Debye structure function

Fig. 2.9. Debye-structure function of an ideal chain with size Ro...

Sd is known as the Debye-structure function of an ideal chain, and a plot is shown in Fig. 2.9. In correspondence to the pair distribution function, the Debye-structure function can also be expressed in a reduced form, with v as general variable. Both the equations for the pair distribution function and for the scattering law indicate that all ideal chains are similar to each other, differing only in the length scale as expressed by Rq. 

According to Eq. (2.61), the Debye-structure function exhibits a characteristic asymptotic behavior... 

The range of the self-similar internal structure ends when r approaches Rq. Around q l/Jf o the curve deviates from the plateau and decays to zero. As mentioned earlier, universal behavior is also limited towards small distances. Specific polymer properties emerge, if one approaches distances in the order of the persistence length. For this reason, for polymers at higher s deviations from the Debye-structure function show up, beginning around q As indicated in Fig. 2.10, one observes an increase in the slope, and... 

Figure 2.13 shows the result of a light scattering experiment on the same system, a dilute solution of polystyrene (Mn = 8.79 10 ) in cyclohexane. The measurement was conducted exactly at the theta-point. As we have learned, ideal chains scatter according to the Debye-structure function, with the asymptotic limit Sd The data display the product... 

On the other hand, for this case the exact form of 5c is known. Since in melts polymer chains are ideal, Sc is given by the Debye-structure function (Eqs. (2.60) and (2.61)), multiplied by the volume fraction in order to account for the dilution... 

We still need an expression for the cross-response coefficients. The Debye-structure functions 5d(-Ra ) and 5D(i B ) are the Fourier-transforms of the pair distribution functions QAAi f ) and gBBi f") for the A- and B-monomers within their blocks. Considering this definition, it is clear how the coefficients for the cross responses and should be calculated. Obviously, they must correspond to the Fourier-transforms of the pair distribution functions gAB r) and gBAi f ) which describe the probability of finding a B- or A-monomer at a distance r from a A- or B-monomer respectively. Actually, both are identical... 

The continuous lines, which give perfect data fits, represent Debye structure functions (with minor corrections to account for polydispersity effects), thus proving the ideal behavior of chains. In addition, intensities were found to be proportional to the weight fraction of deuterated chains... 

By Fourier transforming the EXAFS oscillations, a radial structure function is obtained (2U). The peaks in the Fourier transform correspond to the different coordination shells and the position of these peaks gives the absorber-scatterer distances, but shifted to lower values due to the effect of the phase shift. The height of the peaks is related to the coordination number and to thermal (Debye-Waller smearing), as well as static disorder, and for systems, which contain only one kind of atoms at a given distance, the Fourier transform method may give reliable information on the local environment. However, for more accurate determinations of the coordination number N and the bond distance R, a more sophisticated curve-fitting analysis is required. 

A different analysis of the scattering pattern uses the Debye correlation function (14), derived for a random two-phase structure with sharp interfaces ... 

Where, /(k) is the sum over N back-scattering atoms i, where fi is the scattering amplitude term characteristic of the atom, cT is the Debye-Waller factor associated with the vibration of the atoms, r is the distance from the absorbing atom, X is the mean free path of the photoelectron, and is the phase shift of the spherical wave as it scatters from the back-scattering atoms. By talcing the Fourier transform of the amplitude of the fine structure (that is, X( )> real-space radial distribution function of the back-scattering atoms around the absorbing atom is produced. 

The used S5mbols are K, scale factor n, number of Bragg peaks A, correction factor for absorption P, polarization factor Jk, multiplicity factor Lk, Lorentz factor Ok, preferred orientation correction Fk squared structure factor for the kth reflection, including the Debye-Waller factor profile function describing the profile of the k h reflection. 

Several structure sizes caused by microphase separation occurring in the induction period as well as by crystallization were determined as a function of annealing time in order to determine how crystallization proceeds [9,18]. The characteristic wavelength A = 27r/Qm was obtained from the peak positions Qm of SAXS while the average size of the dense domains, probably having a liquid crystalline nematic structure as will be explained later, was estimated as follows. The dense domain size >i for the early stage of SD was calculated from the spatial density correlation function y(r) defined by Debye and Buche[50]... 

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