Figure 1: Electrochemical potential; electrical neutrality, not equilibrium
To answer various questions that come up from time to time:
| (1) |
where x(,) designates the charge of the first object when contact-electrified against the second. It is even possible to have two objects D and E such that
| (2) |
A related but easier-to-visualize piece of physics is this: given fixed terminals (A1 and A2) made of absolutely identical material (A) plus a moving part of type B, I can create a generator (e.g. as shown in figure 6) which uses contact electrification to move charge from A1 to A2 at a steady rate. Does that imply that A ranks above B and below B in the so-called triboelectric series? I don’t think so.
Mostly it’s just a combination of everyday things:
The following two subsections are parallel to the corresponding sections of reference 1.
The purpose of this section is to introduce the idea of an equilibrium electric field and to quantify it in terms of the work function. This will be a plausibility argument in the form of a Gedankenexperiment.
Start with a chunk of nickel. Make sure it is electrically neutral.
Also take a corresponding chunk of iron. It has the same FCC crystal structure as the nickel, although the lattice spacing will be slightly different. It also starts out electrically neutral.
As we shall see, these two electrically-neutral pieces of metal attract electrons differently. We give a name to this: we say they have different work functions. The work function measures the energy required to take an electron from the surface of the metal to some far-away place.1
The work functions are different because of things like the Pauli exclusion principle.2 You have a different number of fermions in a different-sized box, so the Fermi level will be different. Calculating things like work functions ab initio is a real tour de force, and we need not discuss the details here. We will accept the observed work function values and then make some plausibility arguments.
| Element | Work function | |
| silver | 4.26 eV | |
| aluminum | 4.28 eV | |
| zinc | 4.33 eV | |
| iron | 4.5 eV | |
| copper | 4.65 eV | |
| nickel | 5.15 eV | |
| gold | 5.1 eV |
The fact that a metal has some sort of nonzero work function is not particularly hard to understand. The electrons want to be near the metal nuclei. Even if the metal chunk is slightly negatively charged it will attract electrons. (Indeed even a single neutral hydrogen atom will attract electrons – the H− ion in vacuum has lower energy than a hydrogen atom and electron separately.)
The fact that different materials have different work functions is perfectly understandable also. Different materials have a different spacing between nuclei. So think of it as a simple quantum-mechanical particle-in-a-box problem: The smaller the box, the higher the kinetic energy the electrons must have. You can even make a connection between the work function (a purely electrical property) and the elastic properties of the metal: when you squeeze the chunk of metal you squeeze the electron wavefunctions, and that raises their kinetic energy.
Let’s look at this idea in more detail. The following is accurately true for one metal compared to a compressed piece of the same metal, and true to a first approximation for nickel compared to iron: Since the two pieces of metal have the same number of conduction electrons, and similar shapes, there must be a one-to-one correspondence between their conduction-band wavefunctions. The wavefunctions in the smaller box will (to a first approximation) look the same as those in the larger box, just scaled to a smaller wavelength so they fit in the box. Smaller wavelength λ means higher momentum p = h/λ, hence higher kinetic energy Ek = p2/2m.
To obtain a better approximation, you would need to take into account other contributions to the energy. For starters, there is a contribution to the potential energy from the fact that a nickel atom has more protons than an iron atom. That difference is mostly compensated by an equal difference in the number of non-conduction electrons, but then those extra electrons affect the kinetic energy of the conduction electrons (via the exclusion principle).
Remember, the point of this section was just to make it plausible that different materials have different work functions. For our purposes, we do not need to calculate accurate work function values from first principles. It suffices to accept the experimentally observed values.
To make progress, we must carefully make a distinction: a test charge is not the same as a real electron.
First, arrange for both chunks of metal to be identical in size and shape. File one of them down if necessary. This won’t change the work function. The objective is to make both chunks have the same self-capacitance. They are both still electrically neutral. A test charge placed on the iron chunk creates the same electrical field pattern and has the same energy as a test charge placed on the nickel chunk. So the test charge is equally happy either place.
But test charges do not exist in nature. Equilibrium is not established by the exchange of test charges, or alpha particles, or muons. In practical situations (with a few important exceptions, as we shall see), equilibrium is established by exchange of electrons. Real electrons.
The Pauli exclusion principle involves electrons excluding other electrons. It means that electrons are happier being on the nickel chunk than the iron chunk – even though a test charge would be equally happy.
Figure 1 shows the electron’s "unhappiness function" (i.e. electrochemical potential) when the metal chunks are electrically neutral. This is not the equilibrium situation.
The black line in the diagram represents the energy level of the electron, as a function of position. The green and yellow shaded regions are meant to represent the Fermi sea within the chunks of metal. The bottom of the Fermi sea is not interesting, because it is impossible to inject an electron into this level (by the exclusion principle), and it is energetically unfavorable to extract an electron from this level. Figure 1 does not attempt to accurately portray the bottom of the Fermi sea.
What matters is the top of the Fermi sea. The top of the Fermi sea for iron is labelled t in the diagram. The most easily-added electron will go in just above this level, and the most easily-extracted electron will come out from just below this level. (This corresponds to the LUMO and HOMO in molecular physics – but if that doesn’t mean anything to you, don’t worry about it.)
The length of the vertical steps in the black line represent the work function of the associated chunk of metal, i.e. the energy that the electron loses if it gets ejected from the metal, via the photoelectric effect or whatever. The workfunction for iron is labelled Φ in the figure.
In the figure I have indicated a “zero level” for the potential This choice is arbitrary and has does not affect the physics in any way. In this case I have chosen the energy of an isolated far-away electron in the vacuum to be the reference point for zero potential.
Next, figure 2 shows the electrochemical potential when equilibrium has been established by exchange of electrons:
In equilibrium, the nickel chunks have a definite excess of electrons, and the iron chunks have a definite deficit. In the gaps, there will be an electric field, as indicated by the blue arrows. In the gaps – but not within the metals – the energy per electron can be equated to the conventional electric potential (with the usual minus sign). Within the metals, we won’t even talk about the electric potential, because it is irrelevant. What matters is the electrochemical potential, which contains a huge contribution from the kinetic energy of the electrons.
The magnitude of the electric field in the gaps is equal to the work-function difference divided by the length of the gap, if we assume a nice parallel-plate geometry. Now the surface charge on each plate is proportional to the electric field, so we discover that the amount of charge depends inversely on the gap. (You can wiggle the gap and measure how much current needs to flow in order to maintain equilibrium. This is a way to measure work functions. You can even add a potentiometer and make it a null measurement; this is called a Kelvin bridge.)
Imagine the metal chunks are arranged in a big circle, so we have periodic boundary conditions on the diagram.
Now suppose we hook up certain pairs using aluminum wire in the usual way, the way batteries are hooked up, as shown in figure 3.
The details of the aluminum are not very interesting; its main function is to ensure that the attached Fe and Ni remain in electron-equilibrium. Aluminum conducts electrons quite well; it does not conduct muons or alpha particles or "test charges".
So far there is nothing special about this setup. An electron is equally happy in any of the various chunks of metal. There is an electric field in the gaps. There is a huge "dipole layer" at the Fe/Al interface and also at the Al/Ni interface. You can think of a dipole layer as a near-infinite electric field over a near-zero distance. Let x be the effective thickness of the metal/metal interface. As x becomes small, the electrical field grows like 1/x, so the potential difference remains independent of x and remains equal to the work-function difference. The Fe and Ni pieces are at the same potential here as they were back in figure 2.
Let’s build a simple parallel-plate capacitor; just two plates sitting hear each other, nothing else. One plate is made of Fe (work function 4.63 eV) and the other is made of Ni (work function 5.2 eV). In equilibrium, the two plates will differ in voltage by ΔV = 0.57 volts. We believe, for reasons to be discussed later, that equilibrium is established by exchange of electrons.
As usual, the capacitance is
| C = є0 |
| (3) |
where A is the plate area, and g is the gap between them.
If the gap g is very small, there will be a huge capacitance. We know the capacitance, and we know the voltage, so we can calculate the charge on the capacitor,
| Q = C ΔV (4) |
If we pull the plates apart while allowing the plates to remain in equilibrium, the charge Q will be a decreasing function of the gap-size g, in accordance with equation 3. The voltage ΔV remains constant.
Now, if we have a constant voltage and the capacitance is changing, the charge will be changing, in accordance with equation 4. This changing charge means a current must be flowing through whatever is responsible for maintaining the two plates in equilibrium.
This is the central idea behind the Kelvin bridge, as shown in figure 4, which is used to measure the difference in the work function of two materials.
The blue capacitor plate is made of one material, and the red capacitor plate is made of another material with a different work function. The red plate is forced to move back and forth horizontally at a fairly high frequency. This causes the gap g to change.
Let’s start by ignoring the potentiometer. That is, we slide the potentiometer voltage to zero (slider all the way to the left). Then the wiring simply serves to keep the two capacitor plates in equilibrium, by allowing rapid exchange of electrons. As the gap changes, an AC current is observed on the meter. That’s the essential physics.
As a refinement, we can use the potentiometer to make this a null measurement. That is, we slide the potentiometer to the point where we observe no AC current.
That happens when the potentiometer voltage is just equal to the difference in work functions. The two plates are being held in the non-equilibrium situation where there is no electric field in the capacitor gap.
We read out the voltage on the potentiometer (in the null condition) and call that the result of our measurement of the difference of work functions.
Now let’s consider a slightly different scenario. The plates are in equilibrium, but the process that established equilibrium is no longer operating effectively. That is, the plates are insulated. The charge on each plate is constant. If we now pull the plates apart, the voltage ΔV will grow in proportion to the gap.
To repeat: There is a huge difference between pulling the plates apart at constant voltage (which implies decreasing charge) and pulling them apart at constant charge (which implies increasing voltage).
We can apply what we know to build a powerful static electricity generating machine, as shown in figure 5.
The blue compartment is made of one material and the red compartment is made of another material with a different work function. There also exists a shuttle, shown in black. For definiteness, you can imagine it has a work function intermediate between the other two, but in fact it doesn’t matter; the shuttle work function will drop out of the calculation.
To operate the machine, the shuttle, which has a long insulating handle, is first touched to the inside of the red compartment. This touching brings the shuttle into equilibrium with the red compartment. There is a modest voltage across an infinitesimal gap, which means a huge charge in accordance with equation 4.
Next we withdraw the shuttle. As it moves away from the wall of the compartment, for a while it will remain in equilibrium. There will be field emission, and if the device is operated in air there will be corona discharge, all acting to keep the two objects in equilibrium.
But at some point these processes will stop. In air, corona discharge requires a field on the order of a few megavolts per meter. For a one-volt difference in work function, that corresponds to a gap of less than a micrometer. When the gap is bigger than that, there is no longer any way for equilibrium to be maintained.
Let’s be clear: for very small gaps, there is equilibrium and constant voltage. For gaps larger than a micrometer or so, there is constant charge – the charge is “frozen” onto the shuttle.
We now move the shuttle to the other compartment and follow the same procedure. The shuttle will pick up a different charge, because of the difference in materials.
By shuttling back and forth, we can transfer a constant amount of charge per cycle. We soon develop a huge potential difference across the gap G between the compartments.
|
Note that the shape of the compartments is meant to serve as something
of a Faraday cage, so that the fundamental charge-transfer operation
is not affected by the huge potential across the gap G. That means the machine is not limited by breakdown across the small gap g, but only across the large gap G, which you can make as large as you want. |
This is a nice refinement, in contrast to poorly-engineered or non-engineered contact electrification situations (such as sliding your shoes across a rug in winter) where the generated field can nullify the fundamental charge transfer, much as the potentiometer nulls out the work-function difference in the Kelvin bridge. |
An interesting variation of this machine is shown in figure 6.
The odd thing about this machine is that the two compartments are made of the same material.
This time it is necessary for the shuttle to be made of a material with a different work function. Also, the shuttle has a tricky shape.
The trick is that in one compartment, you choose to use the broad face of the shuttle to make contact with the compartment. Therefore, when you pull the shuttle away, at the time the charge becomes frozen onto the shuttle, there will be a big capacitance because of the big area, in accordance with equation 3.
Then, in the other compartment, you choose to use the very narrow point of the shuttle to make contact. So, at the time the process of equilibration ceases and the charge becomes frozen onto the shuttle, the electric field is about the same and the voltage difference is about the same as in the previous case, but the charge is much less, in proportion to the area.
By shuttling back and forth, you can move a more-or-less constant amount of charge per cycle.
