# Phase transition of a rubber band model

## Homework Statement

In class we studied a phase transition in a model of a rubber band. For small applied tensions, $f < f^*$, the Helmholtz free energy of the band as a function of the length is $A_1(L) = (\kappa_1 /2)L^2$, while at large applied tensions, $f > f^*$, it is $A_2(L) = (\kappa_2 /2)L^2 + a$. The tension at which the two states coexist was given by $f^* = \left (2 a \Delta \kappa \right)^\frac{1}{2}, \Delta \kappa \equiv 1/\kappa_2 - 1/\kappa_1$. This analysis assumes that length and tension are controlled.

Now I have to consider an example where the net length of the system is $L_{total} = L + x$. The net free energy is given by
$$A_i(L) + (k/2)x^2, i = 1 or 2$$

where $i$ refers to the subsystem with variable $L$, called the primary subsystem. The secondary subsystem is that with variable $x$.

I wish to determine the net equilibrium free energy and equilibrium value of $L$ for a fixed value of $L_{total}$.

## The Attempt at a Solution

For the simple rubber band model, we determined the equilibrium free energy by defining a Legendre transform
$$\mathcal{G}_i(L_{total}; f) = A_i(L) + (k/2) x^2 - fL_{total}$$
and finding the minimum value of this by differentiating with respect to the length, setting the derivative to 0 and then evaluating the Legendre transform at this equilibrium length.

I attempted to do the same thing for this problem, and got as far as:
$$\left( \frac{\partial \mathcal{G}_i}{\partial L_{total}} \right)_f = \left[ \kappa_i L_{eq,i} - k \left( L_{total,eq,i} - L_{eq,i} \right) \right] \left( \frac{\partial L}{\partial L_{total}} \right)_f - f = 0$$

I am not sure what to do at this point. How could I go about using that equation to find the minimum of $\mathcal{G}_i$?

Thanks!

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