# American Institute of Mathematical Sciences

April  2013, 6(2): 317-330. doi: 10.3934/dcdss.2013.6.317

## A phase field model for liquid-vapour phase transitions

 1 University of Bologna, Department of Mathemathics, I-40126, Bologna, Italy, Italy, Italy

Received  July 2011 Revised  November 2011 Published  November 2012

We propose a model describing the liquid-vapour phase transition according to a phase-field method. A phase variable $φ$ is introduced whose equilibrium values $φ=0$ and $φ=1$ are associated with the liquid and vapour phases. The phase field obeys Ginzburg-Landau equation and enters the constitutive relation of the density, accounting for the sudden density jump occurring at the phase transition. In this paper we concern ourselves especially with the problems arising in the phase field approach due to the existence of the critical point in the coexistence line, which entails the merging of the phases described by $φ$.
Citation: Valeria Berti, Mauro Fabrizio, Diego Grandi. A phase field model for liquid-vapour phase transitions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 317-330. doi: 10.3934/dcdss.2013.6.317
##### References:
 [1] M. Fabrizio, Ice-water and liquid-vapour phase transitions by a Ginzburg-Landau model,, Journal of Mathematical Physics, 49 (2008). doi: 10.1515/bfup.2008.027. [2] L. P. Kadanoff, et al., Static phenomena near critical points: Theory and experiment,, Reviews of Modern Physics, 39 (1967), 395. [3] P. M. Chaikin and T. C. Lubensky, "Principles of Condensed Matter Physics,", Cambridge University Press, (1995). doi: 10.1016/1053-8127(95)00142-B. [4] A. Berti and C. Giorgi, A phase-field model for liquid-vapor transitions,, J. Non-Equilib. Thermodyn, 34 (2009), 219. [5] M. Fabrizio, Ginzburg-Landau equations and first and second order phase transitions,, Int. J. Eng. Sci., 44 (2006), 529. [6] F. Fried and M. E. Gurtin, Continuum theory of thermally induced phase transitions based on an order parameter,, Physica D, 68 (1993), 326. [7] M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Springer, (1996). [8] M. Fremond, "Non-smooth Thermomechanics,", Springer, (2001). [9] M. Fabrizio, C. Giorgi and A. Morro, A thermodynamic approach to ferromagnetism and phase transitions,, Int J. Eng Sci., 47 (2009), 821. doi: 10.1080/14735780802696351.

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##### References:
 [1] M. Fabrizio, Ice-water and liquid-vapour phase transitions by a Ginzburg-Landau model,, Journal of Mathematical Physics, 49 (2008). doi: 10.1515/bfup.2008.027. [2] L. P. Kadanoff, et al., Static phenomena near critical points: Theory and experiment,, Reviews of Modern Physics, 39 (1967), 395. [3] P. M. Chaikin and T. C. Lubensky, "Principles of Condensed Matter Physics,", Cambridge University Press, (1995). doi: 10.1016/1053-8127(95)00142-B. [4] A. Berti and C. Giorgi, A phase-field model for liquid-vapor transitions,, J. Non-Equilib. Thermodyn, 34 (2009), 219. [5] M. Fabrizio, Ginzburg-Landau equations and first and second order phase transitions,, Int. J. Eng. Sci., 44 (2006), 529. [6] F. Fried and M. E. Gurtin, Continuum theory of thermally induced phase transitions based on an order parameter,, Physica D, 68 (1993), 326. [7] M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Springer, (1996). [8] M. Fremond, "Non-smooth Thermomechanics,", Springer, (2001). [9] M. Fabrizio, C. Giorgi and A. Morro, A thermodynamic approach to ferromagnetism and phase transitions,, Int J. Eng Sci., 47 (2009), 821. doi: 10.1080/14735780802696351.
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