# American Institute of Mathematical Sciences

September  2014, 19(7): 1889-1909. doi: 10.3934/dcdsb.2014.19.1889

## Mathematical modeling of phase transition and separation in fluids: A unified approach

 1 Facoltà di Ingegneria, Università e-Campus, 22060 Novedrate (CO) 2 DICATAM, Università di Brescia, Via Valotti, 9 - 25133 Brescia 3 DIBRIS, Università di Genova, Via Opera Pia 13, 16145 Genova

Received  April 2013 Revised  January 2014 Published  August 2014

A unified phase-field continuum theory is developed for transition and separation phenomena. A nonlocal formulation of the second law which involves an extra-entropy flux gives the basis of the thermodynamic approach. The phase-field is regarded as an additional variable related to some phase concentration, and its evolution is ruled by a balance equation, where flux and source terms are (unknown) constitutive functions. This evolution equation reduces to an equation of the rate-type when the flux is negligible, and it takes the form of a diffusion equation when the source term is disregarded. On this background, a general model for first-order transition and separation processes in a compressible fluid or fluid mixture is developed. Upon some simplifications, we apply it to the liquid-vapor phase change induced either by temperature or by pressure and we derive the expression of the vapor pressure curve. Taking into account the flux term, the sign of the diffusivity is discusssed.
Citation: Alessia Berti, Claudio Giorgi, Angelo Morro. Mathematical modeling of phase transition and separation in fluids: A unified approach. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 1889-1909. doi: 10.3934/dcdsb.2014.19.1889
##### References:
 [1] H. W. Alt and I. Pawlow, On the entropy principle of phase transition models with a conserved parameter, Adv. Math. Sci. Appl., 6 (1996), 291-376. [2] A. Berti and C. Giorgi, A phase-field model for liquid-vapor transitions, J. Non-Equilibrium Thermodyn, 34 (2009), 219-247. doi: 10.1515/JNETDY.2009.012. [3] A. Berti and C. Giorgi, Phase-field modeling of transition and separation phenomena in continuum thermodynamics, AAPP Phys. Math. Nat. Sci., 91 (2013), 21 pp. [4] A. Berti and C. Giorgi, A phase-field model for quasi-incompressible solid-liquid transitions,, to appear in Meccanica., ().  doi: 10.1007/s11012-014-9909-x. [5] A. Berti, C. Giorgi and E. Vuk, Free energies and pseudo-elastic transitions for shape memory alloys, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 293-316. [6] V. Berti, M. Fabrizio and C. Giorgi, Well-posedness for solid-liquid phase transitions with a fourth-order nonlinearity, Physica D, 236 (2007), 13-21. doi: 10.1016/j.physd.2007.07.009. [7] E. Bonetti and M. Frémond, A phase transition model with the entropy balance, Math. Methods Appl. Sci., 26 (2003), 539-556. doi: 10.1002/mma.366. [8] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer New York, 1996. doi: 10.1007/978-1-4612-4048-8. [9] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial energy, J. Chem. Phys., 28 (1958), 258-267. doi: 10.1063/1.1744102. [10] B. D. Coleman and W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Rational Mech. Anal., 13 (1963), 167-178. doi: 10.1007/BF01262690. [11] M. Fabrizio, C. Giorgi and A. Morro, A Thermodynamic approach to non-isotermal phase-field evolution in continuum physics, Phys. D, 214 (2006), 144-156. doi: 10.1016/j.physd.2006.01.002. [12] M. Fabrizio, C. Giorgi and A. Morro, A thermodynamic approach to ferromagnetism and phase transitions, Internat. J. Engrg. Sci., 47 (2009), 821-839. doi: 10.1016/j.ijengsci.2009.05.010. [13] M. Fabrizio, C. Giorgi and A. Morro, Isotropic-nematic phase transitions in liquid crystals, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 565-579. doi: 10.3934/dcdss.2011.4.565. [14] M. Fabrizio, C. Giorgi and A Morro, Phase separation in quasi-incompressible Cahn-Hilliard fluids, Eur. J. Mech. B Fluids, 30 (2011), 281-287. doi: 10.1016/j.euromechflu.2010.12.003. [15] M. Frémond, Non-Smooth Thermomechanics, Springer New york, 2002. doi: 10.1007/978-3-662-04800-9. [16] M. Frémond, Phase Changes in Mechanics, Springer New york, 2012. [17] C. Giorgi, Continuum thermodynamics and phase-field models, Milan J. Math., 77 (2009), 67-100. doi: 10.1007/s00032-009-0101-z. [18] A. E. Green and N. Laws, On a global entropy production inequality, Quart. J. Mech. Appl. Math., 25 (1972), 1-11. doi: 10.1093/qjmam/25.1.1. [19] K. Hutter and Y. Wang, Phenomenological thermodynamics and entropy principles, in Entropy, (eds. A. Greven, G. Keller and G. Warnecke), Princeton University Press Princeton, N.J., 2003. [20] R. A. L. Jones, Soft condensed matter, Eur. J. Phys., 23 (2002), 652. doi: 10.1088/0143-0807/23/6/703. [21] A. Karma and W. J. Rappel, Quantitative phase-field modelling of dendritic growth in two and three dimensions, Phys. Rev. E, 57 (1998), 4323-4349. doi: 10.1103/PhysRevE.57.4323. [22] A. G. Lamorgese, D. Molin and R. Mauri, Phase field approach to multiphase flow modeling, Milan J. Math., 79 (2011), 597-642. doi: 10.1007/s00032-011-0171-6. [23] G. A. Maugin, The Thermomechanics of Nonlinear Irreversible Behaviours. An Introduction, World Scientific Singapore, 1999. doi: 10.1142/3700. [24] G. A. Maugin and W. Muschik, Thermodynamics with internal variables. Part I. General concepts, J. Non-Equilibrium Thermodyn, 19 (1994), 217-249. doi: 10.1515/jnet.1994.19.3.217. [25] A. Morro, A phase-field approach to non-isothermal transitions, Math. Comput. Modelling, 48 (2008), 621-633. doi: 10.1016/j.mcm.2007.11.001. [26] I. Müller, Thermodynamics, Pitman Boston, 1985. [27] O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Phys. D, 43 (1990), 44-62. doi: 10.1016/0167-2789(90)90015-H. [28] O. Penrose and P. C. Fife, On the relation between the standard phase-field model and a "thermodynamically consistent" phase-field model, Phys. D, 69 (1993), 107-113. doi: 10.1016/0167-2789(93)90183-2. [29] I. Singer-Loginova and H. M. Singer, The phase-field technique for modeling multiphase materials, Rep. Prog. Phys., 71 (2008), 106501-106533. doi: 10.1088/0034-4885/71/10/106501. [30] P. Ván, Weakly nonlocal irreversible thermodynamics, Ann. Phys. (8), 12 (2003), 146-173. doi: 10.1002/andp.200310002.

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##### References:
 [1] H. W. Alt and I. Pawlow, On the entropy principle of phase transition models with a conserved parameter, Adv. Math. Sci. Appl., 6 (1996), 291-376. [2] A. Berti and C. Giorgi, A phase-field model for liquid-vapor transitions, J. Non-Equilibrium Thermodyn, 34 (2009), 219-247. doi: 10.1515/JNETDY.2009.012. [3] A. Berti and C. Giorgi, Phase-field modeling of transition and separation phenomena in continuum thermodynamics, AAPP Phys. Math. Nat. Sci., 91 (2013), 21 pp. [4] A. Berti and C. Giorgi, A phase-field model for quasi-incompressible solid-liquid transitions,, to appear in Meccanica., ().  doi: 10.1007/s11012-014-9909-x. [5] A. Berti, C. Giorgi and E. Vuk, Free energies and pseudo-elastic transitions for shape memory alloys, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 293-316. [6] V. Berti, M. Fabrizio and C. Giorgi, Well-posedness for solid-liquid phase transitions with a fourth-order nonlinearity, Physica D, 236 (2007), 13-21. doi: 10.1016/j.physd.2007.07.009. [7] E. Bonetti and M. Frémond, A phase transition model with the entropy balance, Math. Methods Appl. Sci., 26 (2003), 539-556. doi: 10.1002/mma.366. [8] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer New York, 1996. doi: 10.1007/978-1-4612-4048-8. [9] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial energy, J. Chem. Phys., 28 (1958), 258-267. doi: 10.1063/1.1744102. [10] B. D. Coleman and W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Rational Mech. Anal., 13 (1963), 167-178. doi: 10.1007/BF01262690. [11] M. Fabrizio, C. Giorgi and A. Morro, A Thermodynamic approach to non-isotermal phase-field evolution in continuum physics, Phys. D, 214 (2006), 144-156. doi: 10.1016/j.physd.2006.01.002. [12] M. Fabrizio, C. Giorgi and A. Morro, A thermodynamic approach to ferromagnetism and phase transitions, Internat. J. Engrg. Sci., 47 (2009), 821-839. doi: 10.1016/j.ijengsci.2009.05.010. [13] M. Fabrizio, C. Giorgi and A. Morro, Isotropic-nematic phase transitions in liquid crystals, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 565-579. doi: 10.3934/dcdss.2011.4.565. [14] M. Fabrizio, C. Giorgi and A Morro, Phase separation in quasi-incompressible Cahn-Hilliard fluids, Eur. J. Mech. B Fluids, 30 (2011), 281-287. doi: 10.1016/j.euromechflu.2010.12.003. [15] M. Frémond, Non-Smooth Thermomechanics, Springer New york, 2002. doi: 10.1007/978-3-662-04800-9. [16] M. Frémond, Phase Changes in Mechanics, Springer New york, 2012. [17] C. Giorgi, Continuum thermodynamics and phase-field models, Milan J. Math., 77 (2009), 67-100. doi: 10.1007/s00032-009-0101-z. [18] A. E. Green and N. Laws, On a global entropy production inequality, Quart. J. Mech. Appl. Math., 25 (1972), 1-11. doi: 10.1093/qjmam/25.1.1. [19] K. Hutter and Y. Wang, Phenomenological thermodynamics and entropy principles, in Entropy, (eds. A. Greven, G. Keller and G. Warnecke), Princeton University Press Princeton, N.J., 2003. [20] R. A. L. Jones, Soft condensed matter, Eur. J. Phys., 23 (2002), 652. doi: 10.1088/0143-0807/23/6/703. [21] A. Karma and W. J. Rappel, Quantitative phase-field modelling of dendritic growth in two and three dimensions, Phys. Rev. E, 57 (1998), 4323-4349. doi: 10.1103/PhysRevE.57.4323. [22] A. G. Lamorgese, D. Molin and R. Mauri, Phase field approach to multiphase flow modeling, Milan J. Math., 79 (2011), 597-642. doi: 10.1007/s00032-011-0171-6. [23] G. A. Maugin, The Thermomechanics of Nonlinear Irreversible Behaviours. An Introduction, World Scientific Singapore, 1999. doi: 10.1142/3700. [24] G. A. Maugin and W. Muschik, Thermodynamics with internal variables. Part I. General concepts, J. Non-Equilibrium Thermodyn, 19 (1994), 217-249. doi: 10.1515/jnet.1994.19.3.217. [25] A. Morro, A phase-field approach to non-isothermal transitions, Math. Comput. Modelling, 48 (2008), 621-633. doi: 10.1016/j.mcm.2007.11.001. [26] I. Müller, Thermodynamics, Pitman Boston, 1985. [27] O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Phys. D, 43 (1990), 44-62. doi: 10.1016/0167-2789(90)90015-H. [28] O. Penrose and P. C. Fife, On the relation between the standard phase-field model and a "thermodynamically consistent" phase-field model, Phys. D, 69 (1993), 107-113. doi: 10.1016/0167-2789(93)90183-2. [29] I. Singer-Loginova and H. M. Singer, The phase-field technique for modeling multiphase materials, Rep. Prog. Phys., 71 (2008), 106501-106533. doi: 10.1088/0034-4885/71/10/106501. [30] P. Ván, Weakly nonlocal irreversible thermodynamics, Ann. Phys. (8), 12 (2003), 146-173. doi: 10.1002/andp.200310002.
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