- Problems[[Student version, January 17, 2003]] 257
not a minimum of free energy: Its entropy is not maximal. We know that diffusion will eventually
erase the initial order.
Section 7.2.1 argued that for dilute solutions, the dependence of entropy on concentration was
the same as that of an ideal gas. Thus the entropySof our system will be the integral of the entropy
density (Equation 6.36 on page 206) over d^3 r,plus a constant which we can ignore. Calculate the
time derivative ofSin a thermally isolated system, using what you know about the time derivative
ofc.Then comment. [Hint: In this problem you can neglect bulk (convective) flow of water. You
can also assume that the concentration is always zero at the boundaries of the chamber; the ink
spreads from the center without hitting the walls.]
7.9 T 2 Another mean-field theory
The aim of this problem is to gain a qualitative understanding of the experimental data in Fig-
ure 4.8c on page 113, using a mean-field approximation pioneered by P. Flory.
Recall that the figure gives the average size of a random coil of DNA attached (“adsorbed”) to
atwo-dimensional surface—a self-avoiding, two-dimensional random walk. To model such a walk,
wefirst review unconstrained (non-self-avoiding) random walks. Notice that Equation 4.27 on page
128 gives the number ofN-step paths that start at the origin and end in an area d^2 raround the
positionr(in an approximation discussed in Section 4.6.5′on page 134). Using Idea 4.5b on page
104, this number equals e−r
(^2) /(2NL (^2) )
d^2 rtimes a normalization constant, whereLis the step size.
Tofind the mean-square displacement of an ordinary random walk, we compute the average〈r^2 〉,
weighting every allowed path equally. The discussion above lets us express the answer as
〈r^2 〉=−
d
dβ
∣∣
∣∣
β=(2NL^2 )−^1ln∫
d^2 re−βr2
=2L^2 N. (7.42)That’s a familiar result (see Your Turn 4g(c) on page 128).
But wedon’twantto weight every allowed path equally; those that self-intersect should be
penalized by a Boltzmann factor. Flory estimated this factor in a simple way. The effect of self-
avoidance is to swell the polymer coil to a size larger than what it would be in a pure random
walk. This swelling also increases the mean end-to-end length of the coil, so we imagine the coil
as a circular blob whose radius is a constantCtimes its end-to-end distance,r=|r|.The area of
such a blob is thenπ(Cr)^2 .Inthis approximation, then, the average surface density of polymer
segments in the class of paths with end-to-end distancerisN/(πC^2 r^2 ).
Wenext idealize the adsorbed coil as having auniformsurface density of segments, and assume
that each of the polymer’s segments has a probability of bumping into another that depends on
that density.^11 If each segment occupies a surface areaa,then the probability for an area element
to be occupied isNa/(πC^2 r^2 ). The probability that any of theNchain elements will land on a
space that is already occupied is given by the same expression, so the number of doubly occupied
area elements equalsN^2 a/(πC^2 r^2 ). The energy penaltyV equals this number times the energy
penaltypercrossing. Writing ̄=a/(πC^2 kBT)gives the estimateV/kBT= ̄N^2 /r^2.
Adapt Equation 7.42 by introducing the Boltzmann weighting factor e−V/kBT.TakeL=1nm
and ̄=1nm^2 for concreteness, and work at room temperature. Use some numerical software to
evaluate the integral, finding〈r^2 〉as a function ofNfor fixed segment lengthLand overlap cost
̄. Make a log-log plot of the answer and show that for largeN,〈r^2 〉→const×N^2 ν. Find the
exponentνand compare to the experiment.
(^11) Substituting this estimate for the actual self-intersection of the conformation amounts to a mean-field approxi-
mation, similar in spirit to the one in Section 7.4.3 on page 233.