Nucleation Kinetics Byeong-Joo Lee Phase Transformations POSTECH - MSE calphad@postech.ac.kr

Motivation

General Background

Classical theory of nucleation References :    1. K.C. Russell, "Nucleation in Solids" in Phase Transformations, ASM 1970.    2. D. Turnbull, "Phase Changes" in Solid State Physics 3, 226, Academic Press, 1956.    3. J.W. Christian, The Theory of Transformations in Metals and Alloys, Pergamon, 1965.  □ Gibbs (1877) : activation energy ΔG*, critical nucleus size r*.     ※ Understanding of the role of thermal fluctuation ← statistical thermodynamics

Classical theory of nucleation □ Volmer and Weber (1925) :     ▷ formation of larger particle by adding atoms to smaller particles     ▷ van't Hoff's suggestion that the reaction goes in both directions               p1 ⇆ p2 ⇆ p3 ⇆ p4 ⇆ p5   etc.                pi represent the particles of various sizes (number of atoms: i)         @ equilibrium                ni = n1․exp (-ΔGi/kT)            ni : equil. number of particles of size i        for the formation of water droplets in a supersaturated vapor           calculation of the rates of individual reactions           ← calculation of the number of water molecules in the vapor which hit a droplet               per unit time using kinetic gas theory, neglecting reverse reaction             the number of nuclei which grow above the critical size per unit time             I = zA*n* = z·4π(r*)2·n1·exp (-ΔG*/kT)             z : the collision frequency, according to the kinetic gas theory                 z = p/(2πmkT)1/2                     p : vapor pressure        m : mass of molecules

Classical theory of nucleation □ Becker and Döring (1935) :     ▷ Improved the treatment considering accommodation factor and reverse reaction 0 < α < 1 accommodation factor β: correction for reverse reaction I = α βZA*n*     ※ "No one can prove" H.Reiss, J. Chem. Phys. 20, 1216 (1952)

Nucleation Rate in Solids  Collision frequency : → rate by which an atom will jump across the phase interface      : diffusion energy barrier term, exp (-Qdiff/kT), should be appended.     Russell :        Js = Zβ*No* exp (-ΔGn*/kT)  : steady state nucleation rate             Zeldovich factor                                                No* :  number of nucleation site ("per mole" or "per volume")                     J = Js exp (-τ/t) : time-dependent nucleation rate             incubation (induction) time           

Nucleation Rate in Solids Christian :               rate at which atom     number of embryos                will transfer to         of critical size               critical embryo         per unit volume               and make it grow                     : a jump (attempt) frequency              ΔGa        : activation energy for diffusion             Nv         : number of possible nucleation sites per unit volume             ∵  "Observable rate" of 106/m3․sec  requires ΔGc ≲ 70 kT

Homogeneous Nucleation Nucleation of liquid from vapor assume spherical nucleus

Homogeneous Nucleation

Heterogeneous Nucleation Nucleation of solid between liquid and solid mould ※ Physical meaning of f(θ) ※ Application of the concept of f(θ) to non-spherical nuclei            ※ Heterogeneous nucleation in wall crack

Nucleation in Solids

Nucleation in Solids – Effect of Elastic Strain (J.D. Eshelby) with elastic isotropy

Nucleation in Solids

Nucleation in Solids ※ Similar relations for grain boundary edge and corner nucleation can be worked out.          ⇒ for a given θ, ΔG* decreases as the "dimensionality" of the site decreases.   (d = 0, 1, 2, 3 for C, E, B, H respectively)   But although ΔG* decreases, the number of sites available for nucleation also decreases as dimensionality decreases. Set   L : average grain diameter               δ : grain boundary thickness              Nv : number of atoms per unit volume             ⇒ NvB = Nv (δ/L)     # of boundary sites per volume           NvE = Nv (δ/L)2              # of edge sites per volume             NvC = Nv (δ/L)3              # of corner sites per volume

Nucleation in Solids Substituting into general expression for I :

Nucleation in Solids

Nucleation in Solids - nucleation in dislocations (incoherent) Refs. : J.W. Cahn, Acta Met. 5, 169 (1957).            - incoherent nucleation            F.C. Larché, "Nucleation and Precipitation on Dislocations,"            in Dislocations in Metallurgy, Chap.14 of Vol.III of Dislocations in Solids, F.R.N. Nabarro (1979)    - coherent nucleation