Normalized intermediate scattering functions

Fig. 6.18 Normalized intermediate scattering function from centre-labelled 18-arm PI solutions. Collective corresponds to a 4.85% solution of labelled stars, whereas the self data stem from a solution of 1% labelled and 16.6% non-labelled stars. Note the maximum Fourier time of 350 ns (A=1.9 nm), which was obtained at the INI 5 in the case of these strong scattering samples. (Reprinted with permission from [304]. Copyright 2002 Springer)...

Figure 4.20 Normalized intermediate scattering function versus time at a wavenumber near the peak in the static scattering function, for suspensions of hard, noninteracting, spherical particles. The curves are labeled by the volume fraction. The curve for 0 = 0.565 is in the glassy state where the relaxation is arrested after a short time of relaxation. Above the concentration 0 = 0.494 the equilibrium structure would be colloidal crystalline. (From van Megen and Pusey 1991, reprinted with permission from the American Physical Society.)...

Figure 4.24 Normalized intermediate scattering function versus time for NiZr from molecular d)uiamics simulations at wavevectors q = (a) 21.6 nm- and (b) 10.8 nm. The lines are fits using a modified mode-coupling theory. (From Teichler 1996, reprinted with permission from the American Physi-cal Society.)...

Fig. 7 Normalized intermediate scattering function obtained for benzene in Na-Y zeolite (1 molecule per supercage, on average, T = 475K, Q = 0.3A i). The fraction of the intensity due to the zeolite is represented as a dashed line...

Fig. 11 Normalized intermediate scattering functions obtained for isobutane in sUicalite at 550 K, for two different Q values ( ) 0.08 A , ( ) 0.2...

Richter and coworkers [80] have measured normalized intermediate scattering functions I Q,t)/I(Q,0) from the a-process of PB using a neutron spin echo technique in a temperature range between 200 K and 280 K above the glass transition temperature (Tg=181 K). These measurements were performed at a first maximum position (=1.48 A ) of the structure factor S(Q). They found that... 

Figure 4 Plot of the normalized intermediate scattering function [S(Q,f)/S(Q.O)] versus time for hydrogenous polyethylene (12% w/w) in deuterated polyethylene at 509 K for Q = 0.050 and 0.077 A . Also shown are the predictions from the reptation model (solid line) and competing (dashed and dot-dashed lines). (Data recorded with IN15 at the ILL). Reproduced with permission from Schleger P, Farago B, Lartigue C, Kollmar Aand Richter D (1998) Clear evidence for reptation in polyethylene from neutron spin echo spectroscopy. Physical Review Letters 81 124-127.

MCT gives a self-consistent equation of motion of the (normalized) intermediate scattering function. This is the autocorrelation function of coherent density fluctuations with wavevector q, defined by... 

The information on the normalized intermediate scattering function is given by the ratio of the oscillation amplitude and the limiting intensities (lup- fdown). The amplitude a=R S(Q, t) contains information on the instrumental resolution R = t] lo, t) jexp(-2 2o) and S(Q, t) and can be extracted from the results of the symmetry scan... 

The intermediate scattering functions can be expressed in terms of the normalized properties Ulk and U2k ... 

As discussed, it is most appropriate to focus attention directly on the intermediate scattering function F(q, t), which is to be considered as the key random variable. If P[F(q, f)] is the normalized probability distribution of the functional F(q,t), then the modified MCT expression for friction takes the following form ... 

Figure 9. Time dependencies of the single-particle and the collective intermediate scattering functions compared for two different solute sizes at a particular wavenumber q = 6.001 at reduced temperature T" = 0.75 and in the normal density regime (p = 0.89). The solid line represents the collective intermediate scattering function. The long-dashed line is the single-particle intermediate scattering function for solute-solvent size ratio 1.0 and the short-dashed line is for solute-solvent size ratio 0.5. The plots show that the decoupling of the solute motion from the solvent dynamics increases as the solute size is decreased. The time is scaled by rsct where TJC = [mo2/kBT] 2.

Here, i is the cosine of the angle between q and the surface normal, km-m is a cut-off describing the most extended bending mode, which still fits into the persistent surface area with size f, and J0 is the Bessel function of the order 0. This approach was already used to describe experimental intermediate scattering functions [76] and an example will be given in the next subsection. The same method also applies to lamellar phases since only persistent areas are assumed and no further assumptions about the geometry of the surfactant layer are made. 

It is also known that the normalized second-order intensity correlation function is related to the intermediate scattering function following... 

Assuming a log-normal distribution of friction coefficients introduces one new parameter, the distribution width c. Fitting yields cr= 1.6. With that value, the decay of the intermediate scattering function for the three lowest Q values can be perfectly reproduced. The predicted relaxation for Q = 3 nm is somewhat more pronounced than observed.The log-normal distribution with cr= 1.6 describes the broad distribution of local relaxations as observed by the TOF and BSS experiments without further parameters as depirted in Figure 44. 

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 ... 

Fig. 4.1 a Typical time evolution of a given correlation function in a glass-forming system for different temperatures (T >T2>...>T ), b Molecular dynamics simulation results [105] for the time decay of different correlation functions in polyisoprene at 363 K normalized dynamic structure factor at the first static structure factor maximum solid thick line)y intermediate incoherent scattering function of the hydrogens solid thin line), dipole-dipole correlation function dashed line) and second order orientational correlation function of three different C-H bonds measurable by NMR dashed-dotted lines)... 

The directional distribution of the scattering intensity can be described by phase functions. A phase function is defined as the ratio of scattering intensity in a direction to the scattering intensity in the same direction if the scattering is isotropic. Thus, it is a normalized function and is defined over all directions. Typical phase functions for small, intermediate, and large values of f and n, are illustrated in Fig. 4.4, with the spheres assumed to be nonabsorbing [Tien and Drolen, 1987]. It is shown that the phase functions mainly vary... 

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