Reviews in Undergraduate Research - Issue 2
SUMMARY Molecular dynamics (MD) is a widely used tool in condensed matter physics, as well as other disciplines ranging from chemistry to high-energy physics. In MD, one integrates the equations of motion - Newton's second law for classical particles - directly, invoking no approximations. To do so requires an interaction potential energy or forces between atoms. I will discuss both integration schemes and potential types. MD potentially bridges two length scales - macroscopic and atomistic - and also links experimental results with theories. This will be emphasized through a discussion of modern research in solid-state physics, with each research application highlighting a different type of interaction potential. I will discuss fluid flow briefly and highlight some other applications of Lennard-Jones potentials, surface growth using a Stillinger-Weber potential, and defects in silicon using tight-binding potentials. I give some references regarding density-functional theory calculations of these defects. A different avenue of modern MD research
is in the method itself rather than its application. Much research
is towards developing so-called acceleration methods. By taking advantage
of the physics of condensed matter systems, acceleration methods have
been proposed which extend the time scales accessible by MD by orders-of-magnitude
in many cases. In this article, I will focus on A. F. Voter's methods,
giving their motivation, algorithm, and some derivations. INTRODUCTION Molecular dynamics (MD) is a widely used tool in condensed matter physics, as well as other disciplines ranging from chemistry (Kityk et al., 1999) to high-energy physics (Bleicher et al., 1999). This article focuses on applications to solid-state physics; however, the basic concepts presented should be valuable to any newcomer to MD, regardless of their field. One of MD's appeals is that the system in consideration is simulated directly - without assuming the nature of transitions or the types of the structures. Moreover, due to increasing computer power, MD provides atomistic detail for system sizes that are exponentially increasing to mesoscopic scales. Already, with classical potentials, simulations are performed with hundreds of millions of atoms for hundreds of picoseconds (Kadau et al, 2002) or with a more accurate tight-binding potential simulating about 64 atoms for 0.250 microseconds (Richie et al., 2002) - nearly a timescale measurable on a stopwatch! Note that MD bridges two scales - macroscopic and atomistic - and thus links experimental results, say transport measurements, with theories governing the interactions of the simulated matter's constituent atoms. Indeed, conventional "pencil-and-paper" theory generally cannot evaluate the properties that are of direct consequence in experiments. In this light, MD forms another bridge, one between experiments and conventional theory. By iterating the process of performing simulations and experiments, theories of forces between atoms are refined. With MD now appropriately evangelized, the exposition of the fundamentals begins. Traditional (non-accelerated) MD is conceptually simple. Using some potential interaction between particles, Newton's second law updates the particles' velocities and positions; repeating this, the equations of motion are integrated and the system's trajectories are obtained. In order to be relevant at finite temperature, pressure, etc., the system must be modified to ensure it stays at a constant energy, temperature or in whichever thermodynamic ensemble is desired. Methods of simulating the correct ensemble are not given here but can be found in (Rapaport, 1997). Before giving specific examples of research
into new MD methods (acceleration methods), I present the more technical
aspects of MD. Expositions of integration algorithms follow a background
in potentials. Accelerating MD motivates the POTENTIALS To
POSITION INTEGRATION ALGORITHMS Some details of potentials and their calculation are now familiar. With the potential and initial conditions Newton's second law can be integrated for all the particles. The verlet and predictor-corrector (PC) methods are common for performing the integration. The integration algorithm usually does not need to be extremely accurate. Because an extremely slight position displacement at any time (or an equivalent round-off error) can cause huge differences in the atom's trajectory at all later times after a certain time, only quantities which are insensitive to exact trajectories "matter." This is not particular to MD, but is characteristic of the natural process itself. The verlet method gives positions at a short time t after the time corresponding to the supplied positions. Each is derived in a straightforward manner from the Taylor expansion, in time, of the atomic coordinates. The formula for the verlet propagator is (Rapaport, 1997) . The verlet propagator is commonly used for its simplicity and tendency to conserve energy. The PC methods are more accurate than verlet, but they are not used as frequently as the simpler verlet-class propagators. The primary advantage of the PC method over the verlet-like algorithms is in the ability to change Δt on the fly, which may be useful in systems where one set of particles inherently move faster than others. PC is also useful when constraints (on, say, bond length or angles) are placed on the system (Rapaport, 1997). The PC method predicts positions based upon Adams-Bashforth extrapolation, which is exact if they follow monic polynomials. After prediction, correction is made via a different set of formulae. For the (relatively complex) equations see (Rapaport, 1997). BETTER (FASTER!) MOLECULAR DYNAMICS Although MD is an increasingly mature field, there are still continuous advances in methods. Voter et al. have developed several methods for increasing MD's performance by many orders of magnitude. Each method requires some assumptions - usually forms of transition state theory (TST) (Voter, 2002 or Lombardo, 1991) - on the nature of transitions. However, the assumptions are minimal and their validity can be checked. There are three common families of acceleration techniques, namely parallel-replica (PR or par-rep), temperature-accelerated dynamics (TAD), and hyperdynamics. The families can be utilized simultaneously for multiplicative performance boosts. Voter gives an excellent, accessible review concentrating on these acceleration methods (Voter, 2002). The limitation that keeps
one from simulating long time scales is the fact that MD is a multi-scale
problem (for most solid-state systems). Specifically, one must use
a small enough integration time step to reproduce the dynamics of
the fast vibrational modes. Since these vibrations occur on the order
of 10 On the other hand, transitions
occur infrequently; time scales between interesting transitions range
from picoseconds (quickly diffusing surface atoms) to seconds (dislocation
motion under shearing (Haasen, 1996)). One can conceive of watching
interesting behavior for minutes or hours (for example, in crystal
growth), however the longest MD simulations can now run for only microseconds.
It must be kept in mind that the acceleration methods discussed below
only apply to
RECENT SIMULATIONS
CONCLUSIONS This review does not cover all classes of research using MD - references are representative rather than exhaustive. However, it should have given the reader an idea of some typical applications - fluid dynamics, surface growth, and defect dynamics. Through examples, connections between experiment, theory, and MD are emphasized. One now hopefully has a feeling for the variety and advantages of some potential types. Also one should now be
aware of some of the growing number of molecular dynamics acceleration
methods - parallel-replica, temperature-accelerated dynamics, hyperdynamics,
and on-the-fly Monte Carlo. Perhaps the most important information
presented is the necessary background to understand what goes into
an MD simulation. With this and the references, a motivated individual
could probably code a simplistic MD simulator in a matter of a week
(though given the increasing number of sophisticated, fast MD programs,
writing one from scratch is probably not advisable except as a teaching
tool). ABOUT THE AUTHOR Kaden Hazzard is a third-year
undergraduate at The Ohio State University planning on graduating
in the spring of 2004. He plans on pursuing a doctoral degree in condensed
matter physics. He intends to pursue a research career at a university
or at a national lab. He has been involved in computational condensed
matter physics with Professor John Wilkins's research group since
the summer of 2000. He has been researching a variety of topics, mainly
defect evolution in silicon; surface growth in silicon; and structure
recognition, characterization, and data mining of defects in solids.
He has implemented several routines in the group's multi-scale materials
simulator (OHMMS), and a new Monte Carlo method for calculating free
energies. He also spent the summer of 2002 at Los Alamos National
Labs doing low-temperature experimental work. ACKNOWLEDGEMENTS Everyone in Professor John Wilkin's research group who I have had the fortune to work with deserves thanks for their continued guidance, discussions, and ideas. In particular, I would like to thank Professor Wilkins for his continuous suggestions for improving this manuscript and my writing in general. I would also like to thank Angela K. Hartsock for her gracious help with the figures. FURTHER READING A
standard text regarding MD in general, along with many techniques
especially for simulations of liquids is For those interested in
finding out more about density-functional theory, a preprint for a
good review article accessible to undergraduates who have taken some
quantum mechanics, I suggest: REFERENCES Arai, N., Takeda , S., and Kohyama, M. (1997) Self-Interstitial Clustering in Crystalline Silicon Physical Review Letters 78 4265 Baskes, M.I. (1997) Calculation of the behaviour of Si ad-dimers on Si(001) Modelling Simul. Mater. Sci. Eng. 5 p. 149-158. Berthier, L. and Barrat, J.L. (2002) Shearing a Glassy Material: Numerical Tests of Nonequilibrium Mode-Coupling Approaches and Experimental Proposals Physical Review Letters 89 95702 Birner, S., Kim, J., Richie, D.A., Wilkins, J.W., Voter, A.F., and Lenosky, T. (2001) Accelerated dynamics simulations of interstitial-cluster growth. Solid State Communications 120 (7-8) Bleicher, M., Zabrodin, E., Spieles, C., Bass, S.A., Ernst, C., Soff, S., Bravina, L., Belkacem, M., Weber, H., Stöcker, H., and Greiner, W. (1999) Relativistic Hadron-Hadron Collisions in the Ultra-Relativistic Quantum Molecular Dynamics Model (UrQMD) J.Phys. G25 1859-1896 Brooks, B. R., Bruccoleri,
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