Skip to main content
SpringerLink
Log in

Pathways to self-organization: Crystallization via nucleation and growth

  • Topical Review
  • Published:
The European Physical Journal E Aims and scope Submit manuscript

Abstract.

Crystallization, a prototypical self-organization process during which a disordered state spontaneously transforms into a crystal characterized by a regular arrangement of its building blocks, usually proceeds by nucleation and growth. In the initial stages of the transformation, a localized nucleus of the new phase forms in the old one due to a random fluctuation. Most of these nuclei disappear after a short time, but rarely a crystalline embryo may reach a critical size after which further growth becomes thermodynamically favorable and the entire system is converted into the new phase. In this article, we will discuss several theoretical concepts and computational methods to study crystallization. More specifically, we will address the rare event problem arising in the simulation of nucleation processes and explain how to calculate nucleation rates accurately. Particular attention is directed towards discussing statistical tools to analyze crystallization trajectories and identify the transition mechanism.

Graphical abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. McPherson, J. Cryst. Growth 110, 1 (1991)

    Article  ADS  Google Scholar 

  2. P.R. ten Wolde, D. Frenkel, Science 277, 1975 (1997)

    Article  Google Scholar 

  3. U. Gasser, J. Phys.: Condens. Matter 21, 203101 (2009)

    ADS  Google Scholar 

  4. H. Thomas, G.E. Morfill, V. Demmel, J. Goree, B. Feuerbacher, D. Möhlmann, Phys. Rev. Lett. 73, 652 (1994)

    Article  ADS  Google Scholar 

  5. G.E. Morfill, A.V. Ivlev, Rev. Mod. Phys. 81, 1353 (2009)

    Article  ADS  Google Scholar 

  6. D.G. Fahrenheit, Philos. Trans. 33, 78 (1724)

    Article  Google Scholar 

  7. P. Stökel, I.M. Weidinger, H. Baumgärtel, T. Leisner, J. Phys. Chem. A 109, 2540 (2005)

    Article  Google Scholar 

  8. J.A. Sellberg, C. Huang, T.A. McQueen, N.D. Loh, H. Laksmono, D. Schlesinger, R.G. Sierra, D. Nordlund, C.Y. Hampton, D. Starodub, D.P. DePonte, M. Beye, C. Chen, A.V. Martin, A. Barty, K.T. Wikfeldt, T.M. Weiss, C. Caronna, J. Feldkamp, L.B. Skinner, M.M. Seibert, M. Messerschmidt, G.J. Williams, S. Boutet, L.G.M. Pettersson, M.J. Bogan, A. Nilsson, Nature 510, 381 (2014)

    Article  ADS  Google Scholar 

  9. M. Volmer, A. Weber, Z. Phys. Chem. 119, 277 (1926)

    Google Scholar 

  10. R. Becker, W. Döring, Ann. Phys. (Lepzig) 24, 719 (1935)

    Article  ADS  Google Scholar 

  11. D. Turnbull, J.C. Fisher, J. Chem. Phys. 17, 71 (1949)

    Article  ADS  Google Scholar 

  12. P. Geiger, C. Dellago, J. Chem. Phys. 139, 164105 (2013)

    Article  ADS  Google Scholar 

  13. J.W. Cahn, J.E. Hilliard, J. Chem. Phys. 28, 258 (1958)

    Article  ADS  Google Scholar 

  14. K. Binder, Rep. Prog. Phys. 50, 783 (1987)

    Article  ADS  Google Scholar 

  15. D. Frenkel, B. Smit, Understanding Molecular Simulation (Academic Press, San Diego, 2001)

  16. M. Allen, D. Tildesley, Computer Simulation of Liquids (Oxford Science Publications, Oxford, 1987)

  17. M. Salvalaglio, T. Vetter, F. Giberti, M. Mazzotti, M. Parrinello, J. Am. Chem. Soc. 134, 17221 (2012)

    Article  Google Scholar 

  18. M. Salvalaglio, T. Vetter, M. Mazzotti, M. Parrinello, Angew. Chem. Int. Ed. 52, 13369 (2013)

    Article  Google Scholar 

  19. J. Baumgartner, A. Dey, P.H.H. Bomans, C. Le Coadou, P. Fratzl, N.A.J.M. Sommerdijk, D. Faivre, Nat. Mater. 12, 310 (2013)

    Article  ADS  Google Scholar 

  20. D. Gebauer, M. Kellermeier, J.D. Gale, L. Bergström, H. Cölfen, Chem. Soc. Rev. 43, 2348 (2014)

    Article  Google Scholar 

  21. W.M. Jacobs, A. Reinhardt, D. Frenkel, Proc. Natl. Acad. Sci. U.S.A. 112, 6313 (2015)

    Article  ADS  Google Scholar 

  22. S. Whitelam, R.L. Jack, Annu. Rev. Phys. Chem. 66, 143 (2015)

    Article  ADS  Google Scholar 

  23. P.G. Debenedetti, Metastable Liquids: Concepts and Principles (Princeton University Press, Princeton, 1997)

  24. D. Kashchiev, Nucleation: Basic Thory with Applications (Butterworth-Heinemann, Oxford, 2000)

  25. J.W. Gibbs, Trans. Conn. Acad. 3, 108 (1875)

    Google Scholar 

  26. J.W. Gibbs, Trans. Conn. Acad. 3, 333 (1877)

    Google Scholar 

  27. L. Farkas, Z. Phys. Chem. 125, 236 (1927)

    Google Scholar 

  28. I.N. Stranski, R. Kaischew, Z. Phys. Chem. B 26, 100 (1934)

    Google Scholar 

  29. R. Kaischew, I.N. Stranski, Z. Phys. Chem. B 26, 114 (1934)

    Google Scholar 

  30. R. Kaischew, I.N. Stranski, Z. Phys. Chem. 170, 295 (1934)

    Google Scholar 

  31. J. Frenkel, J. Chem. Phys. 7, 200 (1939)

    Article  ADS  Google Scholar 

  32. J. Frenkel, J. Chem. Phys. 7, 538 (1939)

    Article  ADS  Google Scholar 

  33. J.B. Zeldovich, JETP 12, 525 (1942)

    Google Scholar 

  34. D.W. Oxtoby, Acc. Chem. Res. 31, 91 (1998)

    Article  Google Scholar 

  35. R.P. Sear, J. Phys.: Condens. Matter 19, 033101 (2007)

    ADS  Google Scholar 

  36. R.P. Sear, Int. Mater. Rev. 57, 328 (2012)

    Article  Google Scholar 

  37. K.F. Kelton, A.L. Greer, Nucleation in Condensed Matter (Elsevier, Amsterdam, 2010)

  38. H.A. Kramers, Physica 7, 284 (1940)

    Article  ADS  MathSciNet  Google Scholar 

  39. P. Hänggi, P. Talkner, M. Borkovec, Rev. Mod. Phys. 62, 251 (1990)

    Article  ADS  Google Scholar 

  40. R. Zwanzig, Nonequilibrium Statistical Mechanics (Oxford University Press, 2001)

  41. L.S. Pontryagin, A.A. Andronov, A.A. Vitt, JETP 3, 165 (1933)

    Google Scholar 

  42. S. Jungblut, C. Dellago, J. Chem. Phys. 134, 104501 (2011)

    Article  ADS  Google Scholar 

  43. J. Frenkel, Kinetic Theory of Liquids (Clarendon Press, Oxford, 1946)

  44. J.R. Espinosa, E. Sanz, C. Valeriani, C. Vega, J. Chem. Phys. 141, 18C529 (2014)

    Article  Google Scholar 

  45. C.H. Bennett, Molecular Dynamics and Transition State Theory: The Simulation of Infrequent Events, in Algorithms for Chemical Computations, edited by R.E. Christoffersen, Vol. 46 of ACS Symposium Series (American Chemical Society, Washington, D.C., 1977) p. 63

  46. D. Chandler, J. Chem. Phys. 68, 2959 (1978)

    Article  ADS  Google Scholar 

  47. P.R. ten Wolde, M.J. Ruiz-Montero, D. Frenkel, J. Chem. Phys. 104, 9932 (1996)

    Article  ADS  Google Scholar 

  48. M.J. Ruiz-Montero, D. Frenkel, J.J. Brey, Mol. Phys. 90, 925 (1997)

    Article  ADS  Google Scholar 

  49. S. Auer, D. Frenkel, J. Chem. Phys. 120, 3015 (2004)

    Article  ADS  Google Scholar 

  50. S. Auer, D. Frenkel, Nature 409, 1020 (2001)

    Article  ADS  Google Scholar 

  51. M.J. Mandell, J.P. McTague, A. Rahman, J. Chem. Phys. 66, 3070 (1977)

    Article  ADS  Google Scholar 

  52. S. Alexander, J.P. McTague, Phys. Rev. Lett. 41, 702 (1978)

    Article  ADS  Google Scholar 

  53. D. Moroni, P.R. ten Wolde, P.G. Bolhuis, Phys. Rev. Lett. 94, 235703 (2005)

    Article  ADS  Google Scholar 

  54. H. Wang, H. Gould, W. Klein, Phys. Rev. E 76, 031604 (2007)

    Article  ADS  Google Scholar 

  55. G.T. Beckham, B. Peters, J. Phys. Chem. Lett. 2, 1133 (2011)

    Article  Google Scholar 

  56. S. Jungblut, A. Singraber, C. Dellago, Mol. Phys. 111, 3527 (2013)

    Article  ADS  Google Scholar 

  57. W. Ostwald, Z. Phys. Chem. 22, 289 (1897)

    Google Scholar 

  58. I.N. Stranski, D. Totomanow, Z. Phys. Chem. 163, 399 (1933)

    Google Scholar 

  59. D.W. Oxtoby, R. Evans, J. Chem. Phys. 89, 7521 (1988)

    Article  ADS  Google Scholar 

  60. R.C. Tolman, J. Chem. Phys. 17, 333 (1949)

    Article  ADS  Google Scholar 

  61. P.R. ten Wolde, D. Frenkel, J. Chem. Phys. 109, 9901 (1998)

    Article  ADS  Google Scholar 

  62. M. Iwamatsu, J. Phys.: Condens. Matter 6, L173 (1994)

    ADS  Google Scholar 

  63. L. Gránásy, J. Chem. Phys. 109, 9660 (1998)

    Article  ADS  Google Scholar 

  64. M.A. Anisimov, H.J.S. Pierre, Phys. Rev. E 78, 011105 (2008)

    Article  ADS  Google Scholar 

  65. A. Tröster, M. Oettel, B. Block, P. Virnau, K. Binder, J. Chem. Phys. 136, 064709 (2012)

    Article  ADS  Google Scholar 

  66. A. Tröster, K. Binder, J. Phys.: Condens. Matter 24, 284107 (2012)

    Google Scholar 

  67. L.F. Rull, S. Toxvaerd, J. Chem. Phys. 78, 3273 (1983)

    Article  ADS  Google Scholar 

  68. J.Q. Broughton, G.H. Gilmer, J. Chem. Phys. 84, 5759 (1986)

    Article  ADS  Google Scholar 

  69. R.L. Davidchack, B.B. Laird, J. Chem. Phys. 118, 7651 (2003)

    Article  ADS  Google Scholar 

  70. R.L. Davidchack, B.B. Laird, Phys. Rev. Lett. 94, 086102 (2005)

    Article  ADS  Google Scholar 

  71. S.E.M. Lundrigan, I. Saika-Voivod, J. Chem. Phys. 131, 104503 (2009)

    Article  ADS  Google Scholar 

  72. F. Trudu, D. Donadio, M. Parrinello, Phys. Rev. Lett. 97, 105701 (2006)

    Article  ADS  Google Scholar 

  73. S. Prestipino, A. Laio, E. Tosatti, Phys. Rev. Lett. 108, 225701 (2012)

    Article  ADS  Google Scholar 

  74. S. Prestipino, A. Laio, E. Tosatti, J. Chem. Phys. 138, 064508 (2013)

    Article  ADS  Google Scholar 

  75. S. Prestipino, A. Laio, E. Tosatti, J. Chem. Phys. 140, 094501 (2014)

    Article  ADS  Google Scholar 

  76. D.P. Landau, K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University Press, 2000)

  77. A.Z. Panagiotopoulos, J. Phys.: Condens. Matter 12, R25 (2000)

    ADS  Google Scholar 

  78. J.M. Rickman, R. LeSar, Annu. Rev. Mater. Res. 32, 195 (2002)

    Article  Google Scholar 

  79. C. Vega, E. Sanz, J.L.F. Abascal, E.G. Noya, J. Phys.: Condens. Matter 20, 153101 (2008)

    ADS  Google Scholar 

  80. D.L. Mobley, P.V. Klimovich, J. Chem. Phys. 137, 230901 (2012)

    Article  ADS  Google Scholar 

  81. M.B. Sweatman, Mol. Phys. 113, 1206 (2015)

    Article  ADS  Google Scholar 

  82. A.Z. Panagiotopoulos, Mol. Phys. 61, 813 (1987)

    Article  ADS  Google Scholar 

  83. A.Z. Panagiotopoulos, N. Quirke, M. Stapleton, D.J. Tildesley, Mol. Phys. 63, 527 (1988)

    Article  ADS  Google Scholar 

  84. A. Kerrache, J. Horbach, K. Binder, EPL 81, 58001 (2008)

    Article  ADS  Google Scholar 

  85. U.R. Pedersen, J. Chem. Phys. 139, 104102 (2013)

    Article  ADS  Google Scholar 

  86. T. Schilling, F. Schmid, J. Chem. Phys. 131, 231102 (2009)

    Article  ADS  Google Scholar 

  87. D. Frenkel, A.J.C. Ladd, J. Chem. Phys. 81, 3188 (1984)

    Article  ADS  Google Scholar 

  88. J.M. Polson, E. Trizac, S. Pronk, D. Frenkel, J. Chem. Phys. 112, 5339 (2000)

    Article  ADS  Google Scholar 

  89. D. Wilms, N.B. Wilding, K. Binder, Phys. Rev. E 85, 056703 (2012)

    Article  ADS  Google Scholar 

  90. A.D. Bruce, N.B. Wilding, G.J. Ackland, Phys. Rev. Lett. 79, 3002 (1997)

    Article  ADS  Google Scholar 

  91. A.D. Bruce, A.N. Jackson, G.J. Ackland, N.B. Wilding, Phys. Rev. E 61, 906 (2000)

    Article  ADS  Google Scholar 

  92. N.B. Wilding, A.D. Bruce, Phys. Rev. Lett. 85, 5138 (2000)

    Article  ADS  Google Scholar 

  93. A.N. Jackson, A.D. Bruce, G.J. Ackland, Phys. Rev. E 65, 036710 (2002)

    Article  ADS  Google Scholar 

  94. K. Binder, Eur. Phys. J. B 64, 307 (2008)

    Article  ADS  Google Scholar 

  95. G.M. Torrie, J.P. Valleau, Chem. Phys. Lett. 28, 578 (1974)

    Article  ADS  Google Scholar 

  96. B.A. Berg, T. Neuhaus, Phys. Rev. Lett. 68, 9 (1992)

    Article  ADS  Google Scholar 

  97. A. Laio, M. Parrinello, Proc. Natl. Acad. Sci. U.S.A. 99, 12562 (2002)

    Article  ADS  Google Scholar 

  98. A. Barducci, G. Bussi, M. Parrinello, Phys. Rev. Lett. 100, 020603 (2008)

    Article  ADS  Google Scholar 

  99. F. Wang, D.P. Landau, Phys. Rev. Lett. 86, 2050 (2001)

    Article  ADS  Google Scholar 

  100. F. Wang, D.P. Landau, Phys. Rev. E 64, 056101 (2001)

    Article  ADS  Google Scholar 

  101. D.P. Landau, S.-H. Tsai, M. Exler, Am. J. Phys. 72, 1294 (2004)

    Article  ADS  Google Scholar 

  102. A.M. Ferrenberg, R.H. Swendsen, Phys. Rev. Lett. 61, 2635 (1988)

    Article  ADS  Google Scholar 

  103. A.M. Ferrenberg, R.H. Swendsen, Phys. Rev. Lett. 63, 1195 (1989)

    Article  ADS  Google Scholar 

  104. R.H. Swendsen, J.-S. Wang, Phys. Rev. Lett. 57, 2607 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  105. D.J. Earl, M.W. Deem, Phys. Chem. Chem. Phys. 7, 3910 (2005)

    Article  Google Scholar 

  106. P. Virnau, M. Müller, J. Chem. Phys. 120, 10925 (2004)

    Article  ADS  Google Scholar 

  107. M. Chopra, M. Müller, J.J. de Pablo, J. Chem. Phys. 124, 134102 (2006)

    Article  ADS  Google Scholar 

  108. J.P. Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity (Oxford University Press, 2006)

  109. E.E. Santiso, B.L. Trout, J. Chem. Phys. 134, 064109 (2011)

    Article  ADS  Google Scholar 

  110. A.S. Keys, C.R. Iacovella, S.C. Glotzer, J. Comput. Phys. 230, 6438 (2011)

    Article  ADS  Google Scholar 

  111. P.J. Steinhardt, D.R. Nelson, M. Ronchetti, Phys. Rev. B 28, 784 (1983)

    Article  ADS  Google Scholar 

  112. P.R. ten Wolde, M.J. Ruiz-Montero, D. Frenkel, Phys. Rev. Lett. 75, 2714 (1995)

    Article  ADS  Google Scholar 

  113. M.J. Mandell, J.P. McTague, A. Rahman, J. Chem. Phys. 64, 3699 (1976)

    Article  ADS  Google Scholar 

  114. U. Gasser, E.R. Weeks, A. Schofield, P.N. Pusey, D.A. Weitz, Science 292, 258 (2001)

    Article  ADS  Google Scholar 

  115. D.M. Herlach, I. Klassen, P. Wette, D. Holland-Moritz, J. Phys.: Condens. Matter 22, 153101 (2010)

    ADS  Google Scholar 

  116. W. Lechner, C. Dellago, J. Chem. Phys. 129, 114707 (2008)

    Article  ADS  Google Scholar 

  117. W. Lechner, C. Dellago, P.G. Bolhuis, Phys. Rev. Lett. 106, 085701 (2011)

    Article  ADS  Google Scholar 

  118. Y. Chushak, L.S. Bartell, J. Phys. Chem. A 104, 9328 (2000)

    Article  Google Scholar 

  119. J.-M. Leyssale, J. Delhommelle, C. Millot, Chem. Phys. Lett. 375, 612 (2003)

    Article  ADS  Google Scholar 

  120. C. Valeriani, E. Sanz, D. Frenkel, J. Chem. Phys. 122, 194501 (2005)

    Article  ADS  Google Scholar 

  121. J.A. van Meel, A.J. Page, R.P. Sear, D. Frenkel, J. Chem. Phys. 129, 204505 (2008)

    Article  ADS  Google Scholar 

  122. J. Bokeloh, R.E. Rozas, J. Horbach, G. Wilde, Phys. Rev. Lett. 107, 145701 (2011)

    Article  ADS  Google Scholar 

  123. J. Russo, H. Tanaka, Sci. Rep. 2, 505 (2012)

    Article  ADS  Google Scholar 

  124. P.R. ten Wolde, D. Frenkel, Phys. Chem. Chem. Phys. 1, 2191 (1999)

    Article  Google Scholar 

  125. B.A. Klumov, Phys. Usp. 53, 1053 (2010)

    Article  ADS  Google Scholar 

  126. T. Schilling, H.J. Schöpe, M. Oettel, G. Opletal, I. Snook, Phys. Rev. Lett. 105, 025701 (2010)

    Article  ADS  Google Scholar 

  127. T. Kawasaki, A. Onuki, J. Chem. Phys. 135, 174109 (2011)

    Article  ADS  Google Scholar 

  128. D. Frenkel, Simulations: The Dark Side, in Proceedings of the International School of Physics “Enrico Fermi”, Course CLXXXIV “Physics of Complex Colloids”, edited by C. Bechinger, F. Sciortino, P. Ziherl (Società Italiana di Fisica, Bologna

  129. J. Wedekind, R. Strey, D. Reguera, J. Chem. Phys. 126, 134103 (2007)

    Article  ADS  Google Scholar 

  130. J. Wedekind, D. Reguera, J. Phys. Chem. B 112, 11060 (2008)

    Article  Google Scholar 

  131. A.V. Mokshin, B.N. Galimzyanov, J. Chem. Phys. 140, 024104 (2014)

    Article  ADS  Google Scholar 

  132. V.A. Shneidman, J. Chem. Phys. 141, 051101 (2014)

    Article  ADS  Google Scholar 

  133. S. Jungblut, C. Dellago, J. Chem. Phys. 142, 064103 (2015)

    Article  ADS  Google Scholar 

  134. V.G. Baidakov, A.O. Tipeev, Thermochimica Acta 522, 14 (2011)

    Article  Google Scholar 

  135. G. Chkonia, J. Wölk, R. Strey, J. Wedekind, D. Reguera, J. Chem. Phys. 130, 064505 (2009)

    Article  ADS  Google Scholar 

  136. P. Talkner, Z. Phys. B -- Condens. Matter 68, 201 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  137. H. Eyring, J. Chem. Phys. 3, 107 (1935)

    Article  ADS  Google Scholar 

  138. H. Eyring, Trans. Faraday Soc. 34, 41 (1938)

    Article  Google Scholar 

  139. E. Wigner, Trans. Faraday Soc. 34, 29 (1938)

    Article  Google Scholar 

  140. E. Pollak, P. Talkner, CHAOS 15, 026116 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  141. E. Pollak, J. Chem. Phys. 85, 865 (1986)

    Article  ADS  Google Scholar 

  142. J.C. Keck, J. Chem. Phys. 32, 1035 (1960)

    Article  ADS  MathSciNet  Google Scholar 

  143. R.G. Mullen, J.-E. Shea, B. Peters, J. Chem. Phys. 140, 041104 (2014)

    Article  ADS  Google Scholar 

  144. C. Dellago, P.G. Bolhuis, F.S. Csaijka, D. Chandler, J. Chem. Phys. 108, 1964 (1998)

    Article  ADS  Google Scholar 

  145. P.G. Bolhuis, C. Dellago, D. Chandler, Faraday Discuss. 110, 412 (1998)

    Article  ADS  Google Scholar 

  146. P.G. Bolhuis, C. Dellago, P.L. Geissler, D. Chandler, J. Phys.: Condens. Matter 12, A147 (2000)

    ADS  Google Scholar 

  147. C. Dellago, P.G. Bolhuis, P.L. Geissler, Adv. Chem. Phys. 123, 1 (2002)

    Google Scholar 

  148. P.G. Bolhuis, D. Chandler, C. Dellago, P.L. Geissler, Annu. Rev. Phys. Chem. 53, 291 (2002)

    Article  ADS  Google Scholar 

  149. C. Dellago, P.G. Bolhuis, Transition Path Sampling and other Advanced Simulation Techniques for Rare Events, in Advanced Computer Simulation Approaches for Soft Matter Sciences III, edited by C. Holm, K. Kremer, Vol. 221 of Advances in Polymer Science (Springer-Verlag, Berlin, Heidelberg, 2009) p. 167

  150. C. Dellago, P.G. Bolhuis, D. Chandler, J. Chem. Phys. 110, 6617 (1999)

    Article  ADS  Google Scholar 

  151. T.S. van Erp, D. Moroni, P.G. Bolhuis, J. Chem. Phys. 118, 7762 (2003)

    Article  ADS  Google Scholar 

  152. T.S. van Erp, P.G. Bolhuis, J. Comput. Phys. 205, 157 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  153. T.S. van Erp, Phys. Rev. Lett. 98, 268301 (2007)

    Article  ADS  Google Scholar 

  154. P.G. Bolhuis, J. Chem. Phys. 129, 114108 (2008)

    Article  ADS  Google Scholar 

  155. D. Moroni, P.G. Bolhuis, T.S. van Erp, J. Chem. Phys. 120, 4055 (2004)

    Article  ADS  Google Scholar 

  156. D. Moroni, T.S. van Erp, P.G. Bolhuis, Phys. Rev. E 71, 056709 (2005)

    Article  ADS  Google Scholar 

  157. P.L. Geissler, C. Dellago, J. Phys. Chem. B 108, 6667 (2004)

    Article  Google Scholar 

  158. R.J. Allen, P.B. Warren, P.R. ten Wolde, Phys. Rev. Lett. 94, 018104 (2005)

    Article  ADS  Google Scholar 

  159. R.J. Allen, D. Frenkel, P.R. ten Wolde, J. Chem. Phys. 124, 024102 (2006)

    Article  ADS  Google Scholar 

  160. E.E. Borrero, F.A. Escobedo, J. Chem. Phys. 127, 164101 (2007)

    Article  ADS  Google Scholar 

  161. R.J. Allen, C. Valeriani, P.R. ten Wolde, J. Phys.: Condens. Matter 21, 463102 (2009)

    ADS  Google Scholar 

  162. F.A. Escobedo, E.E. Borrero, J.C. Araque, J. Phys.: Condens. Matter 21, 333101 (2009)

    Google Scholar 

  163. T.S. van Erp, Adv. Chem Phys. 151, 27 (2012)

    Google Scholar 

  164. P.G. Bolhuis, C. Dellago, Eur. Phys. J. ST 224, 2409 (2015)

    Article  Google Scholar 

  165. T.J.H. Vlugt, B. Smit, Phys. Chem. Comm. 2, 1 (2001)

    Google Scholar 

  166. R.G. Mullen, J.-E. Shea, B. Peters, J. Chem. Theory Comput. 11, 2421 (2015)

    Article  Google Scholar 

  167. P.G. Bolhuis, J. Phys.: Condens. Matter 15, S113 (2003)

    ADS  Google Scholar 

  168. T.F. Miller III, C. Predescu, J. Chem. Phys. 126, 144102 (2007)

    Article  ADS  Google Scholar 

  169. M. Grünwald, C. Dellago, P.L. Geissler, J. Chem. Phys. 129, 194101 (2008)

    Article  ADS  Google Scholar 

  170. M. Chopra, R. Malshe, A.S. Reddy, J.J. de Pablo, J. Chem. Phys. 128, 144104 (2008)

    Article  ADS  Google Scholar 

  171. B. Peters, B.L. Trout, J. Chem. Phys. 125, 054108 (2006)

    Article  ADS  Google Scholar 

  172. B. Peters, G.T. Beckham, B.L. Trout, J. Chem. Phys. 127, 034109 (2007)

    Article  ADS  Google Scholar 

  173. J. Rogal, P.G. Bolhuis, J. Chem. Phys. 129, 224107 (2008)

    Article  ADS  Google Scholar 

  174. E.E. Borrero, C. Dellago, J. Chem. Phys. 133, 134112 (2010)

    Article  ADS  Google Scholar 

  175. N. Guttenberg, A.R. Dinner, J. Weare, J. Chem. Phys. 136, 234103 (2012)

    Article  ADS  Google Scholar 

  176. E.E. Borrero, M. Weinwurm, C. Dellago, J. Chem. Phys. 134, 244118 (2011)

    Article  ADS  Google Scholar 

  177. W.-N. Du, P.G. Bolhuis, J. Chem. Phys. 139, 044105 (2013)

    Article  ADS  Google Scholar 

  178. D.W.H. Swenson, P.G. Bolhuis, J. Chem. Phys. 141, 044101 (2014)

    Article  ADS  Google Scholar 

  179. N.B. Becker, R.J. Allen, P.R. ten Wolde, J. Chem. Phys. 136, 174118 (2012)

    Article  ADS  Google Scholar 

  180. N.B. Becker, P.R. ten Wolde, J. Chem. Phys. 136, 174119 (2012)

    Article  ADS  Google Scholar 

  181. C., Valeriani R.J. Allen, M.J. Morelli, D. Frenkel, P.R. ten Wolde, J. Chem. Phys. 127, 114109 (2007)

    Article  ADS  Google Scholar 

  182. E.E. Borrero, F.A. Escobedo, J. Chem. Phys. 129, 024115 (2008)

    Article  ADS  Google Scholar 

  183. A.K. Faradjian, R. Elber, J. Chem. Phys. 120, 10880 (2004)

    Article  ADS  Google Scholar 

  184. A.M.A. West, R. Elber, D. Shalloway, J. Chem. Phys. 126, 145104 (2007)

    Article  ADS  Google Scholar 

  185. G.A. Huber, S. Kim, Biophys. J. 70, 97 (1996)

    Article  ADS  Google Scholar 

  186. B.W. Zhang, D. Jasnow, D.M. Zuckerman, Proc. Natl. Acad. Sci. U.S.A. 104, 18043 (2007)

    Article  ADS  Google Scholar 

  187. B.W. Zhang, D. Jasnow, D.M. Zuckerman, J. Chem. Phys. 132, 054107 (2010)

    Article  ADS  Google Scholar 

  188. A. Warmflash, P. Bhimalapuram, A.R. Dinner, J. Chem. Phys. 127, 154112 (2007)

    Article  ADS  Google Scholar 

  189. D.J. Wales, Mol. Phys. 100, 3285 (2002)

    Article  ADS  Google Scholar 

  190. D.J. Wales, Phys. Biol. 2, S86 (2005)

    Article  ADS  Google Scholar 

  191. D.A. Adams, L.M. Sander, R.M. Ziff, J. Chem. Phys. 133, 124103 (2010)

    Article  ADS  Google Scholar 

  192. S.R. Williams, Phys. Rev. E 88, 043301 (2013)

    Article  ADS  Google Scholar 

  193. J.T. Berryman, T. Schilling, J. Chem. Phys. 133, 244101 (2010)

    Article  ADS  Google Scholar 

  194. V.G. Baidakov, K.S. Bobrov, A.S. Teterin, J. Chem. Phys. 135, 054512 (2011)

    Article  ADS  Google Scholar 

  195. S. Jungblut, C. Dellago, Mol. Phys. 113, 2735 (2015)

    Article  ADS  Google Scholar 

  196. L. Onsager, Phys. Rev. 54, 554 (1938)

    Article  ADS  Google Scholar 

  197. D. Ryter, Physica A 142, 103 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  198. A. Berezhkovskii, A. Szabo, J. Chem. Phys. 125, 104902 (2006)

    Article  ADS  Google Scholar 

  199. M.M. Klosek, B.J. Matkowsky, Z. Schuss, Ber. Bunsenges. Phys. Chem. 95, 331 (1991)

    Article  Google Scholar 

  200. E. Pollak, A.M. Berezhkovskii, Z. Schuss, J. Chem. Phys. 100, 334 (1994)

    Article  ADS  Google Scholar 

  201. P. Talkner, Chem. Phys. 180, 199 (1994)

    Article  ADS  Google Scholar 

  202. R. Du, V.S. Pande, A.Y. Grosberg, T. Tanaka, E.S. Shakhnovich, J. Chem. Phys. 108, 334 (1998)

    Article  ADS  Google Scholar 

  203. E. Vanden-Eijnden, Lect. Notes Phys. 703, 439 (2006)

    Google Scholar 

  204. W. E, E. Vanden-Eijnden, J. Stat. Phys. 123, 503 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  205. J.D. Honeycutt, H.C. Andersen, Chem. Phys. Lett. 108, 535 (1984)

    Article  ADS  Google Scholar 

  206. J.D. Honeycutt, H.C. Andersen, J. Phys. Chem. 90, 1585 (1986)

    Article  Google Scholar 

  207. W. E, W. Ren, E. Vanden-Eijnden, Chem. Phys. Lett. 413, 242 (2005)

    Article  ADS  Google Scholar 

  208. R.B. Best, G. Hummer, Proc. Natl. Acad. Sci. U.S.A. 102, 6732 (2005)

    Article  ADS  Google Scholar 

  209. G. Hummer, J. Chem. Phys. 120, 516 (2004)

    Article  ADS  Google Scholar 

  210. P.L. Geissler, C. Dellago, D. Chandler, J. Phys. Chem. B 103, 3706 (1999)

    Article  Google Scholar 

  211. W. Lechner, C. Dellago, P.G. Bolhuis, J. Chem. Phys. 135, 154110 (2011)

    Article  ADS  Google Scholar 

  212. A. Ma, A.R. Dinner, J. Phys. Chem. B 109, 6769 (2005)

    Article  Google Scholar 

  213. B. Peters, Mol. Sim. 36, 1265 (2010)

    Article  Google Scholar 

  214. B. Peters, P.G. Bolhuis, R.G. Mullen, J. Shea, J. Chem. Phys. 138, 054106 (2013)

    Article  ADS  Google Scholar 

  215. C. Leitold, C. Dellago, J. Chem. Phys. 141, 134901 (2014)

    Article  ADS  Google Scholar 

  216. C. Leitold, W. Lechner, C. Dellago, J. Phys.: Condens. Matter 27, 194126 (2015)

    ADS  Google Scholar 

  217. W. Lechner, J. Rogal, J. Juraszek, B. Ensing, P.G. Bolhuis, J. Chem. Phys. 133, 174110 (2010)

    Article  ADS  Google Scholar 

  218. G.T. Beckham, B. Peters, C. Starbuck, N. Variankaval, B.L. Trout, J. Am. Chem. Soc. 129, 4714 (2007)

    Article  Google Scholar 

  219. J.G. Kirkwood, Crystallization as a cooperative phenomenon, in Phase Tranformations in Solids, edited by R. Smoluchowski, J.E. Mayer, W.A. Weyl (Wiley, New York, 1951) p. 67

  220. L. van Hove, Rev. Mod. Phys. 29, 200 (1957)

    Article  ADS  MathSciNet  Google Scholar 

  221. G.E. Uhlenbeck, Round Table in Statistical Mechanics, in The many body problem, edited by J.K. Percus (Interscience Publishers/John Wiley, London, 1963) p. 498

  222. B.J. Alder, T.E. Wainwright, J. Chem. Phys. 27, 1208 (1957)

    Article  ADS  Google Scholar 

  223. W.W. Wood, J.D. Jacobson, J. Chem. Phys. 27, 1207 (1957)

    Article  ADS  Google Scholar 

  224. W.G. Hoover, F.H. Ree, J. Chem. Phys. 49, 3609 (1968)

    Article  ADS  Google Scholar 

  225. D. Frenkel, Physica A 263, 26 (1999)

    Article  ADS  Google Scholar 

  226. M.D. Eldridge, P.A. Madden, D. Frenkel, Nature 365, 35 (1993)

    Article  ADS  Google Scholar 

  227. W.G. Rudd, Z.W. Salsburg, A.P. Yu, F.H. Stillinger, J. Chem. Phys. 49, 4857 (1968)

    Article  ADS  Google Scholar 

  228. P.G. Bolhuis, D. Frenkel, S.-C. Mau, D.A. Huse, Nature 49, 388 (1997)

    Google Scholar 

  229. A.D. Bruce, N.B. Wilding, G.J. Ackland, Phys. Rev. Lett. 79, 3002 (1997)

    Article  ADS  Google Scholar 

  230. S.-C. Mau, D.A. Huse, Phys. Rev. E 59, 4396 (1999)

    Article  ADS  Google Scholar 

  231. T.M. Truskett, S. Torquato, S. Sastry, P.G. Debenedetti, F.H. Stillinger, Phys. Rev. E 58, 3083 (1998)

    Article  ADS  Google Scholar 

  232. B. O'Malley, I. Snook, J. Chem. Phys. 123, 054511 (2005)

    Article  ADS  Google Scholar 

  233. J.M. Brader, J. Chem. Phys. 128, 104503 (2008)

    Article  ADS  Google Scholar 

  234. S. Auer, D. Frenkel, Nature 413, 711 (2001)

    Article  ADS  Google Scholar 

  235. S. Auer, D. Frenkel, Numerical simulation of crystal nucleation in colloids, in Advanced Computer Simulation Approaches for Soft Matter Sciences I, edited by C. Holm, K. Kremer, Vol. 173 of Advances in Polymer Science (Springer-Verlag, Berlin, Heidelberg, 2005) p. 149

  236. M. Radu, T. Schilling, EPL 105, 26001 (2014)

    Article  ADS  Google Scholar 

  237. T. Kawasaki, H. Tanaka, Proc. Natl. Acad. Sci. U.S.A. 107, 14036 (2010)

    Article  ADS  Google Scholar 

  238. T. Palberg, J. Phys.: Condens. Matter 26, 333101 (2014)

    Google Scholar 

  239. L.V. Woodcock, J. Chem. Soc., Faraday Trans. 2 72, 1667 (1976)

    Article  Google Scholar 

  240. J.M. Gordon, J.H. Gibbs, P.D. Fleming, J. Chem. Phys. 65, 2771 (1976)

    Article  ADS  Google Scholar 

  241. L.V. Woodcock, J. Chem. Soc., Faraday Trans. 2 74, 11 (1978)

    Article  Google Scholar 

  242. P.N. Pusey, W. van Megen, Nature 320, 340 (1986)

    Article  ADS  Google Scholar 

  243. W. van Megen, S.M. Underwood, Phys. Rev. Lett. 70, 2766 (1993)

    Article  ADS  Google Scholar 

  244. W. van Megen, S.M. Underwood, Phys. Rev. E 49, 4206 (1994)

    Article  ADS  Google Scholar 

  245. J. Zhu, M. Li, R. Rogers, W. Meyer, R.H. Ottewill, STS-73 Space Shuttle Crew, W.B. Russel, P.M. Chaikin, Nature 387, 883 (1997)

    Article  ADS  Google Scholar 

  246. M.D. Rintoul, S. Torquato, J. Chem. Phys. 105, 9258 (1996)

    Article  ADS  Google Scholar 

  247. S.R. Williams, I.K. Snook, W. van Megen, Phys. Rev. E 64, 021506 (2001)

    Article  ADS  Google Scholar 

  248. H.J. Schöpe, G. Bryant, W. van Megen, Phys. Rev. E 74, 060401(R) (2006)

    Article  ADS  Google Scholar 

  249. E. Zaccarelli, C. Valeriani, E. Sanz, W.C.K. Poon, M.E. Cates, P.N. Pusey, Phys. Rev. Lett. 103, 135704 (2009)

    Article  ADS  Google Scholar 

  250. P.N. Pusey, E. Zaccarelli, C. Valeriani, E. Sanz, W.C.K. Poon, M.E. Cates, Philos. Trans. R. Soc. A 367, 4993 (2009)

    Article  ADS  Google Scholar 

  251. W.-S. Xu, Z.-Y. Sun, L.-J. An, Eur. Phys. J. E 31, 377 (2010)

    Article  Google Scholar 

  252. C. Valeriani, E. Sanz, E. Zaccarelli, W.C.K. Poon, M.E. Cates, P.N. Pusey, J. Phys.: Condens. Matter 23, 194117 (2011)

    ADS  Google Scholar 

  253. E. Sanz, C. Valeriani, E. Zaccarelli, W.C.K. Poon, P.N. Pusey, M.E. Cates, Phys. Rev. Lett. 106, 215701 (2011)

    Article  ADS  Google Scholar 

  254. M. Franke, S. Golde, H.J. Schöpe, Soft Matter 10, 5380 (2014)

    Article  ADS  Google Scholar 

  255. E. Sanz, C. Valeriani, E. Zaccarelli, W.C.K. Poon, M.E. Cates, P.N. Pusey, Proc. Natl. Acad. Sci. U.S.A. 111, 75 (2014)

    Article  ADS  Google Scholar 

  256. H.R. Pruppacher, J.D. Klett, Microphysics of Clouds and Precipitation (Reidel Publishing, Dordrecht, 1978)

  257. D.A. Hegg, M.B. Baker, Rep. Prog. Phys. 72, 056801 (2009)

    Article  ADS  Google Scholar 

  258. C. Gurganus, A.B. Kostinski, R.A. Shaw, J. Phys. Chem. Lett. 2, 1449 (2011)

    Article  Google Scholar 

  259. C. Vega, J.L.F. Abascal, E. Sanz, L.G. MacDowell, C. McBride, J. Phys.: Condens. Matter 17, S3283 (2005)

    ADS  Google Scholar 

  260. M. Matsumoto, S. Saito, I. Ohmine, Nature 416, 409 (2002)

    Article  ADS  Google Scholar 

  261. E.B. Moore, V. Molinero, Nature 479, 506 (2011)

    Article  ADS  Google Scholar 

  262. V. Molinero, E.B. Moore, J. Phys. Chem. B 113, 4008 (2009)

    Article  Google Scholar 

  263. T. Li, D. Donadio, G. Russo, G. Galli, Phys. Chem. Chem. Phys. 13, 19807 (2011)

    Article  Google Scholar 

  264. A. Reinhardt, J.P.K. Doye, J. Chem. Phys. 136, 054501 (2012)

    Article  ADS  Google Scholar 

  265. T. Li, D. Donadio, G. Galli, Nat. Commun. 4, 1887 (2013)

    Article  ADS  Google Scholar 

  266. R. Radhakrishnan, B.L. Trout, J. Am. Chem. Soc. 125, 7743 (2003)

    Article  Google Scholar 

  267. A.V. Brukhno, J. Anwar, R. Davidchack, R. Handel, J. Phys.: Condens. Matter 20, 494243 (2008)

    Google Scholar 

  268. X.-M. Bai, M. Li, J. Chem. Phys. 122, 224510 (2005)

    Article  ADS  Google Scholar 

  269. F. Romano, E. Sanz, F. Sciortino, J. Chem. Phys. 134, 174502 (2011)

    Article  ADS  Google Scholar 

  270. E. Sanz, C. Vega, J.R. Espinosa, R. Caballero-Bernal, J.L.F. Abascal, C. Valeriani, J. Am. Chem. Soc. 135, 15008 (2013)

    Article  Google Scholar 

  271. D. Quigley, P.M. Rodger, J. Chem. Phys. 128, 154518 (2008)

    Article  ADS  Google Scholar 

  272. A. Haji-Akbari, P.G. Debenedetti, Proc. Natl. Acad. Sci. U.S.A. 112, 10582 (2015)

    Article  ADS  Google Scholar 

  273. E.B. Moore, V. Molinero, J. Chem. Phys. 132, 244504 (2010)

    Article  ADS  Google Scholar 

  274. V. Thapar, F.A. Escobedo, Phys. Rev. Lett. 112, 048301 (2014)

    Article  ADS  Google Scholar 

  275. M. Shah, E.E. Santiso, B.L. Trout, J. Phys. Chem. B 115, 10400 (2011)

    Article  Google Scholar 

  276. P. Raiteri, J.D. Gale, J. Am. Chem. Soc. 132, 17623 (2010)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090, Wien, Austria

    S. Jungblut & C. Dellago

Authors
  1. S. Jungblut
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Dellago
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. Dellago.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jungblut, S., Dellago, C. Pathways to self-organization: Crystallization via nucleation and growth. Eur. Phys. J. E 39, 77 (2016). https://doi.org/10.1140/epje/i2016-16077-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epje/i2016-16077-6

Keywords

Search

Navigation