A conversation with Dr. John Cahn

Footage recorded: December 2001

Footage recorded: December 2001

Title: What are some common phenomena that your formulas are able to explain?

Cahn: The first one is a group called — what we call dendrites. It's how alloy metals solidify. But it shows up in snowflakes, it shows up in the frost on the windshield, on patterns that occur on new galvanized iron pails. It's just a very common phenomenon.

The other one is segregation, where like things clump together. And this occurred very importantly in metals—it's what makes them strong and hard, but it also occurred in the early universe. After the big bang, once things began to cool down enough that gravity began to be more important than the random motion due to the high temperature, the early universe began to clump up and we still see this structure.

It is applied widely to things like population dynamics, where you get segregation in cities where people move in the direction of where they are more comfortable, where their neighbors are like them, and it leads to very interesting patterns—geometric patterns—and how they develop in time and when they develop.

So, that's a whole thing. Then another topic is really a study of surfaces near a critical point where the surfaces become very broad and diffuse. And this has led to an amazing insight into the spreading of liquids, which is the thing that we call wetting over surfaces, where we can get a liquid to spread over a surface without detergent. And this is coming into use now in the use of carbon dioxide at high pressure, where the liquid carbon dioxide has a very, very low surface tension and spreads and can be used for dry cleaning without the use of very poisonous chemicals like carbon tetrachloride that have been used in the past.

Title: I understand that you were born in Germany but came to the U.S. when the Nazis came to power. Can you tell us a bit about that?

Cahn: My father had been a young lawyer, and he had taken cases, he had brought suit against the Nazis before they were in power because they were really street gangs that were doing outrageous things. And as soon as they took power, he took the trolley car from his home to his office and was met by a fellow lawyer at the corner, at the street car stop, and he said "Don't go to your office, the SS is waiting for you." So we fled right away, and we were lucky because people who fled in 1933 had a much better chance of surviving than those who waited until later.

Title: What made you want to study in this field of science?

Cahn: Early in my career, I'd become aware of the fact that the theory of diffusion was inadequate. In diffusion, you have a mathematic equation that describes how density of a particular material smoothes out. If you have concentrations where the matter is more dense and places where it's less dense, that the matter diffuses from the regions of high density to low density, and the end result is that everything, in time, becomes quite uniform.

But there were situations in which matter was attracted to the regions where the density of that particular component was higher, so that diffusion instead of going from regions of high density to low density, went the other way. And the mathematical equation for that, the diffusion equation for that situation, had no solution.

And yet I felt it was a very common thing, it was certainly something that we saw in metals. And the discovery that I made was a modification to the diffusion equation, which is now known as the Cahn-Hilliard equation, which deals with this problem. It deals with the case where material diffuses spontaneously from low density to nearby regions of high density.

Title: Is there an every day example where these types of diffusion occur?

Cahn: There had been a mathematics of diffusion, and this is for the problem of, let's say, put a drop of milk in water and it gradually spreads out and in time it becomes quite uniform. There were opposite phenomenon where instead of the spreading out, which is part of this phenomenon, you have a clumping together, like if the milk curdles, if there is acid in there. And the equations of diffusion for this particular situation gave no solution.

Title: Are the Cahn-Hilliard equations complex?

Cahn: They are, to a layman. They are indeed very simple, but they hadn't been studied. They are what the mathematicians call non-linear, fourth-order, partial differential equations, and each of these words, non-linear, partial and fourth-order, were part of their charm for the mathematicians. Here was an equation ... they had not studied many equations of this type, and they knew very little about what interesting things would happen with them. And the equation that I wrote down is the simplest of those kinds of equations that you could possibly write down, and yet it has proved the subject of 50 PhD theses in mathematics alone. It's a fascinating equation to mathematicians.

Title: Were you surprised by how broadly the equation has been applied?

Cahn: Well, you know, having been the father of this equation, it really is very similar to being a father, because it has a life of its own. And I long ago lost the ability to keep guiding it, it's just going and I'm very proud of it, but it's on its own.

This equation that I wrote down was amazingly simple. I had difficulty solving it—I'm not a mathematician. I knew that it would explain what ... the phenomenon in metallurgy that I was looking for. But it turned out to be an extremely rich equation. There were a large number of phenomena that could be explained by it. There were also a large number of different quite general properties that mathematicians could work on and have been working for decades now. They tell me they still have several decades to go.

Title: How long do you think this equation will be useful to scientists?

Cahn: Well, I think, forever. I mean, as long as there's mathematics. There are standard equations that keep coming up, and this is going to be one of them. It's a little bit more complicated.

Let me give you an example: price times the number of things you buy is equal to the cost. The same equation is the rate at which you travel times the number of the time that you travel is the distance covered. Ohm's law is a simple ... mathematically, they're all the same. You multiply two things together to get something else.

So this equation is at its deepest level is a very simple, generic equation for phenomena that occur all the time. And so I think this equation will join those equations which kids will learn in their physics classes or in their economics classes or in their sociology classes. And as we learn more about its properties and how they're solved—and of course, the computer will give you numerical solutions quite easily. It will persist. It's not something that will disappear.

Created June 24, 2011, Updated June 2, 2021