Supercooling and nucleation: Difference between revisions

Latest revision as of 11:09, 17 July 2018

Supercooling, undercooling and nucleation.

Volmer and Weber kinetic model[edit]

Volmer and Weber kinetic model ^[1] results in the following nucleation rate:

I^{VW}=N^{eq}(n^{*})k^{+}(n^{*})=k^{+}(n^{*})N_{A}\exp \left(-{\frac {W(n^{*})}{k_{B}T}}\right)

Szilard nucleation model[edit]

Homogeneous nucleation temperature[edit]

The homogeneous nucleation temperature ( $T_{H}$ ) is the temperature below which it is almost impossible to avoid spontaneous and rapid freezing.

Zeldovich factor[edit]

The Zeldovich factor ^[2] ( $Z$ ) modifies the Volmer and Weber expression, making it applicable to spherical clusters:

Z={\sqrt {\frac {\vert \Delta \mu \vert }{6\pi k_{B}Tn^{*}}}}

Zeldovich-Frenkel equation[edit]

Zeldovich-Frenkel master equation is given by

{\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).

Nucleation theorem[edit]

References[edit]

↑ 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

Books

[1] M. Volmer and A. Weber "Keimbildung in übersättigten Gebilden", Zeitschrift für Physikalische Chemie 119 pp. 277-301 (1926)

[2] J. B. Zeldovich "On the theory of new phase formation, cavitation", Acta Physicochimica URSS 18 pp. 1-22 (1943)

[3] B. Shizgal and J. C. Barrett "Time dependent nucleation", Journal of Chemical Physics 91 pp. 6505-6518 (1989)

[1]

[2]

[3]

@@ Line 1: / Line 1: @@
-{{Stub-general}}
+'''Supercooling''', undercooling and nucleation.
-'''Supercooling''' and undercooling.
+==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> 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) </math>
 ==Szilard nucleation model==
+==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.
 ==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, making it applicable to spherical clusters:
+:<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 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==
 *[[Glass transition]]
 *[[Spinodal decomposition]]
 ==References==
-#[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)]
+<references/>
-#[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)]
+;Related reading
-#[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/10.1063/1.1750413 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/10.1063/1.471721 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/10.1063/1.3506838 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/10.1063/1.4747326   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
+*[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
+*[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://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]]

Supercooling and nucleation: Difference between revisions

Latest revision as of 11:09, 17 July 2018

Contents

Volmer and Weber kinetic model[edit]

Szilard nucleation model[edit]

Homogeneous nucleation temperature[edit]

Zeldovich factor[edit]

Zeldovich-Frenkel equation[edit]

Nucleation theorem[edit]

See also[edit]

References[edit]

Navigation menu

Supercooling and nucleation: Difference between revisions

Latest revision as of 11:09, 17 July 2018

Volmer and Weber kinetic model[edit]

Szilard nucleation model[edit]

Homogeneous nucleation temperature[edit]

Zeldovich factor[edit]

Zeldovich-Frenkel equation[edit]

Nucleation theorem[edit]

See also[edit]

References[edit]

Navigation menu

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