The Number of Chain Conformations

Free energy of a constrained chain It is of considerable interest to count the number of conformations of a polymer chain with some constraints imposed on it. For example, by fixing one end of the chain, the other end is allowed to take an end-to-end distance R. Now, the number of chain conformations Af(R) depends on R, and the chain entropy for a given R is... 

Even in the absence of flow, a polymer molecule in solution is in a state of continual motion set forth by the thermal energy of the system. Rotation around any single bond of the backbone in a flexible polymer chain will induce a change in conformation. For a polyethylene molecule having (n + 1) methylene groups connected by n C — C links, the total number of available conformations increases as 3°. With the number n encompassing the range of 105 and beyond, the number of accessible conformations becomes enormous and the shape of the polymers can only be usefully described statistically. 

The number of possible conformers increases with chain length and can be shown statistically to equal 2 " where n is the number of units. Thus, when n= 1000, the number of possible conformers of HDPE is 2 °°° or 10 °, more than the grains of sand at all of our beaches combined. Four of these possible conformers are shown in Figure 2.12. 

The virtue of the RIS approach lies in the form of Eq. (3) A global exact average (subject, of course to the simplification of factorizability as set forth in Eq. (l),Eq. (2), and Eq. (3)) over a very large number of chain conformations can be obtained with a simple matrix product, calculable with a trivial computational effort that is provided by any modern Personal Computer. And since each U,- and each F,- can be defined separately, practically any chemical structure will yield to an RIS treatment. 

The configurational characteristics of PTE are estimated by calculation based on the information acquired through the analysis of the polythlopropylene chain [Abe, A. Macromolecules 1980, 13, 541). Results suggest that the polymer chain is quite flexible. The flexibility of the chain estimated in terms of the number of allowed conformations for a monomeric residue may be arranged in the order, PTE > PTP > PO E > POP. The results of the present analysis are consistent with the view presented by Takahashi, Tadokoro, and Charani U Macromol. Sci.-Phys. 1968,... 

All proteins are made in the same way but as the growing peptide chains peel off from the ribosome, each of the thousands of different proteins in a living cell folds into its own special tertiary structure.883 The number of possible conformations of a protein chain is enormous. Consider a 300-residue polypeptide which could stretch in fully extended form for -100 nm. If the chain were folded back on itself about 13... 

The number of possible conformations n for each mobile molecular segment is reciprocal to the density of cross-links AT (defined for 100 C-C groups in the polymer chain). 

Hoeve44,45) extended his theory further by considering not only interactions between the train segments but also interactions among the loops, and found that the latter lead to a decrease in the number of possible conformations of adsorbed polymer chains. He assumed that the segment density distribution in any loop is uniformly expanded in one dimension by a factor of at as a result of loop-loop interactions. The volume fraction of segments at a distance z > 6 is then given by... 

In order to find the number of chains Nd in a particular conformation d, In Q is differentiated with respect to Nd, and we get... 

Sample deformations modify the number of accessible conformations of intercross-link chains (cf. Figure 7.7), so that they can be detected by analysis of relaxation and residual dipolar couplings. This is illustrated for strained rubber bands with a cut in Figures 7.6 and 7.11. 

Conformational maps show values of phi and psi that fall into two regions the first region contains points that are always allowable and the second region contains points that are sometimes allowed. As shown in Figure 2.11, all points within the inner lines are always allowed and those within the outer solid lines are sometimes allowed. Hence the entropy (or rotational freedom of a peptide) is obtained from the area surrounded by the solid lines on a conformational map (Figure 2.11) and is proportional to the number of allowable conformations of a dipeptide unit. Please note a flexible chain with a lot of rotational freedom is easier to stretch than a rigid chain. We define the flexibility of a polypeptide as the natural logarithm of the number of allowable conformations,, times Boltzmann s constant (Equation (2.1)). 

---