Photoelastic Property of Cross-Linked Amorphous Polyethylene

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By assuming the additivity of bond polarizabilities, a formal expression is derived for the difference Δγ in principal polarizabilities of a real polymer chain having constant end-to-end distance r. The expression is given as a series in powers of r2, whose coefficients are functions of the differences α1κ—α2κ in principal polarizabilities of constituent bonds, 〈r2k〉, and 〈r2k cos2Φiκ〉, (k≥1). Here, Φiκ is the angle between the end-to-end vector and the unit vector along the κth bond of the ith structural unit, and the averages refer to those about a polymer chain in free state. It is shown that if r2 is in magnitude of the same order as 〈r2〉, at the limit of infinite chain length the expression reduces to Δγ=310〈r2〉*−2 ∑ i,κ(α1κ−α2κ)(3〈r2cosΦiκ〉−〈r2〉)r2,where 〈r2〉* is such a part of 〈r2〉 as proportional to chain length. The difference in principal polarizabilities of the ``equivalent random link'' is shown to be ΔΓ=12〈r2〉*−1 ∑ i,κ(α1κ−α2κ)(3〈r2cos2Φiκ〉−〈r2〉).An expression in a matrix form is derived for ΔΓ of the polyethylene chain. Numerical computations are made by a digital computer, using two geometrical models for this molecule recently proposed by Hoeve and by Nagai and Ishikawa, and also using Denbigh's and Bunn and Daubeny's values for bond polarizabilities. It is found that a good agreement is obtained between predictions of the theory and experimental values if Denbigh's values are used.