Solids dynamic structures

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)...

Figure 6 shows the measured dynamic structure factors for different momentum transfers. The solid lines display a fit with the dynamic structure factor of the Rouse model, where the time regime of the fit was restricted to the initial part. At short times the data are well represented by the solid lines, while at longer times deviations towards slower relaxations are obvious. As it will be pointed out later, this retardation results from the presence of entanglement constraints. Here, we focus on the initial decay of S(Q,t). The quality of the Rouse description of the initial decay is demonstrated in Fig. 7 where the Q-dependence of the characteristic decay rate R is displayed in a double logarithmic plot. The solid line displays the R Q4 law as given by Eq. (29). 

Fig. 6. Dynamic structure factor as observed from PI for different momentum transfers at 468 K. ( Q = 0.038 A"1 Q = 0.051 A-1 A Q = 0.064 A-1 O Q = 0.077 A"1 Q= 0.102 A-1 O Q = 0.128 A 1 Q = 0,153 A "" 11. The solid lines display fits with the Rouse model to the initial decay. (Reprinted with permission from [39]. Copyright 1992 American Chemical Society, Washington)...

Fig. 12a, b. Dynamic structure factor for two polyethylene melts of different molecular mass a Mw = 2 x 103 g/mol b Mw = 4.8 x 103 g/mol. The momentum transfers Q are 0.037, 0.055, 0.077, 0.115 and 0.155 A-1 from top to bottom. The solid lines show the result of mode analysis (see text). (Reprinted with permission from [36]. Copyright 1994 American Chemical Society, Washington)... 

Fig. 40a, b. NSE spectra of a dilute solution under 0-conditions (PDMS/ d-bromobenzene, T = = 357 K). a S(Q,t)/S(Q,0) vs time t b S(Q,t)/S(Q,0) as a function of the Zimm scaling variable ( t(Q)t)2/3. The solid lines result from fitting the dynamic structure factor of the Zimm model (s. Tablet) simultaneously to all experimental data using T/r s as adjustable parameter. 

In contrast to -conditions a large number of NSE results have been published for polymers in dilute good solvents [16,110,115-120]. For this case the theoretical coherent dynamic structure factor of the Zimm model is not available. However, the experimental spectra are quite well described by that derived for -conditions. For example, see Fig. 42a and 42b, where the spectra S(Q, t)/S(Q,0) for the system PS/d-toluene at 373 K are shown as a function of time t and of the scaling variable (Oz(Q)t)2/3. As in Fig. 40a, the solid lines in Fig. 42a result from a common fit with a single adjustable parameter. No contribution of Rouse dynamics, leading to a dynamic structure factor of combined Rouse-Zimm relaxation (see Table 1), can be detected in the spectra. Obviously, the line shape of the spectra is not influenced by the quality of the solvent. As before, the characteristic frequencies 2(Q) follow the Q3-power law, which is... 

Fig. 57. Relaxation spectra of the fully labelled star ( , + ) and the star core (, ) at two different Q-values. The solid lines represent the result of a fit for the Zimm dynamic structure factor to the initial relaxation of the fully labelled star. The dashed lines are visual aids showing the retardation of the relaxation for the star core. (Reprinted with permission from [154]. Copyright 1990 American Chemical Society, Washington)...

Fig. 69.oNSE spectra of 2% diblock copolymer (d-PS and h-PI blocks) in deuterated n-decane. The Q/A-1 values are 0.026, V 0.032, 0.038, x 0.051, O 0.064, A 0.089, O 0.115. Experimental data and theoretical dynamic structure for breathing modes are compared (solid lines). (Reprinted with permission from [174]. Copyright 1993 The American Physical Society, Maryland)... 

In the solid, dynamics occurring within the kHz frequency scale can be examined by line-shape analysis of 2H or 13C (or 15N) NMR spectra by respective quadrupolar and CSA interactions, isotropic peaks16,59-62 or dipolar couplings based on dipolar chemical shift correlation experiments.63-65 In the former, tyrosine or phenylalanine dynamics of Leu-enkephalin are examined at frequencies of 103-104 Hz by 2H NMR of deuterated samples and at 1.3 x 102 Hz by 13C CPMAS, respectively.60-62 In the latter, dipolar interactions between the 1H-1H and 1H-13C (or 3H-15N) pairs are determined by a 2D-MAS SLF technique such as wide-line separation (WISE)63 and dipolar chemical shift separation (DIP-SHIFT)64,65 or Lee-Goldburg CP (LGCP) NMR,66 respectively. In the WISE experiment, the XH wide-line spectrum of the blend polymers consists of a rather featureless superposition of components with different dipolar widths which can be separated in the second frequency dimension and related to structural units according to their 13C chemical shifts.63... 

Secondly, for some crystalline systems, the structure obtained by diffraction techniques may be incomplete. For example, in some cases the diffraction data may not reveal dynamic aspects of the solid-state structure (as in the case of fluxional organo-metallics) and in others it may not be possible to distinguish clearly between different atoms (as for example 27A1 and 29Si in zeolites) and a combination of the NMR and x-ray data will yield a more complete and meaningful description of the structure. 

Molecular Motions and Dynamic Structures. Molecular motions are of quite general occurrence in the solid state for molecules of high symmetry (22,23). If the motion does not introduce disorder into the crystal lattice (as, for example, the in-plane reorientation of benzene which occurs by 60° jumps between equivalent sites) it is not detected by diffraction measurements which will find a seemingly static lattice. Such molecular motions may be detected by wide-line proton NMR spectroscopy and quantified by relaxation-time measurements which yield activation barriers for the reorientation process. In addition, in some cases, the molecular reorientation may be coupled with a chemical exchange process as, for example, in the case of many fluxional organometallic molecules. ... 

The long side chains of a homopolypeptide have remarkable motional freedom about multiple bonds, while the main chain forms the secondary regular conformation such as a-helix, /1-sheet, and turn, which are rigid structures. The macroscopic properties of the rigid a-helical polypeptide, therefore, highly depends on the dynamic structure of the side chains so that a lot of studies on the side chain dynamics of the a-helical polypeptides have been carried out in the solid and solution states.12,14,29 66... 

The first half of this chapter concentrates on the mechanisms of ion conduction. A basic model of ion transport is presented which contains the essential features necessary to describe conduction in the different classes of solid electrolyte. The model is based on the isolated hopping of the mobile ions in addition, brief mention is made of the influence of ion interactions between both the mobile ions and the immobile ions of the solid lattice (ion hopping) and between different mobile ions. The latter leads to either ion ordering or the formation of a more dynamic structure, the ion atmosphere. It is likely that in solid electrolytes, such ion interactions and cooperative ion movements are important and must be taken into account if a quantitative description of ionic conductivity is to be attempted. In this chapter, the emphasis is on presenting the basic elements of ion transport and comparing ionic conductivity in different classes of solid electrolyte which possess different gross structural features. Refinements of the basic model presented here are then described in Chapter 3. 

Fig. 3.15 Dynamic structure factors from PE melts at 509 K a M =2,000 [69, 70] and b 12,400 [71]. The solid lines display the predictions of the Rouse model. The Q-values are noted adjacent to the respective lines. Note that the time frame in b is extended by an order of magnitude compared to a. (a Reprinted with permission from [69]. Copyright 1993 The American Physical Society)...

Fig. 3.16 Scaling presentation of the dynamic structure factor from a M =36,000 PE melt at 509 K as a function of the Rouse scaling variable. The solid lines are a fit with the reptation model (Eq. 3.39). The Q-values are from above Q=0.05,0.077,0.115,0.145 A The horizontal dashed lines display the prediction of the Debye-Waller factor estimate for the confinement size (see text)...

Fig. 3.18 Semi-log plot of the time-dependent single chain dynamic structure factor from a PE melt at T=509 K M =36 kg/mol) for various Q. The solid lines are the fit of the repta-tion model (Eq. 3.39). The dashed lines are a fit using the model of des Cloizeaux (Eq. 3.44). (Reprinted with permission from [4]. Copyright 1998 The American Physical Society)...

Fig. 3.19 Single chain dynamic structure factor Schain(Q>0/5(Q)chain from a M =80 kg/mol PEP melt at T=492 K for the following scattering wave vectors Q=0.03 A Q=0.05 A Q=0.068 A" 0=0.077 A (from above), Q=0.09 A Q=0.115 A"LThe solid lines represents a fit with Eq. 3.39. (Reprinted with permission from [43]. Copyright 2003 The American Physical Society)...

Fig. 3.26 Simulated single chain dynamic structure factor Schain(Q>0 for different chain lengths AT=350 (pluses) 700 (crosses) and 10,000 (filled squares) for various Q-values [79] (Q is given in terms of bead size a). Solid lines are fits to Eq. 3.39 and Eq. 3.42. For equal Q-values the plateaus show a strong N-dependence. (Reprinted with permission from [79]. Copyright 2000 EDP Sciences)...

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)... 

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)...

Fig. 4.23 Scaling representation of the dynamic structure factor of PB data at Q=2.71 A" (empty circle 300 K, filled circle 280 K, empty diamond 260 Ky filled diamond 240 K, empty triangle (up) 220 K, filled triangle (up) 205 K, empty square 190 K, filled square 180 K, empty triangle (down) 170 K). Solid lines correspond to KWW functions with =0.41 [133] (Reprinted with permission from [133]. Copyright 1996 The American Physical Society)...

Fig. 5.3 Single chain dynamic structure factor from PIB in the melt at 470 K and Q=0.04 A" (empty circle), 0.06 A (filled triangle), 0.08 A (empty diamond), 0.10 A" (filled circle), 0.15 A (empty triangle), 0.20 A (filled diamond), 0.30 A (empty square), and 0.40 A (plus). The solid lines show the fit of the Rouse model to the data. (Reprinted with permission from [217]. Copyright 1999 American Institute of Physics)...

Fig. 5.13 Dynamic structure factor measured on deuterated PIB. The symbols (full incoming wavelength A=6 A, empty A=10 A) correspond to the different Q-values indicated. Lines are the resulting KWW fit curves (Eq. 4.21) (solid A=6 A, dotted A=10 A). (Reprinted with permission from [147]. Copyright 2002 The American Physical Society)...

Fig. 5.20 Temporal evolution of the dynamic structure factor of PVE at the first maximum of the static structure factor at 418 K. The solid line represents a KWW fit (p=0.5). Insert S(Q) of PVE measured at 320 K. (Reprinted with permission from [39]. Copyright 2004 EDP Sciences)...

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)...

Fig. 6.1 NSE results at 418 K on the single chain dynamic structure factors from PVE, PVE in PI/PVE, PI in PI/PVE and PI (from above). The different symbols correspond to the following Q-values dash 0.05 A empty square 0.077 k, plus 0.10 A cross 0.13 A" empty diamond 0.15 A empty square 0.18 A empty circle 0.20 A ). Solid lines Rouse structure factors. (Reprinted with permission from [47]. Copyright 2000 The American Physical Society)...

Fig. 6.5 Comparison between NSE spectra at Q=0.1 A and T=473 K of both binary blends (filled circle and filled triangle), and those of the isotopic PDMS blends filled square). The solid lines result from a fit t with the dynamic structure factor for the Rouse model filled square) and of a spatially limited Rouse dynamics, as derived in [256] filled square and filled triangle). (Reprinted with permission from [255]. Copyright 2003)...

Fig. 6.9 NSE spectra from the hPE-dPEE diblock copolymer melt (sample IV) at 473 K. Q values in each case from above are Q/A =0.05,0.08,0.10,0.121,0.187. The solid lines are the result of a fit with the Rouse dynamic structure factor. Their extent marks the fitting range the dashed lines extrapolate the Rouse structure factor. (Reprinted with permission from [44]. Copyright 1999 American Institute of Physics)...

---