Chain end separation

The freely-jointed chain considered previously has no internal restraint, and hence, its internal energy is zero regardless of its present configuration. The entropy (S) is not constant, however, since the number of available configurations decreases with the chain end separation distance. The variation which follows from chain length change by a small amount (dr) at constant temperature (T) is given by the Boltzmann rule of statistical thermodynamics ... 

The extended or contour length of the freely oriented macromolecule will heal. Its rms end-to-end distance will be from the above equation. It can be seen that the ratio of the average end-to-end separation to the extended length is ct Since a will be of the order of a few hundred even for moderately sized macromolecules, d will be on the average much smaller than the chain end separation in the fully extended conformation. 

The average chain end separation which has been calculated gives little information about the magnitudes of this distance for a number of macromolecules at any instant. When this distribution of end-to-end distances is calculated, it is found, not surprisingly, that it is very improbable that the two ends of a linear molecule will be very close or very far from each other. It can also be shown lhat the density of chain segments is greatest near the center of a macromolecule and decreases toward the outside of the random coil. 

The force to hold chain ends separated by a general vector R is linear in R, like a simple elastic spring ... 

Recent theory suggests that cyclization equilibria are doubly sensitive to excluded volume effects (3). Not only does chain expansion increase the mean chain end separation, but a second factor due to pair correlations also acts to decrease the probability of the chain ends being in proximity The two chain... 

The essential requirement for a substance to be rubbery is that it consist of long flexible chainlike molecules. The molecules themselves must therefore have a backbone of many noncolinear single valence bonds, about which rapid rotation is possible as a result of thermal agitation. Some representative molecular subunits of rubbery polymers are shown in Fig. 1 thousands of these units linked together into a chain constitute a typical molecule of the elastomers listed in Fig. 1. Such molecules change their shape readily and continuously at normal temperatures by Brownian motion. They take up random conformations in a stress-free state but assume somewhat oriented conformations if tensile forces are applied at their ends (Fig. 2). One of the first questions to consider, then, is the relationship between the applied tension / and the mean chain end separation r, averaged over time or over a large number of chains at one instant in time. 

The above comments should not be taken to imply that the average distance between the chain ends is zero. This is clearly a nonsense since if the average distance apart were zero all the chain end separation distances would be zero. The chain ends are of course normally at a distance to each other it is simply a matter of averaging the values of the coordinates of the other end which gives the average value coordinate as (0,0,0). To avoid such complications it is useful to consider the root mean square value of the chain end-to-end distance. 

Since the fully extended length is equal to na it follows that the average chain end separation is 1/Vn of the fully extended length. That is to say for a molecule with 10 000 links of length a the average... 

The particular distribution function of the chain end separation r is obtained by transferring eqn (3.13) to polar coordinates to give ... 

Fig. 3.2. Probability distribution function. Distribution of chain end separation distances, (a) Schematic arrangement with one end at the origin and the other in a volume element 4irr dr (b) probability distribution with one end at the origin and the other in a volume element 4irr dr.

If there are n segments with an average -component of (a ) the average x-component of chain end separation will be ... 

Since there is no tension in the y- and z-directions this result is equal to the actual average chain end separation, r, where ... 

This result is interesting in that it indicates that not only is the force proportional to the chain end separation but also proportional to the temperature. This latter relationship reflects the fact that at higher temperatures molecular movement is more violent and alignment is therefore much more difficult. It is also to be noted that the longer the molecule, as represented by na, the lower is the force required to increase the separation to a distance r. 

The only parameter, b, is the inverse of the most probable chain end separation and is under the above conditions Z3/n. The extended length of the chain is, of course, equal to nC, whereas the average chain end separation (the root-mean-square length /W equals /n. 

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