Elasticity of a Rubber Band

Elasticity of a rubber band

The elasticity of a rubber band can be described by a one dimensional model of a polymer, involving N molecules of length a each, linked together end-to-end. The angle between successive links can be taken as 0˚ or 180˚ and the joints can turn freely. The distance between the end points is x, and the temperature is T. Find the force (tension) f which is necessary to maintain the distance x. Does the polymer try to expand or to contract? You can use the canonical formalism in order to solve the problem. Explain why the effect is of “entropic” nature.

Tension of a chain molecule

A three dimensional chain molecule consists of N units, each having length a. The units are joined so as to permit free rotation about the joints. The distance between the two ends is L, and the temperature is T. Find the tension f acting between both ends of the chain molecule.

Tension of chain molecule

N monomeric units are arranged along a straight line to form a chain molecule. each unit can be either in a state  (with length a and energy E) or in a state  (with length b and energy E). Derive the relation between the length L of the chain molecule and the tension f applied between at the ends of the molecule.

Find the compressibility T=(∂L/∂f)T. Plot schematically L(fa/kBT) and T(fa/kBT) and interpret the shape of the plots.

The zipper model for DNA molecule

The DNA molecule forms a double stranded helix with hydrogen bonds stabilizing the double helix. Under certain conditions the two strands get separated resulting in a sharp "phase transition" (in the thermodynamic limit). As a model for this unwinding, use the "zipper model" consisting of N parallel links which can be opened from one end (see figure). If the links 1, 2, 3, ..., p are all open the energy to open to p+1 link is  and if the earlier links are closed the energy to open the link is infinity. The last link p=N cannot be opened. Each open link can assume G orientations corresponding to the rotational freedom about the bond.

Construct the canonical partition function.

Find then the average number of open links <p> as function of x=Gexp[/kBT].

Plot <p> as function of x (assuming N very large).

What is the value of x at the transition?

Study <p> near the transition: what is its slope as N ∞ ?

Derive the entropy S. What is it at the transition region and at the transition?

Do the same for the heat capacity. What is the order of the transition?