1. Byeong-Joo Lee www.postech.ac.kr/~calphad

2. Byeong-Joo Lee www.postech.ac.kr/~calphad Motivation

3. Byeong-Joo Lee www.postech.ac.kr/~calphad General Background

4. Byeong-Joo Lee www.postech.ac.kr/~calphad 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

5. Byeong-Joo Lee www.postech.ac.kr/~calphad □ 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: i) @ 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

6. Byeong-Joo Lee www.postech.ac.kr/~calphad 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)

7. Byeong-Joo Lee www.postech.ac.kr/~calphad 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

8. Byeong-Joo Lee www.postech.ac.kr/~calphad 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 :

9. Byeong-Joo Lee www.postech.ac.kr/~calphad Homogeneous Nucleation Nucleation of liquid from vapor assume spherical nucleus

10. Byeong-Joo Lee www.postech.ac.kr/~calphad Homogeneous Nucleation

11. Byeong-Joo Lee www.postech.ac.kr/~calphad 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

12. Byeong-Joo Lee www.postech.ac.kr/~calphad Nucleation in Solids

13. Byeong-Joo Lee www.postech.ac.kr/~calphad Nucleation in Solids – Effect of Elastic Strain (J.D. Eshelby) with elastic isotropy

14. Byeong-Joo Lee www.postech.ac.kr/~calphad Nucleation in Solids

15. Byeong-Joo Lee www.postech.ac.kr/~calphad 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

16. Byeong-Joo Lee www.postech.ac.kr/~calphad Nucleation in Solids Substituting into general expression for I :

17. Byeong-Joo Lee www.postech.ac.kr/~calphad Nucleation in Solids

18. Byeong-Joo Lee www.postech.ac.kr/~calphad 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