Mean Square Displacement of Monomers

3 Mean Square Displacement of Monomers Now we trace the motion of the same bead (m = n) in all time scales. From Eq. 3.224, the displacement of each 

In the intermediate time range, we need to deal with 1 - exp(-t/T,) as it is. 

In the following, we consider ([r (t)] ) for the Rouse model and the Zimm model in the theta solvent separately. We will also briefly consider for the 

In the Zimm model for the good solvent, our discussion is limited to power relationships. For the short-time behavior, we note k T/= Then, in 

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The quantities that are compared in Fig. 5.2 are mean square displacements, gi(t), of inner monomers in the laboratory frame and analogous quantities, g2 (t), in the center of the mass frame of each chain, the center of mass mean square displacement, g3(t), and mean square displacement of monomers at chain ends, (g4(t), gs(t). The precise definitions of these mean square displacements are as follows [12,20] ... 

Chain sections containing N/p monomers move a distance of order of their size b Njp) during the mode relaxation time Xp. The position vector of monomer j at time t is f)(f). The mean-square displacement of monomer j during time Xp is of the order of the mean-square size of the sections involved in coherent motion on this time scale ... 

At longer times, monomers participate in collective motion of larger sections with smaller effective diffusion coefficient D(t). Therefore the mean-square displacement of monomers is not a linear function of time, but Instead subdiffusive ---------------------------------------------... 

For the Zimm model the mean-square displacement of monomers is faster [Eq. (8.70)] leading to the logarithm of the Zimm dynamic structure factor scaling as the 2/3 power of time for tq < r < zz-... 

By definition, the mean-square displacement of monomers in the time period of Tp is comparable to the mean-square end-to-end distances of sub-molecules (see (5.2)). From (5.29), we have... 

Fig. 5.3 Double logarithmic plot of the mean-square displacement of monomers versus the time, illustrating the scaling laws of the Rouse chain. Monomers are moving slower than simple fluids due to their chain connection...

Similar to the derivation of the scaling law of the Rouse chain, the mean-square displacement of monomers within the time period of the characteristic time Xpz for p-mode sub-molecules is... 

Fig. 5.4 Double logarithmic plot of the mean-square displacements of monomers versus the time for the scaling laws for a short chain in semi-dilute solutions. The Zimm chains are slightly faster than the Rouse chains due to their less frictional barrier in the non-draining mode...

Motion of the Monomers Here, we consider the mean square displacement of monomers on the entangled chains. Over a long time, t > the mean square displacement of the monomer, ([r (0 r (0)] ), becomes identical to the... 

The test chain would follow the dynamics of the unrestricted Rouse chain if the entanglements were absent, as would the primitive chain at f > t. In Section 3.4.9, we considered the mean square displacement of monomers on the Rouse chain. We found that the dynamics is diffusional at r < and Tj < t, where % is the relaxation time of the Mth normal mode but not in between. When the motion of the Rouse chain is resnicted to the tube, the mean square displacement of monomers along the tube, ([ (t) - x(0)] ), will follow the same time dependence as the mean square displacement of the unrestricted Rouse chain in three dimensions. Thus, from Eqs. 3.240 and 3.243,... 

Figure 4.43. Mean square displacement of monomers on entangled chains, [r (i) - r (0)] ) plotted as a function of time t. The plot has four sections with distinct slopes. They are indicated adjacent to the plot. The boundaries of the four sections are specified by their values of t and <[r (f) - r (0)]4.

In the time scale of t > Ij, the effect of finding the new direction by the chain ends becomes dominant, and the mean square displacement of monomers will become equal to that of the center of mass. In the time scale of f< the motion of the monomers is complicated. At sufficiently short times (t < r ), the monomers will make a diffusional motion without feeling the presence of other monomers, as we have seen for both the Rouse chain and the Zimm model. We can at least say that the dependence of ([r (f) - r (0)] ) on t is sharper at t <... 

Thus, we have the mean square displacement of monomers on the test chain for t> t.d ... 

In order to get information about dynamic properties of the system various quantities have been monitored with time at equilibrium states corresponding to various temperatures [38 0] the mean squared displacement of monomers, (r ), the mean squared displacement of the center of mass of chains, the bond autocorrelation function, the end-to-end... 

The diffusion in the systems studied has been detected by monitoring in time the mean squared displacements of monomers and centers of mass of chains. Typical results for the diblock copolymer system are shown in Fig. 8. They indicate that the short time displacement rates are not sensitive to temperature but the long time displacements are influenced slightly by the microphase separation. The self-diffusion constants of chains determined at the long time limit are shown in the inset of Fig. 8a, where the effects of the microphase separation in the diblock can be clearly noticed. The slowing down observed at the microphase separation of the system is, however, rather small and indicates a considerable mobility of chains left even when the chains are confined at interfaces. The nature of this mobility has been analyzed by monitoring the correlation between orientation of chain axes and directions of chain displacements (Fig. 8b). It is established that... 

FIG. 8 (a) Mean square displacements of monomers and mean square displacements of chain centers of mass vs. time for the diblock copolymer system at various temperatures. Temperature dependence of the self-diffusion constant of block copol5mier chains is shown in the inset, (b) Orientation correlation factor between the end-to-end vector and the center of mass displacement vector of copolymer chains at various temperatures above and below the ODT. 

Finally, mean-square displacements of monomers at the free ends of the chains are defined as... 

Fig. 4.8 Variation of the mean-square displacements of monomers as a function of their position along the chains for N= 150, from Ref. 54. The data progressively give monomers from the chain ends (upper curve) inward with an increment of 10.

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