Computers in Physics Entropy-driven phase transitions Harvey Gould, Jan Tobochnik, and Louis Colonna-Romano Citation: Computers in Physics 11, 157 (1997); doi: 10.1063/1.168596 View online: http://dx.doi.org/10.1063/1.168596 View Table of Contents: http://scitation.aip.org/content/aip/journal/cip/11/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Absolute entropy and free energy of fluids using the hypothetical scanning method. I. Calculation of transition probabilities from local grand canonical partition functions J. Chem. Phys. 119, 12084 (2003); 10.1063/1.1625919 Entropy and thermalization of particles in liquids J. Chem. Phys. 118, 6989 (2003); 10.1063/1.1560933 The effect of ions on solid–liquid phase transition in small water clusters. A molecular dynamics simulation study J. Chem. Phys. 118, 6380 (2003); 10.1063/1.1557523 Entropy driven phase transitions in colloid–polymer suspensions: Tests of depletion theories J. Chem. Phys. 116, 2201 (2002); 10.1063/1.1426413 Simulation of phase coexistence for complex molecules Comput. Phys. 11, 246 (1997); 10.1063/1.168607
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COMPUTER SIMULATIONS
ENTROPY-DRIVEN PHASE TRANSITIONS Harvey Gould, Jan Tobochnik, and Louis Colonna-Romano
In this column we discuss some of the applications of molecular dynamics and Monte Carlo methods to simple models of fluids . Because of the extensive literature on the applications of these methods to fluids and the publication of two recent monographs 3.4 on the subject, we discuss only a few of the applications and attempt to "discover" some of the properties of these systems. For computational simplicity, our discussion will be for two-dimensional systems. Furthermore, we will restrict most of this column to models of hard-core particles for which temperature plays no role, and phase transitions are driven by entropic changes. The simplest interparticle interaction u ( r) is given by
+ co
Department Editors:
Harvey Gould
[email protected]
Jan Tobochnik
[email protected]
uch of the progress in our understanding of liquids has come from the insight gained from computer simulations. 1•2 The two most fundamental simulation methods, molecular dynamics and Monte Carlo, are designed to generate a representative sample of the configurations. In molecular dynamics a trajectory in phase space is generated by solving Newton ' s equations of motion numerically and computing time averages of the physical quantities of interest. In recent years the method of molecular dynamics has been extended so that simulations can be done at constant pressure rather than at constant volume and at constant temperature rather than at constant energy. However, molecular dynamics is limited by the smallest temporal process in the system. In a dense liquid this process corresponds to the very short time intervals between successive molecular collisions. This fundamental limitation of molecular dynamics is one reason why we are also interested in Monte Carlo methods. The latter generate a representative sample of configurations according to the desired probability for the ensemble of interest. Averages are done over the states of the ensemble, and it is assumed that ensemble and time averages yield equivalent results. Although Monte Carlo methods yield only equilibrium averages of time-independent quantities, we can interpret the trial moves as a fictitious dynamics. Because Monte Carlo methods allow biased and even unphysical changes to be generated, this class of methods makes it possible for averages to converge more quickly than in molecular dynamics.
M
Louis Co/anna-Romano is a graduate student in physics at Clark University, Worcester, MA 01610. Harvey Gould is aprofessor ofphysics at Clark University, and Jan Toboclmik is an associmeprofessor ofphysics and computer science at Kalamazoo College, Kalamazoo, M/49(}(}6.
r< (J"
u(r)= { O,'
(1)
where (J" is the diameter of the hard di~ks in two dimensions ( d = 2) . Because the interaction (1) is purely repulsive, a system of hard disks does not have distinct gas and liquid phases, and the disordered phase of the system is referred to as a fluid. It is common to express the density in several ways. We will express p in terms of the dimensionless quantity p * = p(J" 2 . (Other common ways are in terms of the density relative to closest packing p/ Pep, where Pcp(J" 2 = 2/ J3 = 1.155, and as bp, where b = 1/27T(J"2 is the second virial coefficient for hard disks.) Although realistic interparticle potentials are more complicated than the interaction (1), the structure of dense fluids in d =2 and 3 is remarkably similar to that of hard disks and hard spheres, respectively. The first application of molecular dynamics was to the simulation of a system of hard spheres by Alder and Wainwright. 5 Molecular-dynamics computations for the interaction (1) are very different from those for continuous potentials, because hard-core particles move with constant velocity between collisions and the change in velocity due to a collision can be computed exactly (within round-off errors). A program to carry out this simulation can be found in Refs. 2 and 3. The main static quantities of interest are the radial distribution function g( r) and the pressure P. The function g(r) is a measure of the correlations between particles. In d = 2, g( r) is determined by (n(r))
pg(r)= - 2 " ' 7Trur
(2)
where ( n(r) ) is the mean number of particles in a ring of width dr a distance r from a given particle, and 27Trdr is the area of the ring. For a uniform system the right-hand side of (2) equals the number density p, and hence g(r) = 1 for an ideal gas. In a real fluid g( r) has several oscillations before decaying to unity at large r. In a crystal g( r) exhibits relative maxima for all r corresponding to the periodicity of the crystal structure. The existence of a fluid-solid transition for hard disks (and hard spheres) at sufficiently high density was one of the first major discoveries from a computer simulation.5•6 In Fig. 1 we show our results, which are similar to those found in Ref. 6, for the dependence of P * ==PA 0 /NkT on p*,
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~
'I!J!!!iii''d'
where N!A 0 =Pep. We note that P* is an increasing function of p * until p * = 0.88. From the observation of the change in the slope of the P * vs. p * curve as indicated in Fig. 1, we can conclude that a transition has occurred. The particle trajectories averaged over approximately 500 collisions per particle are shown in Fig. 2. Note that the particles appear to coexist in a disordered (fluid) phase and an ordered (solid) phase. On the basis of similar data, Alder and Wainwright6 concluded that the hard-disk melting transition is discontinuous (first-order). They also found that the highest possible density at which the fluid phase can exist is = 0.880, and the lowest density of the solid phase is p;=0.912. For pi.;1 are uniform random numbers, and dr max controls the maximum size of a move. If Id rl > d r max• then compute another d r until Id r I:; ; d r max· The trial position is now given by r new= r old+ d r normalized so that Ir new I = R. This procedure ensures that we obtain a circularly symmetric distribution of moves about the old position. The distances r ij between two points are measured along the surface of the -1 sphere and can be computed from r;j= R cos r;·rij. Do a series of simulations using at least 100 particles and slowly increase p* from 0.75 to about 0.93. The reduced density p * can be increased at each ~ep by increasing u so that no overlaps occur. Measure P and g(r). At high densities a split second peak in g( r) should appear
indicating the appearance of an amorphous solid. Compare your results for f3P I p to results obtained using the same value of N with periodic boundary conditions. (6) Simulations of an electron gas confined to a twodimensional surface have been of much interest because of the relation of this model to electrons on the surface of liquid helium. The electrons interact via the Coulomb interaction, u(r)=e 2 1r, and charge neutrality is ensured by a uniform positive background. Because of the long-range nature of the Coulomb potential, the usual approach is to use periodic boundary conditions and Ewald summations of the interaction of a given particle with the infinite number of periodic images of the other particles and the background. One way to avoid the time-consuming Ewald sums is to consider the particles to be on a spherical surface. 24 In this case the modified Coulomb interaction between electrons i and j can be written as e 2I r ij, where r ij is defined in problem (5). The original simulation used molecular dynamics, but implementing a Metropolis Monte Carlo is more straightforward. Do a Monte Carlo simulation and compute the equation of state and g(r) for an electron gas. Consider at least N = 100 electrons. (7) The trial moves in the Monte Carlo algorithms we have discussed so far are local. However, local moves lead to long decorrelation times near a phase transition. For example, Lee and Strandburg 12 found that the time necessary for a system of hard disks to fluctuate between the competing fluid and solid states increased rapidly with system size using the constant-pressure Monte Carlo algorithm. Dress and Krauth 25 have devised a clever Monte Carlo algorithm that allows groups of disks (spheres) to be moved at one time. The idea is to choose an arbitrary pivot point in the simulation box and rotate a copy of the current configuration by 180° about this point. The rotated and original configurations are superimposed and clusters of the superimposed disks are formed. If a disk in the rotated configuration c ~ overlaps a disk in the original configuration c 1, the two disks are in the same cluster. The clusters either have an equal or an unequal number of disks. As an example, we show in Fig. 3 a typical configuration with two clusters of four disks, two clusters of three disks, and three clusters with twu disks. A move consists of swapping the disks in a cluster from c ~ to c 1 and vice versa so that the number of disks in the original configuration c 1 is unchanged. One possibility is to make such a move for an individual even cluster, for example, the cluster of size two with disks 5 and 9. However, this move is local, and for this reason moves involving small, even clusters are ignored. A better move involves pairs of odd clusters. For example, we can swap disks in the two clusters of three disks shown in Fig. 3(c). This move generates a new configuration c 2 as shown in Fig. 3(d). To improve your understanding of the algorithm, write a simple program that allows you to determine the clusters visually. It is sufficient to consider systems of 10-20 disks. What is the approximate density at which a spanning cluster in the superimposed configuration first appears? In analogy
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8~88
00
®8
3.
4. (b)
5. 6. 7.
8. 9. Figure 3. (a) The original configuration c I and the pivot point at x =2.54, y =2.01. The linear dimension is L =4.71. (b) The rotated copy c; of c I obtained by rotating c I by 180° about the pivot point. (c) The clusters of the superimposed configurations. (d) A new configuration c 2 obtained by swapping disks in the two clusters of three particles. See problem (8).
to the cluster algorithms for Ising models, 26 we expect that the cluster algorithm is most efficient at the density at which a spanning cluster first appears. However, at the densities of interest, p* = 0.9, it is difficult to find a sufficient number of small odd clusters to generate nonlocal moves. A generalization of the cluster algorithm is claimed to overcome this difficulty. 25 From the editors. We acknowledge the partial support of NSF Grant No. PHY-9301019. We wish to thank William Swope and Dennis Rapaport for their comments on a draft of our manuscript. Users of our computer simulation text might be interested in knowing that Version 5 of True Basic is now available for the Macintosh, OS/2, and Windows 3.1/95/NT operating systems. More information is available from http://truebasic.com/. We call attention to the language F, a subset of Fortran 90 that retains only its modern features. More information can be found at http:// www.imaginel.com/imaginel/. Several examples of programs in F associated with this column can be downloaded from the Computers in Physics FTP server at the URL ftp://ftp.aip.org/cip/cip_sourcecode. Please send us your comments and suggestions for future columns.
10. 11. 12. 13.
14.
15. 16.
17. 18. 19. 20. 21. 22. 23.
References 1. See, for example, D. Frenkel and J.-P. Hansen, Phys. World 9(4), 35 (1996) for a recent overview of the application of simulation methods to liquids. 2. See, for example, H. Gould and J. Tobochnik, Introduction to Computer Simulation Methods, 2nd ed.
24. 25. 26.
(Addison-Wesley, Reading, MA, 1996) for an introduction to Monte Carlo and molecular-dynamics methods. D. C. Rapaport, The Art of Molecular Dynamics Simulation (Cambridge University Press, 1996). Program listings can be downloaded from http://www.cup.cam. ac. uk/on linepubs/ArtMolecular/ArtMoleculartop.html. D. Frenkel and B. Smit, Understanding Molecular Simulation (Academic, 1996). Program listings are available at http://www.hpcn.tudelft.nl/frenkel_smit. html. B. J. Alder and T. E. Wainwright, J. Chern. Phys. 33, 1439 (1960). B. J. Alder and T. E. Wainwright, Phys. Rev. 127, 359 (1962). J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 (1973); D. R. Nelson and B. I. Halperin, Phys. Rev. B 19, 2457 (1979); A P. Young, ibid. 19, 1855 (1979). J. A Zollweg and G. V. Chester, Phys. Rev. B 46, 11186 (1992). W. W. Wood, J. Chern. Phys. 48, 415 (1968); 52, 729 (1970). I. R. MacDonald, Mol. Phys. 23, 41 (1972). J. Lee and J. M. Kosterlitz, Phys. Rev. Lett. 65, 137 (1990); Phys. Rev. B 43, 3265 (1991). J. Lee and K. J. Strandburg, Phys. Rev. B 46, 11190 (1992). H. Weber, D. Marx, and K. Binder, Phys. Rev. B 51, 14636 (1995); H. Weber and D. Marx, Europhys. Lett. 27, 593 (1994). K. Binder, K. Vollmayr, H.-P. Deutsch, J.D. Reger, M. Scheucher, and D. P. Landau, Int. J. Mod. Phys. C 3, 1025 (1992). M. S. S. Challah and D. P. Landau, Phys. Rev. B 33, 437 (1986). K. Bagchi, Hans C. Andersen, and W. Swope, Phys. Rev. E 53, 3794 (1996); Phys. Rev. Lett. 76, 255 (1996); W. Swope and Hans C. Andersen, J. Chern. Phys. 102, 2851 (1995). B. Widom and J. S. Rowlinson, J. Chern. Phys. 52, 1670 (1970). See, for example, Chap. 8 in Ref. 4. C.-Y. Shew and A Yethiraj, J. Chern. Phys. 104, 7665 (1996). G. Johnson, H. Gould, L. X. Chayes, and J. Machta (unpublished). P. Meakin and R. Jullien, Phys. Rev. A 46, 2029 (1992). J. W. Evans, Rev. Mod. Phys. 65, 1281 (1993). W. Schreiner and K. Kratky, J. Chern. Soc. Faraday Trans. 2 78, 379 (1982); J. Tobochnik and P. M. Chapin, J. Chern. Phys. 88, 5824 (1988); S. P. Giarritta, M. Ferrario, and P. V. Giaquinta, Physica A 187, 456 (1992). J. P. Hansen, D. Levesque, and J. J. Weiss, Phys. Rev. Lett. 43, 979 (1979). C. Dress and W. Krauth, J. Phys. A 28, L597 (1995). See, for example, H. Gould and J. Tobochnik, Comput. Phys. 3, 82 (1989).
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