# What is “surface fluid adhesion energy”?

This is related to my previous question. Pardon me for asking so many questions recently. My physics knowledge is not that good, and some answers are hard to find.

In the question in the link, I asked what was the correct interpretation for $\beta_i$ in the energy formulation:

$$\mathcal{F}(S_1,S_2,S_3)= \sum_{1\leq i<j\leq 3}\sigma_{ij}P_\Omega(S_i \cap S_j)+\sum_{i=1}^3 \beta_i P_{\partial \Omega}(S_i)+\sum_{i=1}^3 g \rho_i\int_{S_i} z dV$$

I got an answer which said that the correct interpretation for $\beta_i$ is surface fluid adhesion energy. I tried googling the term, but I did not find anything clear. My question is

What is surface fluid adhesion energy and how is this related to the fluid-solid interfacial tension? What references are there on this subject?

Here is the context: 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.

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No need to apologise for asking many questions--if they're good questions, we want MORE! (i dunno if your questions are good or bad--im not knowledgeable in this area of physics--but they look good) – Manishearth♦ Mar 26 '12 at 14:17
This could really have been a comment on the answer. The "energies" in this case are really "free energies" (meaning logarithm of probabilities times the temperature in energy units), and adhesion energy is extra probability for making a fluid stick to a surface. This can physically by a binding energy between the surface and the molecules, since, when you are looking microscopically, the log probability is just the ordinary energy. – Ron Maimon Mar 26 '12 at 18:41
@RonMaimon: Does an inequality of the type $|\beta_i-\beta_j| \leq \sigma_{ij}$ can be for this surface adhesion energies? – Beni Bogosel Mar 29 '12 at 7:19
I meant 'can be proved for the surface adhesion energies'. – Beni Bogosel Mar 29 '12 at 8:05

## 1 Answer

Surface fluid adhesion energy is the free energy per unit area of a fluid in contact with a surface. It can be defined by having a given bulk of fluid in contact with a container, and asking how much work does it take to add surface, for example by tilting a non-symmetric container and measuring how much work it takes (very precisely).

You can understand this in a microscopic model by having a two-state lattice (an Ising model) with a box boundary, with extra energy for lattice sites which are "1" near the edge. The surface fluid adhesion energy at any temperature is the difference between the free energy of the lattice with an edge from the free energy of the lattice with periodic boundaries.

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