Non-Gaussian Behavior of the Polyethylene Chain of Finite Length

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Abstract
The second and fourth moments 〈r2〉 and 〈r4〉 of the end-to-end distance r of the polyethylene chain of length n are calculated, based on expressions previously derived. The model employed is the familiar three-state [trans (θ(T)=0°), gauche (θ(G)=120°), and gauche prime (θ(G′)=−120°)] rotational—isomeric model. An approximate expression for the reduced distribution function Pn(ξ)=4π〈r2〉*32ξ2Wn(〈r2〉*12ξ), which is readily obtainable from 〈r2〉 and 〈r4〉, is calculated as a function of the reduced end-to-end distance ξ=〈r2〉*−½r. Here Wn(r) is the distribution function in the usual sense and 〈r2〉*=nlin lim n→∞n−1〈r2〉.(An asterisked quantity denotes the corresponding Gaussian analog.) Approximate expressions for Wn(0)/Wn*(0) and 〈r−1〉/〈r−1〉* are also calculated as functions of n. For the GG′-eliminated trans-preferred chain with reasonable values for parameters, which approximates the polyethylene chain: (i) 〈r2〉 and 〈r4〉 are smaller than their respective Gaussian analogs, the deviation being more pronounced with the latter, (ii) the approximate distribution function becomes sharp and somewhat shifted toward ξ=0 compared with the Gaussian distribution, and (iii) as n decreases from infinity, both Wn(0)/Wn*(0) and 〈r−1〉/〈r−1〉* first decrease gradually from unity, then pass through minima, and subsequently increase rapidly, largely exceeding unity. The GG′-eliminated gauche-preferred chain behaves like the chain just mentioned above in all respects. For the gauche-preferred independent-rotation chain, the opposite trends result for Items (i) and (ii) above; for Item (iii), both quantities monotonously decrease as n decreases. It is shown that for the polyethylene chain, Kuhn's equivalent random chain does not correctly predict (considerably underestimates) the departure from the Gaussian distribution.