Short Chain Dynamics

Note that all these formulas also contain the result for the limiting case of short chains dynamics described by the Rouse model [139,140] if we formally put Ne N in these equations. As will be discussed later (Sect. 2.5), there occurs a crossover in the static critical behavior from mean-field-like behavior where ocR e-1/2 with e= 1 — x/X rit> Scon(0)ccN e to the nonclassical critical behavior with Ising model [73, 74] critical exponents cce-v, S, ii(0) oceT, vw0.63, 1.24. This crossover occurs, as predicted by the Ginzburg... 

Fig. 13 shows this autocorrelation function where the time is scaled by mean square displacement of the center of mass of the chains normalized to Ree)- All these curves follow one common function. It also shows that for these melts (note that the chains are very short ) the interpretation of a chain dynamics within the Rouse model is perfectly suitable, since the time is just given within the Rouse scaling and then normalized by the typical extension of the chains [47]. 

Fig. 5.3. Log-log plot of the self-diffusion constant D of polymer melts vs. chain length N. D is normalized by the diffusion constant of the Rouse limit, DRoUse> which is reached for short chain lengths. N is normalized by Ne, which is estimated from the kink in the log-log plot of the mean-square displacement of inner monomers vs. time [gi (t) vs. t]. Molecular dynamics results [177] and experimental data on PE [178] are compared with the MC results [40] for the athermal bond fluctuation model. From [40]...

In summary, the chain dynamics for short times, where entanglement effects do not yet play a role, are excellently described by the picture of Langevin dynamics with entropic restoring forces. The Rouse model quantitatively describes (1) the Q-dependence of the characteristic relaxation rate, (2) the spectral form of both the self- and the pair correlation, and (3) it establishes the correct relation to the macroscopic viscosity. 

Under these conditions, Eq. (32) indicates the maximum extent to which a particular mode p can reduce S(Q,t) as a function of the momentum transfer Q. Figure 10 presents the Q-dependence of the mode contributions for PE of molecular weights Mw = 2000 and Mw = 4800 used in the experiments to be described later. Vertical lines mark the experimentally examined momentum transfers. Let us begin with the short chain. For the smaller Q the internal modes do not influence the dynamic structure factor. There, only the translational diffusion is observed. With increasing Q, the first mode begins to play a role. If Q is further increased, higher relaxation modes also begin to influence the... 

Fig. 21. Comparison of the dynamic structure factors from long PTHF chains in a matrix of long chains (x) with that in a matrix of short chains ( ). The Q-value of the experiment was Q = 0.09 A-1, the temperature T = 418 K (Reprinted with permission from [57]. Copyright 1985 Royal Society of Chemistry, Cambridge, UK)...

To follow the crystallization of kebabs around a shish, the dynamics of 10 short chains (N = 180) near a preformed shish (from 7 chains of length N = 500) were followed at T = 9.0, by fixing the center of mass of the shish. The initial position of the short free chains was chosen at random in a cylinder around the shish, with radius 30ro and a height of 60ro. Each rim started with different initial conditions. Figure 29 shows one such initial state. 

This author is perfectly aware that he could add very little to the work done by these workers if an attempt was made to focus on intramolecular catalysis phenomena or on the relevance to cyclisation of available models of chain conformation and chain dynamics instead, the aim will be the presentation of a general treatment of the subject, namely, one that includes the cyclisation of very short chains as well as that of very long chains of, say, 100 atoms or more. With a subject as vast as this, an encyclopaedic review would be a hopeless task. Therefore, the subject will be treated in a systematic and critical way, with more concentration on reaction series with regular and wide variations in structure, rather than on scattered examples. The aim will be to show that the field of intramolecular reactions is a mature area in which the merging of concepts from both physical organic chemistry and polymer chemistry leads to a unified treatment of cyclisation rates and equilibria in terms of a few simple generalisations and theories. 

PET contains about 2-3 % of short chain oligomers, which cause problems in the processing of the polymer. Oligomers can occur as linear or cyclic molecules and can be extracted by suitable solvents. Different compounds have been identified depending on the solvent and the analysis technique used [49-52], After their extraction from the polymer, oligomers will reform by thermal treatment of the extracted sample [49], and a dynamic equilibrium between polymer and oligomers has been proposed. 

Fig. 3.2 Development of Schain(Q>0 for different times (a) and the normalized relaxation function 5chain(Q>0/ chain(Q) ( ) for QRg=l, 2,... 6. The dashed lines contain only the intrachain relaxation whereas the solid lines include the centre-of-mass diffusion. Note that for short chains and for small Q the diffusion dominates the observed dynamics (Reprinted with permission from [40]. Copyright 2003 Springer, Berlin Heidelberg New York)...

The mean square segment displacements, which are the key ingredient for a calculation of the dynamic structure factor, are obtained from a calculation of the eigenfunctions of the differential Eq. 5.13. After retransformation from Fourier space to real space B k,t) is given by Eq. 41 of [213]. For short chains the integral over the mode variable q has to be replaced by the appropriate sum. Finally, for observation times mean square displacements can be expressed in... 

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. 

In the low Q-regime RPA describes well the static structure factor for the short chain melt, where the ODT is sufficiently far away (kN 7). In the dynamics we would expect the diblock breathing mode to take over around QRg 2 (Q=0.04 A ). Instead, deviations from Rouse dynamics are already observed at Q values as high as QR =5. At QJ g=3 a crossover to a virtually Q-independent relaxation rate about four to five times faster than the predicted breathing mode is found. This phenomenon is only visible under h-d labelUng. Under single chain contrast (see below) these deviations from RPA are not seen. Thus, the observed fast relaxation mode must be associated with the block contrast. 

The attenuation of die dipole of the repeat unit owing to thermal oscillations was modeled by treating the dipole moment as a simple harmonic oscillator tied to the motion of the repeat unit and characterized by the excitation of a single lattice mode, the mode, which describes the in-phase rotation of the repeat unit as a whole about the chain axis. This mode was shown to capture accurately the oscillatory dynamics of the net dipole moment itself, by comparison with short molecular dynamics simulations. The average amplitude is determined from the frequency of this single mode, which comes directly out of the CLD calculation ... 

The lability of organic S to oxidation and recycling is not understood. Short-chain thiols appear to be dynamically cycled in marine and saltmarsh sediments (120, 211). In contrast, sulfones may represent a stable oxidized product that could provide a useful signature of sediment oxygenation (121, 184). Sulfones have been identified in saltmarsh and ancient marine sediments (32, 207). Other (unidentified) forms of organic S appear to yield acidity and sulfate upon oxidation (38, 192). Much more research is needed to elucidate the mechanisms of organic S fonnation and the relationships between specific organic S compounds and lake conditions. 

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