Abstract
A systematic procedure is given for obtaining the asymptotic late-time behavior of the Becker-Döring equations describing the time evolution of a population of clusters of particles. In lowest order of approximation, the distribution of the sizes of the largest clusters satisfies the equations of the Lifshitz-Slyozov-Wagner theory of coarsening.
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References
A. J. Ardell,The mechanics of phase transitions in crystalline solids, Institute of metals monograph Series, vol. 33, (London, 1969), page 111.
A. J. Ardell and R. B. Nicholson,J. Phys. Chem. Solids 27:1793–1804 (1966).
J. M. Ball, J. Carr and O. Penrose,Commun. Math. Phys. 104:657–692(1986). The Becker-Döring cluster equations: basic properties and asymptotic behavior of solutions.
R. Becker and W. Döring,Ann. Phys. (Leipzig)24:719–752 (1935). Kinetische Behandlung der Keimbildung in übersättigten Dämpfen.
A Buhagiar,Kinetics of phase segregation in a quenched alloy, Thesis, Open University, Milton Keynes, U.K. (1980).
J. J. Burton, Nucleation theory. pp. 195–234 ofStatistical Mechanics, Part A: Equilibrium techniques, edited by B. J. Berne, Plenum, 1977.
J. Carr and O. Penrose, unpublished.
A. DeMasi and E. Presutti,Mathematical methods for hydrodynamic limits, Springer lecture notes in mathematics, no. 1501, Berlin/Heidelberg, 1991.
P. Fratzl and O. Penrose,Phys. Rev. B, to appear (1997). Competing mechanisms for precipitate coarsening in phase separation with vacancy dynamics.
B. D. Gaulin, S. Spooner, and Y. Morii,Phys. Rev. Lett. 59:668–671 (1987). Kinetics of phase separation in Mn0.67Cu0.33.
F. Leyvraz and H. R. Tschudi,J. Phys A: Math. Gen. 14:3389–3405 (1981). Singularities in the kinetics of coagulation processes.
I. M. Lifshitz and V. V. Slyozov,J. Phys. Chem. Sol. 19:35–50 (1961). Kinetics of precipitation from supersaturated solid solutions.
E. M. Hendriks, M. H. Ernst, and R. M. Ziff,J. Stat. Phys. 31:519–563 (1983). Coagulation equations with gelation.
S. C. Hardy and P. W. Voorhees,Metall. Trans. 19A:2713–2721 (1988). Ostwald ripening in a system with a high volume fraction of coarsening phase.
R. Pego,Proc. Roy. Soc. Lond. A 422: 261–278 (1989). Front migration in the nonlinear Cahn-Hilliard equation.
O. Penrose and A. Buhagiar,J. Stat. Phys. 30:219–241 (1983). Kinetics of nucleation in a lattice gas model: microscopic theory and simulation compared.
O. Penrose, J. L. Lebowitz, J. Marro, M. H. Kalos, and A. Sur,J. Stat. Phys. 19:243–267 (1978). Growth of Clusters in a First-order Phase Transition.
J. R. Rubinstein, P. Sternberg, and J. B. Keller,SIAM J. Appl. Math. 49:116–133 (1989). Fast reaction, slow diffusion, and curve shortening.
P. W. Voorhees,J. Stat. Phys. 38:231–252 (1985). The theory of Ostwald ripening.
P. W. Voorhees,Ann. Rev. Mater. Sci. 22:197–215 (1992).
J. Velazquez, preprint (1997). The Becker-Döring equations and the Lifshitz-Slyozov theory of coarsening.
C. Wagner,Z. Elektrochem. 65:581–591 (1961).
J. Wimmel and A. J. Ardell,J. Alloys and Compounds 205:215–223 (1994). Coarsening kinetics and microstructure of Ni3Ga precipitates in aged Ni-Ga alloys.
S. Q. Xiao and P. Haasen,Acta Met. Mat. 39:651–659 (1991). HREM investigation of homogeneous decomposition in a Ni-12 at % Al alloy.
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Penrose, O. The Becker-Döring equations at large times and their connection with the LSW theory of coarsening. J Stat Phys 89, 305–320 (1997). https://doi.org/10.1007/BF02770767
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DOI: https://doi.org/10.1007/BF02770767