Debye-Bueche equation

Based on this equation, one can make a Debye-Bueche plot by plotting [i (q)] versus q and detemiine the slope and the intercept of the curve. The correlation length thus can be calculated as [21]... 

This is the three dimensional form of the Debye-Bueche equation which may be used to describe the scattering of oriented systems. For samples... 

Equation 4.130 is the usual form of the Debye-Bueche equation. This equation generally describes the scattering from a spherically symmetrical system provided a suitable expression can be found for the correlation function. For many systems, an exponential correlation... 

The Debye-Bueche equation is more general than just for light scattering and applies also to x-ray and neutron scattering, but in these cases, the p s and the K s will be different. 

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 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. 

In the second stage, we assume that the correlation between the neighboring domains increases. We assume the correlation function y(r) is y r) = exp(-r/( following Debye and Bueche [71], where is the averaged electron density. Substituting this y(r) into Equation (5.32) gives the Debye-Bueche equation ... 

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. 

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... 

Now we consider the electrokinetic behavior of soft particles, i.e., colloidal particles covered with a polymer layer (Figure 2.2). A number of theoretical studies have been made [34-46] on the basis of the model of Debye and Bueche [47], which assumes that the polymer segments are regarded as resistance centers distributed in the polymer layer, exerting frictional forces y on the liquid flowing in the polymer layer, where u the liquid flow velocity and y a frictional coefficient. The Navier-Stokes equation for the liquid flow inside the polymer layer is thus given by... 

Debye and Bueche (1960) have employed Einstein s (1910) expression for the turbidity of a two-component solution to calculate the turbidity of a monomolecular polymer solution over the whole concentration range (c in g/cm ) (sec Equations 2.1- 32, 2.4-7,-2d)... 

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