On the Evolution of Large Clusters in the Becker-Döring Model

SummaryWe consider the Becker-Döring equations for large times. It is well-known [2] that if the total density of monomers exceeds a critical value, the excess density is contained in larger and larger clusters as time proceeds. We rigorously derive for general coefficients that the evolution of these large clusters is described by a nonlocal transport equation, which is for specific coefficients the classical coarsening model by Lifshitz, Slyozov, and Wagner (LSW). Our proof exploits the estimate of the energy and the energy dissipation rate given by the Lyapunov functional for the Becker-Döring equations. We also provide a detailed asymptotic expansion of the higher-order dynamics.

[1]  R. Pego,et al.  The LSW Model for Domain Coarsening: Asymptotic Behavior for Conserved Total Mass , 2001 .

[2]  Philippe Laurençot,et al.  The Lifshitz-Slyozov-Wagner Equation with Conserved Total Volume , 2002, SIAM J. Math. Anal..

[3]  Juan J. L. Velázquez,et al.  The Becker–Döring Equations and the Lifshitz–Slyozov Theory of Coarsening , 1998 .

[4]  Felix Otto,et al.  Ostwald ripening: The screening length revisited , 2001 .

[5]  S. Mischler,et al.  From the Becker–Döring to the Lifshitz–Slyozov–Wagner Equations , 2002 .

[6]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[7]  Oliver Penrose,et al.  Kinetics of nucleation in a lattice gas model: Microscopic theory and simulation compared , 1983 .

[8]  Barbara Niethammer,et al.  Derivation of the LSW‐Theory for Ostwald Ripening by Homogenization Methods , 1999 .

[9]  O. Penrose,et al.  Growth of clusters in a first-order phase transition , 1978 .

[10]  Oliver Penrose,et al.  Metastable states for the Becker-Döring cluster equations , 1989 .

[11]  Thierry Goudon,et al.  The Beker-Döring System and Its Lifshitz-Slyozov Limit , 2002, SIAM J. Appl. Math..

[12]  Jack Carr,et al.  The Becker-Döring cluster equations: Basic properties and asymptotic behaviour of solutions , 1986 .

[13]  M. Kreer.  Classical Becker-Döring cluster equations: rigorous results on metastability and long-time behaviour , 1993 .

[14]  M. Slemrod.  The Becker-Döring Equations , 2000 .

[15]  R. Becker,et al.  Kinetische Behandlung der Keimbildung in übersättigten Dämpfen , 1935 .

[16]  Barbara Niethammer,et al.  Non-Self-Similar Behavior in the LSW Theory of Ostwald Ripening , 1999 .

[17]  Barbara Niethammer,et al.  On the Initial-Value Problem in the Lifshitz-Slyozov-Wagner Theory of Ostwald Ripening , 2000, SIAM J. Math. Anal..

[18]  Peter V. Coveney,et al.  Becker–Döring model of self-reproducing vesicles , 1998 .

[19]  J. Carr,et al.  Asymptotic behaviour of solutions to the Becker-Döring equations , 1999, Proceedings of the Edinburgh Mathematical Society.

[20]  D. Duncan,et al.  METASTABILITY IN THE CLASSICAL, TRUNCATED BECKER–DÖRING EQUATIONS , 2002, Proceedings of the Edinburgh Mathematical Society.

[21]  P. Coveney,et al.  Analysis of a generalized Becker—Döring model of self-reproducing micelles , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[22]  Oliver Penrose,et al.  The Becker-Döring equations at large times and their connection with the LSW theory of coarsening , 1997 .

[23]  Peter W Voorhees,et al.  The theory of Ostwald ripening , 1985 .

[24]  J. Carr,et al.  Numerical approximation of a metastable system , 1995 .

[25]  O. Penrose,et al.  Kinetics of a first-order phase transition: computer simulations and theory , 1984 .

[26]  Marshall Slemrod,et al.  Trend to equilibrium in the Becker-Doring cluster equations , 1989 .

[27]  I. Lifshitz,et al.  The kinetics of precipitation from supersaturated solid solutions , 1961 .