Coherent scattering function

The bond fluctuation model not only provides a good description of the diffusion of polymer chains as a whole, but also the internal dynamics of chains on length scales in between the coil size and the length of effective bonds. This is seen from an analysis of the normalized intermediate coherent scattering function S(q,t)/S(q,0) of single chains ... 

The Fourier transform of this quantity, the dynamic structure factor S(q, ffi), is measured directly by experiment. The structural relaxation time, or a-relaxation time, of a liquid is generally defined as the time required for the intermediate coherent scattering function at the momentum transfer of the amorphous halo to decay to about 30% i.e., S( ah,xa) = 0.3. 

Figure 12 Test of the factorization theorem of MCT for the intermediate coherent scattering function for the bead-spring model and a range of -values indicated in the Figure. Data taken from Ref. 132 with permission.

Structural relaxation in glass-forming polymers has been studied for many years using chemically realistic simulations. Most of the early work that examined incoherent, as well as coherent scattering functions, is more of a qualitative nature because of the unsatisfactory quality of the force fields employed and the severe limitations on the length of the MD simulations performed. Roe studied the slowdown of structural relaxation in a PE-like model140,141 as well as for polystyrene.142 More recently Okada et al.143,144... 

Fig. 4.27 Q-dependence of the amplitude of the relative quasi-elastic contribution of the -process to the coherent scattering function S (Q)/S(Q) obtained for PB from the hopping model solid line) with dp=l.5 A. The static structure factor S(Q) at 160 K [123] is shown for comparison dashed-dotted line)...

Fig. 4.28 a Form factor associated to the ds-unit of PB, which is schematically represented in the inset, b and c show the Q-dependence of the amplitude of the relative quasi-elastic contribution of the j -process to the coherent scattering function obtained for rotations of the ds-unit around an axis through the centre of mass of the unit and through the main chain, respectively, for different angles 30° (empty diamond), 60° (filled diamond), 90° (empty triangle) and 120° (filled triangle). The static structure factor S(Q) at 160 K [123] is shown for comparison (dashed-dotted line) (Reprinted with permission from [133]. Copyright 1996 The American Physical Society)... 

To evaluate the function, we omit the correlation between particles with different labels and consider the non-coherent scattering function... 

J. Kurkijarvi, A molecular dynamics investigation of the coherent scattering function of simple fluids, (Suomalainen Tiedeakatemia, 1970)... 

The incoherent and coherent scattering functions are related to the self- and pair-correlation functions... 

Coherent quasi-elastic neutron scattering due to the pair-correlation of atomic motions is more complex than incoherent scattering from single particle motion. Therefore, the theory of coherent scattering function is not yet fully developed. However, approximate expressions for the coherent scattering function exist (incoherent approximation). 

A)(Q) = elastic incoherent structure factor (EISF) B = rotational constant Dchem = chemical diffusion coefficient D, = tracer diffusion coefficient Jq x) = zeroth-order spherical Bessel function L(< ) = Lorentzian line shape = mass of cfth atom n(p) = atomic momentum distribution Q = momentum transfer 5 " (Q, to) = coherent scattering function tt>) = incoherent scattering function ... 

Interference of Waves. The coherent scattering property of x-rays is used in x-ray diffraction appHcations. Two waves traveling in the same direction with identical wavelengths, X, and equal ampHtudes (the intensity of a wave is equal to the square of its ampHtude) can interfere with each other so that the resultant wave can have anywhere from zero ampHtude to two times the ampHtude of one of the initial waves. This principle is illustrated in Figure 1. The resultant ampHtude is a function of the phase difference between the two initial waves. 

Fig. 6.5. Coherent structure function S(q) in absolute units in comparison to amorphous cell simulations [194] and neutron scattering data [185]...

The prerequisite for an experimental test of a molecular model by quasi-elastic neutron scattering is the calculation of the dynamic structure factors resulting from it. As outlined in Section 2 two different correlation functions may be determined by means of neutron scattering. In the case of coherent scattering, all partial waves emanating from different scattering centers are capable of interference the Fourier transform of the pair-correlation function is measured Eq. (4a). In contrast, incoherent scattering, where the interferences from partial waves of different scatterers are destructive, measures the self-correlation function [Eq. (4b)]. 

In the case of coherent scattering, which observes the pair-correlation function, interference from scattering waves emanating from various segments complicates the scattering function. Here, we shall explicitly calculate S(Q,t) for the Rouse model for the limiting cases (1) QRe -4 1 and (2) QRe > 1 where R2 = /2N is the end-to-end distance of the polymer chain. 

From Fig. 35, where the normalized coherent scattering laws S(Q, t)/S(Q,0) are plotted as a function of 2 (Q)t for Zimm as well as for Rouse relaxation, one sees that hydrodynamic interaction results in a much faster decay of the dynamic structure factor. 

The decay of the structural correlations measured by the static structure factor can be studied by dynamic scattering techniques. From the simulations, the decay of structural correlations is determined most directly by calculating the coherent intermediate scattering function, which differs from Eq. [1] by a time shift in one of the particle positions as defined in Eq. [2] ... 

In the discussion on the dynamics in the bead-spring model, we have observed that the position of the amorphous halo marks the relevant local length scale in the melt structure, and it is also central to the MCT treatment of the dynamics. The structural relaxation time in the super-cooled melt is best defined as the time it takes density correlations of this wave number (i.e., the coherent intermediate scattering function) to decay. In simulations one typically uses the time it takes S(q, t) to decay to a value of 0.3 (or 0.1 for larger (/-values). The temperature dependence of this relaxation time scale, which is shown in Figure 20, provides us with a first assessment of the glass transition... 

Figure 20 Temperature dependence of the a-relaxation time scale for PB. The time is defined as the time it takes for the incoherent (circles) or coherent (squares) intermediate scattering function at a momentum transfer given by the position of the amorphous halo (q — 1.4A-1) to decay to a value of 0.3. The full line is a fit using a VF law with the Vogel-Fulcher temperature T0 fixed to a value obtained from the temperature dependence of the dielectric a relaxation in PB. The dashed line is a superposition of two Arrhenius laws (see text).

Figure 21 Coherent intermediate scattering functions at the position of the amorphous halo versus time scaled by the a time, which is the time it takes the scattering function to decay by 70%. The thick gray line shows that the a-process can be fitted with a Kohlrausch-Williams-Watts (KWW) law.

Figure 22 von Schweidler fits (dotted lines) to the plateau decay of the coherent intermediate scattering function in the temperature interval 198-253 K. 

Polymer Melts Coherent Scattering and van Hove Correlation Functions. Part I Dynamics in the p-Relaxation Regime. 

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