January  2014, 19(1): 73-88. doi: 10.3934/dcdsb.2014.19.73

Phase transition and separation in compressible Cahn-Hilliard fluids

1. 

Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40127 Bologna, Italy

2. 

DICATAM, Università di Brescia, Via Valotti 9, 25133 Brescia, Italy

3. 

DIBRIS, Università di Genova, Via Opera Pia 13, 16145 Genova, Italy

Received  October 2012 Revised  July 2013 Published  December 2013

The paper provides a scheme for phase separation and transition by accounting for diffusion, dynamic equations and consistency with thermodynamics. The constituents are compressible fluids thus improving the model of a previous approach. Moreover a possible saturation effect for the concentration of a constituent is made explicit. The mass densities of the constituents are independent of temperature. The evolution of concentration is described by the standard equation for mixtures but the balance of energy and entropy of the mixture are stated as for a single constituent. However, due to the non-simple character of the mixture, an extra-energy flux is allowed to occur. Also motion and diffusion effects are considered by letting the stress in the mixture have additive viscous terms and, remarkably, the chemical potential contains a quadratic term in the stretching tensor. As a result a whole set of evolution equations is set up for the concentration, the velocity, and the temperature. Shear-induced mixing and demixing are examined. A maximum theorem is proved which implies that the concentration of the mixture has values from 0 to 1 as is required from the physical standpoint.
Citation: Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Phase transition and separation in compressible Cahn-Hilliard fluids. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 73-88. doi: 10.3934/dcdsb.2014.19.73
References:
[1]

V. Berti, C. Giorgi and M. Fabrizio, Well-posedness for solid-liquid phase transitions with a fourth-order nonlinearity,, Physica D, 236 (2007), 13.   Google Scholar

[2]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions,, Springer, (1996).  doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[3]

I. Steinbach and M. Apel, Multiphase field model for solid state transformation with elastic strain,, Physica D, 217 (2006), 153.  doi: 10.1016/j.physd.2006.04.001.  Google Scholar

[4]

I. Singer-Loginova and H. Singer, The phase field technique for modeling multiphase materials,, Rep. Prog. Phys., 71 (2008).  doi: 10.1088/0034-4885/71/10/106501.  Google Scholar

[5]

C. Giorgi, Continuum thermodynamics and phase-field models,, Milan J. Math., 77 (2009), 67.  doi: 10.1007/s00032-009-0101-z.  Google Scholar

[6]

J. D. van der Waals, Thermodynamique de la capillarité dans l'hypothèse d'une variation continue de densité,, Arch. Néerlandaises, 28 (): 1894.   Google Scholar

[7]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy,, J. Chem. Phys., 28 (1958), 258.  doi: 10.1063/1.1744102.  Google Scholar

[8]

M. E. Gurtin, D. Polignone and J. Viñals, Two-phase binary fluids and immiscible fluids described by an order parameter,, Math. Mod. Meth. Appl. Sci., 6 (1996), 815.  doi: 10.1142/S0218202596000341.  Google Scholar

[9]

P. C. Hohenberg and B. I. Halperin, Theory of dynamical critical phenomena,, Rev. Modern Physics, 49 (1977), 435.  doi: 10.1103/RevModPhys.49.435.  Google Scholar

[10]

D. Jasnow and J. Viñals, Coarse-grained description of thermo-capillary flow,, Phys. Fluids, 8 (1996), 660.  doi: 10.1063/1.868851.  Google Scholar

[11]

J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions,, Proc. R. Soc. Lond., A 454 (1998), 2617.  doi: 10.1098/rspa.1998.0273.  Google Scholar

[12]

A. Onuki, Phase transitions of fluids in shear flow,, J. Phys.: Condens. Matter, 9 (1997), 6119.  doi: 10.1088/0953-8984/9/29/001.  Google Scholar

[13]

M. Fabrizio, C. Giorgi and A. Morro, A thermodynamic approach to non-isothermal phase-field evolution in continuum physics,, Physica D, 214 (2006), 144.  doi: 10.1016/j.physd.2006.01.002.  Google Scholar

[14]

M. Fabrizio, C. Giorgi and A. Morro, Phase separation in quasi-incompressible Cahn-Hilliard fluids,, Eur. J. Mech., 30 (2011), 281.  doi: 10.1016/j.euromechflu.2010.12.003.  Google Scholar

[15]

I. Müller, Thermodynamics of mixtures of fluids,, J. Mécanique, 14 (1975), 267.   Google Scholar

[16]

A. Morro, Governing equations in non-isothermal diffusion,, Int. J. Non-Lin. Mech., 55 (2013), 90.  doi: 10.1016/j.ijnonlinmec.2013.04.010.  Google Scholar

[17]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance,, Physica D, 92 (1996), 178.  doi: 10.1016/0167-2789(95)00173-5.  Google Scholar

[18]

J. D. Clayton and J. Knap, A phase field model of deformation twinning: Nonlinear theory and numerical simulations,, Physica D, 240 (2011), 841.  doi: 10.1016/j.physd.2010.12.012.  Google Scholar

[19]

E. Fried and M. E. Gurtin, Continuum theory of thermally induced phase transitions based on an order parameter,, Physica D, 68 (1993), 326.  doi: 10.1016/0167-2789(93)90128-N.  Google Scholar

[20]

C. G. Gal and M. Grasselli, Instability of two-phase flows: A lower bound on the dimension of the global attractor of the Cahn-Hilliard-Navier-Stokes system,, Physica D, 240 (2011), 629.  doi: 10.1016/j.physd.2010.11.014.  Google Scholar

[21]

C. M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility,, SIAM J. Math. Anal., 27 (1996), 404.  doi: 10.1137/S0036141094267662.  Google Scholar

[22]

M. Fabrizio, Ginzburg-Landau equations and first and second order phase transitions,, Int. J. Engng Sci., 44 (2006), 529.  doi: 10.1016/j.ijengsci.2006.02.006.  Google Scholar

[23]

M. Fabrizio, C. Giorgi and A. Morro, A continuum theory for first-order phase transitions based on the balance of structure order,, Math. Meth. Appl. Sci., 31 (2008), 627.  doi: 10.1002/mma.930.  Google Scholar

show all references

References:
[1]

V. Berti, C. Giorgi and M. Fabrizio, Well-posedness for solid-liquid phase transitions with a fourth-order nonlinearity,, Physica D, 236 (2007), 13.   Google Scholar

[2]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions,, Springer, (1996).  doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[3]

I. Steinbach and M. Apel, Multiphase field model for solid state transformation with elastic strain,, Physica D, 217 (2006), 153.  doi: 10.1016/j.physd.2006.04.001.  Google Scholar

[4]

I. Singer-Loginova and H. Singer, The phase field technique for modeling multiphase materials,, Rep. Prog. Phys., 71 (2008).  doi: 10.1088/0034-4885/71/10/106501.  Google Scholar

[5]

C. Giorgi, Continuum thermodynamics and phase-field models,, Milan J. Math., 77 (2009), 67.  doi: 10.1007/s00032-009-0101-z.  Google Scholar

[6]

J. D. van der Waals, Thermodynamique de la capillarité dans l'hypothèse d'une variation continue de densité,, Arch. Néerlandaises, 28 (): 1894.   Google Scholar

[7]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy,, J. Chem. Phys., 28 (1958), 258.  doi: 10.1063/1.1744102.  Google Scholar

[8]

M. E. Gurtin, D. Polignone and J. Viñals, Two-phase binary fluids and immiscible fluids described by an order parameter,, Math. Mod. Meth. Appl. Sci., 6 (1996), 815.  doi: 10.1142/S0218202596000341.  Google Scholar

[9]

P. C. Hohenberg and B. I. Halperin, Theory of dynamical critical phenomena,, Rev. Modern Physics, 49 (1977), 435.  doi: 10.1103/RevModPhys.49.435.  Google Scholar

[10]

D. Jasnow and J. Viñals, Coarse-grained description of thermo-capillary flow,, Phys. Fluids, 8 (1996), 660.  doi: 10.1063/1.868851.  Google Scholar

[11]

J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions,, Proc. R. Soc. Lond., A 454 (1998), 2617.  doi: 10.1098/rspa.1998.0273.  Google Scholar

[12]

A. Onuki, Phase transitions of fluids in shear flow,, J. Phys.: Condens. Matter, 9 (1997), 6119.  doi: 10.1088/0953-8984/9/29/001.  Google Scholar

[13]

M. Fabrizio, C. Giorgi and A. Morro, A thermodynamic approach to non-isothermal phase-field evolution in continuum physics,, Physica D, 214 (2006), 144.  doi: 10.1016/j.physd.2006.01.002.  Google Scholar

[14]

M. Fabrizio, C. Giorgi and A. Morro, Phase separation in quasi-incompressible Cahn-Hilliard fluids,, Eur. J. Mech., 30 (2011), 281.  doi: 10.1016/j.euromechflu.2010.12.003.  Google Scholar

[15]

I. Müller, Thermodynamics of mixtures of fluids,, J. Mécanique, 14 (1975), 267.   Google Scholar

[16]

A. Morro, Governing equations in non-isothermal diffusion,, Int. J. Non-Lin. Mech., 55 (2013), 90.  doi: 10.1016/j.ijnonlinmec.2013.04.010.  Google Scholar

[17]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance,, Physica D, 92 (1996), 178.  doi: 10.1016/0167-2789(95)00173-5.  Google Scholar

[18]

J. D. Clayton and J. Knap, A phase field model of deformation twinning: Nonlinear theory and numerical simulations,, Physica D, 240 (2011), 841.  doi: 10.1016/j.physd.2010.12.012.  Google Scholar

[19]

E. Fried and M. E. Gurtin, Continuum theory of thermally induced phase transitions based on an order parameter,, Physica D, 68 (1993), 326.  doi: 10.1016/0167-2789(93)90128-N.  Google Scholar

[20]

C. G. Gal and M. Grasselli, Instability of two-phase flows: A lower bound on the dimension of the global attractor of the Cahn-Hilliard-Navier-Stokes system,, Physica D, 240 (2011), 629.  doi: 10.1016/j.physd.2010.11.014.  Google Scholar

[21]

C. M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility,, SIAM J. Math. Anal., 27 (1996), 404.  doi: 10.1137/S0036141094267662.  Google Scholar

[22]

M. Fabrizio, Ginzburg-Landau equations and first and second order phase transitions,, Int. J. Engng Sci., 44 (2006), 529.  doi: 10.1016/j.ijengsci.2006.02.006.  Google Scholar

[23]

M. Fabrizio, C. Giorgi and A. Morro, A continuum theory for first-order phase transitions based on the balance of structure order,, Math. Meth. Appl. Sci., 31 (2008), 627.  doi: 10.1002/mma.930.  Google Scholar

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