338 Chapter 9. Cooperative transitions in macromolecules[[Student version, January 17, 2003]]
Idea 7.28 on page 237). As long as we consider length scales longer than the Debye screening
length of the salt solution, however, our phenomenological argument remains valid; we can simply
incorporate the electrostatic effects into an effective value of the bend stiffness.
9.1.3′
- We can make the interpretation ofAas a persistence length, and the passage from Equation 9.2
to a corresponding FJC model, more explicit. Recalling thatˆt(s)isthe unit vector parallel to the
rod’s axis at contour distancesfrom one end, we first prove that for a polymer under no external
forces,
〈ˆt(s 1 )·ˆt(s 2 )〉=e−|s^1 −s^2 |/A. (to be shown) (9.28)
Heres 1 ands 2 are two points along the chain;Ais the constant appearing in the elastic rod model
(Equation 9.3 on page 304). Once we prove it, Equation 9.28 will make precise the statement that
the polymer “forgets” the directionˆtof its backbone over distances greater than its bend persistence
lengthA.
Toprove Equation 9.28, consider three pointsA, B, Clocated at contour distancess,s+sAB,
ands+sAB+sBCalong the polymer. We will first relate the desired quantityˆt(A)·ˆt(C)to
ˆt(A)·ˆt(B)andˆt(B)·ˆt(C). Set up a coordinate frameˆξ,ηˆ,ˆζwhoseˆζ-axis points alongˆt(B). (We
reserve the symbolsˆx,yˆ,ˆzfor a frame fixed in the laboratory.) Let (θ, φ)bethe corresponding
spherical polar coordinates, takingζˆas the polar axis. Writing the unit operator as (ˆξˆξ+ˆηηˆ+ˆζˆζ)
gives
ˆt(A)·ˆt(C)=ˆt(A)·(ξˆξˆ+ηˆˆη+ζˆˆζ)·ˆt(C)
= t⊥(A)·t⊥(C)+(ˆt(A)·ˆt(B))(ˆt(B)·ˆt(C)).In the first term, the symbolt⊥represents the projection ofˆtto theξηplane. Choosing theˆξaxis
to be alongt⊥(A)gives thatˆt⊥(A)=sinθ(A)ˆξ,so
ˆt(A)·ˆt(C)=sinθ(A)sinθ(C)cosφ(C)+cosθ(A)cosθ(C). (9.29)So far we have only done geometry, not statistical physics. We now take the average of both
sides of Equation 9.29 over all possible conformations of the polymer, weighted by the Boltzmann
factor as usual. The key observations are that:
- The first term of Equation 9.29 vanishes upon averaging. This result follows because
the energy functional, Equation 9.2, doesn’t care which direction the rod bends—
it’s isotropic. Thus, for every conformation with a particular value ofφ(C), there
is another with the same energy but a different value ofφ(C), and so our averaging
overconformations includes integrating the right hand side of Equation 9.29 over all
values ofφ(C). But the integral
∫ 2 π
0 dφcosφequals zero.
10- The second term is the product of two statistically independent factors. The shape
of our rod betweenAandBmakes a contributionABto its elastic energy, and
also determines the angleθ(A). The shape betweenBandCmakes a contribution
BCto the energy, and determines the angleθ(C). The Boltzmann weight for this
conformation can be written as the product of e−AB/kBT(which does not involve
θ(C)), times e−BC/kBT(which does not involveθ(A)), times other factors involving
neitherθ(A)norθ(C). Thus the average of the product cosθ(A)cosθ(C)equals the
product of the averages, by the multiplication rule for probabilities.
(^10) Weused the same logic to discard the middle term of Equation 4.3 on page 103.