Statistics of Orientation Effects in Linear Polymer Molecules

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Abstract
This paper is concerned with the effects of orientation on the combinatorial term g for the number of ways to pack together Nx linear polymers (x mers). Accordingly g is evaluated as a function of the number of molecules in each permitted direction for the case of straight rigid rods. The permitted directions can be continuous so that g is derived as a function of the continuous function f(r) which gives the density of rods lying in the solid angle Δr, or the permitted directions can be discrete so that g is the number of ways to pack molecules onto a lattice. To illustrate the usefulness of the orientation dependent combinatorial terms, liquid crystals are discussed. Another phase is found to exist in addition to the previously predicted nematic phase. This phase is tentatively identified with the cholesteric phase. A procedure is developed for the calculation of the orientation dependent combinatorial term associated with the packing together of molecules of arbitrary shape. A very approximate application of this procedure results in an approximate expression for the combinatorial term which allows one to predict qualitatively the change in the entropy of packing as a function of stretch. It is found that the entropy of packing has the proper behavior to explain the initial deviation of the experimental stress‐strain curve from the previous theoretical predictions.

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