Self-correlation functions

In Eq. (19) we used the fact that the mean square displacement of the center-of-mass provides the diffusion constant according to DR = (l/6t)<(x0(t) — Xo(0))2>. For the special case of the self correlation function (n = m) Arnm(t) reveals the mean square displacement of a polymer segment. For t < xR we obtain... 

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... 

We observe that in spite of the complicated functional form, S(Q,t), like the self-correlation function, depends only on one variable, the Rouse variable... 

Measurements of the self-correlation function with neutrons are normally performed on protonated materials since incoherent scattering is particularly strong there. This is a consequence of the spin-dependent scattering lengths of hydrogen. Due to spin-flip scattering, which leads to a loss of polarization, this... 

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],... 

Thus, while the experimental situation has matured significantly with consistent results for pair and self correlation functions, in the area of simulations fully atomistic MD-calculations on long-enough chains are needed in order to resolve the existing discrepancies between different simulation approaches and experiment. 

Fig. 4.14 Results on fully protonated PIB by means of NSE [147]. a Time evolution of the self-correlation function at the Q-values indicated and 390 K. Lines are the resulting KWW fit curves (Eq. 4.9). b Momentum transfer dependence of the characteristic time of the KWW functions describing Sseif(Q,t) at 335 K (circles), 365 K (squares) and 390 K (triangles). In the scaling representation (lower part) the 335 K and 390 K data have been shifted to the reference temperature 365 K applying a shift factor corresponding to an activation energy of 0.43 eV. Solid (dotted) lines through the points represent (q-2 power laws. Full...

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 ... 

A reasonable approximation for the pair correlation function of the j8-process may be obtained in the following way. We assume that the inelastic scattering is related to imcorrelated jumps of the different atoms. Then all interferences for the inelastic process are destructive and the inelastic form factor should be identical to that of the self-correlation function, given by Eq. 4.24. On... 

The value of the jump distance in the )0-relaxation of PIB found from the study of the self-motion of protons (2.7 A) is much larger than that obtained from the NSE study on the pair correlation function (0.5-0.9 A). This apparent paradox can also be reconciled by interpreting the motion in the j8-regime as a combined methyl rotation and some translation. Rotational motions aroimd an axis of internal symmetry, do not contribute to the decay of the pair correlation fimction. Therefore, the interpretation of quasi-elastic coherent scattering appears to lead to shorter length scales than those revealed from a measurement of the self-correlation function [195]. A combined motion as proposed above would be consistent with all the experimental observations so far and also with the MD simulation results [198]. 

There is a fundamental question concerning the nature of the self-motion of protons in glass-forming polymers. In Sect. 4.1 we have shown that the existing neutron scattering results on the self-correlation function at times close to the structural relaxation time r (Q-region 0.2t) with a KWW-like functional form and stretching exponents close to jSsO.5. 

Figure 5.24 shows that this approach fails not only quantitatively but also qualitatively. Neither is the strong increase of the collective times relative to the self-motion in the peak region of Spair(Q) explained (this is the quantitative failure) nor is the low Q plateau of tpair(Q) predicted (this is the quaUtative shortcoming). We note that for systems hke polymers an intrinsic problem arises when comparing the experimentally accessible timescales for self- and collective motions the pair correlation function involves correlations between all the nuclei in the deuterated sample and the self-correlation function relates only to the self-motion of the protons. As the self-motion of carbons is experimentally inaccessible (their incoherent cross section is 0), the self counterpart of the collective motion can never be measured. For PIB we observe that the self-correlation function from the protonated sample decays much faster than the pair... 

Q-dependent Rouse rates were obtained by fitting each spectrum separately. For a comparison with the BS data (see below) the obtained values for Wf (Q) were transformed into average relaxation times for the Rouse self-correlation function by (r (Q))=18TiQ V[ Wf (Q)] (Eq. 3.18). 

G. Van Hove Self-Correlation Functions from Computer Experiments... 

We shall now discuss the non-Gaussian behavior of our self-correlation functions. Rahman32 pointed out that it is convenient to do this by introducing the coefficients a N(t) which for Gs(r, t) are defined as... 

These coefficients are strongly dependent on the number of molecules used in the simulations. For example, Figures 37 and 38 present the coefficients from the Stockmayer simulation using 216 and 512 molecules, respectively. The corresponding coefficients from the 216 and 512 molecule systems differ substantially from each other. Therefore, we feel that these coefficients from our simulations are only qualitative indications of the non-Gaussian behavior of our self-correlation functions. Figure 41 presents the coefficients from the modified Stockmayer simulation. Comparing the results for the two simulations we see ... 

None of the self-correlation functions is a Gaussian for all time. 

The self-correlation functions from the Stockmayer simulation are closer to Gaussians than those from the modified Stockmayer simulation. 

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