- Autor(in)
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- Seitenbereich
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0398 - 0417
- Schlagwort(e)
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<KWD>Dynamics of first-order phase transition
Classical nucleation theory
Cluster growth
Metastable states
Spectral properties of infinite tridiagonal matrices
- Zusammenfsg.
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(ii) For times greater than the lifetime <I>T</I><sub><I>M</I></sub> this metastable state breaks down in the following sense: as <I>t</I>→∞ each of the <I>c</I><sub><I>l</I></sub>(<I>t</I>) converges towards the Becker-Döring steady-state solution <I>f</I><sub><I>l</I></sub>(<I>z</I>) like <I>c</I><sub><I>l</I></sub>(<I>t</I>)-<I>f</I><sub><I>l</I></sub>(<I>z</I>) = O (exp (- | λ<sub>1</sub> | <I>t</I>)) (where λ<sub>1</sub> < 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 <I>l</I> > <I>l</I>*) diverges in this limit. For large times the cluster-number increases linearly in time in the sense that lim<sub><I>t</I>→∞</sub> <I>n</I> (<I>t</I>)/<I>t</I> = <I>J</I>(<I>z</I>), where <I>J</I>(<I>z</I>) > 0 is the Becker-Döring steady-state current. For the average cluster size <I>l</I> = σ<sub><sub><I>l</I></sub><sup>∞</sup>=1</sub> <I>lc</I><sub><I>l</I></sub>(<I>t</I>)/σ<sub>l</sub><sup>∞</sup><sub><I>t</I>=1</sub> <I>c</I>(<I>t</I>), we find for sufficiently large times algebraic growth in time <I>t</I>, that is, μ<sub>1</sub><I>t</I><sup>1/(1-α)</sup> < <I>l</I>(<I>t</I>) < μ<sub>2</sub> <I>t</I><sup>1(1-α)</sup> (where 0 < α < 1 is the algebraic growth exponent of the <I>a</I><sub><I>l</I></sub> ~ <I>l</I><sup>α</sup> and μ<sub>1</sub>, μ<sub>2</sub> are suitable positive constants). This bound covers previous suggestions due to computer simulations and heuristic calculations.
We consider the classical Becker-Döring cluster equations with constant monomer concentration <I>c</I><sub>1</sub> = <I>z</I> > 0 and as a model which describes the kinetics of a first-order phase transition. For a large class of positive coefficients <I>a</I><sub><I>l</I></sub> and <I>b</I><sub><I>l</I></sub> (including the ones commonly used in physics and chemistry) we prove the following: (i) When the monomer concentration <I>z</I> is slightly greater than <I>z</I><sub><I>s</I></sub> = lim<sub><I>l</I>→∞</sub> <I>b</I><sub><I>l</I></sub>/<I>a</I><sub><I>l</I></sub> then all initial states - containing only subcritical clusters of size <I>l</I> < <I>l</I>* (where <I>l</I>* denotes the critical size of a nucleus and depends on the supersaturation <I>z</I>-<I>z</I><sub><I>s</I></sub> > 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 <I>T</I><sub><I>M</I></sub> ~ exp (<I>C</I>(<I>z</I>-<I>z</I><sub><I>s</I></sub>)<sup>-ω</sup>) (where <I>C</I> and ω are some positive constants).
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