9.1. Elasticity models of polymers[[Student version, January 17, 2003]] 305
length” forA: Only at separations less thanA will the molecule remember which way it was
pointing.^1
This analysis suggests that the “paperclip-chain” model of the conformation of polymers should
bebroadly applicable. Since a polymer is rigid on the scale of a monomer, yet flexible on length
scales much longer thanA,it’s reasonable to make the idealization that its conformation is a chain
ofperfectly straightsegments, joined byperfectly freejoints. We take theeffective segment length,
Lseg,tobeaphenomenological parameter of the model. (Many authors refer toLsegas theKuhn
length.) We expectLsegto be roughly the same asA;sinceAis itself an unknown parameter we
lose no predictive power if we instead phrase the model in terms ofLseg.^2 Toshow our increased
respect for this model, we now rename it thefreely jointed chain(orFJC)model. Section 9.2
will show that for DNA,Lseg≈ 100 nm;conventional polymers like polyethylene have much shorter
segment lengths, closer to 1nm.Since the value ofLsegreflects the bend stiffness of the molecule,
DNA is often called a “stiff,” orsemiflexiblepolymer.
The FJC model is a reduced form of the underlying elastic rod model (Equation 9.3). We will
improve its realism later. But it at least incorporates the insight of Idea 9.5, and it will turn out
to be mathematically simpler to solve than the full elastic rod model.
In short, we propose to study the conformation of a polymer as a random walk with step size
Lseg.Before bringing any mathematics to bear on the model, let us first see if we find any qualitative
support for it in our everyday experience.
The elasticity of rubber Atfirst sight the freely jointed chain may not seem like a promising
model for polymer elasticity. Suppose we pull on the ends of the chain until it’s nearly fully
stretched, then release it. If you try this with a chain made of paperclips, the chain stays straight
after you let go. And yet a rubber band, which consists of many polymer chains, will insteadrecoil
when stretched and released. What have we missed?
The key difference between a macroscopic chain of paperclips and a polymer is scale: The
thermal energykBTis negligible for macroscopic paperclips, but significant for the nanometer-scale
monomers of a macromolecule. Suppose we pull our paperclip chain out straight, then place it
on a vibrating table, where it gets random kicks many times larger thankBT:Then its endswill
spontaneously come closer together, as its shape gradually becomes random. Indeed, we would
have to place the ends of the chain under constant, gentle tension topreventthis shortening, just
as we must apply a constant force to keep a rubber band stretched.
Wecan understand the retracting tendency of a stretched polymer using ideas from Chap-
ters 6–7. A long polymer chain can consist of hundreds (or millions) of monomers, with a huge
number of possible conformations. If there’s no external stretching, the vast majority of these
conformations are sphere-like blobs, with mean-square end-to-end lengthzmuchshorter thanLtot
(see Section 4.3.1 on page 110). That’s because there’sonly one wayto be straight, but many ways
to be coiled up. Thus if we hold the ends a fixed distancezapart, the entropy decreases whenz
goes up. According to Chapter 7, there must then be an entropic force opposing such stretching.
That’s why a stretched rubber band spontaneously retracts:
The retracting force supplied by a stretched rubber band is entropic in origin. (9.6)(^1) The situation is quite different for two-dimensional elastic objects, for example membranes. We already found in
Section 8.6.1 that the energy cost to bend a patch of membrane into, say, a hemisphere is 4πκ,aconstantindependent
of the radius. Hence membranes donotrapidly lose their planar character on length scales larger than their thickness.
(^2) T 2 Section 9.1.3′on page 338 shows that the precise relation isLseg=2A.