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Supercooling and nucleation

2012-09-04T09:58:07Z

137.224.252.10: /* References */

'''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''' 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>
==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 \eqref{eq_IVW}, 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==
<references/>
;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/10.1063/1.471721 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)]
;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]]

Phase transitions

2012-09-04T09:54:20Z

137.224.252.10: /* References */

A '''phase transition''' occurs at points on the [[Phase diagrams | phase diagram]] where the [[Helmholtz energy function]] is a singular function of the thermodynamic variables within it
<ref>[http://dx.doi.org/10.1007/s10955-009-9814-1 Leo P. Kadanoff "More is the Same; Phase Transitions and Mean Field Theories", Journal of Statistical Physics '''137''' pp. 777-797 (2009)]</ref>.
The following is (an incomplete) list of phase transitions. The change from one state to another is usually monitored by some sort of [[order parameters | order parameter]].
*[[1-dimensional phase transitions]]
*[[First-order transitions]]
*[[Gas-liquid phase transitions]]
*[[Glass transition]]
*[[Isotropic-nematic phase transition]]
*[[Kosterlitz-Thouless transition]]
*[[Landau theory of second-order phase transitions]]
*[[Liquid-liquid phase transitions]]
*[[Martensitic phase transitions]]
*[[Nematic-smectic phase transition]]
*[[Nonequilibrium phase transitions]]
*[[Order-disorder transitions]]
*[[Quantum phase transitions]]
*[[Second-order transitions]]
*[[Solid-liquid phase transitions]]
*[[Topological phase transitions]]
*[[Gas-liquid phase transitions | Vapour-liquid phase transitions]]
'''Other transitions'''
*[[Coil-globule transition]]
*[[Helix-coil transition]]
==See also==
*[[Computation of phase equilibria]]
==References==
<references/>
'''Recommended reading'''
*Paul Ehrenfest "Phase conversions in a general and enhanced sense, classified according to the specific singularities of the thermodynamic potential", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen '''36''' pp. 153-157 (1933)
*[http://www.oup.com/uk/catalogue/?ci=9780195053166 H. Eugene Stanley "Introduction to Phase Transitions and Critical Phenomena", Oxford University Press (1988)]
*[http://www.oup.com/uk/catalogue/?ci=9780198517306 J. M. Yeomans "Statistical Mechanics of Phase Transitions", Oxford University Press (1992)]
*[http://dx.doi.org/10.1021/jp9536460 Benjamin Widom "Theory of Phase Equilibrium", Journal of Physical Chemistry '''100''' pp 13190-13199 (1996)]
*[http://dx.doi.org/10.1070/PU1999v042n06ABEH000543 Georgii A. Martynov "The problem of phase transitions in statistical mechanics", Physics-Uspekhi '''42''' pp. 517-543 (1999)]
*[http://www.nature.com/nature/journal/v402/n6761supp/full/402c73a0.html Philip Ball "Transitions still to be made", Nature '''402''' C73-C76 (1999)]
* [[Hagen Kleinert|H. Kleinert]] and V. Schulte-Frohlinde, ''Critical Properties of φ<sup>4</sup>-Theories'', [http://www.worldscibooks.com/physics/4733.html World Scientific (Singapore, 2001)]; Paperback ISBN 981-02-4659-5 (Read online at [http://www.physik.fu-berlin.de/~kleinert/b8])''
[[category: phase transitions]]
*[http://arxiv.org/abs/1208.3377 R. Ni "Entropy-Driven Phase Transitions in Colloidal Systems", PhD Thesis, Utrecht University (2012)]

Phase transitions

2012-09-04T09:53:05Z

137.224.252.10: /* References */

A '''phase transition''' occurs at points on the [[Phase diagrams | phase diagram]] where the [[Helmholtz energy function]] is a singular function of the thermodynamic variables within it
<ref>[http://dx.doi.org/10.1007/s10955-009-9814-1 Leo P. Kadanoff "More is the Same; Phase Transitions and Mean Field Theories", Journal of Statistical Physics '''137''' pp. 777-797 (2009)]</ref>.
The following is (an incomplete) list of phase transitions. The change from one state to another is usually monitored by some sort of [[order parameters | order parameter]].
*[[1-dimensional phase transitions]]
*[[First-order transitions]]
*[[Gas-liquid phase transitions]]
*[[Glass transition]]
*[[Isotropic-nematic phase transition]]
*[[Kosterlitz-Thouless transition]]
*[[Landau theory of second-order phase transitions]]
*[[Liquid-liquid phase transitions]]
*[[Martensitic phase transitions]]
*[[Nematic-smectic phase transition]]
*[[Nonequilibrium phase transitions]]
*[[Order-disorder transitions]]
*[[Quantum phase transitions]]
*[[Second-order transitions]]
*[[Solid-liquid phase transitions]]
*[[Topological phase transitions]]
*[[Gas-liquid phase transitions | Vapour-liquid phase transitions]]
'''Other transitions'''
*[[Coil-globule transition]]
*[[Helix-coil transition]]
==See also==
*[[Computation of phase equilibria]]
==References==
<references/>
'''Recommended reading'''
*Paul Ehrenfest "Phase conversions in a general and enhanced sense, classified according to the specific singularities of the thermodynamic potential", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen '''36''' pp. 153-157 (1933)
*[http://www.oup.com/uk/catalogue/?ci=9780195053166 H. Eugene Stanley "Introduction to Phase Transitions and Critical Phenomena", Oxford University Press (1988)]
*[http://www.oup.com/uk/catalogue/?ci=9780198517306 J. M. Yeomans "Statistical Mechanics of Phase Transitions", Oxford University Press (1992)]
*[http://dx.doi.org/10.1021/jp9536460 Benjamin Widom "Theory of Phase Equilibrium", Journal of Physical Chemistry '''100''' pp 13190-13199 (1996)]
*[http://dx.doi.org/10.1070/PU1999v042n06ABEH000543 Georgii A. Martynov "The problem of phase transitions in statistical mechanics", Physics-Uspekhi '''42''' pp. 517-543 (1999)]
*[http://www.nature.com/nature/journal/v402/n6761supp/full/402c73a0.html Philip Ball "Transitions still to be made", Nature '''402''' C73-C76 (1999)]
* [[Hagen Kleinert|H. Kleinert]] and V. Schulte-Frohlinde, ''Critical Properties of φ<sup>4</sup>-Theories'', [http://www.worldscibooks.com/physics/4733.html World Scientific (Singapore, 2001)]; Paperback ISBN 981-02-4659-5 (Read online at [http://www.physik.fu-berlin.de/~kleinert/b8])''
[[category: phase transitions]]
*[http://arxiv.org/abs/1208.3377 Ran Ni "Entropy-Driven Phase Transitions in Colloidal Systems", PhD Thesis, Utrecht University (2012)]