Bueche model

The Rouse-Bueche model (97,98) replaces the real molecule of n main chain atoms by a mechanical chain of N +1 beads joined in sequence by N linear springs. The frictional interactions with the medium, which are distributed uniformly along the length of the real molecule to give a molecular frictional coefficient n(0, ate concentrated at regular intervals in the beads. The frictional... 

The Bueche model is based on random-flight statistics of freely draining polymer molecules. Accordingly, two possible relations exist between g and g". (1) For the Zimm-Kilb model, combining Equations 2, 3, 8, and 11 the relation obtained is given by ... 

The combined Yarusso/Debye-Bueche model provides an excellent fit to the full range of the ASAXS data, as shown in Figure 7 for the difference pattern. The fit parameters are listed in Table III. 

Figure 7. Yarusso/Debye-Bueche model fit to the difference pattern. Solid line is the fit circles are data. (Reprinted from ref. 12. Copyright 1988 American Chemical Society.)...

Table III. Yarusso/Debye-Bueche Model Fit Parameters for NiSPS 3 2...

Rouse-Bueche model, the relaxation rate constants of a segment containing n ions is K/n2. Such an assumption is also justified by the relaxation behavior of ionomers of various degrees of neutralization [7]. The model detailed below was applied to the data of Navratil in order to quantify it [4,8],... 

The Guinier, Debye-Bueche, Invariant and Porod analyses are all based on the assumption of well defined phases with sharp interfacial boundaries. In addition, the Guinier approach is based on the assumption that the length distribution function (23.15), or probability Poo(r) that a randomly placed rod (length, r) can have both ends in the same scattering particle (phase) is zero beyond a well defined limit. For example, for monodisperse spheres, diameter D, Poo = 0, for r > D. In the Debye-Bueche model, Poo has no cut off and approaches zero via an exponential correlation function only in the limit r oo [45,46]. 

However, due to experimental limitations, only a limited/finite (j-range is acces-sible/measured, and hence extrapolation of measured intensity data to both a low-and a high-g region is necessary before the Fourier transform. The Debye-Bueche model [26,27] defined by... 

Figure 5.12 Sketch of steady-state compliance versus molecular weight for samples of a monodisperse, linear polymer. Below the linear increase is in accord with the Rouse-Bueche model (Eq. 5.9), while above this critical molecular weight, a further increase is suppressed by entanglements (Eq. 5.10).

As the arm length increases above M q, we expect the onset of entanglement to cause marked deviations from this relationship. However, for star polymers it is observed that continues to increase linearly with M, in accord with the Rouse-Bueche model for Hnear, unentangled polymers. This is in contrast to the behavior of entangled, linear, monodisperse melts, for which 7s° is independent of M at large M as shown by Eq. 5.10. Figure 5.21 shows data of Graessley and Roovers for four and six arm polystyrenes [90]. The horizontal line is based on... 

Generally, the two most useful approaches dealing with the scattering of phase separated systems are those due to Debye-Bueche and Porod. For a non-homoge-neous blend where the two phases have random shape and size with sharp phase boundaries, the scattering is described by the Debye-Bueche model [85-87] ... 

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