End segment distribution

Fig. 28). Clearly the profile becomes flatter with decreasing solvent quality. Note, however, that the self-consistent potential and the end-segment distributions still satisfy Eqs. (109) and (110). 

We can easily generalize this procedure to longer chains by relating G(z s-i-l) to the end segment distribution G(z s) of sequences that are one segment shorter. We have the following recurrence relation ... 

In Fig. 16, the average size R of an annealing PE star (specified as the first moment of the end segment distribution) is plotted as a function of the salt concentration, 0ion (Ps, for different values of the bulk pH. A maximum is observed for pH < pK. ... 

To calculate the single chain partition function Q/, it is useful to define the end segment distribution, qj(r, t), which describes the probability of finding the end of a chain fragment—containing the segments [0 f] and being exposed to the fields Wa(oi = A, B) —at position r [9] ... 

The crossover from the Gaussian chain behavior on large length scales to the rod-Hke behavior on the length scale of the statistical segment length can be described by the worm-like chain model [18]. In this case, the end-segment distribution q depends both on the spatial coordinate r as well as on... 

To calculate the segment densities it is useful to introduce the end segment distribution... 

End segment distribution 29 End-to-end distance 24 Energy barrier, free 165 Energy of mixing 241 Ensemble, microcanonical 18, 108... 

The equations of motion (75) can also be solved for polymers in good solvents. Averaging the Oseen tensor over the equilibrium segment distribution then gives = l/ n — m Y t 1 = p3v/rz and Dz kBT/r sNY are obtained for the relaxation times and the diffusion constant. The same relations as (80) and (82) follow as a function of the end-to-end distance with slightly altered numerical factors. In the same way, a solution of equations of motion (75), without any orientational averaging of the hydrodynamic field, merely leads to slightly modified numerical factors [35], In conclusion, Table 4 summarizes the essential assertions for the Zimm and Rouse model and compares them. 

Fig. 4. Backfolding in dendrimers as predicted by analytical theory [12]. Free end probability distribution function of the radial distance for generations 2-7. All data has been calculated assuming a realistic excluded volume parameter of the segments of the dendrimer (see [12] for further details). Reproduced with permission from [12]...

Figure 5.5 Distribution of mineral elements within potato tubers. Data show longitudinal profiles of elements from the distal (Segment 1) to the stem end (Segment 5) of tubers from Stirling plants (N. Subramanian, G. Ramsay, M. R. Broadley and P.J. White, unpublished data).

If we neglect the distortion of the segment distribution in the fuzzy cylinder by the shear flow, we can apply Doi s stress expression, Eq. (61), to fuzzy cylinder systems as it stands. The neglect of the distortion may be justified when the shear-rate is low. Equation (61) expresses the contribution of the end-over-end rotation of the chain to asegment distribution is not distorted, the orientational entropy term Sor in the static free energy expression contains only the orientational entropy loss of the entire chain, but not the conformational entropy loss cf. Sect. 2.3. 

Hesselink39,40 derived relations between the root-mean-square thickness and the loop or tail size. For the case in which no intrasegment interaction exists, he derived the segment distribution p4(z) for a single loop of size i by considering all posable configurations of the chain that starts at the interface and returns at its end to the interface. His expression for p4(z) reads... 

While a chain within a branched molecule is by no means Gaussian in this real world (d = 3 note that we are considering the end-to-end distance distribution, and not the segment-density distribution about the center of gravity), by virtue of the perturbation expansion with respect to one can apply the... 

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