Phase transition of a rubber band model

Ultimately, both methods should give you the same result for the equilibrium values of L and x. In summary, to find the net equilibrium free energy and equilibrium value of L for a fixed value of L_{total}, you can use either method to solve for the minimum of \mathcal{G}_i.
  • #1
mooshasta
31
0

Homework Statement


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

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

where [itex]i[/itex] refers to the subsystem with variable [itex]L[/itex], called the primary subsystem. The secondary subsystem is that with variable [itex]x[/itex].

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

Homework Equations



The Attempt at a Solution



For the simple rubber band model, we determined the equilibrium free energy by defining a Legendre transform
[tex]\mathcal{G}_i(L_{total}; f) = A_i(L) + (k/2) x^2 - fL_{total}[/tex]
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:
[tex]\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[/tex]

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

Thanks!
 
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  • #2


To find the equilibrium value of L, you can solve for L in the equation you have derived:
\kappa_i L_{eq,i} - k \left( L_{total,eq,i} - L_{eq,i} \right) = f

Then, you can substitute this value of L into the expression for the net free energy:
\mathcal{G}_i(L_{total}; f) = A_i(L) + (k/2) x^2 - fL_{total}

This will give you the net equilibrium free energy as a function of x. To find the minimum value of this function, you can take the derivative with respect to x and set it equal to 0, then solve for x. This will give you the equilibrium value of x, which you can then use to find the equilibrium value of L using the equation you derived earlier.

Alternatively, you can also use the method of Lagrange multipliers to find the minimum of \mathcal{G}_i. This involves introducing a Lagrange multiplier \lambda and solving the following set of equations:
\frac{\partial \mathcal{G}_i}{\partial L_{eq,i}} = \lambda \frac{\partial L}{\partial L_{total}}
\frac{\partial \mathcal{G}_i}{\partial x} = \lambda \frac{\partial L}{\partial L_{total}}
\frac{\partial \mathcal{G}_i}{\partial L_{total}} = 0

Solving these equations will give you the equilibrium values of L and x, as well as the value of the Lagrange multiplier \lambda, which can be used to find the minimum of \mathcal{G}_i.
 

Related to Phase transition of a rubber band model

1. What is a phase transition in the context of a rubber band model?

A phase transition in a rubber band model refers to a sudden change in the behavior or properties of the rubber band, such as its length, elasticity, or shape, due to a change in external conditions like temperature or stress.

2. What factors influence the phase transition of a rubber band model?

The phase transition of a rubber band model is influenced by various factors, including the material of the rubber band, the temperature, the amount of stress applied, and the rate at which the stress is applied.

3. How does the temperature affect the phase transition of a rubber band model?

Temperature plays a significant role in the phase transition of a rubber band model. As the temperature increases, the rubber band becomes more flexible and can stretch further before breaking. However, at extremely high temperatures, the rubber band may lose its elasticity and become permanently deformed.

4. What is the difference between a first-order and a second-order phase transition in a rubber band model?

A first-order phase transition in a rubber band model is characterized by a sudden and discontinuous change in the properties of the rubber band, such as its length or elasticity. On the other hand, a second-order phase transition is a gradual and continuous change in these properties, without a sudden jump.

5. Can the phase transition of a rubber band model be reversed?

Yes, the phase transition of a rubber band model can be reversed by changing the external conditions, such as decreasing the temperature or removing the stress applied. This can cause the rubber band to return to its original state before the phase transition occurred.

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