Supercooling and nucleation: Difference between revisions
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'''Supercooling''', undercooling and nucleation. | '''Supercooling''', undercooling and nucleation. | ||
==Volmer and Weber kinetic model== | ==Volmer and Weber kinetic model== | ||
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Zeldovich-Frenkel [[master equation]] is given by | Zeldovich-Frenkel [[master equation]] is given by | ||
:<math>\frac{\partial N(n, t)}{\partial t} = \frac{\partial }{\partial n} \left( k^+ (n) N^{eq} (n) \frac{\partial }{\partial n} \left( \frac{N(n, t)}{N^{eq}(n)} \right) \right)</math> | :<math>\frac{\partial N(n, t)}{\partial t} = \frac{\partial }{\partial n} \left( k^+ (n) N^{eq} (n) \frac{\partial }{\partial n} \left( \frac{N(n, t)}{N^{eq}(n)} \right) \right).</math> | ||
See also Shizgal and Barrett <ref>[http://dx.doi.org/10.1063/1.457366 B. Shizgal and J. C. Barrett "Time dependent nucleation", Journal of Chemical Physics '''91''' pp. 6505-6518 (1989)]</ref>. | |||
==See also== | ==See also== | ||
*[[Glass transition]] | *[[Glass transition]] | ||
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*[http://dx.doi.org/10.1063/1.2779036 Lawrence S. Bartell and David T. Wu "Do supercooled liquids freeze by spinodal decomposition?", Journal of Chemical Physics '''127''' 174507 (2007)] | *[http://dx.doi.org/10.1063/1.2779036 Lawrence S. Bartell and David T. Wu "Do supercooled liquids freeze by spinodal decomposition?", Journal of Chemical Physics '''127''' 174507 (2007)] | ||
*[http://dx.doi.org/10.1063/1.471721 Pieter Rein ten Wolde, Maria J. Ruiz-Montero and Daan Frenkel "Numerical calculation of the rate of crystal nucleation in a Lennard-Jones system at moderate undercooling", Journal of Chemical Physics '''104''' pp. 9932-9947 (1996)] | *[http://dx.doi.org/10.1063/1.471721 Pieter Rein ten Wolde, Maria J. Ruiz-Montero and Daan Frenkel "Numerical calculation of the rate of crystal nucleation in a Lennard-Jones system at moderate undercooling", Journal of Chemical Physics '''104''' pp. 9932-9947 (1996)] | ||
*[http://dx.doi.org/10.1063/1.2800001 Richard C. Flagan "A thermodynamically consistent kinetic framework for binary nucleation", Journal of Chemical Physics '''127''' 214503 (2007)] | *[http://dx.doi.org/10.1063/1.2800001 Richard C. Flagan "A thermodynamically consistent kinetic framework for binary nucleation", Journal of Chemical Physics '''127''' 214503 (2007)] | ||
*[http://dx.doi.org/10.1063/1.3506838 Laura Filion, Michiel Hermes, Ran Ni and Marjolein Dijkstra "Crystal nucleation of hard spheres using molecular dynamics, umbrella sampling, and forward flux sampling: A comparison of simulation techniques", Journal of Chemical Physics '''133''' 244115 (2010)] | *[http://dx.doi.org/10.1063/1.3506838 Laura Filion, Michiel Hermes, Ran Ni and Marjolein Dijkstra "Crystal nucleation of hard spheres using molecular dynamics, umbrella sampling, and forward flux sampling: A comparison of simulation techniques", Journal of Chemical Physics '''133''' 244115 (2010)] | ||
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;Books | ;Books | ||
*[http://dx.doi.org/10.1016/S0081-1947(08)60604-9 David T. Wu "Nucleation Theory", Solid State Physics '''50''' pp. 37-187 (1996)] | *[http://dx.doi.org/10.1016/S0081-1947(08)60604-9 David T. Wu "Nucleation Theory", Solid State Physics '''50''' pp. 37-187 (1996)] | ||
*[http://www.amolf.nl/publications/theses/valeriani/valeriani.html Chantal Valeriani "Numerical studies of nucleation pathways of ordered and disordered phases", PhD Thesis (2007)] | |||
* Dimo Kashchiev "Nucleation", Butterworth-Heinemann (2000) ISBN 978-0-7506-4682-6 | * Dimo Kashchiev "Nucleation", Butterworth-Heinemann (2000) ISBN 978-0-7506-4682-6 | ||
*[http://dx.doi.org/10.1016/j.physrep.2009.03.003 Andrea Cavagna "Supercooled liquids for pedestrians", Physics Reports '''476''' pp. 51-124 (2009)] | |||
*[http://www.sciencedirect.com/science/bookseries/14701804/15 Ken F. Kelton and Alan Lindsay Greer "Nucleation in Condensed Matter: Applications in Materials and Biology", Pergamon Materials Series Volume 15 (2010)] ISBN 978-0-08-042147-6 | *[http://www.sciencedirect.com/science/bookseries/14701804/15 Ken F. Kelton and Alan Lindsay Greer "Nucleation in Condensed Matter: Applications in Materials and Biology", Pergamon Materials Series Volume 15 (2010)] ISBN 978-0-08-042147-6 | ||
[[category: Phase transitions]] | [[category: Phase transitions]] | ||
Revision as of 17:00, 1 February 2012
Supercooling, undercooling and nucleation.
Volmer and Weber kinetic model
Volmer and Weber kinetic model [1] results in the following nucleation rate:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I^{VW} = N^{eq}(n^*) k^+(n^*) = k^+(n^*) N_A \exp \left( -\frac{W(n^*)}{k_BT} \right) \label{eq_IVW} }
Szilard nucleation model
Homogeneous nucleation temperature
The homogeneous nucleation temperature (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_H} ) is the temperature below which it is almost impossible to avoid spontaneous and rapid freezing.
Zeldovich factor
The Zeldovich factor [2] (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z} ) modifies the Volmer and Weber expression \eqref{eq_IVW}, making it applicable to spherical clusters:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z= \sqrt{\frac{ \vert \Delta \mu \vert }{6 \pi k_B T n^*}} }
Zeldovich-Frenkel equation
Zeldovich-Frenkel master equation is given by
See also Shizgal and Barrett [3].
See also
References
- ↑ M. Volmer and A. Weber "Keimbildung in übersättigten Gebilden", Zeitschrift für Physikalische Chemie 119 pp. 277-301 (1926)
- ↑ J. B. Zeldovich "On the theory of new phase formation, cavitation", Acta Physicochimica URSS 18 pp. 1-22 (1943)
- ↑ B. Shizgal and J. C. Barrett "Time dependent nucleation", Journal of Chemical Physics 91 pp. 6505-6518 (1989)
- Related reading
- J. Frenkel "Statistical Theory of Condensation Phenomena", Journal of Chemical Physics 7 pp. 200-201 (1939)
- Lawrence S. Bartell and David T. Wu "Do supercooled liquids freeze by spinodal decomposition?", Journal of Chemical Physics 127 174507 (2007)
- Pieter Rein ten Wolde, Maria J. Ruiz-Montero and Daan Frenkel "Numerical calculation of the rate of crystal nucleation in a Lennard-Jones system at moderate undercooling", Journal of Chemical Physics 104 pp. 9932-9947 (1996)
- Richard C. Flagan "A thermodynamically consistent kinetic framework for binary nucleation", Journal of Chemical Physics 127 214503 (2007)
- Laura Filion, Michiel Hermes, Ran Ni and Marjolein Dijkstra "Crystal nucleation of hard spheres using molecular dynamics, umbrella sampling, and forward flux sampling: A comparison of simulation techniques", Journal of Chemical Physics 133 244115 (2010)
- Ran Ni, Simone Belli, René van Roij, and Marjolein Dijkstra "Glassy Dynamics, Spinodal Fluctuations, and the Kinetic Limit of Nucleation in Suspensions of Colloidal Hard Rods", Physical Review Letters 105 088302 (2010)
- Andrea Cavagna "Supercooled liquids for pedestrians", Physics Reports 476 pp. 51-124 (2009)
- Books
- David T. Wu "Nucleation Theory", Solid State Physics 50 pp. 37-187 (1996)
- Chantal Valeriani "Numerical studies of nucleation pathways of ordered and disordered phases", PhD Thesis (2007)
- Dimo Kashchiev "Nucleation", Butterworth-Heinemann (2000) ISBN 978-0-7506-4682-6
- Andrea Cavagna "Supercooled liquids for pedestrians", Physics Reports 476 pp. 51-124 (2009)
- Ken F. Kelton and Alan Lindsay Greer "Nucleation in Condensed Matter: Applications in Materials and Biology", Pergamon Materials Series Volume 15 (2010) ISBN 978-0-08-042147-6