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Parallel Replica Exchange Monte Carlo Applied to Hard Systems

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High Performance Computer Applications (ISUM 2015)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 595))

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Abstract

In this chapter we show how the replica exchange Monte Carlo algorithm can be used to study hard systems, i.e. systems composed by hard particles such as spheres, ellipsoids, disks, and ellipses, among others. The method is based on the definition of an extended ensemble which usually uses temperature as the expansion variable. This way, the low temperature replicas perform local sampling on the configuration space while high temperature replicas produce large jumps. The replica swap moves allow for a low temperature replica to heat, reach another region of configuration space, and cool back, enhancing the sampling of low-temperature, uneven free energy landscapes. Each replica is handled by a single thread making the parallelization straightforward. On the other hand, hard particles cannot overlap and do not contribute to the potential energy. In this case we carry out a pressure expansion of the isothermal-isobaric ensemble. Here we show how this expansion is able to resolve the phase diagrams of hard systems in two and three dimensions.

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References

  1. Lyubartsev, A.P., Martsinovski, A.A., Shevkunov, S.V., Vorontsov-Velyaminov, P.N.: New approach to Monte Carlo calculation of the free energy: method of expanded ensembles. J. Chem. Phys. 96, 1776 (1991)

    Article  Google Scholar 

  2. Marinari, E., Parisi, G.: Simulated tempering: a new Monte Carlo scheme. Europhys. Lett. 19, 451 (1992)

    Article  Google Scholar 

  3. Yan, Q.L., de Pablo, J.J.: Hyper parallel tempering Monte Carlo: application to the lennard-jones fluid and the restricted primitive model. J. Chem. Phys. 111, 9509 (1999)

    Article  Google Scholar 

  4. van der Spoel, D., Lindahl, E., Hess, B., Groenhof, G., Mark, A.E., Berendsen, H.J.C.: Gromacs: fast, flexible and free. J. Comput. Chem. 26, 1701 (2005)

    Article  Google Scholar 

  5. Hess, B., Kutzner, C., van der Spoel, D., Lindahl, E.: Gromacs 4: algorithms for highly efficient, load-balanced, and scalable molecular simulation. 4, 435 (2008)

    Google Scholar 

  6. Falcioni, M., Deem, M.W.: A biased Monte Carlo scheme for zeolite structure solution. J. Chem. Phys. 110, 1754 (1999)

    Article  Google Scholar 

  7. Hernández-Rojas, J., Llorente, J.M.G.: Microcanonical versus canonical analysis of protein folding. Phys. Rev. Lett. 100, 258104 (2008)

    Article  Google Scholar 

  8. Fiore, C.E.: First-order phase transitions: a study through the parallel tempering method. Phys. Rev. E. 78, 041109 (2008)

    Article  Google Scholar 

  9. Imperio, A., Reatto, L.: Microphase separation in two-dimensional systems with competing interactions. J. Chem. Phys. 124, 164712 (2006)

    Article  Google Scholar 

  10. Arnold, A., Holm, C.: Interactions of like-charged rods at low temperatures: analytical theory vs. simulations. Eur. Phys. J. E 27, 21 (2008)

    Article  Google Scholar 

  11. Odriozola, G., Berthier, L.: Equilibrium equation of state of a hard sphere binary mixture at very large densities using replica exchange Monte Carlo simulations. J. Chem. Phys. 134, 054504 (2011)

    Article  Google Scholar 

  12. Fortini, A., Dijkstra, M.: Phase behaviour of hard spheres confined between parallel hard plates: manipulation of colloidal crystal structures by confinement. J. Phys. Condens. Matter 18, L371 (2006)

    Article  Google Scholar 

  13. Damasceno, P.F., Engel, M., Glotzer, S.C.: Crystalline assemblies and densest packings of a family of truncated tetrahedra and the role of directional entropic forces. ACS Nano 6, 609 (2012)

    Article  Google Scholar 

  14. van Anders, G., Ahmed, N.K., Smith, R., Engel, M., Glotzer, S.C.: Entropically patchy particles: engineering valence through shape entropy. ACS Nano 8, 931 (2014)

    Article  Google Scholar 

  15. Wilding, N.B., Bruce, A.D.: Freezing by Monte Carlo phase switch. Phys. Rev. Lett. 85, 5138 (2000)

    Article  Google Scholar 

  16. Noya, E.G., Vega, C., de Miguel, E.: Determination of the melting point of hard spheres from direct coexistence simulation methods. J. Chem. Phys. 128, 154507 (2008)

    Article  Google Scholar 

  17. Frenkel, D., Smit, B.: Understanding Molecular Simulation. Academic, New York (1996)

    MATH  Google Scholar 

  18. Rosenbluth, M.N., Rosenbluth, A.W.: Further results on Monte Carlo equations of state. J. Chem. Phys. 22, 881 (1954)

    Article  Google Scholar 

  19. Wood, W.W., Jacobson, J.D.: Preliminary results from a recalculation of the Monte Carlo equation of state of hard spheres. J. Chem. Phys. 27, 1207 (1957)

    Article  Google Scholar 

  20. Alder, B.J., Wainwright, T.E.: Phase transition for a hard sphere system. J. Chem. Phys. 27, 1208 (1957)

    Article  Google Scholar 

  21. Parisi, G., Zamponi, F.: Mean-field theory of hard sphere glasses and jamming. Rev. Mod. Phys. 82, 789 (2010)

    Article  Google Scholar 

  22. Speedy, R.J.: On the reproducibility of glasses. J. Chem. Phys. 100, 6684 (1994)

    Article  Google Scholar 

  23. van Blaaderen, A., Wiltzius, P.: Real-space structure of colloidal hard-sphere glasses. Science 270, 1177 (1995)

    Article  Google Scholar 

  24. Speedy, R.J.: Pressure of the metastable hard-sphere fluid. J. Phys. Condens. Matter 9, 8591 (1997)

    Article  Google Scholar 

  25. Angelani, L., Foffi, G.: Configurational entropy of hard spheres. J. Phys. Condens. Matter 19, 256207 (2007)

    Article  Google Scholar 

  26. Pusey, P.N., van Megen, W.: Observation of a glass transition in suspensions of spherical colloidal particles. Phys. Rev. Lett. 59, 2083 (1987)

    Article  Google Scholar 

  27. van Megen, W., Mortensen, T.C., Williams, S.R., Müller, J.: Measurement of the self-intermediate scattering function of suspensions of hard spherical particles near the glass transition. Phys. Rev. E 58, 6073 (1998)

    Article  Google Scholar 

  28. Cheng, Z., Zhu, J., Chaikin, P.M., Phan, S.-E., Russel, W.B.: Nature of the divergence in low shear viscosity of colloidal hard-sphere dispersions. Phys. Rev. E 65, 041405 (2002)

    Article  Google Scholar 

  29. Brambilla, G., El Masri, D., Pierno, M., Berthier, L., Cipelletti, L., Petekidis, G., Schofield, A.B.: Probing the equilibrium dynamics of colloidal hard spheres above the mode-coupling glass transition. Phys. Rev. Lett. 102, 085703 (2009)

    Article  Google Scholar 

  30. Hermes, M., Dijkstra, M.: Thermodynamic signature of the dynamic glass transition in hard spheres. J. Phys. Condens. Matter 22, 104114 (2010)

    Article  Google Scholar 

  31. Lyubartsev, A.P., Martinovski, A.A., Shevkunov, S.V., Vorontsov-Velyaminov, P.N.: New approach to Monte Carlo calculation of the free energy: method of expanded ensembles. J. Chem. Phys. 96, 1776 (1992)

    Article  Google Scholar 

  32. Hukushima, K., Nemoto, K.: Exchange Monte Carlo method and application to spin glass simulations. J. Phys. Soc. Jpn. 65, 1604 (1996)

    Article  Google Scholar 

  33. Gallicchio, E., Xia, J., Flynn, W.F., Zhang, B., Samlalsingh, S., Mentes, A., Levy, R.M.: Asynchronous replica exchange software for grid and heterogeneous computing. Comput. Phys. Commun. (2015). doi:10.1016/j.cpc.2015.06.010

    Google Scholar 

  34. Odriozola, G.: Replica exchange Monte Carlo applied to hard spheres. J. Chem. Phys. 131, 144107 (2009)

    Article  Google Scholar 

  35. Okabe, T., Kawata, M., Okamoto, Y., Mikami, M.: Replica-exchange Monte Carlo method for the isobaric-isothermal ensemble. Chem. Phys. Lett. 335, 435 (2001)

    Article  Google Scholar 

  36. Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. J. Chem. Phys. 81, 2340 (1994)

    Article  Google Scholar 

  37. Donev, A., Torquato, S., Stillinger, F.H.: Neighbor list collision-driven molecular dynamics simulation for nonspherical hard particles. I. Algorithmic details. J. Comput. Phys. 202, 737 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  38. Donev, A., Torquato, S., Stillinger, F.H.: Neighbor list collision-driven molecular dynamics simulation for nonspherical hard particles. II. Applications to ellipses and ellipsoids. J. Comput. Phys. 202, 765 (2005)

    MathSciNet  MATH  Google Scholar 

  39. Rathore, N., Chopra, M., de Pablo, J.J.: Optimal allocation of replicas in parallel tempering simulations. J. Chem. Phys. 122, 024111 (2005)

    Article  Google Scholar 

  40. Speedy, R.J.: Pressure and entropy of hard-sphere crystals. J. Phys. Condens. Matter 10, 4387 (1998)

    Article  Google Scholar 

  41. Ferrenberg, A.M., Swendsen, R.H.: New Monte Carlo technique for studying phase transitions. Phys. Rev. Lett. 61, 2635 (1988)

    Article  Google Scholar 

  42. Ferrenberg, A.M., Swendsen, R.H.: Optimized Monte Carlo data analysis. Phys. Rev. Lett. 63, 1195 (1989)

    Article  Google Scholar 

  43. Steinhardt, P.J., Nelson, D.R., Ronchetti, M.: Bond-orientational order in liquids and glasses. Phys. Rev. B 28, 784 (1983)

    Article  Google Scholar 

  44. Rintoul, M.D., Torquato, S.: Computer simulations of dense hard sphere systems. J. Chem. Phys. 105, 9258 (1996)

    Article  Google Scholar 

  45. Hales, T.C., Ferguson, S.P.: The Kepler Conjecture: The Hales-Ferguson Proof. Springer, New York (2011)

    Google Scholar 

  46. Pusey, P.N.: The effect of polydispersity on the crystallization of hard spherical colloids. J. Phys. France 48, 709 (1987)

    Article  Google Scholar 

  47. Ogarko, V., Luding, S.: Prediction of polydisperse hard-sphere mixture behavior using tridisperse systems. Soft Matter 9, 9530 (2013)

    Article  Google Scholar 

  48. O’Toole, P.I., Hudson, T.S.: New high-density packings of similarly sized binary spheres. J. Phys. Chem. C 115, 19037 (2011)

    Article  Google Scholar 

  49. Santos, A., Yuste, S.B., López de Haro, M., Odriozola, G., Ogarko, V.: Simple effective rule to estimate the jamming packing fraction of polydisperse hard spheres. Phys. Rev. E 89, 040302(R) (2014)

    Article  Google Scholar 

  50. Berthier, L., Witten, T.A.: Glass transition of dense fluids of hard and compressible spheres. Phys. Rev. E 80, 021502 (2009)

    Article  Google Scholar 

  51. Perera, D.N., Harrowell, P.: Stability and structure of a supercooled liquid mixture in two dimensions. Phys. Rev. E 59, 5721 (1999)

    Article  Google Scholar 

  52. Biben, T., Hansen, J.P.: Phase separation of asymmetric binary hard-sphere fluids. Phys. Rev. Lett. 66, 2215 (1991)

    Article  Google Scholar 

  53. Santen, L., Krauth, W.: Absence of thermodynamic phase transition in a model glass former. Nature 405, 550 (2000)

    Article  Google Scholar 

  54. Frenkel, D., Mulder, B.M., McTague, J.P.: Phase diagram of a system of hard ellipsoids. Phys. Rev. Lett. 52, 287 (1984)

    Article  Google Scholar 

  55. Frenkel, D., Mulder, B.M.: The hard ellipsoid-of-revolution fluid. I. Monte Carlo simulations. Mol. Phys. 55, 1171 (1985)

    Article  Google Scholar 

  56. Odriozola, G.: Revisiting the phase diagram of hard ellipsoids. J. Chem. Phys. 136, 134505 (2012)

    Article  Google Scholar 

  57. Bautista-Carbajal, G., Moncho-Jordá, A., Odriozola, G.: Further details on the phase diagram of hard ellipsoids of revolution. J. Chem. Phys. 138, 064501 (2013)

    Article  Google Scholar 

  58. Perram, J.W., Wertheim, M.S., Lebowitz, J.L., Williams, G.O.: Monte Carlo simulation of hard spheroids. Chem. Phys. Lett. 105, 277 (1984)

    Article  Google Scholar 

  59. Perram, J.W., Wertheim, M.S.: Statistical mechanics of hard ellipsoids. I. Overlap algorithm and the contact function. J. Comput. Phys. 58, 409 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  60. Paramonov, L., Yaliraki, S.N.: The directional contact distance of two ellipsoids: coarse-grained potentials for anisotropic interactions. J. Chem. Phys. 123, 194111 (2005)

    Article  Google Scholar 

  61. Vesely, F.J.: Nematic-smectic transition of parallel hard spheroellipsoids. J. Chem. Phys. 141, 064109 (2014)

    Article  Google Scholar 

  62. Berne, B.J., Pechukas, P.: Gaussian model potentials for molecular interactions. J. Chem. Phys. 56, 4213 (1972)

    Article  Google Scholar 

  63. Rickayzen, G.: A model for the study of the structure of hard molecular fluids. Mol. Phys. 95, 393 (1998)

    Article  Google Scholar 

  64. de Guevara-Rodríguez, F.J., Odriozola, G.: Hard ellipsoids: analytically approaching the exact overlap distance. J. Chem. Phys. 135, 084508 (2011)

    Article  Google Scholar 

  65. Donev, A., Stillinger, F.H., Chaikin, P.M., Torquato, S.: Unusually dense crystal packings of ellipsoids. Phys. Rev. Lett. 92, 255506 (2004)

    Article  Google Scholar 

  66. Pfleiderer, P., Schilling, T.: Simple monoclinic crystal phase in suspensions of hard ellipsoids. Phys. Rev. E 75, 020402 (2007)

    Article  Google Scholar 

  67. Radu, M., Pfleiderer, P., Schilling, T.: Solid-solid phase transition in hard ellipsoids. J. Chem. Phys. 131, 164513 (2009)

    Article  Google Scholar 

  68. Herod, T.E., Duran, R.S.: Two and three-dimensional nanoparticles of liquid-crystals prepared at the air liquid interface. Langmuir 14, 6606 (1998)

    Article  Google Scholar 

  69. Kim, F., Kwan, S., Akana, J., Yang, P.D.: Langmuir-Blodgett nanorod assembly. J. Amm. Chem. Soc. 123, 4360 (2001)

    Article  Google Scholar 

  70. Davies, G.B., Krüger, T., Coveney, P.V., Harting, J., Bremse, F.: Interface deformations affect the orientation transition of magnetic ellipsoidal particles adsorbed at fluid-fluid interfaces. Soft Matter 10, 6742 (2014)

    Article  Google Scholar 

  71. Zheng, Z., Wang, F., Han, Y.: Glass transitions in quasi-two-dimensional suspensions of colloidal ellipsoids. Phys. Rev. Lett. 107, 065702 (2011)

    Article  Google Scholar 

  72. Constantin, D., Davidson, P., Chanéac, C.: Lyotropic lamellar phase doped with a nematic phase of magnetic nanorods. Langmuir 26, 4586 (2010)

    Article  Google Scholar 

  73. Frenkel, D., Eppenga, R.: Evidence for algebraic orientational order in a two-dimensional hard-core nematic. Phys. Rev. A 31, 1776 (1985)

    Article  Google Scholar 

  74. Cuesta, J.A., Frenkel, D.: Monte Carlo simulation of two-dimensional hard ellipses. Phys. Rev. A 42, 2126 (1990)

    Article  Google Scholar 

  75. Donev, A., Burton, J., Stillinger, F.H., Torquato, S.: Tetratic order in the phase behavior of a hard-rectangle system. Phys. Rev. B 73, 054109 (2006)

    Article  Google Scholar 

  76. Avendaño, C., Escobedo, F.A.: Phase behavior of rounded hard-squares. Soft Matter 8, 4675 (2012)

    Article  Google Scholar 

  77. Shah, A.A., Kang, H., Kohlstedt, K.L., Ahn, K.H., Glotzer, S.C., Monroe, C.W., Solomon, M.J.: Liquid crystal order in colloidal suspensions of spheroidal particles by direct current electric field assembly. Small 8, 1551 (2012)

    Article  Google Scholar 

  78. Qi, W., de Graaf, J., Qiao, F., Marras, S., Manna, L., Dijkstra, M.: Ordered two-dimensional superstructures of colloidal octapod-shaped nanocrystals on flat substrates. Nano Lett. 12, 5299 (2012)

    Article  Google Scholar 

  79. Qi, W., de Graaf, J., Qiao, F., Marras, S., Manna, L., Dijkstra, M.: Phase diagram of octapod-shaped nanocrystals in a quasi-two-dimensional planar geometry. J. Chem. Phys. 138, 154504 (2013)

    Article  Google Scholar 

  80. Quan, Z., Fang, J.: Superlattices with non-spherical building blocks. Nano Today 5, 390 (2010)

    Article  Google Scholar 

  81. Schmitt, J., Gruenewald, T., Decher, G., Pershan, P., Kjaer, K., Losche, M.: Internal structure of layer by layer adsorbed polyelectrolyte films - a neutron and x-ray reflectivity study. Macromolecules 26, 7058 (1993)

    Article  Google Scholar 

  82. Decher, G.: Fuzzy nanoassemblies: toward layered polymeric multicomposites. Science 277, 1232 (1997)

    Article  Google Scholar 

  83. Rycenga, M., Camargo, P.H.C., Xia, Y.: Template-assisted self-assembly: a versatile approach to complex micro- and nanostructures. Soft Matter 5, 1129 (2009)

    Article  Google Scholar 

  84. Kosterlitz, J.M., Thouless, D.J.: Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C 6, 1181 (1973)

    Article  Google Scholar 

  85. Halperin, B.I., Nelson, D.R.: Theory of two-dimensional melting. Phys. Rev. Lett. 41, 121 (1978)

    Article  MathSciNet  Google Scholar 

  86. Straley, J.P.: Liquid crystals in two dimensions. Phys. Rev. A 4, 675 (1971)

    Article  Google Scholar 

  87. Bates, M.A., Frenkel, D.: Phase behavior of two-dimensional hard rod fluids. J. Chem. Phys. 112, 10034 (2000)

    Article  Google Scholar 

  88. Zheng, Z., Han, Y.: Self-diffusion in two-dimensional hard ellipsoid suspensions. J. Chem. Phys. 133, 124509 (2010)

    Article  Google Scholar 

  89. Xu, W.S., Li, Y.W., Sun, Z.Y., An, L.J.: Hard ellipses: Equation of state, structure, and self-diffusion. J. Chem. Phys. 139, 024501 (2013)

    Article  Google Scholar 

  90. Jaster, A.: Computer simulations of the two-dimensional melting transition using hard disks. Phys. Rev. E 59, 2594 (1999)

    Article  Google Scholar 

  91. Bernard, E.P., Krauth, W.: Two-step melting in two dimensions: first-order liquid-hexatic transition. Phys. Rev. Lett. 107, 155704 (2011)

    Article  Google Scholar 

  92. Bautista-Carbajal, G., Odriozola, G.: Phase diagram of two-dimensional hard ellipses. J. Chem. Phys. 140, 204502 (2014)

    Article  Google Scholar 

  93. Vieillard-Baron, J.: Phase transitions of the classical hard-ellipse system. J. Chem. Phys. 56, 4729 (1972)

    Article  Google Scholar 

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Authors and Affiliations

  1. Academia de Matemáticas, Universidad Autónoma de la Ciudad de México, 07160, México, D.F., Mexico

    Gustavo Bautista-Carbajal

  2. Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana-Azcapotzalco, Av. San Pablo 180, 02200, México D.F., Mexico

    Carlos A. Vargas, Eduardo Basurto & Gerardo Odriozola

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  1. Gustavo Bautista-Carbajal
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  2. Carlos A. Vargas
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  3. Eduardo Basurto
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  4. Gerardo Odriozola
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Correspondence to Gerardo Odriozola .

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Editors and Affiliations

  1. ABACUS Centro de Matemáticas Aplicadas, CINVESTAV-IPN, La Marquesa, Mexico

    Isidoro Gitler

  2. Instituto Nacional de Investigaciones Nu, La Marquesa, Mexico

    Jaime Klapp

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Bautista-Carbajal, G., Vargas, C.A., Basurto, E., Odriozola, G. (2016). Parallel Replica Exchange Monte Carlo Applied to Hard Systems. In: Gitler, I., Klapp, J. (eds) High Performance Computer Applications. ISUM 2015. Communications in Computer and Information Science, vol 595. Springer, Cham. https://doi.org/10.1007/978-3-319-32243-8_28

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