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# Through the Theoretical Glass

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It’s difficult to envision what dimensions beyond 3D are, and why physicists, chemists, and mathematicians want to study them. Duke University chemist Patrick Charbonneau studies the theory behind the formation of glass, tackling questions about an area of research called the glass problem. His research has helped progress this field to a new paradigm.
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American Scientist
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associate editor Katie L. Burke interviewed him in September 2013.

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Photo credit: Les Todd/Duke Photography.
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Tell me about the glass problem—its history and how you fit into it.
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The glass problem can actually be subdivided into two glass problems. The first one we could consider to be an engineering glass problem. The general recipe for making any type of glass, whether it be window glass or metallic glasses or organic glasses, is always the same: You start from a high temperature liquid that you cool down, avoiding crystallization. One needs to get a super-cooled liquid to get a glass, so as an important step to achieve this, one needs to avoid the crystal. So, the engineering problem to making glasses is finding good mixtures that don't crystallize too easily, or crystallize very slowly, or ideally never, so that you can obtain the glass. So, that's one facet of the glass problem.

The other facet of the glass problem is the physics one: how to explain how a super-cooled liquid becomes so sluggish as to become solid—so we can use it as a building material. Both problems have a slightly different history. The history of glassmaking itself is thousands of years old and goes back to almost the dawn of civilization. How to control the materials and how to find the right mixtures that will have the right mechanical properties—that will be glasses and not crystalline. But, the glass problem in its physics formulation is a bit younger. Its heyday was from the 1960s on, with different generations of scientists thinking about possible explanations as to why a system that seems to have so little change in structure becomes so sluggish so rapidly.

One can separate the pre-1980s from the post-1980s as being two very different phases in the history of the glass problem. And the reason is that there's a number of very important theoretical developments that were made in the 1980s, both from a dynamics of liquid point-of-view, and from a physics of disordered systems point-of-view. So, on the one hand, what's called the Mode-Coupling Theory and related approaches, and on the other hand the Spin Glass–type descriptions. These two paradigm shifts, or the advent of these two descriptions, really changed the way people thought about glasses. And, they were not embraced, and they're still not embraced—one or the other all have supporters and detractors—but you could say that it was a major advance at least in the conversation about glasses. And now, if we fast-forward to the last 10 years, there are other descriptions that have come about that have stimulated the conversation quite extensively, and there's been a flurry of activity. The advance in the speed of computers has allowed us to study glasses with a completely different angle. One of the problems, up until the late 80s and early 90s, was that the only glasses we had were the materials ones. These are typically very small systems; they are atomic- or molecule-based. And the information we can get about them is not sufficiently detailed to be able to discriminate between different descriptions.

The simulation of glasses on computers has completely changed the way we can start questioning the theories about the glass transition. There was a major advance in the mid- to late 90s, when people started to realize that you could get information that was otherwise inaccessible. And in parallel to this in the late 90s and early 2000s, there were experimental systems developed that were actually a lot closer to what people could do on a computer than the previous generations of glasses—an example of that is the colloidal glasses. There's a number of scientists, especially around the group of Professor David Weitz at Harvard, who started to control sufficiently the properties of colloidal suspension to model the behavior of glasses with objects that could be seen under a microscope. So, you could have access to the same type of microscopic description with this experiment compared to what was available in numerical simulations. And, it's the advent of those two techniques that completely changed the nature of the questions that could be asked and answered about the glass transition, and helped stimulate the discussion and, in a sense, a new type of paradigm shift.

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How did you begin studying this, and what do you find compelling about this field of research?
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So, I started working on glasses as a graduate student. This was a suggestion from my PhD supervisor that I spend my research efforts working on this problem. And it's a very frustrating problem to work on for many reasons, and one of them is that there are so many different descriptions that compete to describe it. So, just to get a grasp of what those descriptions are and what they mean and their impact takes months if not years of work, or investment, just to understand what has been done. When you're new in an old field, there's all this baggage that you're supposed to master just to be able to do something remotely new. And, when I finished my PhD, I was never going to work on this problem again. I'd spent years, and I thought, you know, I'll never be good enough, or I'll never have good enough ideas after seeing all this mayhem. And I said, "Let me focus on other problems.

But, it's a problem that's so haunting that it turns out that if you've worked on it for a few years, every time there's a new subject you study, you see the glass problem popping up again. It filters your view of the rest of science, because you think about the rest of science just as a way to study glasses, suddenly. When I was applying for faculty positions, I never ever mentioned the word glasses because I generally was not thinking I was going to work on glasses. I thought I was done. And yet, there's a side project that came up, and it was an interesting question about the crystallization in higher dimensions, and then I started working and finding collaborators. One thing led to another, and before I knew it, I had a body of work that was resulting in something intellectually coherent, to the point where it had grown so big that it was even larger than what I could just do as a side project. I needed to get help. I needed to get actual students or postdocs working on this project because I had too many ideas, suddenly. So, it was not a long, thought-out inquiry where I knew where I was going to go, and this was where I was heading. It was really something that I was doing because it was haunting me, and kept on growing larger and larger, year after year, to the point where now, it's a ridiculously large fraction of my research effort.

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You mentioned that you started looking at this problem in higher dimensions. Can you talk a little bit about the difference between liquid and crystal in 3D compared to 4D, and talk about what 4D is?
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OK, let's start with the last question. So, what fourth dimension is: In the case in which we're interested, dimensions are all spatial dimensions. They're mathematical abstractions that just give one more orthogonal axis to the three earlier dimensions, the same way that one can go from two dimensions on a plane, let's say the surface of a table, to three dimensions, by looking at the space above the table. One just goes to the fourth dimension by looking at the hyperspace perpendicular to the space above the table. And it doesn't really exist, but it's a mathematical trick that allows us to access different geometries, to begin with. The initial motivation was to try to understand how geometry affects the formation of glasses. And, we thought it was a very convenient way to question geometry by changing the dimension of space. If we change the dimension of space, we change the geometry, the same way that the Platonic, or the perfect, shapes in 2D do not necessarily correspond to perfect Platonic solids in 3D. If we change the dimension, there's a change in the identity of the regular forms, and therefore, we can check by going from one dimension to the next, the role that those forms or shapes play or not in a given phenomenon.

The types of packings one can have changes with dimension, and crystals can be understood as just an ideal way to pack objects. This one way that's very regular, we can repeat in all directions, presumably up to infinity, to fill all space. And, if we look at the structure of the liquid, it does not have that same regularity. There's a lot of people who have tried to describe the liquid using shapes and forms, but it's never quite as satisfying as looking at the order of the crystal. So, by changing dimension, we wanted to be able to get access to how those descriptions of the liquid order made or did not make sense when they were based on geometry, and what we found is that as we go to higher and higher dimensions, the structure of the liquid becomes in a sense more and more boring. There's really no clear geometry, no clear shapes, that they fill up. But the structure of the crystal always remains beautifully geometric, meaning there's always nice ordered packing of spheres, or whatever other objects—in our case we were interested in packing of spheres. So, the fact that geometry plays less and less of a role in the liquid and remains a strong determinant in the structure of the crystal allowed us to differentiate what role geometry could play in going from the liquid to the crystal.

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What have you found in your research about the role of geometry in glassiness? What did you expect first, and then what did you find after you started looking into it?
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So, what I first expected, as many other people had said before me, is that geometry played a very important role in understanding the development of or the slow-down of the liquid, so that it becomes very viscous and then glassy. At first, I was working really hard trying to detect that geometrical order, on trying to visualize what those structures should be and to try to identify them in the liquids we had—up until the day that we realized that the reason we were having difficulty finding them was not because we were incapable of conceiving them but because they were not there. And, that's the day when we accepted the fact that maybe geometry was not such a determinant in understanding the slow-down. And that changed the way we were thinking about the problem and opened the floodgates to a number of other results. Identifying that geometry was not a determinant, or as powerful of a determinant, as we had once thought allowed us to comfortably study the glass formation at higher dimensions without having to worry about the geometrical details, which become more and more difficult to capture as you increase dimension, obviously. There's often mention of how negative results don't get published. Well, this is a negative result that we managed to get published and got a fairly high amount of impact, because it helped people realize that maybe the way they were thinking about the glass problem was just from the wrong end of the telescope. We should be looking maybe not at the geometrical details but at how irrelevant geometry is in that context.

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Do you have any idea why the glass at higher dimensions isn't geometrically interesting, as you say? Why it doesn't have geometric patterns that you might expect, given what other materials do?
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OK, one way in which you can understand geometry in higher dimensions is that if you think, let's say, about the surface of a sphere: A sphere is the same diameter in all dimensions, but as you increase the dimension, the space at the surface is much larger to fill, meaning that there's many more ways in which you can place objects at the surface as you increase the dimension. And, a way maybe to understand why disorder is more important, or this lack of geometry is more important, as you increase dimension is because the increasing amount of ways that you can reorganize objects at the surface end up "winning” over the efficient ways in which you can place objects at the surface. This is sort of an entropic description. Entropy favors situations where there are many more possibilities than one, and because that number of possibilities grows with dimension, that ends up dominating.If you take objects in 2D, like if you take coins on a table—take quarters, let's say—and you press them against each other. They will just spontaneously form hexagonal-type packing—nice little triangles that are regular. But, if you go to higher dimensions, just even 3D, and you start pressing objects at random into each other—let's say, tennis balls. You will actually need to do a little rearrangement with your fingers for them to pack nicely. And, the reason is that there are so many more ways in which these objects can come together in 3D. This ends up “winning” over the dense packing that you're trying to generate. And, as you increase dimension, this only becomes more pronounced.

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Most of our relationships with glass are the glass in the window or glassware in our kitchen. How can you bring your understanding of the higher dimensions of glass to that kind of understanding all of us have of the glass we use everyday?
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That's a very good question. The question is, in a sense: Does understanding higher dimensional glasses allow us to make better glasses, better glasses that we can use? The simple answer is it doesn't really help you, in the sense that it mostly satisfies people who've been wondering why it gets slow. It doesn't necessarily help people getting it to form in the first place. But, that's just a simple answer.It turns out that there are some insights that come in from higher dimensions. By studying higher dimensions, we're in the process of trying to bridge what is observed experimentally with the theoretical descriptions that exist. And, those theoretical descriptions are often developed for very high dimensional systems where the theories are easy—or at least easier—but have a hard time making contact with what happens in real-life experiments. And, establishing this bridge and validating some of those theories does give up insights into how one can go about and form new types of glasses.A new way people have been thinking about making glasses in the last few years is not by taking a liquid and cooling it, but by slowly depositing a glass layer by layer at the atomic level. And it turns out that this is a very different algorithm, a very different procedure, for generating glasses, and the resulting glasses have very different properties. This description came about independently—it was not motivated by theory—but you can imagine that if we had a better connection with the theoretical descriptions, we may be able to devise new ways to build glasses that have more robustness or more interesting mechanical behavior, or maybe have our equivalent to glasses tempered for thousands of years instead of the few months or few hours in which we can typically industrially produce them.

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It’s really interesting work, thank you so much for coming to talk to us, Patrick.
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Thank you very much; it was my pleasure.

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