On a modified Becker–Döring model for two-phase materials

  • Thomas BlesgenEmail author
  • Ada Amendola
  • Fernando Fraternali
Original Article


This work reconsiders the Becker–Döring model for nucleation under isothermal conditions in the presence of phase transitions. Based on thermodynamic principles, a modified model is derived where the condensation and evaporation rates may depend on the phase parameter. The existence and uniqueness of weak solutions to the proposed model are shown and the corresponding equilibrium states are characterized in terms of response functions and constitutive variables.


Becker–Döring equations Nucleation Phase transitions Reaction diffusion equations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



TB acknowledges the support by the Hausdorff Institute of Mathematics, Bonn. AA and FF gratefully acknowledge financial support from the Italian Ministry of Education, University and Research (MIUR) under the ‘Departments of Excellence’ Grant L.232/2016.


  1. 1.
    Ask, A., Forest, S., Appolaire, B., Ammar, K.: A Cosserat-phase-field theory of crystal plasticity and grain boundary migration at finite deformation. Contin. Mech. Thermodyn. (2018).
  2. 2.
    Ball, J.M., Carr, J., Penrose, O.: The Becker–Döring cluster equations: basic properties and asymptotic behaviour of solutions. Commun. Math. Phys. 104, 657–692 (1986)ADSCrossRefzbMATHGoogle Scholar
  3. 3.
    Becker, R., Döring, W.: Kinetische Behandlung der Keimbildung in übersättigten Dämpfen. Ann. Phys. 24, 719–752 (1935)CrossRefzbMATHGoogle Scholar
  4. 4.
    Blesgen, T.: A revised model for diffusion induced segregation processes. J. Math. Phys. 46, 022702 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Blesgen, T., Luckhaus, S.: On the role of lattice defects close to phase transitions. Math. Methods Appl. Sci. 29, 525–536 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Blesgen, T., Weikard, U.: Multi-component Allen–Cahn equation for elastically stressed solids. Electron. J. Differ. Equ. 89, 1–17 (2005)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Blesgen, T.: A variational model for dynamic recrystallization based on Cosserat plasticity. Compos. B Eng. 115, 236–243 (2017)CrossRefGoogle Scholar
  8. 8.
    Burton, J.J.: Nucleation theory. In: Berne, B.J. (ed.) Statistical Mechanics, Modern Theoretical Chemistry, vol. 5, pp. 195–234. Springer, Boston (1977)Google Scholar
  9. 9.
    Dell’Isola, F., Gouin, H., Rotoli, G.: Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations. Eur. J. Mech. B Fluids 15, 545–568 (1996)zbMATHGoogle Scholar
  10. 10.
    Dell’Isola, F., Hutter, K.: What are the dominant thermomechanical processes in the basal sediment layer of large ice sheets? Proc. R. Soc. Lond. A 454, 1169–1195 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dell’Isola, F., Romano, A.: A phenomenological approach to phase transition in classical field theory. Int. J. Eng. Sci. 25, 1469–1475 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    De Masi, A., Dirr, N., Presutti, E.: Interface instability under forced displacements. Ann. Henri Poincaré 7, 471–511 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dreyer, W., Duderstadt, F.: Towards the thermodynamic modeling of nucleation and growth of liquid droplets in single crystals. Int. Ser. Numer. Math. 147, 113–130 (2004)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Dreyer, W., Duderstadt, F.: On the Becker/Döring theory of nucleation of liquid droplets in solids. J. Stat. Phys. 123, 55–87 (2006)ADSCrossRefzbMATHGoogle Scholar
  15. 15.
    Duval, P., Ashby, M.F., Anderman, I.: Rate-controlling processes in the creep of polycrystalline ice. J. Phys. Chem. 87(21), 4066–4074 (1983)CrossRefGoogle Scholar
  16. 16.
    Frenkel, J.I.: A general theory of heterogeneous fluctuations and pretansition phenomena. J. Chem. Phys. 7, 538–547 (1939)ADSCrossRefGoogle Scholar
  17. 17.
    Haumesser, P.-H.: Nucleation and Growth of Metals: From Thin Films to Nanoparticles, pp. 1–194. Elsevier, Amsterdam (2016)CrossRefGoogle Scholar
  18. 18.
    Herrmann, M., Naldzhieva, M., Niethammer, B.: On a thermodynamically consistent modification of the Becker–Döring equations. Physica D 222, 116–130 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hingant, E., Yvinec, R.: Deterministic and stochastic Becker–Dring equations: past and recent mathematical developments. In: Holcman, D. (ed.) Stochastic Processes, Multiscale Modeling and Numerical Methods for Computational Cellular Biology, pp. 175–204. Springer, Berlin (2017)CrossRefGoogle Scholar
  20. 20.
    Hong, B.Z., Keong, L.K., Shariff, A.M.: CFD modelling of most probable bubble nucleation rate from binary mixture with estimation of components’ mole fraction in critical cluster. Contin. Mech. Thermodyn. 28(3), 655–668 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kashchiev, D., Rosmalen, G.M.: Review: nucleation in solutions revisited. Cryst. Res. Technol. 38, 555–574 (2003)CrossRefGoogle Scholar
  22. 22.
    Kamb, B.: Experimental Recrystallization of Ice Under Stress, vol. 16, pp. 211–241. American Geophysical Union Geophysical Monograph Series, Washington DC (1972)Google Scholar
  23. 23.
    Kirkaldy, J.S., Young, D.J.: Diffusion in the Condensed State. The Institute of Metals, London (1987)Google Scholar
  24. 24.
    Niethammer, B.: On the dynamics of the Becker–Döring equations. Habilitation Thesis, University of Bonn (2002)Google Scholar
  25. 25.
    Niethammer, B.: On the evolution of large clusters in the Becker–Döring model. J. Nonlinear Sci. 13, 115–155 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Pathria, R.K.: Statistical Mechanics, 2nd edn. Butterworth-Heinemann Publications, Oxford (1996)zbMATHGoogle Scholar
  27. 27.
    Penrose, O.: Metastable states for the Becker–Döring cluster equations. Commun. Math. Phys. 124, 515–541 (1989)ADSCrossRefzbMATHGoogle Scholar
  28. 28.
    Penrose, O.: The Becker–Döring equations at large times and their connection with the lsw theory of coarsening. Commun. Math. Phys. 189, 305–320 (1997)zbMATHGoogle Scholar
  29. 29.
    Placidi, L., Faria, S.H., Hutter, K.: On the role of grain growth, recrystallization and polygonization in a continuum theory for anisotropic ice sheets. Ann. Glaciol. 39, 49–52 (2004)ADSCrossRefGoogle Scholar
  30. 30.
    Placidi, L., Greve, R., Seddik, H., Faria, S.H.: Continuum-mechanical, anisotropic flow model for polar ice masses, based on an anisotropic flow enhancement factor. Contin. Mech. Thermodyn. 22, 221–237 (2010)ADSCrossRefzbMATHGoogle Scholar
  31. 31.
    Puglisi, G.: Nucleation and phase propagation in a multistable lattice with weak nonlocal interactions. Contin. Mech. Thermodyn. 19(5), 299–319 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Roters, F., Raabe, D., Gottstein, G.: Work hardening in heterogenoeus alloys - a microstructural approach based on three internal state variables. Acta Mater. 48, 4181–4189 (2000)CrossRefGoogle Scholar
  33. 33.
    Shi, R., Shen, C., Dregia, S.A., Wang, Y.: Form of critical nuclei at homo-phase boundaries. Scr. Mater. 146, 276–280 (2018)CrossRefGoogle Scholar
  34. 34.
    Tutcuoglu, A.D., Vidyasagar, A., Bhattacharya, K., Kochmann, D.M.: Stochastic modeling of discontinuous dynamic recrystallization at finite strains in HCP metals. J. Mech. Phys. Solids 122(2019), 590–612 (2019)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Tutyshkin, N.D., Lofink, P., Mller, W.H., Wille, R., Stahn, O.: Constitutive equations of a tensorial model for strain-induced damage of metals based on three invariants. Contin. Mech. Thermodyn. 29, 251–269 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Zecevic, M., Lebensohn, R.A., Mc Cabe, R.J., Knezevic, M.: Modelling recrystallization textures driven by intragranular fluctuations implemented in the viscoplastic self-consistent formulation. Acta Mater. 164, 530–546 (2019)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Applied SciencesBingenGermany
  2. 2.Department of Civil EngineeringUniversity of SalernoFiscianoItaly

Personalised recommendations