Byeong-Joo Lee

Byeong-Joo Lee Motivation

Byeong-Joo Lee General Background

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

Byeong-Joo Lee □ 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 p 1 ⇆ p 2 ⇆ p 3 ⇆ p 4 ⇆ p 5 etc. p i represent the particles of various sizes (number of atoms: equilibrium n i = n 1 ․ exp (-ΔG i /kT) n i : 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 · n 1 · 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

Byeong-Joo Lee 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)

Byeong-Joo Lee Nucleation Rate in Solids Collision frequency : → rate by which an atom will jump across the phase interface : diffusion energy barrier term, exp (-Q diff /kT), should be appended. Russell : J s = Zβ * N o * exp (-ΔG n * /kT) : steady state nucleation rate Zeldovich factor N o * : number of nucleation site ("per mole" or "per volume") J = J s exp (-τ/t) : time-dependent nucleation rate incubation (induction) time

Byeong-Joo Lee Nucleation Rate in Solids 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 ΔG a : activation energy for diffusion N v : number of possible nucleation sites per unit volume ∵ "Observable rate" of 10 6 /m 3 ․ sec requires ΔG c ≲ 70 kT Christian :

Byeong-Joo Lee Homogeneous Nucleation Nucleation of liquid from vapor assume spherical nucleus

Byeong-Joo Lee Homogeneous Nucleation

Byeong-Joo Lee 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

Byeong-Joo Lee Nucleation in Solids

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

Byeong-Joo Lee Nucleation in Solids

Byeong-Joo Lee 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 N v : number of atoms per unit volume ⇒ N v B = N v (δ/L) # of boundary sites per volume N v E = N v (δ/L) 2 # of edge sites per volume N v C = N v (δ/L) 3 # of corner sites per volume

Byeong-Joo Lee Nucleation in Solids Substituting into general expression for I :

Byeong-Joo Lee Nucleation in Solids

Byeong-Joo Lee 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