Single-chain dynamics

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.)...

Under -conditions the situation is more complex. On one side the excluded volume interactions are canceled and E,(c) is only related to the screening length of the hydrodynamic interactions. In addition, there is a finite probability for the occurrence of self-entanglements which are separated by the average distance E,i(c) = ( (c)/)1/2. As a consequence the single chain dynamics as typical for dilute -conditions will be restricted to length scales r < (c) [155,156],... 

Transition in the Single Chain Dynamics owing to the Screening of Hydrodynamic Interactions... 

The observation of single-chain dynamics in semi-dilute solutions requires contrast matching and labelling. In the case of PDMS this can be achieved using... 

Then we address the dynamics of diblock copolymer melts. There we discuss the single chain dynamics, the collective dynamics as well as the dynamics of the interfaces in microphase separated systems. The next degree of complication is reached when we discuss the dynamic of gels (Chap. 6.3) and that of polymer aggregates like micelles or polymers with complex architecture such as stars and dendrimers. Chapter 6.5 addresses the first measurements on a rubbery electrolyte. Some new results on polymer solutions are discussed in Chap. 6.6 with particular emphasis on theta solvents and hydrodynamic screening. Chapter 6.7 finally addresses experiments that have been performed on biological macromolecules. 

Prominent examples of successful application of contrast variation are the investigations of the single chain dynamics of polymers in melts. Here a mixture of about 10% h-polymer in a matrix of d-polymer is used. Further details are obtained by investigating d-polymers that contain only a h-labelled section, i.e. at the ends, at branching points or at its centre in a fully deuterated matrix. 

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. 3.26 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...

The single chain dynamics of one given block or of one chain in a diblock copolymer melt is observed if a matched deuterated diblock is mixed with a small amount of labelled diblocks, where the label could be a protonated a or b block or a protonated chain. In terms of the dynamic RPA such a system is a four-component polymer mixture. It is characterized by four different relaxation modes A1-A4 which - depending on the contrast conditions - appear with... 

As pointed out above, the RPA theory predicts that the dynamics of the respective homopolymers should be observed at high Q in the Rouse regime. While the experiment shows that the predicted Q dependencies are reproduced well by the data, the absolute values for the observed relaxation rates disagree with the predictions (see Table 6.2). In particular the observed Rouse factors for PE are considerably smaller than predicted, (Wf )expt=2xl0 s" compared to Wf pa=3.8x 10 A s at T=473 K. At low Q values, the two blocks display the same single chain dynamics. 

In this paper we will concentrate on the diffraction techniques (SANS and reflectometry), and hence static measurements. However, it should be pointed out that through inelastic scattering, aspects of polymer dynamics are accessible. In particular, it has been possible to access single chain dynamics in bulk systems, deformation and relaxation of polymer melts under shear, shed new light on viscoelasticity in polymer melts, and obtain direct information on polymer reputation and particle fluctuations. 

Show that a stress relaxation modulus of an entangled but non-concate-nated melt of rings on the basis of the single chain dynamic modes described in Problem 9.31 is... 

We conclude this section by drawing attention to various theories considering the dynamics of block copolymer melts rheology of these systems has been considered [340-342], single chain dynamics and selfdiffusion [343, 344], nu-cleation of the ordered phase [61], ordering kinetics [345,346], and dynamics of concentration fluctuations [347]. These topics are not under consideration here, just as other extensions of the theory random copolymer melts [348, 349], multiblock copolymer melts [350] etc. 

An alternative idea proposed by Edwards and Freed is to consider the motion of a single chain in an effective medium which includes the effect of the other chains. The property of the effective medium is determined self-consistently from the single chain dynamics. Though this method fails to describe the entanglement effect appropriately, it indicates an important aspect of the hydrodynamic interaction in the concentrated system, which is the screening of the hydrodynamic interaction. 

Bearing in mind that the connectivity of monomeric units along the backbone of the polymer is an essential ingredient of the single chain dynamics it is clear that a non-local coupling should lead to a better description. In the Rouse model forces acting on a monomer caused by the chain connectivity are additionally taken into account [88,89]. This leads to a kinetic factor that is proportional to the intramolecular pair-correlation function [90-92], P(r,r )... 

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