Segment displacement, mean square

In Eq. (19) we used the fact that the mean square displacement of the center-of-mass provides the diffusion constant according to DR = (l/6t)<(x0(t) — Xo(0))2>. For the special case of the self correlation function (n = m) Arnm(t) reveals the mean square displacement of a polymer segment. For t < xR we obtain... 

The self-correlation function leads directly to the mean square displacement of the diffusing segments Ar2n(t) = <(rn(t) — rn(0))2>. Inserting Eq. (20) into the expression for Sinc(Q,t) [Eq. (4b)] the incoherent dynamic structure factor is obtained... 

Tube confinement leads to strong alterations of the mean square segment displacements as compared to the Rouse model. 

Fig. 18. Mean square displacement of a chain segment undergoing reptational motion as a function of time...

In addition, analogously to (20), for t < xz the mean square displacement of a diffusing polymer segment becomes... 

In contrast to normal diffusion, in the segmental regime the mean-square displacement does not grow linearly, but with the square route of time. For the... 

Fig. 3.4 Time-dependent mean-square displacement of a PEP segment in the melt at 492 K. The solid line indicates the prediction of the Rouse model. (Reprinted with permission from [43]. Copyright 2003 The American Physical Society)...

We now consider the predictions of the reptation model for the mean-square displacement of the chain segments. Figure 3.13 gives an overview. 

We conclude this brief discussion of recent results on the segment mean-square displacement (msd) with a brief look at the latest atomistic simulation put forward by Harmandaris et al. [53]. In this simulation fully atomistic PE... 

Fig. 3.24 Mean-square displacement g(f) of the innermost atomistic chain segments vs. time f in a log-log plot for the PE melt systems of different chain length (see insert). (Reprinted with permission from [53]. Copyright 2003 American Chemical Society)...

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... 

The mean-square displacement of the chain segments of BR swollen with deuterated benzene was observed to be independent of diffusion time, indicating restricted diffusion around an attractive centre. The mean-squared displacement decreased with increasing crosslinking density and was approximately equal to the mean-squared collective fluctuations calculated for these polymers. 

For times shorter than the Rouse time of the chain (tcoherent motion of a chain segment consisting of a/t/to neighbouring monomers. The time-dependent curvilinear coordinate of a monomer along the contour of the tube is s t) (Fig. 9.19). The mean-square monomer displacement along the tube is of the order of the mean-square size of this section in three-dimensional space [Eq. (8.58)] ... 

In this set of equations fx is the force on the ith bead in the x direction, X, is the amount by which bead i has been displaced from its equilibrium position, and a2 is the mean square end-to-end distance of the submolecule. The form of the equation results from the fact that the -directed force on the ith bead reflects the difference between the x-directed forces on the ith and i + 1st segments. An additional force acts on the molecule due to the viscous nature of the medium in which it is immersed. Under the assumption that the beads... 

Fig. 12 Mean-square displacement of a polymer segment in a time interval t is plotted against the mean z position of the segment for various temperatures N = 100). The average is taken for each layer of thickness Icr...

An example from recent works is the study of the dynamics of unentangled PEO chains in 35% PEO/65% PMMA by quasielastic neutron scattering (Niedzwiedz et al., 2007), and in 25% PEO/75% PMMA by molecular dynamics simulations (Brodeck et al., 2010). The Rouse model has the mean-square displacement, (r (r)>, of a chain segment, which increases proportionally to the square root of time according to... 

Stochastic equation for reptation dynamics Although the above probabilistic description is quite useful in understanding the essence of reptation dynamics, it becomes progressively more difficult to proceed with the calculation for other types of time correlation function. For example, it is not easy to calculate the mean square displacement of a primitive chain segment (R(s, t)-R(s, 0)) ) by this method. In this section we shall describe a convenient method" for calculating general time correlation functions. 

This result is easily derived, for if r the polymer segment remains in the initial tube, so that if the chain moves (r) dong the tube, the mean square displacement in the three-dimensional space is given by a( (0l) (see eqn (6.2)). Hence 0(s, s t) is given by... 

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