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A hybrid kinetic-thermodynamic Monte Carlo model for simulation of homogeneous burst nucleation

  • Karl K. Sabelfeld ORCID logo EMAIL logo and Georgy Eremeev

Abstract

We develop in this paper a hybrid kinetic Monte Carlo and continuous thermodynamically based model for the simulation of homogeneous nucleation under burst regime when a long incubation time is followed by rapid nucleation of stable nuclei. In this model we assume that the kinetics of particle nucleation and disaggregation is governed by a Smoluchowski equation while the size of a stable nuclei is taken from the thermodynamic theory of nucleation with varying supersaturation under metastable conditions. We show that the Smoluchowski equations without the metastable conditions cannot describe the regime of burst nucleation showing the following general feature: the longer the incubation time, the slower the nucleation rate even if a multiple disaggregation is assumed. In contrast, a combined hybrid Monte Carlo and metastable thermodynamic model suggested is able to predict a long incubation time followed by rapid nucleation regime. A series of numerical simulations presented supports this conclusion.

MSC 2010: 65C05; 65C20; 65Z05

Award Identifier / Grant number: 14-11-00083

Funding statement: Support of the Russian Science Foundation under Grant 14-11-00083 is kindly acknowledged.

References

[1] D. J. Aldous, Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists, Bernoulli 5 (1999), no. 1, 3–48. 10.2307/3318611Search in Google Scholar

[2] H. Babovsky, On a Monte Carlo scheme for Smoluchowski’s coagulation equation, Monte Carlo Methods Appl. 5 (1999), no. 1, 1–18. 10.1515/mcma.1999.5.1.1Search in Google Scholar

[3] J. M. Ball and J. Carr, The discrete coagulation-fragmentation equations: Existence, uniqueness, and density conservation, J. Stat. Phys. 61 (1990), no. 1–2, 203–234. 10.1007/BF01013961Search in Google Scholar

[4] A. Baronov, K. Bufkin and D. W. Shaw, A simple model of burst nucleation, Phys. Chem. Chem. Phys. 17 (2015), Article ID 20846. 10.1039/C5CP01745ASearch in Google Scholar

[5] R. Becker and W. Döring, Kinetische Behandlung der Keimbildung in übersättigten Dämpfen, Ann. Phys. 415 (1935), no. 8, 719–752. 10.1002/andp.19354160806Search in Google Scholar

[6] D. B. K. Chu, J. S. Owen and B. Peters, Nucleation and growth kinetics from LaMer burst data, J. Phys. Chem. A 121 (2017), no. 40, 7511–7517. 10.1021/acs.jpca.7b08368Search in Google Scholar PubMed

[7] S. Fernandez-Garrido, V. M. Kaganer, K. K. Sabelfeld, T. Gotschke, J. Grandal, E. Calleja, L. Geelhaar and O. Brandt, Self-regulated radius of spontaneously formed GaN nanowires in molecular beam epitaxy, Nano Lett. 13 (2013), no. 7, 3274–3280. 10.1021/nl401483eSearch in Google Scholar PubMed

[8] S. K. Friedlander, Smoke, Dust and Haze: Fundamentals of Aerosol Behaviour, John Wiley & Sons, New York, 1977. 10.1063/1.3037714Search in Google Scholar

[9] F. Guiaş, A Monte Carlo approach to the Smoluchowski equations, Monte Carlo Methods Appl. 3 (1997), no. 4, 313–326. 10.1515/mcma.1997.3.4.313Search in Google Scholar

[10] V. M. Kaganer, W. Braun and K. K. Sabelfeld, Ostwald ripening of faceted two-dimensional islands, Phys. Rev. B 76 (2007), 10.1103/PhysRevB.76.075415. 10.1103/PhysRevB.76.075415Search in Google Scholar

[11] A. A. Kolodko and K. K. Sabelfeld, Stochastic Lagrangian model for spatially inhomogeneous Smoluchowski equation governing coagulating and diffusing particles, Monte Carlo Methods Appl. 7 (2001), no. 3–4, 223–228. 10.1515/mcma.2001.7.3-4.223Search in Google Scholar

[12] A. A. Kolodko and K. K. Sabelfeld, Stochastic particle methods for Smoluchowski coagulation equation: Variance reduction and error estimations, Monte Carlo Methods Appl. 9 (2003), no. 4, 315–339. 10.1515/156939603322601950Search in Google Scholar

[13] A. A. Kolodko, K. K. Sabelfeld and W. Wagner, A stochastic method for solving Smoluchowski’s coagulation equation, Math. Comput. Simulation 49 (1999), no. 1–2, 57–79. 10.1016/S0378-4754(99)00008-7Search in Google Scholar

[14] A. A. Kolodko and W. Wagner, Convergence of a Nanbu type method for the Smoluchowski equation, Monte Carlo Methods Appl. 3 (1997), no. 4, 255–273. 10.1515/mcma.1997.3.4.255Search in Google Scholar

[15] V. K. LaMer, Nucleation in phase transitions, Ind. Eng. Chem. 44 (1952), no. 6, 1270–1277. 10.1021/ie50510a027Search in Google Scholar

[16] K. Liffman, A direct simulation Monte Carlo method for cluster coagulation, J. Comput. Phys. 100 (1992), 116–127. 10.1016/0021-9991(92)90314-OSearch in Google Scholar

[17] A. A. Lushnikov, Some new aspects of coagulation theory (in Russian), Izv. Acad. Nauk SSSR 14 (1978), no. 10, 738–743. Search in Google Scholar

[18] A. H. Marcus, Stochastic coalescence, Technometrics 10 (1968), 133–143. 10.1080/00401706.1968.10490541Search in Google Scholar

[19] S. A. Matveev, P. L. Krapivsky, A. P. Smirnov, E. E. Tyrtyshnikov and N. V. Brilliantov, Oscillations in aggregation-shattering processes, Phys. Rev. Lett. 119 (2017), no. 26, Article ID 260601. 10.1103/PhysRevLett.119.260601Search in Google Scholar PubMed

[20] D. T. Robb and V. Privman, Model of nanocrystal formation in solution by burst nucleation and diffusional growth, Langmuir 24 (2008), no. 1, 26–35. 10.1021/la702097gSearch in Google Scholar PubMed

[21] K. K. Sabelfeld and A. A. Kolodko, Monte Carlo simulation of the coagulation processes governed by Smoluchowski equation with random coefficients, Monte Carlo Methods Appl. 3 (1997), no. 4, 275–311. 10.1515/mcma.1997.3.4.275Search in Google Scholar

[22] K. K. Sabelfeld and O. Kurbanmuradov, Stochastic Lagrangian models for two-particle motion in turbulent flows, Monte Carlo Methods Appl. 3 (1997), no. 1, 53–72. 10.1515/mcma.1997.3.1.53Search in Google Scholar

[23] K. K. Sabelfeld, A. Levykin and T. Privalova, A fast stratified sampling simulation of coagulation processes, Monte Carlo Methods Appl. 13 (2007), no. 1, 71–88. 10.1515/MCMA.2007.004Search in Google Scholar

[24] K. K. Sabelfeld, S. V. Rogasinsky, A. A. Kolodko and A. I. Levykin, Stochastic algorithms for solving Smolouchovsky coagulation equation and applications to aerosol growth simulation, Monte Carlo Methods Appl. 2 (1996), no. 1, 41–87. 10.1515/mcma.1996.2.1.41Search in Google Scholar

[25] T. Vetter, M. Iggland, D. R. Ochsenbein, F. S. Hanseler and M. Mazzotti, Modeling nucleation, growth, and ostwald ripening in crystallization processes: A comparison between population balance and kinetic rate equation, Cryst. Growth Design 13 (2013), no. 11, 4890–4905. 10.1021/cg4010714Search in Google Scholar

[26] M. M. R. Williams and S. K. Loyalka, Aerosol Science. Theory and Practice, Pergamon, New York, 1991. Search in Google Scholar

Received: 2018-02-09
Accepted: 2018-06-26
Published Online: 2018-07-07
Published in Print: 2018-09-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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