Random chain

The molecules used in the study described in Fig. 2.15 were model compounds characterized by a high degree of uniformity. When branching is encountered, it is generally in a far less uniform way. As a matter of fact, traces of impurities or random chain transfer during polymer preparation may result in a small amount of unsuspected branching in samples of ostensibly linear molecules. Such adventitious branched molecules can have an effect on viscosity which far exceeds their numerical abundance. It is quite possible that anomalous experimental results may be due to such effects. 

Note that, while the random chain twists always decrease the hopping amplitudes (all ()/ , + are negative), // (a) can be both positive and negative, as it is the alternating part of the fluctuations. As in the FCM, we consider white noise disorder with a correlation function given by Eq. (3.22). This corresponds to independent random variations of the hopping amplitudes <5/ on different bonds. 

Melting of ECC involving transition into the isotropic melt was shown by Flory to be a first-order process. It can be seen in Fig. 18 b that there occurs a transition from a complete order to a fully random chain arrangement in the isotropic melt (Fig. 16, point 4). 

Various studies have been made on the effects of radiation on lactide/glycolide polymers (24,38,58). Gilding and Reed (24) reported the effect of y rays on Dexon sutures. Those results confirmed that deterioration of the sutures occurs but that random chain scission is not the primary mechanism. Number average-molecular weight Mn showed a dramatic decrease at doses above 1.0 Mrad. Thus, unzipping of the polymer chain appeared to be the more dominant process, at least in the case of polyglycolide. 

Derivation of the Gaussian Distribution for a Random Chain in One Dimension.—We derive here the probability that the vector connecting the ends of a chain comprising n freely jointed bonds has a component x along an arbitrary direction chosen as the x-axis. As has been pointed out in the text of this chapter, the problem can be reduced to the calculation of the probability of a displacement of x in a random walk of n steps in one dimension, each step consisting of a displacement equal in magnitude to the root-mean-square projection l/y/Z of a bond on the a -axis. Then... 

Equation (10) directs attention to a number of important characteristics of the molecular expansion factor a. In the first place, it is predicted to increase slowly with molecular weight (assuming t/ i(1 — 0/T) >0) and without limit even when the molecular weight becomes very large. Thus, the root-mean-square end-to-end distance of the molecule should increase more rapidly than in proportion to the square root of the molecular weight. This follows from the theory of random chain configuration according to which the unperturbed root-mean-square end- o-end distance is proportional to (Chap. X), whereas /r = ay/rl. 

It is important to realize that the random-chain model need not imply an absence of residual structure in the unfolded population. Formative articles—many of them appearing on the pages of Advances in Protein Chemistry—recognized this fact. Kauzmann s famous review raised the central question about structure in the unfolded state (Kauzmann, 1959) ... 

Meanwhile, evidence continues to mount that the unfolded state is far from a random chain at length scales of interest, even under strongly... 

These results strongly suggest that unstructured peptides have definite backbone conformations and that the concept of a denatured protein as a structureless random chain breaks down when backbone conformations of individual residues are described, although the random chain concept may still be useful when describing the overall chain conformation. 

Figure 30 Mechanism of random chain copolymerisation of two comonomers (Mj and M2).

A discontinuous coil to stretch transition is evident at sc = 0.000725. The transition point sc was found by using two different initial conformations as described above. For values lower than ec> the random chain will eventually coil, form a folded chain crystalline structure and stay in that conformation until the end of the run for relatively long run times. On the other hand, a prestretched chain would fluctuate and eventually form a crystallized folded chain that is stable. Similarly, for flow rates higher than sCy a pre-stretched chain will never coil and a random chain will eventually stretch. 

Silicone paints are formed by controlled hydrolysis and condensation of alkyl alkox-ysilanes, and may be encountered either alone or in formulations with other synthetic resins. The typical structural unit in the polymer chain is dimethyl siloxane, and pyrolysis of such resins takes place with random chain scission and the extended formation of stable cyclic fragments. In Figure 12.14 the pyrogram of a silicone resin is shown, with cyclic siloxane oligomers eluting at the shorter retention times, followed by the linear siloxane fragments. 

The two polymer substrates investigated as part of the study of DBDPO mixtures were polypropylene (PP) and linear high density polyethylene (HDPE). while both PP and HDPE decompose by similar random chain scission, radical mechanisms, chain transfer occurs much more teadily during the pyrolysis of PP because of the presence of the tertiary hydrogens. In addition, only primary chain end radicals are formed when the HDPE chain cleaves homolytically. Therefore, a comparison of the PP/DBDPO and the HDPE/DBDPO mixtures volatile product distributions was undertaken. 

We appropriate the random chain and notify as nt the numbers of steps of the end of chain random walk along /-directions of d-dimensional lattice. At this,... 

Distribution (3) describes the RW trajectory of one random chain whereas the expression (4) assigns the links distribution of all m chains. That is why, the probability 6j(s) of common event consisting of the fact that the RW trajectory of random chain is also the SARW trajectory and at given Z, n, N and , will turned out by its own last step in one among 2d equiprobable cells M(s) will be equal to... 

Determination (7) means, that the (/-dimensional space consisting of Z cells is disposable for any random chain this demands of their full mixing. 

Expression (20) determines not only the conformational radius of one random chain, but due to the chains intertwining effect also the conformational radius of all m chains. Thereby... 

We can see from the comparison of (20) and (1), that the conformational radius Rm of m-ball and respectively of any random chain in it is more than the conformational radius Rt of random chain in w—ball the chains are stretched but are not twisted. The presence of other chains diminishes the number of free cells of (/-dimensional lattice accessible for SAR W trajectory of presented chains enforcing it to encroach more volume of the space. 

---