Centre of mass diffusion

As in Sect. 2.1, Dj is the curvilinear centre-of-mass diffusion constant of the chain, and is given in terms of the monomeric friction constant by the Einstein relation Dj =kT/Nl. L is as before the length of the primitive path, or tube length of the chain, which is Finally, we need the initial condition on p(s,t), which... 

Unless the molecular weight of the star arm is rather low, this period of relaxation by fluctuation is generally interrupted by the reptation of the linear polymers. This happens when the centre of mass diffusion of the linear chains is able to renew the tube of all (central) portions of the chain not relaxed by the star-like modes. So the reptation time is given by... 

For small Q(QRpf) describes the centre of mass diffusion of the chain ... 

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 prediction of a time-dependent centre of mass diffusion coefficient has recently been corroborated by a combined atomistic simulation and an NSE approach on PB ([55]). The dynamic structure factor from simulation and experiment obtained at 353 K are displayed in Fig. 3.11. 

For small chains in solution the translational diffusion significantly contributes to the overall decay of Schain(Q>0- Therefore precise knowledge of the centre of mass diffusion is essential. Combing dynamic light scattering (DLS) and NSE revealed effective collective diffusion coefficients. Measurements at different concentrations showed that up to a polymer volume fraction of 10% no concentration dependence could be detected. All data are well below the overlap volume fraction of (p =0.23. Since no -dependence was seen, the data may be directly compared with the Zimm prediction [6] for dilute solutions ... 

Thus, the initial slope corresponds to that of an A arm performing translational diffusion without knowing about the slower B arm. Similarly the prediction for a small fraction of labelled B arms leads to the unphysical result IXQ)=k TQ /Ni, i again a single arm performing centre of mass diffusion. 

Non-spherical micelles of poly(ethylene)(PE)-poly(ethylene-propylene)(PEP) in decane are self-assembhng in the form of extended platelets that have a crystalline PE-core and a planar PEP brush on both sides. Due to the large size of the platelets the centre of mass diffusion is extremely slow and allows a clear separation of the density fluctuation in the brush. NSE experiments [301] have been analysed in terms of the model of de Gennes [300]. The friction coefficient and modulus of the brush were found to be similar to those of a typical gel. 

As already pointed out the first work directly measuring the deformation dynamics in an o/w-droplet microemulsion using NSE was published by Huang et al. [45]. In this work, a microemulsion based on the surfactant AOT was studied and it was shown that the intermediate scattering functions contain information about the centre of mass diffusion and in addition also contributions from the deformation dynamics. The intermediate scattering functions obtained in this work are shown in Fig. 2.3. 

Figure 2.14 Scheme of the important processes and relationships in quasi-elastic scattering experiments. The used radiation exhibits an initial line shape, which is changed (broadened) due to the energy exchange with the thermally excited modes in the sample, e.g. centre of mass diffusion. 

Reptation involves a one-dimensional (head-first) diffusion of the polymer in whidi lateral motion of the diain is restricted as though the chain were emfined to a tube whose effective dimensions reflect the mean entanglement spadng in the polymer melt It is widely appreciated that for reptating chains, the dqrendence of the centre-of-mass diffusion coefiBcient on molar mass (Af) is Dcm oc who eas for Rouse chains, D a oc M in addition, the melt viscosity (and terminal relaxation time) increase as for entangled chains [37]. 

There will be two important differences between interfacial broadening on these small length scales and more conventional centre-of-mass diffusion. Firstly, because polymer chain conformations will be perturbed in the vicinity... 

Beside centre-of-mass diffusion and segmental reorientation, the description of the diffusion behaviour of propagating polymer coils can be further complicated by other modes of diffusion, such as reaction diffusion [41], which is the diffusive motion of the polymer-chain-end radical as a result of propagation, and reptation [42-45]. All these different modes of diffusion are, however, not equally important over the entire range of conversion and the discussion of free-radical termination is therefore normally divided into three conversion regimes [15]. In the low conversion regime (approx. 0-10% conversion in a bulk polymerization), prior to the onset of the gel-effect, polymer chains may overlap but are... 

Translational centre-of-mass diffusion (TD) of both species towards each other travelling through the reaction medium. 

The SD control of (kt) in the initial stage of a polymerization is often called plateau-regime, since (kt) remains more or less constant with increasing conversion. The plateau level depends on (mostly the viscosity of) monomer and solvent. SD control is characterized by fast centre-of-mass diffusion of macroradicals through the environment of mostly monomer and solvent and subsequent segmental re-orientation, which also occurs against the friction of... 

Because monomeric radicals are so small, their termination A t(l,l) must be via centre-of-mass diffusion. This situation can adequately be described by the Smoluchowski equation (1.27)... 

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