Strong Stretching of a Freely-Jointed Chain . 139

If a freely-jointed chain is confined in a narrow tube, each of its straight segments makes a small angle with the tube axis. That is unfavorable in terms of entropy: as the tube gets more narrow, every segment has fewer

140 Giant Molecules: Here, There, and Everywhere

orientations available to it. We can apply Boltzmann principle to analyze this “orientational entropy”. Since we talk about orientations only, we can imagine that one end of the segment is fixed in some point in space, then the other end chooses any point on the sphere of radius `, its surface is 4⇡`². But if the segment is confined in a tube, its end is restricted to the much smaller surface proportional to D², as it should be clear from the lower part of Figure 7.11 a. Therefore, dropping as usual the numerical coefficients³, we can say the entropic price of confining one segment into a tube is about kBln(D²/`<sup>2</sup>), and for all N segments we get S = N kBln(D<sup>2</sup>/`²). The physical meaning of this quantity is revealed by noticing that T S is the minimal amount of work necessary to confine polymer into the tube, and

S < 0 means that the chain resists being squeezed.

Interestingly, it does not matter how we are going to perform work:

we can squeeze polymer from the sides or we can pull it by the ends, the amount of work should not depend on that (the beauty of entropy and Boltzmann principle!). But if we are talking about pulling the ends, it is more convenient to use end-to-end distance R instead of D. How are they related? For each segment, its projection along the tube axis is `² D²

` D<sup>2</sup>/2` (since D `). Therefore, total end-to-end distance is R N ` D²/2` = L LD<sup>2</sup>/2`². By doing now simple algebra we can re-express D in terms of R and then re-write entropy or the corresponding free energy for all L/` segments as an explicit function of R:

F kBTL

The necessary pulling force is found from here by di↵erentiation: f =

@ F/@R:

f L

` kBT

L R (freely jointed, strong stretching), L R L . (7.35) We see that force blows up as we approach full stretching. It is always difficult to devoid any system of its last pieces of freedom, which is why it is impossible to reach absolute zero of temperature, and which is why it is impossible to reach complete stretching of a polymer with finite force; in fact, the chain will break far before it reaches complete stretching.

This is all good, but formula (7.35) did not agree with experiment on dsDNA.

3We drop numerical coefficients inside the ln, such as 4⇡ etc., because, for instance, ln(4⇡`<sup>2</sup>/D<sup>2</sup>) = ln(`²/D²) + ln(4⇡) and, since ` D, we neglect the second term.

7.12.3 Strong Stretching of a Worm-Like Chain

Let’s try to analyze the worm-like chain model in the tube. We will do it using the idea of Figure 7.6, namely, finding the fluctuation with energy about kBT . As it is seen in Figure 7.11 b, conformation of a worm-like chain in a tube represents a kind of succession of arcs. Let’s concentrate on one such arc. If is the length of it, then simple geometry, shown in the lower part of Figure 7.11 b suggests that curvature radius ⇢ of the arc is about

⇢

2

D, (7.36)

(again dropping numerical coefficients). What is then its energy?

Well, we never said so far how to find bending energy of worm-like chain.

As a matter of fact, we can borrow the prescription for this task from mechanical engineers, for they know everything about bending of elastic

(ℓ2-D2)1/2

ℓ

D

D/2 D

R L

a b

Fig. 7.11 Confined in a narrow tube freely-jointed chain (a) and worm-like chain (b). The lower figures represent auxiliary geometric constructions explained in the text. Notice that worm-like chain hits the walls in more points than freely-jointed.

142 Giant Molecules: Here, There, and Everywhere

beams. But we can also guess ourselves: bending energy should be pro-portional to the squared curvature 1/⇢² (because it should vanish for the straight shape with no curvature, and it should not depend on the sign of curvature); it should be proportional to the length of the piece in question

; and there should be some coefficient describing the rigidity of the ma-terial. If we take bending energy in the units of kBT , the result should be then unitless and, therefore, the material coefficient in this case can be nothing else but persistence length or e↵ective segment (up to a numerical coefficient). Thus, bending energy of the arc reads Ebend kBT ` /⇢². Since bending energy should be just about kBT (see Figure 7.6), we get that

`

⇢² 1 . (7.37)

Equations (7.36)–(7.37) together determine the important length scale — the so-called undulation length (also called Odijk length after the author of the concept — Theo Odijk of Leiden University in The Netherlands):

`^1/3D^2/3 . (7.38)

Notice that `; that means, worm-like chain hits the tube walls more frequently, in more places, than freely-jointed chain; one can say, worm-like chain requires more guidance to keep going straight.

And now we can estimate confinement entropy. We will argue that this entropy is of order unity per each arc of the length . Indeed, we can say each arc can bend to either right or left, which means the relevant number of conformations is estimated as 2^L/ , where L/ is the number of arcs.

According to the Boltzmann principle, we obtain then entropy proportional to L/ . Transforming this to the function of end-to-end distance R (which is geometrically related to in the same way as R was related to ` for freely jointed chain, i.e., R L LD²/2 ², or given formula (7.38), R L

L /` ), we arrive at the free energy F kBT L²

(L R)` (7.39)

and pulling force:

f kBT L²

(L R)²` (worm-like, strong stretching), L R L . (7.40) As in case of freely-jointed chain, Equation (7.35), the force also blows up at the approach to complete stretching, but does so much stronger — as

(L R) ²instead of (L R) ¹. And this di↵erence in power does the trick

— formula (7.40) agrees with the data very well.

It is important to understand the physical meaning of the di↵erence:

why stretching worm-like chain becomes much more difficult at large elon-gations than stretching a seemingly similar freely-jointed polymer? This is actually yet another demonstration of entropic nature of polymer elasticity.

Indeed, to keep the freely jointed chain in the confines of a thin tube it is necessary and sufficient to control the direction of every segment to within proper accuracy, but the segment has to be oriented at one point only. By contrast, the direction of worm-like chain has to be controlled everywhere:

as we decrease the diameter of the confining tube, we have to correct the orientation at an increasing number of points; in other words, as we squeeze the chain in an increasingly narrow tube, or pull it by the ends with an in-creasing force, we must suppress inin-creasingly short wave length motions of the chain, which requires more and more entropy.

A prepared student would be well advised to think through the analogy of this situation with the well-known Einstein and Debye models of heat capacity in solid state physics. The analogy is surely incomplete, and in some ways even reverted, but still very instructive. Let’s remind that in Einstein model a solid body is represented as a set of oscillators, all hav-ing the same frequency; this is similar to freely-jointed chain, in which all possible ways to bend the chain have one and the same wavelength — the segment length `. Einstein model played an important historical role in physics, for it showed how heat capacity can violate the classical mechanics prediction (which says that heat capacity does not depend on temperature);

but Einstein model did not agree with experiments. Debye model proved immensely more successful, because it predicts a certain universal (for all bodies) behavior of heat capacity at low absolute temperature, proportional to T³ (for regular three-dimensional crystals), which agrees very well with numerous experiments. The key feature of Debye model is the realization that oscillators in the crystal are nothing else but sound waves, or phonons in quantum language. As we lower temperature, we remain with increas-ingly long wave length waves (their frequencies and energies are lower), and these are universal because long wave envelops many atoms and hides the di↵erence between them. In this sense strong stretching of worm-like chain is somewhat opposite to lowering temperature of a solid, because in the former case we suppress the long wave lengths, while in the latter case we suppress the short ones. (As a matter of precaution, we warn the reader that these “waves” in a worm-like polymer, unlike regular solid, are just

144 Giant Molecules: Here, There, and Everywhere

a mathematical way to think about bending fluctuations, these waves are overdamped by the friction in the viscous solvent, and they would not exist if not for the constant excitation by the surrounding molecules.)