Debye functions

Debye first calculated the form factor for scattering from an ideal chain in 1947. This form factor will be useful in interpretation of a wide variety of scattenng experiments on polymers. The form factor for an ideal linear chain is obtained by averaging the form factor of isotropic scatterers [Eq. (2.143)] over the probability distribution for distances 7 ,y between monomers i and j on the ideal chain  

The integral over Ry can be evaluated by converting it into a Gaussian integral (by writing the complex representation of sine and completing the square in the exponent)  

Replacing the summations over monomer indices by integrations provides the integral form of the form factor of an ideal chain  

This form factor of an ideal linear polymer is called the Debye function and can be rewritten in terms of the product of the square of scattering wavevector q and the mean-square radius of gyration of the chain RI)  

In the limit of small scattering angles, where qRg 1, the exponential can be expanded to simplify the Debye function  

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The discovery of a transition which we identify with this has been reported by Simon, Mendelssohn, and Ruhemann,16 who measured the heat capacity of hydrogen with nA = 1/2 down to 3°K. They found that the heat capacity, after following the Debye curve down to about 11°K, rose at lower temperatures, having the value 0.4 cal/deg., 25 times that of the Debye function, at 3°K. The observed entropy of transition down to 3°K, at which the transition is not completed, was found to be about 0.5 E.U. That predicted by Eq. (15) for the transition is 2.47 E.U. 

To better understand the nature of the two types of dynamics, the Cole-Cole plots were plotted, and they clearly indicate the evolution from FR to SR with the changing temperature. Strikingly, the plots can be nicely fitted by the sum of two modified Debye functions (Figure 3.8b, inset), which is used to explain such a complex relaxation process. Here, the two separated relaxation processes are most likely associated with distinct anisotropic centres, that is, two Dy ions with... 

The shape of the Debye function corresponds to the uppermost curve in Fig. 3.2. 

Further note that for t=0 Eq. 3.24 does not resemble the Debye function but yields its high Q-limiting behaviour i.e. it is only valid for QR >1. In that regime the form of Dr immediately reveals that the intra-chain relaxation increases in contrast to normal diffusion ocQ, Finally, Fig. 3.2 illustrates the time development of the structure factor. 

Below the ODT such a label highlights the polymer-polymer interface. A main peak around Q" =0.02 A" corresponding to a lamellar periodocity 2 n/diain with di j =3l5 A is observed. Its visibility results from the asymmetric nature of the diblock. We note the existence of a second order peak, which is well visible at Todt=433 K. At large Q>Q the scattering is dominated by the form factor of the PEP-label in the environment of the deuterated monomers at the interface. This form factor may be described by a Debye function A)ebye( ) (Eq. 3.23). The absolute cross-section for these labels is given by ... 

Fig. 6.24 Comparison of the scattering from a semidilute PDMS solution under normal polymer contrast (I) revealing the correlation length with the single chain scattering (JJ as obtained by a zero average contrast preparation (see text). The line through 4 represents a Debye function with R =7 nm whereas the line through I corresponds to a Lorentzian (Orn-stein-Zernike) with a correlation length f=l nm. (Reprinted with permission from [325]. Copyright 1991 EDP Sciences)...

Fig. 3. Temperature variation of the cubic lattice parameter of Gdlnj measured by x-ray powder diffraction (this work). The full line corresponds to a fit of a Debye model to the whole temparature range, the dashed line shows an extrapolation from the paramagnetic range obtained from fitting the Debye function only to the data points...

Fig. 9. Anisotropic thermal expansion of Gd2ln measured by x-ray powder diffraction (Gratz and Lindbaum 1998). The lines are the result of fitting Debye functions. The arrows indicate the two magnetic transitions at...

In the last line we used Eq. (3.22) to introduce the Debye function. We will find in Sect. 5.4 that t/o(q) shows all the aspects of screening. Here we want to show that uo(q) can be introduced to replace consistently all vertices... 

In many perturbative results there naturally occurs the Debye function averaged over polydispersity. We thus introduce... 

We indeed encountered additive renormalization before. In our calculation of f.ip(n) (Sect. 5.4.3) we had to subtract the single chain part from the one loop correction, before we could replace the sum over discrete segments by the Debye function, which is calculated for a continuous chain. (See Eq. (3.22).) In the result (5.6 ) this introduced the term -n/ c (f) to shift the chemical potential. All the... 

The simple expression (14,5) could lead to the erroneous conclusion that the momentum dependence of the autocorrelation function in tree approximation follows the Debye function, reproducing the result for a noninteracting chain. This conclusion is false, since the uncritical surface and thus both Nr and q2 nontrivially depend on q3. The effects show up in the region of large momenta q2 L... 

Equation (13.24) shows that q2i 2 > 1 for / bounded necessarily implies q2 Nr 1, so that we can replace the Debye function by the asymptotic... 

In the dilute region w < 1 the autocorrelation function directly crosses over from Debye-type behavior on scales q2Ji2 < 1 to the large momentum behavior as discussed above. For semidilute systems w intermediate regime, governed by the asymptotics of the Debye function,... 

This form for PG(an) is the widely used Debye function. Note that for homopolymer chains, n is large (large degree of polymerization) and a is small (SANS instruments do not see monomer chemistry) so that these last expressions can be used. However, for short block copolymers, n is not necessarily large and the more general equations are more appropriate to use. 

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