Amorphous scattering function

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 11 MCT P-scaling for the amplitudes of the von Schweidler laws fitting the plateau decay in the incoherent intermediate scattering function for a R value smaller than the position of the amorphous halo, q = 3.0, at the amorphous halo, q = 6.9, and at the first minimum, q = 9.5. Also shown with filled squares is the P time scale. All quantities are taken to the inverse power of their predicted temperature dependence such that linear laws intersecting the abscissa at Tc should result. 

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.

Different methods of measuring the scatter function of corpuscular as well as wave radiations are used for indirect measurements of microscopic heterogeneities or changes in morphology, which develop under the action of load, especially development, distribution and size of crazes and craze fibrils in amorphous and ruhher-toughened polymers, and change of lamellae in semiciystalline polymers. 

Unlike the solid state, the liquid state cannot be characterized by a static description. In a liquid, bonds break and refomi continuously as a fiinction of time. The quantum states in the liquid are similar to those in amorphous solids in the sense that the system is also disordered. The liquid state can be quantified only by considering some ensemble averaging and using statistical measures. For example, consider an elemental liquid. Just as for amorphous solids, one can ask what is the distribution of atoms at a given distance from a reference atom on average, i.e. the radial distribution function or the pair correlation function can also be defined for a liquid. In scattering experiments on liquids, a structure factor is measured. The radial distribution fiinction, g r), is related to the stnicture factor, S q), by... 

Figure 4.7c illustrates how x-ray diffraction techniques can be applied to the problem of evaluating 6. If the intensity of scattered x-rays is monitored as a function of the angle of diffraction, a result like that shown in Fig. 4.7c is obtained. The sharp peak is associated with the crystalline diffraction, and the broad peak, with the amorphous contribution. If the area A under each of the peaks is measured, then... 

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

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