How Solids Melt: ORNL Simulations Support Theory
By Mark Mostoller, Ted Kaplan, and Kun Chen


Mark Mostoller (right), Ted Kaplan, and Kun Chen (sitting) view a computer visualization of two-dimensional melting based on their computer simulation using ORNL’s Intel Paragon XP/S 35. Photograph by Tom Cerniglio.

Chocolate melts in our mouths, but how it turns from solid to liquid isn’t clear in our minds. Recently, three ORNL researchers have used the Intel Paragon XP/S 35 to help confirm a theory developed in the 1970s about how substances melt in two dimensions. They simulated two-dimensional systems containing 576, 4,096, 16,384, 36,864, and 102,400 atoms with an interatomic force model often used for rare-gas solids. Results for the two largest systems show the existence of a new “hexatic” phase between the solid and liquid phases, as predicted by the theory. This two-dimensional simulation helps explain the melting process.

Every solid melts if taken to a high enough temperature. For example, hydrogen melts at about 14 Kelvin (or K, the number of degrees above absolute zero); ice at 273 K, iron at 1810 K, and tungsten at nearly 3700 K, making it very useful as the filament material in light bulbs. Melting in three dimensions (3D), or the conversion from solid to liquid, is a first-order phase transition in which a latent heat must be provided. Thus, ice keeps a drink cold by taking in its latent heat, which causes the ice gradually to melt.

Melting is a fundamental phenomenon that all of us have witnessed and think we understand, but, paradoxically, there is no generally accepted picture of how solids melt at an atomistic level. Instead, there are empirical rules such as the Lindemann criterion, named after the man who presented it 85 years ago. The atoms in a crystal vibrate around fixed positions, just like a guitar string when plucked. The size, or amplitude, of the vibrations increases with increasing temperature. According to the Lindemann criterion, a crystal will melt when the vibrational amplitudes of its constituent atoms reach some critical magnitude, say 10 to 15% of the interatomic spacing.

Physics is a search for the elegance and simplicity that underlie nature’s often apparently chaotic and complex behavior. The Lindemann picture of melting is simple, but it certainly is not elegant. It proposes that the disorganized vibrations of the atoms in a solid somehow cause the solid to melt at a very precise temperature. Other phase transitions do not occur in such a disorganized fashion. Superconductivity, for example, occurs because electrons with opposite spins form pairs that can move through the lattice without resistance. A number of famous physicists, including Richard Feynman, have proposed theories for melting in 3D that involved more efficient mechanisms to mediate the process, but none of these theories has proven true.

In two dimensions (2D), the situation is different. In the 1970s, an elegant theory for melting was developed by John Kosterlitz, David Thouless, Burt Halperin, David Nelson, and Peter Young (the KTHNY theory, after the initials of its authors’ last names). Many experiments and computer simulations have since been done to prove or disprove the theory, with ambivalent results. The experiments have often been quite imaginative. They have used layers of rare gas on graphite, charged polystyrene spheres in solution, and monolayers of metal spheres (metal shot to hunters) shaken between confining plates. Experimental results generally have been supportive of the KTHNY theory. In contrast, simulations, in which complications such as interactions of rare gas atoms with the graphite substrate can be avoided by studying ideal 2D systems, have given predominantly negative results. A number of people, including Burt Halperin and David Nelson, thought the reason for these results could lie in the system sizes and time scales employed in the simulations.

Enter ORNL

Here our part of the story begins. In 1992, the Intel Paragon XP/S 150 became a promise that would soon be funded. Mark Mostoller and Ted Kaplan began an effort in numerical simulations, hoping to harness the power the new machine would provide. They persuaded a close friend of long standing, Mark Rasolt, to join them in this new venture. Rasolt was a very creative and productive theorist who periodically spent time at Harvard University, where Halperin and Nelson are professors. Rasolt saw the development of our new capabilities in simulations and the arrival of the Paragon as an opportunity to test the KTHNY theory at the required system sizes and time scales for the first time. He also thought that if the mechanism for 2D melting could be confirmed, the studies could be extended to melting in 3D.

Together we prepared a proposal to the Laboratory Directed Research and Development Program which Rasolt presented in September 1992. Everyone acknowledged that the proposal carried a high risk of failure because it would use a then nonexistent machine to examine a fundamental problem that great physicists had been unable to solve. In November 1992, shortly after learning that the project had been funded, Rasolt died of a massive heart attack at Los Angeles International Airport while returning from a conference in Australia. He was 49 years old.

It was hard to continue the work then, and it is hard to tell the story now. Kaplan took over leadership of the project. In August 1993, Kun Chen joined us as a postdoctoral researcher. The Intel Paragon XP/S 35 came on line in April 1994, and the XP/S 150 began operation in January 1995. Our tools had arrived, and we set to work.

Theoretically Speaking

Fig. 1. A cartoon picture of the order present in the solid, hexatic, and liquid phases in two dimensions.

A principal feature of the KTHNY theory of melting is its prediction of a new phase between solid and liquid. A crystalline solid has two kinds of order, translational and orientational. Translational order means that, if you start at a particular atom and take steps along well-defined paths over long distances, you will arrive in close proximity to another atom. Orientational order means that, if you look at two atoms separated by large distances, their neighbors will be oriented relative to some fixed axis in the same way. A liquid has no long-range order of either kind, but it has short-range order: any atom will have some average number of nearest neighbors clustered around an average neighbor distance with no preferred orientation. KTHNY’s new phase, called the hexatic phase, has no translational order but retains orientational order. In short, you can’t predict where the atoms are, but you can predict their neighbor environment.

Figure 1 shows a cartoon representation of the three phases. The theory predicts that under certain conditions, the system will transform from solid to hexatic to liquid in two continuous phase transitions that do not involve a latent heat. All of these predictions rest on a picture that invokes specific defects (dislocations and disclinations) rather than the action of all atoms vibrating willy-nilly.

Perfected Method for the Paragon

What have we done that is new? Here’s the list.

  1. We chose to do our simulations with a recently perfected method for treating a constant pressure (P) and temperature (T) statistical ensemble. A constant volume and temperature ensemble allows mixed solid and liquid phases that can mimic the hexatic and confuse analysis. The constraint of constant volume can also cause a problem when equilibrating the system because vacancies cannot be introduced freely.

  2. We chose a longer interaction range for the interatomic potentials than previous simulations. For present purposes, this is a detail, but an important one.

  3. Our use of the Paragon supercomputer was the real key to our success. Chen has simulated systems of 576, 4096, 16,384, 36,864, and 102,400 atoms at very long time scales, making him one of the largest users of the new massively parallel processing machines. To give an idea of the computer usage involved, a run of 5 million time steps for the 36,864-atom system at a given P and T takes about 10.5 days running continuously, using 128 nodes on the XP/S 35.
These were our innovations and computational advantages. Our results provide the strongest evidence yet that the KTHNY theory of 2D melting is valid. In small samples like those treated in previous work, we found ambiguous evidence for a first-order transition buried in fluctuations larger than the transition itself. It was not until the system size reached 36,864 atoms that a metastable hexatic phase emerged. This phase lasted for about a million time steps before relaxing down into a liquid. The metastable hexatic phase persists for longer times (3 million time steps) in the larger 102,400-atom system.


We concluded that previous simulations were performed on systems that are too small and that they were carried out over time scales that are too short. The Paragon and our own insights have made it possible to find confirmation of a theory for 2D melting. Our research was supported by ORNL’s Laboratory Directed Research and Development Program and reported in “Melting in Two-Dimensional Lennard-Jones Systems: Observation of a Metastable Hexatic Phase,” Physical Review Letters 74, 4019 (1995). For this paper, we received a Lockheed Martin Awards Night publication award.

What’s left to do? Create a visualization of the 2D melting process at the atomic level and add a dimension so we can understand 3D melting.

What’s the problem? Getting enough Paragon time.

What’s the lesson? National laboratories that have managers who will take risks can continue to solve fundamental problems.


MARK MOSTOLLOER was born in Somerset, Pennsylvania, and grew up in Pittsburgh. He received his Ph.D. degree in applied physics from Harvard University. He was a staff member in ORNL’s Solid State Division from 1969 until spring 1997. He has worked on lattice vibrations, random alloys, electronic and vibrational properties at surfaces and interfaces, and numerical simulations of the structure and properties of materials. He received Martin Marietta technical achievement awards in 1990 and 1992 and a Lockheed Martin publication award in 1995.

TED KAPLAN is a theoretical physicist in ORNL’s Computer Science and Mathematics Division. Born in New York City, he received both his undergraduate and Ph.D. degrees from the Massachusetts Institute of Technology. He has been at ORNL since 1972, when he joined the Solid State Division. His research has included the theory of random alloys, fractal interfaces, chaos, rapid solidification, and thin film growth. Most recently, he has worked on large-scale numerical simulations of the properties of materials. He received a Lockheed Martin publication award in 1995.

KUN CHEN, a native of Xiamen, China, was a postdoctoral researcher in the Solid State Division at ORNL from August 1993 to August 1996. He received his B.S. degree from Beijing University and his M.S. degree from Xiamen University. After teaching at Xiamen for 2 years, he came to the United States in 1988 and received his Ph.D. degree from the University of Georgia in 1993. He received a Lockheed Martin publication award in 1995.


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