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A common, intuitive, way of defining free energy is as being the internal energy of a thermodynamical system that is available to perform work.

In the thermodynamical thery of surface tension, we can relate this quantity with a free energy per area unit: \begin{equation} \Omega_s=\gamma S \end{equation}

Let's say that I, somehow, create a gradient $\nabla\gamma$ of surface tension on the surface. In the region of "less" surface tension, we could then conclude that there's less internal energy available to perform work on this region, while in the regions of "more" surface tension we have more internal energy available to perform work.

One well-known phenomena where this takes place is the Marangoni Effect. There, the particles tend to move to the region of higher surface tension.

And that's what I want to understand: Why does the particles tend to go to these regions? That is, why does the particles tend to go to the regions where there's more energy available to perform work?

I thought about some chemical aspects of surface tension, but it would be interesting if I could describe this aspect physically, with mathematics.

Johann Wagner
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I'm not sure this is what you're asking about, but the Second Law tells us that entropy $S$ tends to be maximized ($dS>0$). Then, we can derive that entropy maximization implies free energy minimization ($dG=dU-TdS+PdV=\gamma dA+\mu dN<0$), where $U=TS-PV+\gamma A+\mu N$ is the internal energy, $T$ is temperature, $P$ is pressure, $V$ is volume, $\gamma$ is surface tension, $A$ is the surface area, $\mu$ is the chemical potential, and $N$ is the amount of material. For a constant amount of material, the Gibbs free energy change is then $dG=\gamma dA<0$, which predicts that area is minimized or that particles move to areas of higher surface tension because occupying these regions ($dA<0$) produces a greater Gibbs free energy reduction.

Chemomechanics
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  • Thank you! But why does the particles occupy the regions where a greater reduction of Gibbs free energy is produced? There's probably a fundamental principle that I'm missing. – Johann Wagner Oct 29 '21 at 22:23
  • The ultimate reason is the Second Law: we tend to more often see arrangements that have more ways of occurring. A high surface tension of material X means that the X bonds are very unsatisfied at the air interface. If instead a particle sits there, then the air–X interface is replaced by a particle–X interface that presumably satisfies the X bonds a little better. (If that's not the case, then the particles will instead move away from the high-surface-tension region.) The energy released by that better bonding ultimately heats the universe, increasing its entropy. Does this make sense? – Chemomechanics Oct 29 '21 at 22:56