Debye-Bueche model

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...

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... 

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] ... 

The origin of the spherical polar coordinate system (r, 9, cp) is held fixed at the center of one particle and the polar axis (9 = 0) is set parallel to E. Let the electrolyte be composed of M ionic mobile species of valence zt and drag coefficient A,-(/ = 1, 2,. . . , M), and let nf be the concentration (number density) of the ith ionic species in the electroneutral solution. We also assume that fixed charges are distributed with a density of pflx. We adopt the model of Debye-Bueche where the polymer segments are regarded as resistance centers distributed in the polyelectrolyte... 

As in the case of static electrophoresis, we adopt the model of Debye-Bueche [22, 23] where the polymer segments are regarded as resistance centers distributed in the polyelectrolyte layer, exerting a frictional force on the liquid flowing in the... 

SANS experiments have indicated that blends of high density (linear) and long-chain branched low density polyethylenes (HDPE/LDPE) are homogeneous in the melt, though the components may separate on slow cooling due to the difference in crystallization mechanisms [43]. The semicrystalline blends form effectively two-phase systems in the solid state, and it was shown [43,44] that the Debye-Bueche (DB) [45,46] model was appropriate to describe the morphology, with a SANS cross section of the form... 

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. 

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... 

The discussion here follows the more quantitative treatment given by Debye and Bueche, although we seek to avoid their model consisting of a sphere throughout which the segment density is uniform, and beyond which it is zero. 

Affected by multiple scattering are, in particular, porous materials with high electron density (e.g., graphite, carbon fibers). The multiple scattering of isotropic two-phase materials is treated by Luzatti [81] based on the Fourier transform theory. Perret and Ruland [31,82] generalize his theory and describe how to quantify the effect. For the simple structural model of Debye and Bueche [17], Ruland and Tompa [83] compute the effect of the inevitable multiple scattering on determined structural parameters of the studied material. 

Comparisons of the data with predictions based on the theory of Brinkman (45) and Debye and Bueche (79) or on the model experiments of Kuhn and Kuhn (153) lead to very similar conclusions. 

Application of the concept of partial shielding by Debye and Bueche, and, independently, by Brinkman,to the pearl-necklace model gave, for viscosity and sedimentation. 

Two early theoretical models to rationalize this result were pursued the porous-sphere model of Debye and Bueche [1948], in which spherical beads representing the monomers are distributed uniformly in a spherical volume, and the more realistic pearl necklace model, proposed by Kuhn and Kuhn [1943], in which the beads are linked together by infinitely thin linkages. For each of these models, the principal challenge was to describe the flow of solvent around and within the volume occupied... 

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... 

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