Skip to main content
SpringerLink
Log in

Mesoscopic Modeling of Surface Processes

  • Conference paper
Dispersive Transport Equations and Multiscale Models

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 136))

Abstract

In this paper we discuss mesoscopic models describing a broad class of pattern formation mechanisms, focusing on a prototypical system of surface processes. These models are in principle stochastic integrodifferential equations and are derived through an exact coarse-graining, directly from microscopic lattice models, and include detailed microscopic information on particle-particle interactions and particle dynamics.

*

The research of MAK is partially supported by NSF DMS-9626804, NSF DMS-9801769 and NSF DMS-0079536, and the research of DGV is partially supported by the NSF CTS-9702615, NSF CTS-9904242 and NETI.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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.

Refrences

  1. R. Imbihl AND G. Ertl, Oscillatory kinetics in heterogeneous catalysis, Chem. Rev. 95, 697 (1995).

    Article  Google Scholar 

  2. G. Ertl, Oscillatory kinetics and spatio-temporal self-organization in reactions at solid surfaces, Science 254, 1750 (1991).

    Article  Google Scholar 

  3. G.H. Gilmer AND P. Bennema, Simulation of crystal growth with surface diffusion, J. Appl. Phys. 43, 1347 (1972).

    Article  Google Scholar 

  4. K. Binder (ed.), Monte Carlo Methods in Statistical Physics. Springer-Verlag, Berlin (1986).

    Book  MATH  Google Scholar 

  5. M.P. Allen AND DJ. Tildesley, Computer Simulation of Liquids. Oxford Science Publications, Oxford (1989).

    Google Scholar 

  6. S. Jakubith, H.H. Rotermund, W. Engel, A. Von Oertzen, AND G. Ertl, Spatiotemporal concentration patterns in a surface reaction: Propagating and standing waves, rotating spirals, and turbulence, Phys. Rev. Letters 65, 3013 (1990).

    Article  Google Scholar 

  7. J.L. Lebowitz, E. Orlandi, AND E. Presutti, A particle model for spinodal decomposition, J. Stat. Phys. 63, 933 (1991).

    Article  MathSciNet  Google Scholar 

  8. A. De Masi, E. Orlandi, E. Presutti, AND L. Triolo, Glauber evolution with Kac potentials 1: mesoscopic and macroscopic limits, interface dynamics, Non-linearity 7, 633 (1994).

    MATH  Google Scholar 

  9. M.A. Katsoulakis AND P.E. Souganidis, Stochastic Ising models and anisotropic front propagation, J. Stat. Phys. 87, 63 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Hildebrand AND A.S. Mikhailov, Mesoscopic modeling in the kinetic theory of adsorbates, J. Phys. Chem. 100, 19089 (1996).

    Article  Google Scholar 

  11. G. Giacomin AND J.L. Lebowitz, Exact macroscopic description of phase segregation in model alloys with long range interactions, Phys. Rev. Letters 76, 1094 (1996).

    Article  Google Scholar 

  12. D.G. Vlachos AND M.A. Katsoulakis, Derivation and validation of mesoscopic theories for diffusion-reaction of interacting molecules, Phys. Rev. Lett. 85, 3898 (2000).

    Article  Google Scholar 

  13. G. Giacomin AND J.L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits, J. Stat. Phys. 87, 37 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  14. G. Giacomin AND J.L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. II. Interface motion, SIAM J. Appl. Math. 58, 1707 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  15. M.A. Katsoulakis AND P.E. Souganidis, Generalized motion by mean curvature as a macroscopic limit of stochastic Ising models with long range interactions and Glauber dynamics, Comm. Math. Phys. 169, 61 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  16. M.A. Katsoulakis AND D.G. Vlachos, Prom microscopic interactions to macroscopic laws of cluster evolution, Phys. Rev. Letters 84, 1511 (2000).

    Article  Google Scholar 

  17. J.W. Cahn AND J.E. Hilliard, Free energy of a nonuniform system I: Interfacial free energy, J. Chem. Phys. 28, 258 (1958).

    Article  Google Scholar 

  18. Q. Tran-Cong AND A. Harada, Reaction-induced ordering phenomena in binary polymer mixtures, Phys. Rev. Letters 76, 1162 (1996).

    Article  Google Scholar 

  19. S.C. Glotzer, E.A. Di Marzio, AND M. Muthukumar, Reaction-controlled morphology of phase-separating mixtures, Phys. Rev. Letters 74, 2034 (1995).

    Article  Google Scholar 

  20. M. Motoyama AND T. Ohta, Morphology of phase-separating binary mixtures with chemical reaction, J. Phys. Soc. Jpn. 66, 2715 (1997).

    Article  Google Scholar 

  21. T. Basak, D.G. Vlachos, AND M.A. Katsoulakis, Mesoscopic theories for diffusion-reaction of interacting molecules: An alternative to Monte Carlo simulations, in preparation.

    Google Scholar 

  22. M.N. Kuperman AND H.E. Troiani, Pore formation during dezincification of Zn-based alloys, Applied Surf. Science 148, 56 (1999).

    Article  Google Scholar 

  23. M. Hildebrand, A.S. Mikhailov, AND G. Ertl, Nonequilibrium stationary microstructures in surface chemical reactions, Phys. Rev. E 58, 5483 (1998).

    Article  Google Scholar 

  24. N. Maurits, P. Altevogt, O. Evers, AND J. Fraaije, Simple numerical quadrature rules for Gaussian chain polymer density functional calculations in 3D and implementation on parallel platforms, Comp. Polymer Sci. 6, 1 (1996).

    Google Scholar 

  25. D.J. Horntrop, M.A. Katsoulakis, AND D.G. Vlachos, Spectral Methods for Mesoscopic Models In Pattern Formation, submitted.

    Google Scholar 

  26. D. Ruelle, Statistical Mechanics: Rigorous Results. W.A. Benjamin, Inc., New York-Amsterdam (1969).

    MATH  Google Scholar 

  27. H. Spohn, Large Scale Dynamics of Interacting Particles. Springer-Verlag, New York (1991).

    Book  MATH  Google Scholar 

  28. H.C. Kang AND W.H. Weinberg, Dynamic Monte Carlo with a proper energy barrier: Surface diffusion and two-dimensional domain orderings, J. Chem. Phys. 90, 2824 (1988).

    Article  Google Scholar 

  29. H.C. Kang AND W.H. Weinberg, Modeling the kinetics of heterogeneous catalysis, Chem. Rev. 95, 667 (1995).

    Article  Google Scholar 

  30. A. De Masi AND E. Presutti, Mathematical Methods for Hydrodynamic Limits. Lecture Notes in Mathematics, 1501. Springer-Verlag, Berlin (1991).

    Google Scholar 

  31. G. Giacomin, J. Lebowitz, AND E. Presutti, Deterministic and stochastic hydrodynamic equations arising from simple microscopic model systems, in Stochastic Partial Differential Equations: Six Perspectives. Edited by R. Carmona and B. Rozovskii, Math. Surveys Monogr., Vol. 64, p. 107, Amer. Math. Soc, Providence, RI (1999).

    Google Scholar 

  32. X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations 2, 125 (1997).

    MathSciNet  MATH  Google Scholar 

  33. A. De Masi, T. Gobron, AND E. Presutti, Travelling fronts in non-local evolution equations. Arch. Rational Mech. Anal. 132, 143 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  34. P. Bates, P. Fife, X. Ren, AND X. Wang, Traveling waves in a convolution model for phase transitions. Arch. Rational Mech. Anal. 138, 105 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  35. D.G. Vlachos, L.D. Schmidt, AND R. Aris, Effect of phase transitions, surface diffusion, and defects on heterogeneous reactions: multiplicities and fluctuations, Surf. Science 249, 248 (1991).

    Google Scholar 

  36. J.W. Evans, Kinetic phase transition in catalytic reaction models, Langmuir 7, 2514–2519 (1991).

    Article  Google Scholar 

  37. D. Horntrop, M. Katsoulakis, D. Vlachos, in preparation.

    Google Scholar 

  38. M.A. Katsoulakis AND A.T. Kho, Stochastic Curvature Flows: Asymptotic Derivation, Level Set Formulation and Numerical Experiments, to appear in J. Interfaces and Free Boundaries.

    Google Scholar 

  39. S.M. Allen AND J. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Act. Metall. 27, 1089 (1979).

    Article  Google Scholar 

  40. J. Fraaije, B. VAN Vlimmeren, N. Maurits, M. Postma, O. Evers, C. Hoffmann, P. Altevogt, G. Goldbeck-Wood, The dynamic mean-field density functional method and its application to the mesoscopic dynamics of quenched block copolymer melts, J. Chem. Phys. 106, 4260 (1997).

    Article  Google Scholar 

  41. C. Kipnis, S. Olla, AND S.R.S. Varadhan, Hydrodynamics and large deviation for simple exclusion processes, Comm. Pure Appl. Math. 42, 115 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  42. A. Asselah AND G. Giacomin, Metastability for the exclusion process with meanfield interaction, J. Statist. Phys. 93, 1051 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  43. S. Renisch, R. Schuster, J. Wintterlin, AND G. Ertl, Dynamics of adatom motion under the influence of mutual interactions: O/Ru(0001), Phys. Rev. Lett. 82, 3839 (1999).

    Article  Google Scholar 

  44. E. Carlen, M. Carvalho, AND E. Orlandi, in preparation.

    Google Scholar 

  45. S. Osher AND J. A. Sethian, Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys. 78, 12 (1988).

    Article  MathSciNet  Google Scholar 

  46. L.C. Evans AND J. Spruck J., Motion of level sets by mean curvature I, J. Diff. Geom. 33, 635 (1991).

    MathSciNet  MATH  Google Scholar 

  47. Y.-G. Chen, Y. Giga Y., AND S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Diff. Geom. 33, 749 (1991).

    MathSciNet  MATH  Google Scholar 

  48. G. B Arles AND P.E. Souganidis, A new approach to front propagation problems: theory and applications, Arch. Rational Mech. Anal. 141, 237 (1998).

    Article  MathSciNet  Google Scholar 

  49. M.. Hildebrand, A.S. Mikhailov, AND G. Ertl, Nonequilibrium stationary microstructures in surface chemical reactions, Phys. Rev. E 58, 5483 (1998).

    Article  Google Scholar 

  50. A.M. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. (London) Ser. B 237, 37 (1952).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, 01003, USA

    Markos A. Katsoulakis

  2. Department of Chemical Engineering, University of Delaware, Newark, DE, 19716, USA

    Dionisios G. Vlachos

Authors
  1. Markos A. Katsoulakis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dionisios G. Vlachos
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Laboratoire MIP, Université Paul Sabatier, Toulouse Cedex 4, 31062, France

    Naoufel Ben Abdallah  & Pierre Degond  & 

  2. Angewandte Mathematik, Universität des Saarlandes, Saarbrucken, D-66041, Germany

    Anton Arnold

  3. Department of Mathematics, University of Texas at Austin, Austin, TX, 78712, USA

    Irene M. Gamba

  4. Department of Mathematics, Indiana University, Bloomington, IN, 47405-5701, USA

    Robert T. Glassey

  5. CSCAMM, University of Maryland, College Park, MD, 20742-4015, USA

    C. David Levermore

  6. Department of Mathematics, Arizona State University, Tempe, AZ, 85287, USA

    Christian Ringhofer

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer Science+Business Media New York

About this paper

Cite this paper

Katsoulakis, M.A., Vlachos, D.G. (2004). Mesoscopic Modeling of Surface Processes. In: Abdallah, N.B., et al. Dispersive Transport Equations and Multiscale Models. The IMA Volumes in Mathematics and its Applications, vol 136. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8935-2_12

Download citation

  • DOI: https://doi.org/10.1007/978-1-4419-8935-2_12

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-6473-6

  • Online ISBN: 978-1-4419-8935-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation