Single-Chain Dynamic Structure Factor

For non-interacting, incompressible polymer systems the dynamic structure factors of Eq. (3) may be significantly simplified. The sums, which in Eq. (3) have to be carried out over all atoms or in the small Q limit over all monomers and solvent molecules in the sample, may be restricted to only one average chain yielding so-called form factors. With the exception of semi-dilute solutions in the following, we will always use this restriction. Thus, S(Q, t) and Sinc(Q, t) will be understood as dynamic structure factors of single chains. Under these circumstances the normalized, so-called macroscopic coherent cross section (scattering per unit volume) follows as... 

Fig. 20. Single-chain dynamic structure factor of the Ronca model as a function of the Rouse variable for different values of Qdt (dt tube diameter dt = d). (Reprinted with permission from [50]. Copyright 1983 American Institute of Physics, Woodbury N.Y.)...

Now we turn to the single-chain dynamic structure factor, which is also strongly affected by the topological tube constraints. Qualitatively we would expect the following behaviour ... 

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. 3.25 Comparison of the experimental dynamic single chain structure factors for PEP at Q=0.135 Qd=6A) and PE at Q=0.128 A (Qd=5.5) with the dynamic structure factors from the computer polymer. The various/w// lines represent MD results for different Qd=3.1 (a), 3.9 (b), 4.6 (c), 6.2 (d), and 7.7 (e). In the upper part the computer results are the structure factors from a fully labelled chain, while in the lower part only the centre 35 monomers are labelled. (Reprinted with permission from [49]. Copyright 1992 American Chemical 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. 4.20 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)...

Recently a very detailed study on the single chain dynamic structure factor of short chain PIB (M =3870) melts was undertaken with the aim to identify the leading effects limiting the applicability of the Rouse model toward short length scales [217]. This study was later followed by experiments on PDMS (M =6460), a polymer that has very low rotational barriers [219]. Finally, in order to access directly the intrachain relaxation mechanism experiments comparing PDMS and PIB in solution were also carried out [186]. The structural parameters for both chains were virtually identical, Rg=19.2 (21.3 A). Also their characteristic ratios C =6.73 (6.19) are very similar, i.e. the polymers have nearly equal contour length L and identical persistence lengths, thus their conformation are the same. The rotational barriers on the other hand are 3-3.5 kcal/mol for PIB and about 0.1 kcal/mol for PDMS. We first describe in some detail the study on the PIB melt compared with the PDMS melt and then discuss the results. 

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. 5.6 Single chain dynamic structure factor measured for PDMS chains at 373 K in the melts compared to the standard Rouse model (lines) at the Q-values (A Q indicated. (Reprinted with permission from [186]. Copyright 2001 American Chemical Society)...

Not only the single chain dynamic structure factor of PIB but also other investigated polymers show deviations from Rouse dynamics when approaching local scales. As already discussed above, in the combined NSE and MD-simu-... 

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. 6.2 Single chain dynamic structure factors at Q=0.15 A vs. the rescaled time From...

As the final example, we shall consider the dynamic structure factor of a single chain ... 

FIG. 6 Comparison of single chain intermediate dynamic structure factor of polyethylene at r= 509K between NSE experiments and chemically realistic MD simulations [21]. 

Single-chain m (k,t) and collective k,t) dynamical structure factors for projected dynamics in Eq. 113 reflect the fact that local density fluctuations around the tagged chain are relaxed by the projected motions both of the tagged chain and the matrix chains. The projected nature of the dynamics is hidden in Eqs. 114 and 115 in the projected mean squared displacement, which qualitatively describes typical distances over which elementary density fluctuations become dispersed around the tagged chain during the time t. 

Single Chain Structure Factor for Star-Polymer Dynamics.221... 

---