# free energy and entropy of 2D soap froth

This is a (exploratory) computational project. The soap froth was created by injecting bubbles into a chamber formed by two rectangular plates which are 0.16cm. From the moment the soap froth was created, a camera is employed to take a snapshot of the froth every 10 minutes. The details of the experiment can be found here: http://link.aip.org/link/doi/10.1063/1.59943. I did not participate in the experiment.

The data from the experiment include, for each bubble on the froth in a particular snapshot, area, perimeter, number of edges, the id of neighbors in unknown order (every bubble has a unique id, but the id was not tracked over time), lengths of edges in unknown order, xy coordinates of the vertices in unknown order.

My task is to find a way to calculate (or estimate) the change of Helmholtz free energy as a function of time and hence also the change of entropy. The 1st problem is how to calculate the internal energy of the soap froth. An obvious part of the internal energy is the internal energy of the gas phase treated as ideal gas. (is it justified to approximate it as ideal gas?) The 2nd part in the internal energy of the soap froth should be the surface energy, which is

$$0.5\sum_i\langle l\rangle_i\,z\lambda A_i,$$ where the sum is over all bubbles in the froth, $\langle l\rangle_i$ is the average edge length for an individual bubble $i$, $z$ is the space between the two plates, $A_i$ is the area of bubble $i$ (such that $zA_i$ is the volume of bubble $i$), and $\lambda$ is the surface tension per unit length.

In the experiment, the excess fluid was allowed to flow into the slots of the chamber. I am not sure if the excess fluid contributes significantly to the internal energy. Is there any other significant contribution to the internal energy? Can it be computed from the data?

The calculation of the entropy part sounds even more remote to me. I only know how to compute the ideal gas entropy.

In the calculation, I assume the pressure is uniform inside individual bubble. Even then the estimation of the pressure is not easy. In principle, we can deduce the pressure difference any two neighboring bubbles given the length of the share curved edge and the two shared vertices. (of course one will also have to which bubble has higher pressure) But the resolution of the data is not very large, so there can be considerable error in the estimation, and what's worse is I have a hard time to identify (from the list of curved edge lengths) the correct edge length between two given bubbles. A very crude way is to assume the pressure outside a given bubble to be the atmospheric pressure, and the pressure inside the given bubble is larger than the atmospheric pressure if its number of sides is less than 6, the pressure lower than the atmospheric pressure if its number of sides is larger than 6. And the magnitude of pressure difference is estimated by its average edge length. That is, $$P(i)=P_0+\dfrac{\lambda\alpha(n_i)}{\langle l\rangle_i},$$

where $\alpha=\dfrac{2\pi}{n}-\dfrac{\pi}{3}$ and $n_i$ is the number of sides of bubble $i$.

In summary, the non-computational questions are:

1) Is the internal energy calculation scheme reasonable? Are there any other contributions to the internal energy?

2) What contributes to the entropy of the soap froth besides the ideal gas entropy?

3) Is the pressure approximation reasonable? Or is there a way to tell whether the approximation scheme is reasonable or not?

• The FAQ states that computational physics questions should be asked at scicomp.stackexchange.com. You might get a better response there. – Shivam Sarodia Jan 1 '13 at 16:57
• Also, you could summarize the (non-computational) physics questions that you have at the end of the post, because even after reading the whole post I'm a little confused about what exactly your problem is. – Kitchi Jan 1 '13 at 18:20
• This sounds like it might be more of a physics question than a computational question, making it appropriate here, but like Kitchi said, it's hard to tell because I'm not entirely sure what you're asking. wdg, could you edit the question to make that more clear? Once you do, if this is a computational question of the sort that would be off topic here, we'll migrate it to Computational Science - no need to repost it there. – David Z Jan 1 '13 at 20:13
• Could you state your questions more explicitly? Are you asking whether your scheme works or not? It looks like an experimental rather than computational, so do you have access to the experimental equipment, i.e. whether you can change the data collected by altering the experiments. Or you are just using the data provided from somewhere. Lastly, is the data you have on hand exactly the same as the paper you cited. – unsym Jan 1 '13 at 22:24
• Hi, I have updated the post to make my questions more explicit. I don't have access to the experimental equipment, and I don't think I am able to do similar experiments. I'm confident that the data is at least very similar to that of the paper. – wdg Jan 2 '13 at 13:02

The following figures show the evolution of the soap forth with total area $26.7 \times 36.8 cm^2$ in $150$ hours with boundary marked by dark lines. FIG. 1. Superimposed snapshots of a coarsening froth up to time t, starting from the initial state shown (left to right) in (a), and after (b) 25.66 h, (c) 75.00 h, and (d) 150.00 h. The white disconnected regions are unswept areas, while the shaded region is the swept area. (d) The black connected lines are the boundaries of the froth at the latest time: They mark the survivors at the latest time. source

The first and most important things to realize is that it is not an equilibrium system since the boundaries are still moving and merging. While comparing the time scale for the movement of boundaries and the gas molecule movements, it is very reasonable to assume the interior of each bubble can be treated as idea gas: $$P_iV_i = N_i k_B T \tag{1}$$ where $i$ is the index of the bubble, and it should follow: $$dU_i = T dS_i - P_i dV_i + \mu dN_i \tag{2}$$ The temperature $T$ is kept constant in the experiment, and all parameters evolve over time.

One thing that I dont understand is that how can the particles from one bubble move to another one when a bubble disappear? The bubble may just break suddenly and all particles belong to another bubble instead of diffuse through the boundary, so the last term in (2) should be no use.

1) Is the internal energy calculation scheme reasonable? Are there any other contributions to the internal energy?

It is reasonable, the internal energy for closed should be the sum of all parts, i.e. the gas in bubbles, the soap froth liquids and the surface tension (I am not quite sure whether it overlap with previous one). However, your first equations does not make sense to me as the dimension is wrong.

The main problem is that your system is not closed as you described there is fluid flow in.

3) Is the pressure approximation reasonable? Or is there a way to tell whether the approximation scheme is reasonable or not?

No, it is not reasonable since it is not a equilibrium system. Any static time snapshot cannot tell you what the pressure $P_i$ is. Considering the simplest case that a chamber is separated by a movable wall. A snapshot cannot give you any information since all pressures difference is possible.

Also, I see no reason why less edges should corresponding to higher pressure. In the figure (d), I only see that a small bubble has less edges. If your assumption is correct, that means the bubble will expand again, i.e. a equilibrium. (Surely I dont know whether the edges will change when expanding).

In general, I dont think it is possible to find Helmholtz free energy using only snapshot of soap froth. You must also be able to keep track of the evolution of each bubble to get the $dV_i(t)$, which should have more information for you to deduce other informations. My suggestions is that it would easier for you to only count the Helmholtz free energy for bubbles instead of the whole system, but you still lea of the information $dV_i(t)$.