Supercooling and nucleation - Revision history

Carl McBride: Slight tidy

2018-07-17T10:09:23Z

Slight tidy

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	The homogeneous nucleation temperature (<math>T_H</math>) is the [[temperature]] below which it is almost impossible to avoid spontaneous and rapid freezing.		The homogeneous nucleation temperature (<math>T_H</math>) is the [[temperature]] below which it is almost impossible to avoid spontaneous and rapid freezing.
	==Zeldovich factor==		==Zeldovich factor==
	The Zeldovich factor <ref>J. B. Zeldovich "On the theory of new phase formation, cavitation", Acta Physicochimica URSS '''18''' pp. 1-22 (1943)</ref> (<math>Z</math>) modifies the Volmer and Weber expression ~~\eqref{eq_IVW}~~, making it applicable to spherical clusters:		The Zeldovich factor <ref>J. B. Zeldovich "On the theory of new phase formation, cavitation", Acta Physicochimica URSS '''18''' pp. 1-22 (1943)</ref> (<math>Z</math>) modifies the Volmer and Weber expression, making it applicable to spherical clusters:

	:<math>Z= \sqrt{\frac{ \vert \Delta \mu \vert }{6 \pi k_B T n^*}} </math>		:<math>Z= \sqrt{\frac{ \vert \Delta \mu \vert }{6 \pi k_B T n^*}} </math>

Carl McBride: /* References */ Added a recent publication

2018-07-17T10:06:30Z

References: Added a recent publication

← Older revision		Revision as of 12:06, 17 July 2018
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	*[http://dx.doi.org/10.1103/PhysRevLett.105.088302 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)]		*[http://dx.doi.org/10.1103/PhysRevLett.105.088302 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)]
	*[http://dx.doi.org/10.1063/1.4747326 M. D. Ediger and Peter Harrowell "Perspective: Supercooled liquids and glasses", Journal of Chemical Physics '''137''' 080901 (2012)]		*[http://dx.doi.org/10.1063/1.4747326 M. D. Ediger and Peter Harrowell "Perspective: Supercooled liquids and glasses", Journal of Chemical Physics '''137''' 080901 (2012)]
			*[https://doi.org/10.1063/1.5034091 Edgar D. Zanotto and Daniel R. Cassar "The race within supercooled liquids—Relaxation versus crystallization", Journal of Chemical Physics '''149''' 024503 (2018)]
	;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)]

Carl McBride: Fixed LaTeX math markup

2017-07-28T10:25:05Z

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← Older revision		Revision as of 12:25, 28 July 2017
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	Volmer and Weber kinetic model <ref>M. Volmer and A. Weber "Keimbildung in übersättigten Gebilden", Zeitschrift für Physikalische Chemie '''119''' pp. 277-301 (1926)</ref> results in the following nucleation rate:		Volmer and Weber kinetic model <ref>M. Volmer and A. Weber "Keimbildung in übersättigten Gebilden", Zeitschrift für Physikalische Chemie '''119''' pp. 277-301 (1926)</ref> results in the following nucleation rate:

	:<math>I^{VW} = N^{eq}(n^) k^+(n^) = k^+(n^) N_A \exp \left( -\frac{W(n^)}{k_BT} \right) ~~\label{eq_IVW}~~ </math>		:<math>I^{VW} = N^{eq}(n^) k^+(n^) = k^+(n^) N_A \exp \left( -\frac{W(n^)}{k_BT} \right) </math>
	==Szilard nucleation model==		==Szilard nucleation model==
	==Homogeneous nucleation temperature==		==Homogeneous nucleation temperature==

Carl McBride: /* References */ Added a recent publication

2012-09-19T13:06:16Z

References: Added a recent publication

← Older revision		Revision as of 15:06, 19 September 2012
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	*[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)]
	*[http://dx.doi.org/10.1103/PhysRevLett.105.088302 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)]		*[http://dx.doi.org/10.1103/PhysRevLett.105.088302 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)]
			*[http://dx.doi.org/10.1063/1.4747326 M. D. Ediger and Peter Harrowell "Perspective: Supercooled liquids and glasses", Journal of Chemical Physics '''137''' 080901 (2012)]
	;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)]

137.224.252.10: /* References */

2012-09-04T09:58:07Z

References

← Older revision		Revision as of 11:58, 4 September 2012
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	*[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://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
			*[http://arxiv.org/abs/1208.3377 R. Ni "Entropy-Driven Phase Transitions in Colloidal Systems", PhD Thesis, Utrecht University (2012)] ISBN 978-90-393-5798-9


	[[category: Phase transitions]]		[[category: Phase transitions]]

Carl McBride: /* References */ Removed duplicate reference

2012-02-01T16:19:03Z

References: Removed duplicate reference

← Older revision		Revision as of 18:19, 1 February 2012
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	*[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)]
	*[http://dx.doi.org/10.1103/PhysRevLett.105.088302 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)]		*[http://dx.doi.org/10.1103/PhysRevLett.105.088302 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)]
	*[http://dx.doi.org/10.1016/j.physrep.2009.03.003 Andrea Cavagna "Supercooled liquids for pedestrians", Physics Reports '''476''' pp. 51-124 (2009)]
	;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)]

Carl McBride: Added a subsection headding

2012-02-01T16:17:58Z

Added a subsection headding

← Older revision		Revision as of 18:17, 1 February 2012
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	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 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>.
			==Nucleation theorem==
	==See also==		==See also==
	*[[Glass transition]]		*[[Glass transition]]

Carl McBride at 16:00, 1 February 2012

2012-02-01T16:00:49Z

← Older revision		Revision as of 18:00, 1 February 2012
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	~~{{Stub-general}}~~
	'''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://www.amolf.nl/publications/theses/valeriani/valeriani.html Chantal Valeriani "Numerical studies of nucleation pathways of ordered and disordered phases", PhD Thesis (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/10.1063/1.2800001 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]]

Carl McBride: Added Zeldovich-Frenkel equation

2012-02-01T15:45:52Z

Added Zeldovich-Frenkel equation

← Older revision		Revision as of 17:45, 1 February 2012
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	'''Supercooling''', undercooling and nucleation.		'''Supercooling''', undercooling and nucleation.
	==Volmer and Weber kinetic model==		==Volmer and Weber kinetic model==
	Volmer and Weber kinetic model <ref>M. Volmer and A. Weber "Keimbildung in übersättigten Gebilden", Zeitschrift für Physikalische Chemie '''119''' pp. 277-301 (1926)</ref>:		Volmer and Weber kinetic model <ref>M. Volmer and A. Weber "Keimbildung in übersättigten Gebilden", Zeitschrift für Physikalische Chemie '''119''' pp. 277-301 (1926)</ref> results in the following nucleation rate:

	:<math>~~\label{eq_IVW}~~I^{VW} = N^{eq}(n^) k^+(n^) = k^+(n^) N_A \exp \left( -\frac{W(n^)}{k_BT} \right)</math>		:<math>I^{VW} = N^{eq}(n^) k^+(n^) = k^+(n^) N_A \exp \left( -\frac{W(n^)}{k_BT} \right) \label{eq_IVW} </math>
	==Szilard nucleation model==		==Szilard nucleation model==
	==Homogeneous nucleation temperature==		==Homogeneous nucleation temperature==
	The homogeneous nucleation temperature (<math>T_H</math>) is the [[temperature]] below which it is almost impossible to avoid spontaneous and rapid freezing.		The homogeneous nucleation temperature (<math>T_H</math>) is the [[temperature]] below which it is almost impossible to avoid spontaneous and rapid freezing.
	==Zeldovich factor==		==Zeldovich factor==
	The Zeldovich factor, <math>Z</math>, modifies the Volmer and Weber expression \eqref{eq_IVW}, making it applicable to spherical clusters:		The Zeldovich factor <ref>J. B. Zeldovich "On the theory of new phase formation, cavitation", Acta Physicochimica URSS '''18''' pp. 1-22 (1943)</ref> (<math>Z</math>) modifies the Volmer and Weber expression \eqref{eq_IVW}, making it applicable to spherical clusters:

	:<math>Z= \sqrt{\frac{ \vert \Delta \mu \vert }{6 \pi k_B T n^*}} </math>		:<math>Z= \sqrt{\frac{ \vert \Delta \mu \vert }{6 \pi k_B T n^*}} </math>
			==Zeldovich-Frenkel equation==
			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>
	==See also==		==See also==
	*[[Glass transition]]		*[[Glass transition]]
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	<references/>		<references/>
	;Related reading		;Related reading
			*[http://dx.doi.org/10.1063/1.1750413 J. Frenkel "Statistical Theory of Condensation Phenomena", Journal of Chemical Physics '''7''' pp. 200-201 (1939)]
	*[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)]

Carl McBride at 15:19, 1 February 2012

2012-02-01T15:19:03Z

← Older revision		Revision as of 17:19, 1 February 2012
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	'''Supercooling''', undercooling and nucleation.		'''Supercooling''', undercooling and nucleation.
	==Volmer and Weber kinetic model==		==Volmer and Weber kinetic model==
	<ref>M. Volmer and A. Weber "Keimbildung in übersättigten Gebilden", Zeitschrift für Physikalische Chemie '''119'''		Volmer and Weber kinetic model <ref>M. Volmer and A. Weber "Keimbildung in übersättigten Gebilden", Zeitschrift für Physikalische Chemie '''119''' pp. 277-301 (1926)</ref>:
	pp. 277-301 (1926)</ref>

	:<math>		:<math>\label{eq_IVW}I^{VW} = N^{eq}(n^) k^+(n^) = k^+(n^) N_A \exp \left( -\frac{W(n^)}{k_BT} \right)</math>
	\label{eq_IVW}
	I^{VW} = N^{eq}(n^) k^+(n^) = k^+(n^) N_A \exp \left( -\frac{W(n^)}{k_BT} \right)</math>
	==Szilard nucleation model==		==Szilard nucleation model==
	==Homogeneous nucleation temperature==		==Homogeneous nucleation temperature==
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	* 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://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]]