Perturbation theory for a polymer chain with excluded volume interaction

Abstract

We present a simple derivation of the mean square end‐to‐end distance 〈R 2〉 of a linear flexible chain as a perturbation series in the dimensionless excluded volume parameter z d . Our results, to orders six and four in space dimension d=3 and 2, respectively, are 〈R 2〉= L l[1+ (4)/(3) z 3−2.075 385 396 z 2 3+6.296 879 676 z 3 3 GFIX−25.057 250 72 z 4 3+116.134 785 z 5 3 GFIX−594.716 63 z 6 3+⋅⋅⋅], d=3, 〈R 2〉= L l[1+ 1/2 z 2−0.121 545 25 z 2 2+0.026 631 36 z 3 2 GFIX−0.132 236 03 z 4 2+⋅⋅⋅], d=2, where z 3=(3/2πl)3 / 2 w L 1 / 2 and z 2=w L/πl with L the contour length of the chain, l the effective or Kuhn segment length, and w l 2 the effective binary cluster integral for a pair of segments. Our method uses in an essential way Laplace transforms with respect to the contour length L; the resulting graphical expansion, when combined with the field theoretical methods, is far simpler than that in the conventional cluster expansion approach. Furthermore, we prove that 〈R 2〉 is free of ln L terms to all orders in the perturbation theory in both d=2 and 3.