Kinetic modelling of late stages of phase separation

  • Guido Manzi
  • Rossana Marra
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


We want to provide some tools for studying the behaviour of a fluid where two phases are present separated by a sharp layer which moves according to the dynamics of the system. We specialize to the situation of a mixture of two fluids 1 and 2, that, if quenched below the coexistence curve, start to segregate in domains, some of which are rich in fluid 1 and the others in fluid 2.ℝℂ


Surface Tension Principal Curvature Free Boundary Problem Sharp Interface Phase Segregation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [BCK]
    M. Biskup, L. Chayes, and R. Kotecky, On the formation/dissolution of equilibrium droplets, Europhys. Lett. 60, 21–27 (2002); Critical region for droplet formation in the two-dimensional Ising model, Comm. Math. Phys. 242, 137–183 (2003).CrossRefGoogle Scholar
  2. [C]
    R. Caflish, The fluid dynamical limit of the nonlinear Boltzmann equation, Commun. Pure and Appl. Math. 33, 651–666 (1980).CrossRefGoogle Scholar
  3. [CF]
    G. Caginalp and P. C. Fife, Phase-field methods for interfacial boundaries, Phys Rev. B 33, 7792–7794 (1986); Dynamics of layered interfaces arising from phase boundaries, SIAM J. Appl. Math. 48, 506–518 (1988).CrossRefMathSciNetGoogle Scholar
  4. [CCELM]
    E. A. Carlen, M. C. Carvahlo, R. Esposito, J. L. Lebowitz, and R. Marra, Free energy minimizers for a two-species model with segregation and liquid-vapor transition, Nonlinearity 16, 1075–1105 (2003).MATHCrossRefMathSciNetGoogle Scholar
  5. [CCELM1]
    E. Carlen, M. C. Carvahlo, R. Esposito, J. L. Lebowitz, and R. Marra, Phase transitions in equilibrium systems: Microscopic models and mesoscopic free energies, Molecular Physics 103, 3141–3151 (2005).CrossRefGoogle Scholar
  6. [CCELM2]
    E. Carlen, M. C. Carvahlo, R. Esposito, J. L. Lebowitz, and R. Marra, Droplet minimizers for the Cahn-Hilliard free energy functional, Journal of Geometric Analysis 16 n. 2 (2006).Google Scholar
  7. [CCO1]
    E. A. Carlen, M. C. Carvalho, and E. Orlandi, Approximate solution of the Cahn-Hilliard equation via corrections to the Mullins-Sekerka motion, Arch. Rat. Mechanics 178, 1–55, (2005).MATHCrossRefMathSciNetGoogle Scholar
  8. [CCO2]
    E. A. Carlen, M. C. Carvalho, and E. Orlandi, Algebraic rate of decay for the excess free energy and stability of fronts for a non local phase kinetics equation, I and II, Jour. Stat. Phys. 95, 1069–1117 (1999) and Comm. PDE 25, 847–886 (2000).MATHCrossRefMathSciNetGoogle Scholar
  9. [Ce]
    C. Cercignani, R. Illner, and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer-Verlag, New York (1994).MATHGoogle Scholar
  10. [Cha]
    S. Chandrasekar, Hydrodynamic and Hydromagnetic Stability, Chapter X, Clarendon Press, Oxford (1961).Google Scholar
  11. [CS]
    D. Coutand and S. Shkoller, Unique solvability of the free-boundary Navier-Stokes equations with surface tension, arXiv:math.AP/0212116 v2 (2003).Google Scholar
  12. [DOPT]
    A. De Masi, E. Orlandi, E. Presutti, and L. Triolo, Stability of the interface in a model of phase separation, Proc. Royal Soc. Edinburgh 124, (1994).Google Scholar
  13. [Ge]
    P. G. de Gennes, Dynamics of fluctuations and spinodal decomposition in polymer blends, J. Chem. Phys. 72, 4756–4763 (1980).MATHCrossRefMathSciNetGoogle Scholar
  14. [GL]
    G. Giacomin and J. L. Lebowitz, Exact macroscopic description of phase segregation in model alloys with long range interactions, Phys. Rev. Lett. 76, 1094–1098 (1996); Phase segregation dynamics in particle systems with long range interaction. I: Macroscopic limits, J. Stat. Phys. 87, 37–61(1997); Phase segregation dynamics in particle systems with long range interaction. II: Interface motion, SIAM J. Appl. Math. 58, 1707–1729 (1998).CrossRefGoogle Scholar
  15. [GLP]
    G. Giacomin, J. L. Lebowitz, and E. Presutti, Deterministic and stochastic hydrodynamic equations arising from simple microscopic model systems, in Stochastic Partial Differential Equations, Six Perspectives, pp. 107–152, Math. Survey Monograph 64, Amer. Math. Soc., Providence, RI (1999).Google Scholar
  16. [G]
    Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J. 53, 1081–1094 (2004).MATHCrossRefMathSciNetGoogle Scholar
  17. [H]
    W. J. Harrison, The influence of viscosity on the oscillations of superposed fluids, Proc. London Math. Soc. 6, 396–405 (1908).CrossRefGoogle Scholar
  18. [LY]
    T.-P. Liu and S.-H. Yu, Boltzmann equation: Micro-Macro decompositions and positivity of shock profiles, Commun. Math. Phys. 246, 133–179 (2004).MATHCrossRefMathSciNetGoogle Scholar
  19. [MM]
    G. Manzi and R. Marra, Phase segregation and interface dynamics in kinetic systems, Nonlinearity 19, 115–147 (2006).MATHCrossRefMathSciNetGoogle Scholar
  20. [OE]
    F. Otto and E. Weinan, Thermodynamically driven incompressible fluid mixtures, J. Chem. Phys. 107(23), 10177–10184 (1997).CrossRefGoogle Scholar
  21. [Sh]
    S. Shkoller, Private communication.Google Scholar
  22. [SG]
    R. M. Strain and Y. Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations 31, 417–429 (2006).MATHCrossRefMathSciNetGoogle Scholar
  23. [wip]
    Work in progress (E. A. Carlen, M. C. Carvahlo, R. Esposito, J. L. Lebowitz, and R. Marra).Google Scholar

Copyright information

© Birkhäuser Boston 2007

Authors and Affiliations

  • Guido Manzi
    • 1
  • Rossana Marra
    • 2
  1. 1.Max-Planck-Institut fÜr Mathematik in den NaturwissenschaftenLeipzigGermany
  2. 2.Dipartimento di FisicaUniversità di Roma Tor Vergata e INFNRomaItaly

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