Self-dynamic structure factor

As mentioned before, due to the coupling between different modes of the solvent, all solvent dynamic quantities are interdependent and need to be calculated self-consistently. An additional self-consistency is present in the calculation of friction, since the self-dynamic structure factor itself depends on the friction. 

From the above discussion, it is obvious that the mode coupling theory calculations are quite involved and numerically formidable. Balucani et al. [16] have made some simple approximations to incorporate the self-consistency between the self-dynamic structure factor and the friction. This required the knowledge of only the zero frequency friction. The full self-consistent calculation is more elaborate and will be discussed later in this chapter. 

The solute dynamic variables required to calculate the density and current contribution are the self-dynamic structure factor, Fs(q, t) and inertial part of the self-intermediate structure factor, Fs0(q, t). Fs0(q, f) is given by... 

The other dynamic variables required to calculate Rpp(t) and Rrr(t) are the dynamic structure factor of the solvent, F(q, t), the inertial part of the dynamic structure factor, Fo(q, t), the transverse current autocorrelation function of the solvent, C (q,t), the inertial part of the same, Ctf0(q, t), the self-dynamic structure factor of the solute, Fs(q, t), and the inertial part of the self-dynamic structure factor of the solute, Fs0(q,t). The expressions for all the above-mentioned dynamic quantities are similar to that given in Section IX but in two dimensions. 

The inconsistency in this approach and all other mode coupling theoretical approaches [9, 37, 57, 176] is that a finite diffusion coefficient has been assumed to define the diffusive behaviour of the self-dynamic structure factor, and then it has been concluded that this diffusion coefficient itself diverges. 

In the present theoretical approach this inconsistency is avoided by not assuming any diffusion coefficient in defining the self-dynamic structure factor. 

The self-dynamic structure factor of the solute at all time is calculated from the mean square displacement (MSD) using the following definition ... 

Calculational procedure of all the dynamic variables appearing in the above expressions—namely, the dynamic structure factor F(q,t) and its inertial part, Fo(q,t), and the self-dynamic structure factor Fs(q,t) and its inertial part, Fq (q, t) —is similar to that in three-dimensional systems, simply because the expressions for these quantities remains the same except for the terms that include the dimensionality. Cv(t) is calculated so that it is fully consistent with the frequency-dependent friction. In order to calculate either VACF or diffusion coefficient, we need the two-particle direct correlation function, c(x), and the radial distribution function, g(x). Here x denotes the separation between the centers of two LJ rods. In order to make the calculations robust, we have used the g(x) obtained from simulations. 

Brodeck et al., 2010). In the Gaussian approximation for displacements, the segmental self-correlation function relates directly to (r (r)), resulting in the intermediate self-dynamic structure factor having the Gaussian form (Niedzwiedz et al., 2007),... 

Inelastic neutron scattering is a technique that has been widely used both in the liquid and in the solid states to measure the stmcture and dynamics at small (that is, molecular) length scales. In an incoherent inelastic neutron-scattering experiment, the measured quantity is the self-dynamic structure factor Ss(Q, (o), which gives information, as in the liquid state, of the self-diSiision coefficient of the water molecules. Ss(Q, (o) is the Fourier transform of the intermediate self-scattering function Fg(Q, t), which is defined by... 

Figure 4 is a comparison of the behaviour of these functions, along with W( ) for a particle moving through a typical liquid. Fourier transforming the Fs(Q, )s gives the corresponding forms for the (self) dynamic structure factors ... 

A substantially similar process beginning with the self-dynamic structure factor, Eq. 4.7, leads to... 

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)]. 

The self-correlation function leads directly to the mean square displacement of the diffusing segments Ar2n(t) = <(rn(t) — rn(0))2>. Inserting Eq. (20) into the expression for Sinc(Q,t) [Eq. (4b)] the incoherent dynamic structure factor is obtained... 

Fig. 4. Neutron spin echo spectra for the self-(above) and pair-(below) correlation functions obtained from PDMS melts at 100 °C. The data are scaled to the Rouse variable. The symbols refer to the same Q-values in both parts of the figure. The solid lines represent the results of a fit with the respective dynamic structure factors. (Reprinted with permission from [41]. Copyright 1989 The American Physical Society, Maryland)...

Though the functional form of the dynamic structure factor is more complicated than that for the self-correlation function, the data again collapse on a common master curve which is described very well by Eq. (28). Obviously, this structure factor originally calculated by de Gennes, describes the neutron data well (the only parameter fit is W/4 = 3kBT/2/C) [41, 44],... 

Fig. 4.20 Temperature dependence of the average relaxation times of PIB results from rheological measurements [34] dashed-dotted line), the structural relaxation as measured by NSE at Qmax (empty circle [125] and empty square), the collective time at 0.4 A empty triangle), the time corresponding to the self-motion at Q ax empty diamond),NMR dotted line [136]), and the application of the Allegra and Ganazzoli model to the single chain dynamic structure factor in the bulk (filled triangle) and in solution (filled diamond) [186]. Solid lines show Arrhenius fitting curves. Dashed line is the extrapolation of the Arrhenius-like dependence of the -relaxation as observed by dielectric spectroscopy [125]. (Reprinted with permission from [187]. Copyright 2003 Elsevier)...

In [189] a simple two state model for the dynamic structure factor corresponding to the Johari-Goldstein jS-process was proposed. In this model the jS-relaxation is considered as a hopping process between two adjacent sites. For such a process the self-correlation function is given by a sum of two contributions ... 

Fig. 5.23 Time evolution of the three functions investigated for PIB at 390 K and Q=0.3 A"h pair correlation function (empty circle) single chain dynamic structure factor (empty diamond) and self-motion of the protons (filled triangle). Solid lines show KWW fitting curves. (Reprinted with permission from [187]. Copyright 2003 Elsevier)...

Note that the expression for Rlpp itself depends on the dynamic structure factor. Thus the calculation of F(q, t) and Rlpf) should be performed self-consistently. We shall discuss in the next section that this self-consistency provides a feedback mechanism and leads to a divergence of the dynamic structure factor at zero frequency. 

In this section we will not discuss the glass transition in every detail. Instead, the merit of the self-consistent calculation of the dynamic structure factor and how it works as a nonlinear feedback mechanism near glass transition will be discussed here. 

In Section XI we discussed the calculational method of the dynamic structure factor in the supercooled regime. We also discussed that the memory function F// needs to be calculated self-consistently with the dynamic structure factor itself. Near the glass transition, the dynamic structure factor is expected to diverge. This leads to an infinite loop numerically formidable calculation. 

Finally, we comment on the difference between the self part and the full density autocorrelation function. The full density autocorreration function and the dynamical structure factor ire experimentally measured, while in the present MD simulation only the self pairt was studied. However, the difference between both correlation functions (dynamical structure factors) is considered to be rather small except that additional modes associated with sound modes appear in the full density autocorrelation. We have previously computed the full density autocorrelation via MD simulations for the same model as the present one, and found that the general behavior of the a relaxation was little changed. General trends of the relaxation are nearly the same for both full correlation and self part. In addition, from a point of numerical calculations, the self pMt is more easily obtained than the full autocorrelation the statistics of the data obtained from MD simulatons is much higher for the self part than for the full autocorrelation. 

Concerning the dynamic structure factor, we shall confine our attention to the incoherent case, where the self-correlation function B(0, ) only is required since we have q -4 1, it may be shown that the results are essentially valid for the coherent case as well (long-time limit) [86]. From,Eqn. (3.1.18) we get... 

Fig. 11 The dynamic structure factor C(, r) of polybutadiene star 12880 (nominally f = 128, Ma = 80kgmol ) in cyclohexane at ci = 0.016gmL and q = 0.035nm , along with the fit (solid line) from the ILT analysis. The corresponding relaxation distribution function L(ln(T)) (shown here for f i and q = 0.023gmL ) embraces the cooperative diffusion (1), the collective apparent diffusion (2), and the self-diffusion (3). The slowing-down of the middle structural mode (2) and the increase of its intensity with q are shown in the upper inset whereas the lower cartoon illustrates the liquid-like ordering [43,189]. The core regions are drawn out of scale (larger) for clarity...

The intermediate scattering function F(q,t) and the dynamic structure factor S(q,co) can also be split into their respective self and distinct parts, as in... 

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