Article Contents
Article Contents

# Phase transition and separation in compressible Cahn-Hilliard fluids

• 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.
Mathematics Subject Classification: Primary: 82B26; Secondary: 82C26, 80A22, 74A15, 74A50, 80A17.

 Citation:

•  [1] V. Berti, C. Giorgi and M. Fabrizio, Well-posedness for solid-liquid phase transitions with a fourth-order nonlinearity, Physica D, 236 (2007), 13-21. [2] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996.doi: 10.1007/978-1-4612-4048-8. [3] I. Steinbach and M. Apel, Multiphase field model for solid state transformation with elastic strain, Physica D, 217 (2006), 153-160.doi: 10.1016/j.physd.2006.04.001. [4] I. Singer-Loginova and H. Singer, The phase field technique for modeling multiphase materials, Rep. Prog. Phys., 71 (2008), 106501.doi: 10.1088/0034-4885/71/10/106501. [5] C. Giorgi, Continuum thermodynamics and phase-field models, Milan J. Math., 77 (2009), 67-100.doi: 10.1007/s00032-009-0101-z. [6] J. D. van der Waals, Thermodynamique de la capillarité dans l'hypothèse d'une variation continue de densité, Arch. Néerlandaises, 28 (1894-1895), 121-219. [7] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.doi: 10.1063/1.1744102. [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), 815-831.doi: 10.1142/S0218202596000341. [9] P. C. Hohenberg and B. I. Halperin, Theory of dynamical critical phenomena, Rev. Modern Physics, 49 (1977), 435-479.doi: 10.1103/RevModPhys.49.435. [10] D. Jasnow and J. Viñals, Coarse-grained description of thermo-capillary flow, Phys. Fluids, 8 (1996), 660-669.doi: 10.1063/1.868851. [11] J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, Proc. R. Soc. Lond., A 454 (1998), 2617-2654.doi: 10.1098/rspa.1998.0273. [12] A. Onuki, Phase transitions of fluids in shear flow, J. Phys.: Condens. Matter, 9 (1997), 6119-6157.doi: 10.1088/0953-8984/9/29/001. [13] M. Fabrizio, C. Giorgi and A. Morro, A thermodynamic approach to non-isothermal phase-field evolution in continuum physics, Physica D, 214 (2006), 144-156.doi: 10.1016/j.physd.2006.01.002. [14] M. Fabrizio, C. Giorgi and A. Morro, Phase separation in quasi-incompressible Cahn-Hilliard fluids, Eur. J. Mech., 30 (2011), 281-287.doi: 10.1016/j.euromechflu.2010.12.003. [15] I. Müller, Thermodynamics of mixtures of fluids, J. Mécanique, 14 (1975), 267-303. [16] A. Morro, Governing equations in non-isothermal diffusion, Int. J. Non-Lin. Mech., 55 (2013), 90-97.doi: 10.1016/j.ijnonlinmec.2013.04.010. [17] M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192.doi: 10.1016/0167-2789(95)00173-5. [18] J. D. Clayton and J. Knap, A phase field model of deformation twinning: Nonlinear theory and numerical simulations, Physica D, 240 (2011), 841-858.doi: 10.1016/j.physd.2010.12.012. [19] E. Fried and M. E. Gurtin, Continuum theory of thermally induced phase transitions based on an order parameter, Physica D, 68 (1993), 326-343.doi: 10.1016/0167-2789(93)90128-N. [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), 629-635.doi: 10.1016/j.physd.2010.11.014. [21] C. M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423.doi: 10.1137/S0036141094267662. [22] M. Fabrizio, Ginzburg-Landau equations and first and second order phase transitions, Int. J. Engng Sci., 44 (2006), 529-539.doi: 10.1016/j.ijengsci.2006.02.006. [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), 627-653.doi: 10.1002/mma.930.