Systematic development of coarse-grained polymer models Patrick ...

1.3. Progression of Coarse-grained Models 7
response in terms of the ratio of the Kuhn length to the bond length, aK/a = C∞/β. If this ratio is
only slightly greater than one, such as 5 to 15, it is generally believed that the freely jointed chain
correctly represents the behavior out of equilibrium. This is the typical range for many synthetic
polymers.
However, if the value of C∞/β is very large, such as about 300 for double stranded DNA, the
freely jointed chain does not represent the behavior of the polymer out of equilibrium. Note that
this is true even if the chain is flexible in that the chain contains many Kuhn lengths, NK ≫ 1. The
typical micromechanical model which is used to describe these polymers is the continuous worm-like
chain instead of the freely jointed chain [16, 17, 18, 19, 20, 21, 22]. The continuous worm-like chain
is a continuous space-curve like a string which has a bending rigidity. Essentially the backbone of
the chain is so stiff that the Kuhn length is much larger than the bond length. Therefore, at the
scales of interest, the bonds look essentially as a continuum.
The worm-like chain micromechanical model is defined by a space curve r(s) where s measures
the position along the contour and runs from 1 to L. A unit tangent vector is defined by
∂r
t(s) = /
∂r
∂s ∂s
. (1.9)
The energy of a configuration is
E = κ
2
L
0
2 ∂t
ds (1.10)
∂s
where κ is the bending modulus. It is common to replace the bending modulus with a persistence
length, which in three dimensional space is defined by κ = kBTAp. Because of the energetic
cost to bending, within short distances along the contour the chain persists in the same direction.
At long distances along the contour, the chain orientation becomes uncorrelated with previous
orientations. The persistence length is the characteristic distance over which this decorrelation
occurs. Mathematically this is written as
Another important property of the worm-like chain is
〈t(s) · t(p)〉 = exp(−|s − p|/Ap) . (1.11)
〈r 2 〉 =2LAp +2A 2 p(exp(−L/Ap) − 1) . (1.12)
In order to be in the limit where 〈r 2 〉∝L, the contour length must be much larger than the
persistence length. The response of the continuous worm-like chain in this limit is
〈r 2 〉 =2LAp . (1.13)
Recall that in this limit, near equilibrium, any polymer can be describe by the freely jointed chain.
By comparing this formula with the formula for a freely jointed chain, we see that for the freely
jointed chain to model a very long continuous worm-like chain near equilibrium, the Kuhn length
must be equal to twice the persistence length. Note however that this whole discussion of the
worm-like chain was motivated by the fact the freely jointed chain does not model the response
correctly away from equilibrium.
Even though this discussion was motivated by flexible polymers which are not modeled correctly