In the last post I talked about energy minimization, and how it constitutes a particular way of viewing the universe and solving physical problems.  It’s an important paradigm in science, and I appreciate all the insightful comments you readers contributed.

Anyway, all the philosophy is in the last post.  Here I just want to give another example of a strange (and correct) conclusion at which you can arrive by believing that the universe will somehow find a way to minimize its total energy.  It is similar in setup to the previous example, and the result (to me) is even more counterintuitive.  It’s called the electrocaloric effect.

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Example #2: a strange, spontaneous cooling

In the last problem, I considered the following question: “Put two oppositely charged plates next to a body of water.  What happens?”  The surprising answer was that some of the water spontaneously jumps up between the metal plates.

Here is perhaps an even simpler question: “Put two oppositely charged plates on either side of an enclosed tank of water.  What happens?”

It seems at first like nothing should happen at all, since the water has no freedom to go anywhere in this problem.  But consider the situation from a microscopic point of view.  The water is composed of individual water molecules, each of which has a positive and a negative end (water is like a tiny electric dipole with charge $\pm e$  separated by a distance $0.4 \textrm{ \AA}$).  So the water molecules will naturally tend to point their positive ends toward the negative plate and their negative ends toward the positive plate.  Something like this:

Of course, water molecules have some thermal energy, so they kick around randomly and don’t align perfectly with the electric field that has been applied across the tank.  But a lot of them do align, and as a result the water molecules greatly reduce the strength of the applied field.  This is why water has such a large dielectric constant: water molecules put their positive/negative ends up against the negative/positive plate, so that all but one-eightieth of the plates’ charge is canceled out.  This is what it means to say that water has a dielectric constant of 80.

Remember, though, that the water molecules are spinning around randomly in the tank of water, and their thermal energy prevents them from perfectly canceling the charge.  What would happen if the water was a little colder?  Well, the water molecules would have less thermal energy, so they would spin around a little less aggressively and consequently would do a better job of canceling out the applied electric field.  In other words, the dielectric constant of water $\epsilon$ increases as the temperature is reduced.

So in some sense, the system has an incentive to cool itself down.  If the water were colder, then the water molecules would do a better job of canceling the electric field, and the total energy of the system would decrease.  As it turns out, this is exactly what happens.

Somehow, when electric charge is applied to either side of the tank, the system spontaneously emits heat to its surroundings so that the water molecules will slow down and be better blockers of electric field.  The resulting energy gain has to be balanced against the entropy that is lost by the water molecules, and at the end of a fairly simple calculation you find that the water spontaneously cools by an amount

$\Delta T = (T V^2/2 C_V \rho d^2)\cdot (d \epsilon/d T)$,

where $T$ is the ambient temperature, $C_V$ is the specific heat of water, $\rho$ is the density of water, $V$ is the voltage you apply across the tank, and $d$ is the distance between the two plates.  This is the electrocaloric effect: the temperature of an insulating material will spontaneously decrease when a voltage is applied across it.

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Now, before you rush home to attempt this experiment yourself, I should point out that there is a reason you probably haven’t heard of this effect before.  That’s because, in almost every material we know of, the change in temperature $\Delta T$ is much too small to notice at all but the largest applied voltages.  For example, if you applied 1000 Volts across a 1 cm slab of water at room temperature, the water would only cool down by about $10^{-6}$ degrees (Celsius).  Microscopically, however, the effect can be quite important.  I’ve read a couple of papers that probe the structure of water on a charged surface, and they find that even at room temperatures the first few water layers are “frozen” in place, effectively forming a very thin ice crystal.  And, apparently, there is some research on materials that demonstrate the electrocaloric effect to a much larger extent.

1. March 23, 2010 7:58 pm

Actually, this is one of those cases where the fluid will have different temperatures in different directions because the distribution of velocities parallel to the field will be different than the distribution of velocities perpendicular. Seems weird but it happens in ionized plasmas all the time.

2. April 8, 2010 4:56 pm

In adiabatic demagnetization the matter cools as more entropy is taken up by spin disorder. In adiabatic magnetization the matter must warm. Similarly for electrification, and for stretching a rubber band, which warms. So if you polarize the water adiabatically it water warms (I think).

3. April 7, 2012 11:15 pm

Is it possible to produce unfreezable water by using powerful magnetic plates. Would the plates be strong enough to not allow water to acheive its crystalline structure it forms when it makes ice? Would the plates lower the freezing temperature of water, or make a solid that is more dense than Liquid Water, due too the non-expansion of the H2O Molecules?

4. April 7, 2012 11:16 pm

Just wondering