Lattice-switch Monte Carlo method

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 61, Issue 1, 2000, Pages 906-919

  Bruce, A D ^a   Jackson, A N ^a   Ackland, G J ^a   Wilding, N B ^a  

^a UNIVERSITY OF EDINBURGH   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

FREE ENERGY;

CLOSE PACKED STRUCTURES; CONSTANT DENSITY; CONSTANT PRESSURES; DIRECT EVALUATIONS; LATTICE SITES; LATTICE VECTORS; SAMPLING BIAS; SINGLE PROCESS;

MONTE CARLO METHODS;

EID: 0001486374     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.61.906     Document Type: Article

References (63)

1. G. Jacucci and N. Quirke, in Complete Simulation in the Physics and Chemistry of Solids, edited by C.R.A. Catlow and W. C. Mackrodt, Vol. 166 of Lecture Notes in Physics (Springer, Berlin, 1982), p. 38.
2. Most, perhaps all, of the methods referred to here may also be realized within the framework of molecular dynamics.
3. K. Binder, Rep. Prog. Phys. 60, 487 (1997).
4. D. Frenkel, in Molecular-Dynamics Simulation of Statistical-Mechanical Systems, edited by G. Ciccotti and W.G. Hoover (North-Holland, Amsterdam, 1986), p. 151.
5. M.P. Allen, in Monte Carlo and Molecular Dynamics of Condensed Matter Systems, edited by K. Binder and G. Ciccotti (Italian Physical Society, Bologna, 1996), p. 255.
6. A path is defined by a sequence of values of either some macroscopic observable or some parameter (Formula presented), which controls a thermodynamic field or a model parameter.
7. In some instances one needs to make use of separate reference systems for each phase.
8. A full survey is not appropriate here; we note only some examples. References 9 and 10 both use multistage (integration) methods involving reference paths—the former to an Einstein solid, and the latter to a single-occupancy-cell model of an ideal gas–to study the fcc-hcp phase behavior of hard spheres. Reference 11 describes a study of the fcc-bcc phase behavior of Lennard-Jones systems, using a multistage (overlap) method applied to a nonphysical interphase path, with the (Formula presented) parameter 6 indexing a configurational energy that interpolates between those of the two structures. This method has been widely used (see, e.g., Ref. 12). Reference 13 describes a physical interphase path linking NaCl and CsCl structures, but does not utilize the path for a free-energy-difference calculation. The “lattice-shear” method described in Ref. 14, and applied to hard spheres, is a natural refinement of Ref. 13 in which a single-stage sampling procedure is used to explore a path constructed out of a series of mutually sheared lattices that interpolate between the lattices of interest.
9. D. Frenkel and A.J.C. Ladd, J. Chem. Phys. 81, 3188 (1984).
10. L.V. Woodcock, Nature (London) 384, 141 (1997).
11. A. Rahman and G. Jacucci, Nuovo Cimento D 4, 357 (1984).
12. M.C. Moody, J.R. Ray, and A. Rahman, J. Chem. Phys. 84, 1795 (1985)
13. B. Kuchta and R.D. Etters, Phys. Rev. 47, 14 691 (1993).
14. M. Parrinello and A. Rahman, J. Phys. (Paris) 42, 511 (1981).
15. S.-C. Mau and D.A. Huse, Phys. Rev. E 59, 4396 (1999).
16. B.A. Berg and T. Neuhaus, Phys. Rev. Lett. 68, 9 (1992)
17. B.A. Berg, J. Stat. Phys. 82, 323 (1996).
18. A.P. Lyubartsev, A.A. Martsinovski, S.V. Shevkunov, and P.N. Vorontsov-Velyaminov, J. Chem. Phys. 96, 1776 (1992).
19. E. Marinari and G. Parisi, Europhys. Lett. 19, 451 (1992).
20. G.M. Torrie and J.P. Valleau, Chem. Phys. Lett. 28, 578 (1974).
21. Since, by definition, the end points of an interphase path lie in different phases, the associated macroscopic property may reasonably be described as an order parameter.
22. The nature of the configurations visited along a path is not generally predictable a priori from the choice of order parameter. One may say only that the configurations sampled at a given point on the path will be those which have measurable canonical probabilities conditioned on that macrostate, and which are, moreover, accessible on the relevant time scales.
23. The interfacial free energy emerges as a by-product—and in some cases 22 may actually be the principal focus of interest.
24. B.A. Berg, U.H.E. Hansmann, and T. Neuhaus, Z. Phys. B: Condens. Matter 90, 229 (1993).
25. N.B. Wilding, Phys. Rev. E 52, 602 (1995).
26. B. Grossmann, M. Laursen, T. Trappenberg, and U.J. Wiese, Phys. Lett. B 293, 175 (1992).
27. It might work acceptably well if the dynamics of the interface between the two phases is favorable: systems with martensitic phase transitions may fall into this category: Z. Nishiyama, Martensitic Transformations (Academic, New York, 1978). Note also that the special case in which the structural phase transition involves no change of symmetry can be handled within the standard multicanonical framework: see Ref. 26.
28. G.R. Smith and A.D. Bruce, Phys. Rev. E 53, 6530 (1996).
29. The conjugate of a given configuration is the configuration associated with the sameset of displacements attached to the other set of lattice vectors.
30. The general defining characteristic of the gateway configurations is that a LS operation, launched from within this set, will be accepted with a probability sufficient to make the attempt worthwhile.
31. A configuration is energy matched (to its conjugate) if the difference between its energy and that of its conjugate is small, on the scale of (Formula presented).
32. A.D. Bruce, N.B. Wilding, and G.J. Ackland, Phys. Rev. Lett. 79, 3002 (1997).
33. S. Pronk and D. Frenkel, J. Chem. Phys. 110, 4589 (1999).
34. P.N. Pusey, in Liquids Freezing and the Glass Transition, edited by J.P. Hansen, D. Leveesque, and J. Zinn-Justin (Elsevier, Amsterdam, 1991) p. 763.
35. In its most general context the LS is the basis for estimating the differences between the free energy (Helmholtz or Gibbs) of two crystalline phases; in the case of hard sphere systems, at constant density, this free energy difference is purely entropic.
36. M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon, Oxford, 1968).
37. The hcp “lattice” is not a “Bravais lattice”; see e.g., Ref. 36, p. 79.
38. N.W. Ashcroft and N.D. Mermin, Solid State Physics (Saunders, Philadelphia, 1976).
39. W.G. Hoover and F.H. Ree, J. Chem. Phys. 49, 3609 (1968).
40. W.G. Rudd, Z.W. Salsburg, A.P. Yu, and F.H. Stillinger, J. Chem. Phys. 49, 4857 (1968).
41. The divergence is an artifact of the classical character of the model.
42. As detailed in Ref. 38 the argument leading to Eq. (10a) actually entails a rescaling of the displacement coordinates by an (Formula presented)-dependent factor.
43. The “visualization” exercise has its limitations. The dodecahedra (and the lattice spacing) should be thought of as “infinitely large” compared to the mean separation of adjacent faces.
44. We subsume a factor of (Formula presented) into the “pressure” (Formula presented).
45. If the interparticle potential is not of the hard-sphere form [Eq. (1)] the “perfect crystal” configurations (classical ground states) of the two structures will generally have different energies. But one may handle the effects of this energy mismatch quite simply, by attaching different multicanonical weights to the gateway configurations of the two structures.
46. The sign convention here has no deep significance. Defining (Formula presented) so that it has different signs in the two phases simply allows us to make a visually clear distinction (Fig. 88) between the contributions which each phase makes to the (Formula presented) distribution.
47. In fact, within the framework of periodic boundary conditions, the second generalization subsumes the first. Thus the various lattice-to-lattice mappings discussed in Sec. IV A can all be thought as a single mapping but with different hcp T matrices [Eq. (19)], chosen to interchange appropriate displacements.
48. In the absence of any phase-defining constraint on the configurational integral specified in Eq. (5), the integral extends over all configurations compatible with the boundary conditions.
49. This density value was chosen to coincide with one of those studied in Ref. 9.
50. This conclusion is in qualitative accord with that of Ref. 14, although we find the effects of next-neighbor encounters to be somewhat smaller than is reported there.
51. D. Frenkel, in Computer Modelling of Fluids, Polymers and Solids, edited by C.R.A. Catlow, C.S. Parker, and M.P. Allen (Kluwer Academic, Dordrecht, 1990), p. 83.
52. G.R. Smith and A.D. Bruce, J. Phys. A, 28, 6623 (1995).
53. M. Fitzgerald, R.R. Picard, and R.N. Silver, Europhys. Lett. 46, 282 (1999).
54. These approximations are surprisingly good, except close to melting. See inter alia D.A. Young and B.J. Alder, J. Chem. Phys. 60, 1254 (1974)
55. J. Piasecki, and L. Peliti, J. Phys. A 26, 4819 (1993)
56. R. Ohnesorge, H. Löwen, and H. Wagner, Europhys. Lett. 22, 245 (1993).
57. P.G. Bolhuis, D. Frenkel S.-C. Mau, and D.A. Huse, Nature (London) 388, 235 (1997).
58. L.V. Woodcock, Nature (London) 388, 236 (1997).
59. R.J. Speedy, J. Phys.: Condens. Matter 10, 4387 (1998).
60. Y. Choi, T. Ree, and F.H. Ree, J. Chem. Phys. 99, 9917 (1993). show that predictions for the phase diagram of a Lennard Jones solid depend extremely sensitively on the fcc-hcp hard-sphere entropy difference.
61. P.N. Pusey, W. van Megen, P. Bartlett, B.J. Anderson, J.G. Rarity, and S.M. Underwood, Phys. Rev. Lett. 63, 2753 (1989).
62. J. Zhu, M. Li, R. Rogers, W. Meyer, R.H. Ottewill, W.B. Russel, and P.M. Chaikin, Nature (London) 387, 883 (1997).
63. D.A. Young, Phase Diagrams of the Elements (University of California, Berkeley, 1991).