Debye-Bueche approach

The application of the Porod equation or of the Debye-Bueche approach are particularly attractive because they offer the possibility to evaluate the interfacial area between the phases of the blend, and they are probably the only way to quantify such feature in polymer blends and composites. In fact, when the two polymers are mixed together in a blend, traditional methods based on the adsorption of small molecules, i.e. the BET approach, are inapplicable. Image analysis of TEM micrographs can in principle be an option, but it is extremely time consuming and it suffers from a number of limitations, such as dependence on sample preparation, on projection effects, and on image defocus. The validity of SAXS for the study of interpenetrating networks has been shown for several systems. ... 

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]. 

A further approach stems from the Debye-Bueche description of scattering from random heterogeneous media, which gives, for spherically symmetrical systems... 

There are a number of works on rubber-based blends which exploited a SAXS data analysis based on approximations such as the Guinier, Porod or Debye-Bueche. These approaches are very interesting because they offer valuable information on the size of dispersed domains within the matrix of a blend, without the need of intensive calculation and without having to develop complex theoretical models for the fitting of SAXS patterns. 

The success of such data analysis approach is necessarily linked to the reliability of the model chosen to describe the system. This limited the use of this method of interpretation in the study of blends, in favour of more model-independent methods, like the Porod and Debye-Bueche described above. However, some examples of the use of Equation (21.13) may be found in the literature. Micellar systems of block copolymers dispersed in a polyisoprene matrix were modelled by Pavlopoulos et al7 with the form factor of a homogeneous sphere, multiplied by a function accounting for the poly-dispersity in the micelles. In this case, the structure factor was neglected, due to the extreme dilution of the system. 

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] ... 

Our approach in this chapter is to alternate between experimental results and theoretical models to acquire familiarity with both the phenomena and the theories proposed to explain them. We shall consider a model for viscous flow due to Eyring which is based on the migration of vacancies or holes in the liquid. A theory developed by Debye will give a first view of the molecular weight dependence of viscosity an equation derived by Bueche will extend that view. Finally, a model for the snakelike wiggling of a polymer chain through an array of other molecules, due to deGennes, Doi, and Edwards, will be taken up. 

In most cases polymer solutions are not ideally dilute. In fact they exhibit pronounced intermolecular interactions. First approaches dealing with this phenomenon date back to Bueche [35]. Proceeding from the fundamental work of Debye [36] he was able to show that below a critical molar mass Mw the zero-shear viscosity is directly proportional to Mw whereas above this critical value r 0 is found to be proportional to (Mw3,4) [37,38]. This enhanced drag has been attributed to intermolecular couplings. Ferry and co-workers [39] reported that the dynamic behaviour of polymeric liquids is strongly influenced by coupling points. 

The approaches of colloidal optics are required for analysis of light scattering on heterogeneous gels. In particular, Rayleigh-Debye s approximation with the correlation function formedism finds application (see subsection 2.1.2) (Gallacher and Betterheim, 1962 Bueche, 1970 Pokrovski et al., 1976 Volkova et al., 1987). 

---