Numerical tests of nucleation theories for the Ising models

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 82, Issue 1, 2010, Pages

  Ryu, Seunghwa ^a   Cai, Wei ^b  

^a STANFORD UNIVERSITY   (United States)

^b Stanford University   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

2D ISING MODEL; 3D ISING MODEL; CLASSICAL NUCLEATION THEORY; COARSE GRAINING; DROPLET SIZES; FIELD THEORY; FREE ENERGY FUNCTION; LOGARITHMIC CORRECTIONS; MARKOV CHAIN; NUCLEATION RATE; NUCLEATION THEORY; NUMERICAL RESULTS; NUMERICAL TESTS; POTENTIAL PROBLEMS; REACTION COORDINATES; RING THEORY; SPIN FLIP; SURFACE ENERGIES; TEMPERATURE DEPENDENCE; THREE-DIMENSIONAL (3D);

DROP FORMATION; FREE ENERGY; ISING MODEL; MARKOV PROCESSES; NUCLEATION; NUMERICAL ANALYSIS; SPIN DYNAMICS; SURFACE CHEMISTRY; SURFACE TENSION; TWO DIMENSIONAL;

THREE DIMENSIONAL;

EID: 77954825434     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.82.011603     Document Type: Article

References (59)

1. J. E. McDonald, Am. J. Phys. 30, 870 (1962). 10.1119/1.1941841
2. F. F. Abraham, Homogeneous Nucleation Theory (Academic, New York, 1974).
3. R. J. Speedy and C. A. Angell, J. Chem. Phys. 65, 851 (1976) 10.1063/1.433153
4. P. H. Poole, F. Sciortino, U. Essmann, and H. E. Stanley, Nature (London) 360, 324 (1992). 10.1038/360324a0
5. E. Mendez-Villuendas and R. K. Bowles, Phys. Rev. Lett. 98, 185503 (2007). 10.1103/PhysRevLett.98.185503
6. R. J. Young, Introduction to Polymers (CRC Press, New York, 1981).
7. M. Gleiser and E. Kolb, Int. J. Mod. Phys. C 3, 773 (1992). 10.1142/S0129183192000464
8. R. Becker and W. Döring, Ann. Phys. (N.Y.) 24, 719 (1935).
9. Ya. B. Zeldovich, Ann. Phys. (N.Y.) 18, 1 (1943).
10. J. Frenkel, Kinetic Theory of Liquids (Oxford University Press, Oxford, 1946).
11. L. Farkas, Z. Phys. Chem. (Munich) 125, 236 (1927).
12. A. Laaksonen, V. Talanquer, and D. W. Oxtoby, Annu. Rev. Phys. Chem. 46, 489 (1995). 10.1146/annurev.pc.46.100195.002421
13. J. Lothe and G. Pound, J. Chem. Phys. 36, 2080 (1962). 10.1063/1.1732832
14. J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 31, 688 (1959). 10.1063/1.1730447
15. J. S. Langer, Ann. Phys. (N.Y.) 41, 108 (1967) 10.1016/0003-4916(67) 90200-X
16. J. S. Langer, Phys. Rev. Lett. 21, 973 (1968) 10.1103/PhysRevLett.21.973
17. J. S. Langer, Ann. Phys. (N.Y.) 54, 258 (1969). 10.1016/0003-4916(69) 90153-5
18. X. C. Zeng and D. W. Oxtoby, J. Chem. Phys. 94, 4472 (1991). 10.1063/1.460603
19. C. Valeriani, E. Sanz, and D. Frenkel, J. Chem. Phys. 122, 194501 (2005). 10.1063/1.1896348
20. E. Sanz, C. Valeriani, T. Vissers, A. Fortini, M. E. Leunissen, A. van Blassderen, D. Frenkel, and M. Dijkstra, J. Phys.: Condens. Matter 20, 494247 (2008). 10.1088/0953-8984/20/49/494247
21. S. Auer and D. Frenkel, Annu. Rev. Phys. Chem. 55, 333 (2004). 10.1146/annurev.physchem.55.091602.094402
22. P. G. Bolhuis, D. Chandler, C. Dellago, and P. L. Geissler, Annu. Rev. Phys. Chem. 53, 291 (2002). 10.1146/annurev.physchem.53.082301.113146
23. P. A. Rikvold, H. Tomita, S. Miyashita, and S. W. Sides, Phys. Rev. E 49, 5080 (1994). 10.1103/PhysRevE.49.5080
24. V. A. Shneidman, K. A. Jackson, and K. M. Beatty, J. Chem. Phys. 111, 6932 (1999). 10.1063/1.479985
25. D. Stauffer, A. Coniglio, and D. W. Heermann, Phys. Rev. Lett. 49, 1299 (1982). 10.1103/PhysRevLett.49.1299
26. M. Acharyya and D. Stauffer, Eur. Phys. J. B 5, 571 (1998). 10.1007/s100510050480
27. K. Brendel, G. T. Barkema, and H. van Beijeren, Phys. Rev. E 71, 031601 (2005). 10.1103/PhysRevE.71.031601
28. A. C. Pan and D. Chandler, J. Phys. Chem. B 108, 19681 (2004). 10.1021/jp0471249
29. R. J. Allen, C. Valeriani, S. Tanase-Nicola, P. R. ten Wolde, and D. Frenkel, J. Chem. Phys. 129, 134704 (2008). 10.1063/1.2981052
30. L. Maibaum, Phys. Rev. Lett. 101, 256102 (2008). 10.1103/PhysRevLett.101. 256102
31. S. Wonczak, R. Strey, and D. Stauffer, J. Chem. Phys. 113, 1976 (2000). 10.1063/1.482003
32. D. W. Heermann, A. Coniglio, W. Klein, and D. Stauffer, J. Stat. Phys. 36, 447 (1984). 10.1007/BF01010991
33. H. Vehkamaki and I. J. Ford, Phys. Rev. E 59, 6483 (1999). 10.1103/PhysRevE.59.6483
34. S. Ryu and W. Cai, Phys. Rev. E 81, 030601 (R) (2010). 10.1103/PhysRevE.81.030601
35. R. J. Allen, D. Frenkel, and P. R. ten Wolde, J. Chem. Phys. 124, 024102 (2006) 10.1063/1.2140273
36. M. Volmer and A. Weber, Z. Phys. Chem. (Munich) 119, 227 (1926).
37. N. J. Gunther, D. J. Wallace, and D. A. Nicole, J. Phys. A 13, 1755 (1980). 10.1088/0305-4470/13/5/034
38. M. J. Lowe and D. J. Wallace, J. Phys. A 13, L381 (1980). 10.1088/0305-4470/13/10/008
39. D. J. Wallace, in Phase Transitions, Proceedings of a Summer Institute, Cargese, Corsica, 1980, edited by, M. Levy,,, J. C. Le Guillou,, and, J. Ziin-Justin, (Plenum, New York, 1982), p. 423.
40. C. C. A. Gunther, P. A. Rikvold, and M. A. Novotny, Physica A 212, 194 (1994). 10.1016/0378-4371(94)90147-3
41. G. Jacucci, A. Perini, and G. Martin, J. Phys. A 16, 369 (1983). 10.1088/0305-4470/16/2/019
42. A. Perini, G. Jacucci, and G. Martin, Phys. Rev. B 29, 2689 (1984). 10.1103/PhysRevB.29.2689
43. R. K. P. Zia and D. J. Wallace, Phys. Rev. B 31, 1624 (1985). 10.1103/PhysRevB.31.1624
44. D. Stauffer and C. S. Kiang, Phys. Rev. Lett. 27, 1783 (1971). 10.1103/PhysRevLett.27.1783
45. C. K. Harris, J. Phys. A 17, L143 (1984). 10.1088/0305-4470/17/3/009
46. R. K. P. Zia and J. E. Avron, Phys. Rev. B 25, 2042 (1982). 10.1103/PhysRevB.25.2042
47. M. Hasenbusch and K. Pinn, Physica A 203, 189 (1994). 10.1016/0378-4371(94)90152-X
48. T. S. van Erp, D. Moroni, and P. G. Bolhuis, J. Chem. Phys. 118, 7762 (2003). 10.1063/1.1562614
49. E. E. Borrero and F. A. Escobedo, J. Chem. Phys. 129, 024115 (2008). 10.1063/1.2953325
50. D. Moroni, P. G. Bolhuis, and T. S. van Erp, J. Chem. Phys. 120, 4055 (2004). 10.1063/1.1644537
51. See supplementary material at http://link.aps.org/supplemental/10.1103/ PhysRevE.82.011603 for free energy curves for cluster sizes up to n = 1950 in a wide temperature range.
52. V. A. Shneidman and G. M. Nita, J. Chem. Phys. 121, 11232 (2004). 10.1063/1.1814080
53. G. Wilemski, J. Chem. Phys. 103, 1119 (1995). 10.1063/1.469822
54. Here, the word "prefactor" means the factor in front of the exponential term.
55. Crowding of droplets, or interaction among droplets, have been investigated by Pan and Chandler using very high field up to h = 1.18 J. They have found for the 3D Ising model, even for fields up to h ∼ 0.8 J at k B T = 0.6 T c, the critical nucleus size n c obtained by umbrella sampling and the critical nucleus size n k obtained by commitor distribution matches well, indicating the crowding effect is negligible. Because in this paper, we consider field values much lower than 0.8 J, the crowding effect should be very small, as also confirmed by our own data on the two critical sizes. Indeed, the population of up-spins is at most 4% for the highest temperature and highest field conditions considered in our study, and is usually less than 2%. The interaction among those sparse up-spins would not affect free energy significantly.
56. 2 √ π n is the circumference of a circle with area n. However, a real droplet is not compact but consists of a mixture of up and down spins. The percentage of down-spins in the droplet as a function of T is also absorbed in σ eff (T).
57. This is against the expected temperature dependence of τ due to suppression of shape fluctuation below the roughening temperature, see Sec..
58. Perini originally introduced the "ledge" term in order to improve the quality of the fit in the range of n < 19. However, we found that fitting to the data in the range of n < 19 will lead to large discrepancies in the range of n > 100. Given that the droplet theory is supposed to work better in the continuum limit of large n, we believe the function should be fitted to data at large n.
59. Eq. also fits the data at non-zero h. However, the resulting σ eff 3 D from the fit slightly increases with h. For example, σ eff 3 D increase by about 3% as h changes from 0 to 0.5.