is permeated by a continuous network of chains. This result was anticipated in the discussion of the Bueche theory for chain entanglements in the last chapter, when we observed that viscosity would be infinite with entanglements if there were no slippage between chains. 

It should be noted that the Bueche theory predicts a slightly different form for the variation of (Mc)soln with concentration than that inferred earlier on the basis of experimental data ... 

Tables IV and V show the dimensions calculated from the various theories of viscosity and sedimentation, respectively, in comparison with those obtained from light-scattering. It will be seen that, in the case of the viscosity data, the agreement between the calculated and experimental values is, with the exception of the Brinkman-Debye-Bueche theory, very reasonable. The values of (ro) calculated from sedimentation data are not, in general, in such good agreement with the light-scattering results.

The Debye-Bueche theory, on the other hand, considers the partially draining coil as a sphere that is more or less permeable, within which a number of smaller beads is homogeneously distributed. The beads correspond to the monomeric units. The drag which one bead produces on the others is calculated, and this resistance is then expressed in terms of a length L, which corresponds to the distance from the surface of the sphere to where the flow rate of the solvent is reduced to 1 /e times what it is at the surface of this sphere. The shielding ratio, or shielding factor, is given by... 

The molecular theories of Bueche and Graessley are similar in that both theories relate the pseudoplastic nature of polymer solutions as a function of a dimensionless Deborah number which represents a ratio of the response time of the polymer molecules in solution, X, to the time scale of the flow process (18). In simple shearing the time scale of the flow process is inversely proportional to the shear rate. Thus both equation (5), developed from the Bueche theory, and equation (14), developed from the Graessley theory, can be expressed in terms of the Deborah number,... 

In conclusion, both the Graessley and Bueche theories confirm the general experimental observation that the reduced solution viscosity, ti/Doj function of X. Therefore, solution rheological behavior can be described by two material parameters, the zero shear viscosity and polymer response time. In turn, these parameters are functions of macromolecular structure and solvent properties. 

The Rouse-Bueche theory is useful especially below 1% concentration. However, only poor agreement is obtained on studies of the bulk melt. The theory describes the relaxation of deformed polymer chains, leading to advances in creep and stress relaxation. While it does not speak about the center-of-mass diffusional motions of the polymer chains, the theory is important because it serves as a precursor to the de Gennes reptation theory, described next. 

Compare the Rouse-Bueche theory with the de Gennes theory. How do they model molecular motion ... 

Viscosity enhancement in branched polystyrenes would be expected to be considerably smaller than in branched diene polymers. A smaller Z factor and a higher Me for this polymer combine to reduce the V value. The star-branched polymers shown in Fig. 11 would not therefore be expected to show viscosity enhancement at a concentration of 0.25 g/ml. Experiments reported on melt viscosities of star-branched polystyrenes having f 3 on the other hand would have been expected to have shown some enhancement at least at the h est molecular weight. None was in fact foimd, the viscosities beii approximately predictable from the Bueche theory i. e. were all lower than their linear counterparts. Viscosity enhancements have been reported for multibranch star polystyrenes (f = 7 to 13) in melt flow experiments. These are of comparable magnitude to the values found for four-brandi star pedybutadienes at the same value of M /Mc. Specific effects of multiple brandling were not considered in the model described above but there is evidence that the major enhancement effect is produced at f = 3. Increase of the branch nuniber to four at constant increases the enhancement by about a factor of two but subsequent increases in f have only small effect . ... 

Debye-Bueche Theory Debye and Bueche (1948) criticize the theory of Kirkwood and Riseman as being unrealistic in assuming that the hydrodynamic... 

M at which the slope changes is a characteristic value, Me, which according to the Bueche theory is related to the average molecular weight spacing between entanglement points, Me, in a rather complicated manner approximately, Me s 2 Me- Because the change in slope is of course not really a discontinuity, there has been some skepticism of a real qualitative difference between the r ons M < Me and M > Me- However, in the relaxation and retardation spectra there is a clear difference only for M > Me do two maxima appear. 

The Rouse, Zimm and Bueche theories are satisfaetory for the longer relaxation times, which involve movement of submolecules. This has been eonfirmed for dilute polymer solutions, where the theory would be expeeted to be most appropriate [29,30]. More remarkably, it also holds for solid amorphous polymers (Referenee 11, Chapter 13), provided that the friction coefficient is suitably modified. 

Mendelson et al. 1970). The Bueche theory predicts that the viscosity of branched flexible homopolymers is lower than that of linear homopolymers of the same The reduction of melt viscosity in the presence of long-chain branching may be attributed to (1) a reduction in relaxation times and (2) a reduction in chain entanglements or other interactions between polymer molecules in the bulk. It appears that the reduction in relaxation times may be related to g given in Eq. (6.40), and the reduction in chain entanglements or other interactions between polymer molecules may be related to E(g) given in Eq. (6.41). 

Combining this with the prediction of the Rouse-Bueche theory for Hnear, imentangled melts (Eq. 6.16), we obtain ... 

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