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I have studied for a couple of months now a mathematical model of the energy of a configuration of immiscible fluids situated in a fixed container such that the fluids fill the container. In other terms, I considered a partition of the container into $2,3,...n$ sets. I will present the $3$ fluid case here.

So, in my hypothesis the container $\Omega$ is partitioned in $S_1,S_2,S_3$, the three fluids, with prescribed volumes $v_i$ and prescribed densities $\rho_i$. I took into account in the formulation of the energy the interfacial tensions, the gravity, and the contact of the fluids with the container $\Omega$ (these are not my ideas; they are taken from other similar mathematical articles). I will denote $P_\Omega(S)$ the perimeter of $S$ situated in the interior of $\Omega$ and $P_{\partial \Omega}(S)$ the perimeter of $S$ which is situated on the boundary of $\Omega$. I will not be very rigorous in what I'm about to write: I will write, for example $P_\Omega(S_i\cap S_j)$ the perimeter of the intersection $S_i\cap S_j$ even if as set theory intersection, this is void. Still, I think that the idea will be clear.

So, the formula for the energy of the configuration, which I found in other articles too is:

$$\mathcal{F}(S_1,S_2,S_3)= \sum_{1\leq i<j\leq 3}\sigma_{ij}P_\Omega(S_i \cap S_j)+\sum_{i=1}^3 \beta_i P_{\partial \Omega}(S_i)+\sum_{i=1}^3 g \rho_i\int_{S_i} z dV$$ where $\sigma_{ij}$ is the interfacial tension between $S_i$ and $S_j$ and $\beta_i$ is something I do not understand entirely, but it should take into account the effect of the walls of $\Omega$ (which in some cases, like thin tubes are not negligible). The last term is the potential gravitational energy (which was explained here).

My goal is that using this mathematical model of the phenomenon to prove the existence result of the minimal energy configuration, and to be able to obtain some numeric results, which will show how the final configuration looks like.

In the mathematical model, the inequalities $\sigma_{ij}+\sigma_{jk} > \sigma_{ik}$ are assumed. These assumptions seem physically reasonable, after reading the following article of J. Cahn

My confusion is about what the $\beta_i $ mean. In my intuition, $\beta_i$ should express that fluid $S_i$ 'likes' to be in contact with the wall or 'doesn't' like to be in contact with the wall. There are two things I have in mind for $\beta_i$.

  1. Can we consider $\beta_i$ the interfacial tension between the wall and the fluid $S_i$? In this case $\beta_i$ would be positive and very large, verifying the same triangle inequality conditions $|\beta_i-\beta_j| \leq \sigma_{ij}$. Given these inequalities, a proof can be given for the existence in the case of $3$ fluids (and probably for $n$ fluids; work in progress). Can we consider $\beta_i$ the interfacial tension between the wall and the fluid $S_i$? In this case $\beta_i$ would be positive and very large, verifying the same triangle inequality conditions $|\beta_i-\beta_j| \leq \sigma_{ij}$. Given these inequalities, a proof can be given for the existence in the case of $3$ fluids (and probably for $n$ fluids; work in progress). It seems reasonable that surface tensions between the wall and the fluids determine somehow the way the fluids interact with the wall: If the interfacial tension is very large, then the wall tries to reject the fluid and the equilibrium configuration will try to minimize the wall contact with such fluids, and maximize contact with fluids which have less interfacial tension. My confusion comes also from the fact that $\beta_i$ is not considered to be positive in the mathematical model; I'm not sure why is this. Maybe it is just a way of saying that "the mathematical model works even if $\beta_i$ are negative".
  2. $\beta_i$ could be the wetting coefficient, $\beta_i=\cos \theta$ (ref: Molecular theory of Capillarity, pag 9) where $\theta$ is the equilibrium contact angle.

My questions are:

  1. Is this mathematical model valid?
  2. Is the energy formula physically reasonable? If not, what changes should be made?
  3. What is the correct interpretation for $\beta_i$?
  4. Can we expect that the equilibrium configuration (which exists in experiments) must be a minimizer for the energy of the system?
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1 Answer 1

This problem of three phases in contact was recently studied by the elder Widom (the one from Cornell University), whose stuff is very deep. Here is a link to something related: http://128.84.158.119/abs/1111.2884v1

The quick answers are:

  1. yes
  2. not entirely (no line tension, but zero line-tension is ok physically, just not negative line tension)
  3. surface fluid adhesion energy
  4. yes.

I will elaborate on point 2: the formula for the energy that you give is missing the line tension (and less important point tension) which is present when different phases meet. If the regions are $S_1,S_2,S_3$, and you take the closure, and the boundary of the container is $S_4$, the energies are the bulk gravitational potential energy, which is the integral of the height times the density over each region, the surface area of contact $S_i\cap S_j$ times the mutual surface tension for each pair of fluids and each fluid and the boundary, and the line-tension times the length of each of the lines $S_i\cap S_j \cap S_k$, and the point tension times the number of points in $S_1\cap S_2 \cap S_3 \cap S_4$.

The mutual surface tensions must be positive if the fluids are immiscible, otherwise the area of contact grows without bound, leading to an unsharp separation of the two phases. The inequality of surface tensions is required to prevent fluid 3 from forming a thin lubrication layer between fluid 1 and 2 at the contact region, which would wreck any attempted proof that there is an equilibrium, since a thinner and thinner lubricating layer will be lower energy, until you get to zero thickeness, so there will be no minimum energy configuration independent of molecular considerations.

The point tension is not as important, because the 4-fluid contact point will only move around under small deformation, it has no length to change, so it contributes a constant to any class of configurations, depending on whether these points are present or not. It is important when comparing two different configurations, where a four-S point is present, vs. one where it isn't.

You would like the line tensions to be non-negative as well, otherwise, the contact line length can grow without bound to minimize the energy, while letting the area not grow too much, by having the line wriggle around on very small scales, and straightening out the surface after a very thin skin. This will lower the energy without bound (of course, physically there is an atomic scale cutoff, but this means there will be mixing of the fluids localized at the 3-fluid interface, or at the two fluid/wall interface if the line-tension is negative--- this is a weird situation, and I do not know if it is realized in nature)

There is no condition of positivity on the point free energy.

The minimizer will be the unique equilibrium configuration. You can try to argue that it exists using a Potts model, where each point on the interior takes the values 1,2,3 (representing occupation with each of the three fluids) and you define the energy such that you reproduce the 3-phase properties above, and restrict the amount of 1,2,3 sites with the appropriate chemical potential. Then you can show that the least energy state exists for any lattice, and it shouldn't be too hard to show that the limit is the calculus of variations answer.

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This is a great answer. Thank you very much. Can you please provide a reference for the work of Widom you say in the first part? Also, if you can, say a few words on the 'surface fluid adhesion energy' and how is this related to interfacial tension between the solid and the fluid. Also I am interested in the last part, where you say how existence is proved. –  Beni Bogosel Mar 26 '12 at 8:38
    
google.com/… –  Ron Maimon Mar 30 '12 at 4:00

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