Worm-like chain

<r^Tl₁>/l = l [1 - (- cos theta)ⁿ]/(1 + cos theta)

The persistence length, a, is the limit for an infinitely long chain.

a = lim_{nl -> infinity} <r^Tl₁>/l = l/(1 + cos theta)

The worm-like chain has the same value of a. It is obtained by letting l approach zero, simultaneously increasing n so that the contour length, nl, remains constant. The bond angle is opened so that 1 + cos theta will vanish, causing a to retain the desired finite value. The result is a chain of continuous curvature, with a random direction for the curvature at any point on the chain. The chain is described by two parameters, one of which specifies the stiffness (a) and the other the contour length (r_max).

<r²>₀ = 2ar_max {1 - (a/r_max)[1 - exp (-r_max/a)]}

• The dimensionless ratio <r²>₀/ar_max approach a limit as a/r_max goes to zero. This limit describes a long flexible chain, where

  <r²>₀/r_max = 2a(1 - a/r_max) = 2a

• A stiff chain has a small value of r_max/a, and <r²>₀ is proportional to r_max².

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