Skip to main content
Log in

Modeling Aggregation Processes of Lennard-Jones particles Via Stochastic Networks

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We model an isothermal aggregation process of particles/atoms interacting according to the Lennard-Jones pair potential by mapping the energy landscapes of each cluster size N onto stochastic networks, computing transition probabilities from the network for an N-particle cluster to the one for \(N+1\), and connecting these networks into a single joint network. The attachment rate is a control parameter. The resulting network representing the aggregation of up to 14 particles contains 6427 vertices. It is not only time-irreversible but also reducible. To analyze its transient dynamics, we introduce the sequence of the expected initial and pre-attachment distributions and compute them for a wide range of attachment rates and three values of temperature. As a result, we find the configurations most likely to be observed in the process of aggregation for each cluster size. We examine the attachment process and conduct a structural analysis of the sets of local energy minima for every cluster size. We show that both processes taking place in the network, attachment and relaxation, lead to the dominance of icosahedral packing in small (up to 14 atom) clusters.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Notes

  1. The dataset for LJ\(_{13}\) found in [43] containing 28970 Morse-index one saddles significantly oversamples the set of transition states in comparison with our networks LJ\(_N\), \(N\ge 8\). Therefore, we computed the set of transition states for LJ\(_{13}\) using our technique, so that it is sampled consistently with our networks LJ\(_{N}\), \(6\le N\le 12\) and \(N=14\).

  2. S. Sousa Castellanos (East Carolina University) was M. Cameron’s MAPS-REU student in Summer 2016.

  3. Courtesy of David Wales.

References

  1. Arkus, N., Manoharan, V., Brenner, M.P.: Minimal energy clusters of hard spheres with short ranged attractions. Phys. Rev. Lett. 103, 118303 (2009)

    Article  ADS  Google Scholar 

  2. Baletto, F., Rapallo, A., Rossi, G., Ferrando, R.: Dynamical effects in the formation of magic cluster structures. Phys. Rev. B 69, 235421 (2004)

    Article  ADS  Google Scholar 

  3. Becker, O.M., Karplus, M.: The topology of multidimensional potential energy surfaces: theory and application to peptide structure and kinetics. J. Chem. Phys. 106(4), 1495 (1997)

    Article  ADS  Google Scholar 

  4. Calvo, F., Schebarchov, D., Wales, D.J.: Grand and semigrand canonical basin-hopping. J. Chem. Theory Comput. 12, 902–909 (2016)

    Article  Google Scholar 

  5. https://www.math.umd.edu/~mariakc/lennard-jones.html

  6. Cameron, M., Vanden-Eijnden, E.: Flows in complex networks: theory, algorithms, and application to Lennard-Jones cluster rearrangement. J. Stat. Phys. 156(3), 427–454 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Cameron, M.K.: Metastability, spectrum, and eigencurrents of the Lennard-Jones-38 network. J. Chem. Phys. 141, 184113 (2014)

    Article  Google Scholar 

  8. Cameron, M.K., Gan, T.: Spectral analysis and clustering of large stochastic networks. Application to the Lennard-Jones-75 cluster. Mol. Simul. 42(16), 1410–1428 (2016)

    Article  Google Scholar 

  9. Carr, J.M., Wales, D.J.: Folding pathways and rates for the three-stranded -sheet peptide Beta3s using discrete path sampling. J. Phys. Chem. B 112, 8760–8769 (2008)

    Article  Google Scholar 

  10. Doye, J.P.K., Wales, D.J., Miller, M.A.: Thermodynamics and the global optimization of Lennard-Jones clusters. J. Chem. Phys. 109, 8143 (1998)

    Article  ADS  Google Scholar 

  11. Doye, J.P.K., Wales, D.J., Miller, M.A.: The double-funnel energy landscape of the 38-atom Lennard-Jones cluster. J. Chem. Phys. 110, 6896 (1999)

    Article  ADS  Google Scholar 

  12. Weinan, E., Ren, W., Vanden-Eijnden, E.: String method for the study of rare events. Phys. Rev. B 66, 052301 (2002)

    ADS  Google Scholar 

  13. Weinan, E., Ren, W., Vanden-Eijnden, E.: Simplified and improved string method for computing the minimum energy paths in barrier-crossing events. J. Chem. Phys. 126, 164103 (2007)

    Article  ADS  Google Scholar 

  14. Weinan, E., Zhou, X.: The gentlest ascend dynamics. Nonlinearity 24, 1831–1842 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Echt, O., Sattler, K., Recknagel, E.: Magic numbers for sphere packings: experimental verification in free xenon clusters. Phys. Rev. Lett. 47(16), 1121–1124 (1981)

    Article  ADS  Google Scholar 

  16. Echt, O., Kandler, O., Leisner, T., Mlechle, W., Recknagel, E.: Magic numbers in mass spectra of large van der Waals clusters. J. Chem. Soc. Faraday Trans. 86, 2411 (1990)

    Article  Google Scholar 

  17. Farges, J., de Feraudy, M.F., Raoult, B., Torchet, G.: Structure and temperature of rare gas clusters in a supersonic expansion. Surf. Sci. 106, 95 (1981)

    Article  ADS  Google Scholar 

  18. Fejer, S.N., Wales, D.J.: Helix self-assembly from anisotropic molecules. Phys. Rev. Lett. 99, 086106 (2007)

    Article  ADS  Google Scholar 

  19. Gao, W., Leng, J., Zhou, X.: Iterative minimization algorithm for efficient calculations of transition states. J. Comp. Phys. 309, 6987 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Harris, I.A., Kidwell, R.S., Northby, J.A.: Structure of charged argon clusters formed in free jet expansion. Phys. Rev. Lett. 53, 2390 (1984)

    Article  ADS  Google Scholar 

  21. Harris, I.A., Norman, K.A., Mulkern, R.V., Northby, J.A.: Icosahedral structure of large charged argon clusters. Chem. Phys. Lett. 130, 316 (1986)

    Article  ADS  Google Scholar 

  22. Henkelman, G., Jonsson, H.: A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives. J. Chem. Phys. 111, 7010 (1999)

    Article  ADS  Google Scholar 

  23. Holmes-Cerfon, M., Gortler, S.J., Brenner, M.P.: A geometrical approach to computing free-energy landscapes from short-ranged potentials. Proc. Natl. Acad. Sci. USA 110(1), E5–E14 (2013)

    Article  ADS  Google Scholar 

  24. Holmes-Cerfon, M.: Sticky-sphere clusters. Annu. Rev. Condens. Matter Phys. In press (expected 8, March 10, 2017)

  25. Jonsson, H., Mills, G., Jacobsen, K.W.: Nudged elastic band method for finding minimum energy paths of transitions. In: Berne, B.J., Ciccotti, G., Coker, D.F. (eds.) Classical and Quantum Dynamics in Condensed Phase Simulations, vol. 385. World Scientific, Singapore (1998)

    Google Scholar 

  26. Kakar, S., Bjoerneholm, O., Weigelt, J., de Castro, A.R.B., Troeger, L., Frahm, R., Moeller, T.: Size-dependent K-edge EXAFS study of the structure of free Ar clusters. Phys. Rev. Lett. 78(9), 1675–1678 (1997)

    Article  ADS  Google Scholar 

  27. Kovalenko, S.I., Solnyshkin, D.D., Verkhovtseva, E.T., Eremenko, V.V.: Experimental detection of stacking faults in rare gas clusters Chem. Phys. Lett. 250, 309 (1996)

    Google Scholar 

  28. Langer, J.S.: Statistical theory of the decay of metastable states. Ann. Phys. 54, 258–275 (1969)

    Article  ADS  Google Scholar 

  29. Mandelshtam, V.A., Frantsuzov, P.A.: Multiple structural transformations in Lennard-Jones clusters: generic versus size-specific behavior. J. Chem. Phys. 124, 204511 (2006)

    Article  ADS  Google Scholar 

  30. Meng, G., Arkus, N., Brenner, M.P., Manoharan, V.: The free energy landscape of hard sphere clusters. Science 327, 560 (2010)

    Article  ADS  Google Scholar 

  31. Miller, M.A., Wales, D.J.: Isomerization dynamics and ergodicity in Ar 7. J. Chem. Phys. 107, 8568 (1997)

    Article  ADS  Google Scholar 

  32. Munro, L.J., Wales, D.J.: Defect migration in crystalline silicon. Phys. Rev. B 59, 3969–3980 (1999)

    Article  ADS  Google Scholar 

  33. Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, New York (2006)

    MATH  Google Scholar 

  34. Picciani, M., Athenes, M., Kurchan, J., Taileur, J.: Simulating structural transitions by direct transition current sampling: the example of LJ38. J. Chem. Phys. 135, 034108 (2011)

    Article  ADS  Google Scholar 

  35. Stillinger, F.H.: Exponential multiplicity of inherent structures. Phys. Rev. E 59(1), 48–51 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  36. Wallace, D.C.: Statistical mechanics of monatomic liquids. Phys. Rev. E 56(4), 4179–4186 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  37. Wales, D.J., Doye, J.P.K.: Global optimization by basin-hopping and the lowest energy structures of Lennard-Jones clusters containing up to 110 atoms. J. Phys. Chem. A 101, 5111–5116 (1997)

    Article  Google Scholar 

  38. Wales, D.J.: Discrete path sampling. Mol. Phys. 100, 3285–3306 (2002)

    Article  ADS  Google Scholar 

  39. van de Waal, B.W.: No evidence for size-dependent icosahedral \(\rightarrow \) fcc structural transition in rare-gas clusters. Phys. Rev. Lett. 76(7), 1083–1086 (1996)

    Article  ADS  Google Scholar 

  40. Wales, D.J.: Energy Landscapes: Applications to Clusters, Biomolecules and Glasses. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  41. Wales, D.J.: Energy landscapes: calculating pathways and rates. Int. Rev. Chem. Phys. 25(1–2), 237–282 (2006)

    Article  Google Scholar 

  42. Wales, D.J.: Energy landscapes of clusters bound by short-ranged potentials. ChemPhysChem 11(12), 2491–2494 (2010)

    Article  Google Scholar 

  43. http://www-wales.ch.cam.ac.uk/CCD.html

  44. Zhang, J., Du, Q.: Shrinking dimer dynamics and its applications to saddle point search. SIAM J. Numer. Anal. 50(4), 1899–1921 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was partially supported by the NSF Grant DMS1554907 and the NSF REU Grant DMS1359307 at the University of Maryland, College Park.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Maria Cameron.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Forman, Y., Cameron, M. Modeling Aggregation Processes of Lennard-Jones particles Via Stochastic Networks. J Stat Phys 168, 408–433 (2017). https://doi.org/10.1007/s10955-017-1794-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-017-1794-y

Keywords

Navigation