Classical Becker-Döring cluster equations: Rigorous results on metastability and long-time behaviour


  • Markus Kreer

    1. Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, United Kingdom
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    • This research was carried out under a Heriot-Watt University scholarship.


We consider the classical Becker-Döring cluster equations with constant monomer concentration c1 = z > 0 and

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as a model which describes the kinetics of a first-order phase transition. For a large class of positive coefficients al and bl (including the ones commonly used in physics and chemistry) we prove the following: (i) When the monomer concentration z is slightly greater than zs = liml→∞bl/al then all initial states - containing only subcritical clusters of size l < l* (where l* denotes the critical size of a nucleus and depends on the supersaturation z-zs > 0) - converge within a fairly short time towards a metastable state. In this metastable state only subcritical clusters are present. The “metastable equilibrium” has an exponentially long lifetime TM ∼ exp (C(z-zs)−ω) (where C and ω are some positive constants).

(ii) For times greater than the lifetime TM this metastable state breaks down in the following sense: as t→∞ each of the cl(t) converges towards the Becker-Döring steady-state solution fl(z) like cl(t)-fl(z) = O (exp (- | λ1 | t)) (where λ1 < 0 is the eigenvalue closest to 0 of a certain infinite transition matrix) and the total mass of supercritical clusters (i.e. clusters of size l > l*) diverges in this limit. For large times the cluster-number increases linearly in time in the sense that limt→∞n (t)/t = J(z), where J(z) > 0 is the Becker-Döring steady-state current. For the average cluster size l = Σmath imagelcl(t)/Σmath imaget=1c(t), we find for sufficiently large times algebraic growth in time t, that is, μ1t1/(1-α) < l(t) < μ2t1(1-α) (where 0 < α < 1 is the algebraic growth exponent of the allα and μ1, μ2 are suitable positive constants). This bound covers previous suggestions due to computer simulations and heuristic calculations.