Configurations and dynamics of real chains. III. The excluded volume effect on inner segments

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Abstract
The expansion effect experienced by the inner segments within an infinitely long polyethylene chain is investigated. Following Flory [J. Chem. Phys. 17, 303 (1949)], the chain free energy is expressed as the sum of an elastic and an excluded volume component, within the Gaussian approximation. Introducing the Fourier coordinates l̃ (q) = ∑ k l (k)e iqk [ l (k) = general bond vector] allows us to formulate the latter component as a sum of pairwise contributions where the appropriate expansion is considered for each atom pair, unlike the mean‐field approaches. Minimization of the free energy over the expansion factor α̃ 2 (q) of the squared amplitude 〈‖ l̃ (q)‖ 2 〉 produces an integral equation that eventually gives α 2 (k), i.e., the expansion of the square distance between kth neighboring atoms. In addition to asymptotic expressions for very small and very large expansions, numerical results are given in a wide range of segment lengths for different values of the excluded volume parameter and of the smallest number of bonds between two atoms in contact. It turns out that, even at comparatively long range, the chain expansion is sensitive to the latter parameter, which effectively embodies the detailed pattern of short‐range interactions. On the average, the inner expansion factor α 2 (k) is more than twice larger than for the terminal atoms of a (k+1)‐ atom chain, using the classical Flory equation α 5 −α 3 = C√k and the value of C suggested by Fixman on the basis of exact calculations [J. Chem. Phys. 23, 1656 (1955)]. Asymptotic analysis shows that α 2 (k)∝k 1/5 ⋅( log k) 2/5 … for large k, thus reobtaining Flory’s well‐known power law except for ’’weak’’ logarithmic factors. In the free‐draining case, the half‐peak time t 1/2 of the incoherent dynamic structure factor scales as Q −11/3 [Q = 4π sin (θ/2)/λ] in the long‐time limit, as already predicted, although different exponents appear at intermediate times.