Self-motion

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 0.2 Q 1 A ) are well described in terms of sublinear diffusion, (r (t))ocfP. This qualification rests on the observation of Sseif(Q t) with a KWW-like functional form and stretching exponents close to jSsO.5. 

Turning to the self-motion of PVE protons, Fig. 5.18 shows the NSE data obtained for Sseif(Q,t). If indeed there existed a second Gaussian subdiffusive regime at length scales shorter than that of the Rouse dynamics, then following Eq. 3.26 also for Q Qr it should be possible to construct a Q-independent (r t)). This is shown in Fig. 5.19. For Q 0.20 A a nearly perfect collapse of the experimental data into a single curve is obtained. The time dependence of the determined can be well described by a power law with an exponent of 

The time needed by a proton to move as far as the average interchain distance dchain  

From Spair(Q) measurements (see the insert in Fig. 5.20) Qmax=0.9 A reveal-ing dchain=27i/Qmax 7 A. Such a distance is covered by the protons in about 1 ns, just in the middle of the time region where the sublinear diffusive regime has 

2 A (empty square), 0.3 A (filled diamond) and 0.4 A (empty triangle). The solid line represents r (t))-f -. The dashed lines mark the characteristic times and lengths discussed in the text. The thick solid line in the upper right corner displays the mean squared displacement in the Rouse regime. The two dotted lines extrapolate into the crossover regime. (Reprinted with permission from [39]. Copyright 2004 EDP Sciences) 

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Phys. Condens. Matter, 15, S1127 (2003). Self-Motion and the a-Relaxation in Glass-Forming Polymers. Molecular Dynamic Simulation and Quasielastic Neutron Scattering Results in Polyisoprene. 

To date, incoherent quasi-elastic neutron scattering experiments on the a-relaxation regime of glass-forming polymers have revealed the following main features for the self-motion of hydrogens ... 

Fig. 4.13 Momentum transfer dependence of the characteristic time associated to the self-motion of protons in the a-relaxation regime Master curve (time exponentiated to p) constructed with results from six polymers polyisoprene (340 K, p=0.57) (filled square) [9] polybutadiene (280 K, p=0Al) (filled circle) [146] polyisobutylene (390 K, p=0.55) (empty circle) [147] poly (vinyl methyl ether) (375 K, f=0A4) (filled triangle) [148] phenoxy (480 K, p=0A0) (filled diamond) [148] and poly(vinyl ethylene) (340 K, p=0A3) (empty diamond) [ 146]. The data have been shifted by a polymer dependent factor Tp to obtain superposition. The solid line displays a Q -dependence corresponding to the Gaussian approximation (Eq. 4.11). (Reprinted with permission from [149]. Copyright 2003 Institute of Physics)...

We may now discuss the imphcations of the results foimd for the self-motion of hydrogens in the a-relaxation regime by neutron scattering. It is well known that for some simple cases - free nuclei in a gas, harmonic crystals. 

These two facts motivated a critical check of the validity of Eq. 4.11 in a wide Q-range [9,105,154,155]. For this purpose the information obtainable from fully atomistic MD simulations was essential. The advantage of MD simulations is that, once they are validated by comparison with results on the real system, magnitudes that cannot be accessed by experiments can be calculated, as for example the time dependence of the non-Gaussian parameter. The first system chosen for this goal was the archetypal polymer PL The analysis of the MD simulations results [105] on the self-motion of the main chain hydrogens was performed in a similar way to that followed with experimental data. This led to a confirmation of Eq. 4.11 beyond the uncertainties for Q<1.3 A (see Fig. 4.15). However, clear deviations from the Q-dependence of the Gaussian behaviour... 

Finally we compare the temperature dependencies reported for the structural relaxation and the self-motion of hydrogens studied by NSE. For PI, the shift factors used for the construction of the master curve on Q,T) (Fig. 4.17) are identical to those observed for the structural relaxation time [8]. This temperature dependence also agrees with DS and rheological studies. The case of PIB is more complex [ 147]. The shift factors obtained from the study of Teif(Q>T) (Fig. 4.14b) reveal an apparent activation energy close to that reported from NMR results (-0.4 eV) [136]. This temperature dependence is substantially weaker than that observed for the structural relaxation time (=0.7 eV, coinciding with rheological measurements) in the same temperature range (see Fig. 4.20). 

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)...

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]. 

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)...

We now address the connection between the self-motions of protons and the collective motions. Figure 5.24 shows that the structural relaxation time is about one order of magnitude slower than the characteristic time for proton self-motion at However, the power-law increase of towards low Q slows... 

The relation between collective and self-motion in simple monoatomic liquids was theoretically deduced by de Gennes [233] applying the second sum rule to a simple diffusive process. Phenomenological approaches like those proposed by Vineyard [ 194] and Skbld [234] also relate pair and single particle motions and may be applied to non-exponential functions. The first clearly fails to describe the PIB results since it considers the same time dependence for both correlators. Taking into account the stretched exponential forms for Spair(Q.t) (Eq. 4.21) and Sseif(Q>0 (Eq 4.9), the Skold approximation ... 

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... 

Finally, it is worthy of remark that, though the comparison between the timescales leads to an almost perfect agreement between the predictions of the Allegra and Ganazzoli model and the collective and self-motion results, it is evident that clear differences appear when comparing the spectral shapes of the respective functions. The model delivers close to exponential decays for both correlators while experimentally one observes significantly stretched relaxation function (j0=O.5). 

Stages of Self-Motion Processing in Primate Posterior Parietal Cortex... 

Neural Mechanisms for Self-Motion Perception in Area MST... 

For hydrocarbons in zeolites, only incoherent scattering has to be considered because of the large incoherent cross section of hydrogen. The neutron intensity scattered follows the incoherent scattering law 5i c(Q, (o), which is related to the self-motion of protons, where AQ and Aw denote the neutron momentum transfer and the neutron energy transfer, respectively. 

The virtual mass coefficient for a sphere in an invicid fluid is thus Cy = The basic model (5.111) is often slightly extended to take into account the self-motion of the fluid. In general the added mass force is expressed in terms of the relative acceleration of the fluid with respect to the particle acceleration. 

The quantity rj (t) — r, (0) introduces sensitivity to relative motion of two scattering centres, so that the interpretation of coherent is more complicated than the self-motion measured by 5incoherent and by PFG NMR. 

These results show that the main contribution to the scattered j.ntensity comes from the water molecules. For Q > 0.4 A, it is reasonable to assume that the corresponding scattering is incoherent since the broad peak aroud 1.2 A comes mainly from the polymer. Consequently, the analysis of the quasielastic spectra may be arried out in terms of self motion of the protons. For Q intense small angle scattering due to water means that the coherent scattering is no more negligeable and that it may be important to include collective effects in the analysis. In the present work, we hav restricted the NQES study to Q values between 0.4 and 1.2 A so that the latter effects can be neglected. 

The time dependence is the stretched exponential function of time, exp[-(r/TK) ], with fS = 0.5. The experiments performed at the backscattering instruments by Niedzwiedz et al. (2007) are capable of measuring the self-motion of PEG chains up to 1 ns. The mean square displacements of PEG chains in the 35% PEG/65% PMMA sample were obtained by Fourier transformation of backscattering spectra at G = 2.4 and 3.2 nm and temperatures from 350 K to 400 K. The results show the PEG chain dynamics are in agreement with the Rouse model prediction for f < 1 ns (see Figure 1 in Niedzwiedz etal. (2007)). 

The longitudinal current spectra are calculated from Eqs. (5.89) and (5.116) with the exponential model for K. k,t) given by Eq. (5.120), and the results for A-A and B-B pairs are presented as solid lines in Figs. 5.4 and 5.5, respectively. Also added as dashed lines in these figures are the single-particle contributions (i.e., contributions from the self-motions) calculated from a similar theory [54] with the aim of elucidating the collective nature of the excitations in the small-A region. 

Because of its sensitivity to self-motion of the labelled chain, as indicated in equation (9.7), PGSE NMR has proven of especial value in studying the... 

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