Skip to main content

Kinetic modelling of late stages of phase separation

  • Chapter
Transport Phenomena and Kinetic Theory

Abstract

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

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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).

    Article  Google Scholar 

  2. R. Caflish, The fluid dynamical limit of the nonlinear Boltzmann equation, Commun. Pure and Appl. Math. 33, 651–666 (1980).

    Article  Google Scholar 

  3. 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).

    Article  MathSciNet  Google Scholar 

  4. 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).

    Article  MATH  MathSciNet  Google Scholar 

  5. 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).

    Article  Google Scholar 

  6. 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. 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).

    Article  MATH  MathSciNet  Google Scholar 

  8. 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).

    Article  MATH  MathSciNet  Google Scholar 

  9. C. Cercignani, R. Illner, and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer-Verlag, New York (1994).

    MATH  Google Scholar 

  10. S. Chandrasekar, Hydrodynamic and Hydromagnetic Stability, Chapter X, Clarendon Press, Oxford (1961).

    Google Scholar 

  11. 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. 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. P. G. de Gennes, Dynamics of fluctuations and spinodal decomposition in polymer blends, J. Chem. Phys. 72, 4756–4763 (1980).

    Article  MATH  MathSciNet  Google Scholar 

  14. 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).

    Article  Google Scholar 

  15. 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. Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J. 53, 1081–1094 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  17. W. J. Harrison, The influence of viscosity on the oscillations of superposed fluids, Proc. London Math. Soc. 6, 396–405 (1908).

    Article  Google Scholar 

  18. T.-P. Liu and S.-H. Yu, Boltzmann equation: Micro-Macro decompositions and positivity of shock profiles, Commun. Math. Phys. 246, 133–179 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  19. G. Manzi and R. Marra, Phase segregation and interface dynamics in kinetic systems, Nonlinearity 19, 115–147 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  20. F. Otto and E. Weinan, Thermodynamically driven incompressible fluid mixtures, J. Chem. Phys. 107(23), 10177–10184 (1997).

    Article  Google Scholar 

  21. S. Shkoller, Private communication.

    Google Scholar 

  22. R. M. Strain and Y. Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations 31, 417–429 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  23. Work in progress (E. A. Carlen, M. C. Carvahlo, R. Esposito, J. L. Lebowitz, and R. Marra).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Birkhäuser Boston

About this chapter

Cite this chapter

Manzi, G., Marra, R. (2007). Kinetic modelling of late stages of phase separation. In: Cercignani, C., Gabetta, E. (eds) Transport Phenomena and Kinetic Theory. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4554-0_9

Download citation

  • DOI: https://doi.org/10.1007/978-0-8176-4554-0_9

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-0-8176-4489-5

  • Online ISBN: 978-0-8176-4554-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics