6.16 Chemical Potential and Diffusion

The chemical potential, or Fermi level, that appears in the Fermi-Dirac distribution is very important for solids in contact. If two solids are put in electrical contact, at first electrons will diffuse to the solid with the lower chemical potential. It is another illustration that differences in chemical potential cause particle diffusion.

Of course the diffusion cannot go on forever. The electrons that transfer to the solid with the lower chemical potential will give it a negative charge. They will also leave a net positive charge behind on the solid with the higher chemical potential. Therefore, eventually an electrostatic force builds up that terminates the further transfer of electrons. With the additional electrostatic contribution, the chemical potentials of the two solids have then become equal. As it should. If electrons can transfer from one solid to the other, the two solids have become a single system. In thermal equilibrium, a single system should have a single Fermi-Dirac distribution with a single chemical potential.

The transferred net charges will collect at the surfaces of the two solids, mostly where the two meet. Consider in particular the contact surface of two metals. The interiors of the metals have to remain completely free of net charge, or there would be a variation in electric potential and a current would flow to eliminate it. The metal that initially has the lower Fermi energy receives additional electrons, but these stay within an extremely thin layer at its surface. Similarly, the locations of missing electrons in the other metal stay within a thin layer at its surface. Where the two metals meet, a “double layer” exists; it consists of a very thin layer of highly concentrated negative net charges next to a similar layer of highly concentrated positive net charges. Across this double layer, the mean electrostatic potential changes almost discontinuously from its value in the first metal to that in the second. The step in electrostatic potential is called the “Galvani potential.”

Galvani potentials are not directly measurable; attaching voltmeter leads to the two solids adds two new contact surfaces whose potentials will change the measured potential difference. More specifically, they will make the measured potential difference exactly zero. To see why, assume for simplicity that the two leads of the voltmeter are made of the same material, say copper. All chemical potentials will level up, including those in the two copper leads of the meter. But then there is no way for the actual voltmeter to see any difference between its two leads.

Of course, it would have to be so. If there really was a net voltage in thermal equilibrium that could move a voltmeter needle, it would violate the second law of thermodynamics. You cannot get work for nothing.

Note however that if some contact surfaces are at different temperatures than others, then a voltage can in fact be measured. But the physical reason for that voltage is not the Galvani potentials at the contact surfaces. Instead diffusive processes in the bulk of the materials cause it. See section 6.28.2 for more details. Here it must suffice to note that the usable voltage is powered by temperature differences. That does not violate the second law; you are depleting temperature differences to get whatever work you extract from the voltage.

Similarly, chemical reactions can produce usable electric power. That is the principle of the battery. It too does not violate the second law; you are using up chemical fuel. The chemical reactions do physically occur at contact surfaces.

Somewhat related to Galvani potentials, there is an electric field in the gap between two different metals that are in electrical contact elsewhere. The corresponding change in electric potential across the gap is called the “contact potential” or “Volta potential.”

As usual, the name is poorly chosen: the potential does not occur at the contact location of the metals. In fact, you could have a contact potential between different surfaces of the same metal, if the two surface properties are different. “Surface potential difference” or gap potential would have been a much more reasonable term. Only physicists would describe what really is a gap potential as a “contact potential.”

The contact potential is equal to the difference in the work functions of the surfaces of the metals. As discussed in the previous section, the work function is the energy needed to take a Fermi-level electron out of the solid, per unit charge. To see why the contact potential equals the difference in work functions, imagine taking a Fermi-level electron out of the first metal, moving it through the gap, and putting it into the second metal. Since the electron is back at the same Fermi level that it started out at, the net work in this process should be zero. But if the work function of the second metal is different from the first, putting the electron back in the second metal does not recover the work needed to take it out of the first metal. Then electric work in the gap must make up the difference.


Key Points
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When two solids are brought in contact, their chemical potentials, or Fermi levels, must line up. A double layer of positive and negative charges forms at the contact surface between the solids. This double layer produces a step in voltage between the interiors of the solids.

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There is a voltage difference in the gap between two metals that are electrically connected and have different work functions. It is called the contact potential.