Byeong-Joo Lee Byeong-Joo Lee POSTECH - MSE Nucleation Kinetics.

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

Byeong-Joo LeeByeong-Joo Lee

POSTECH - MSEPOSTECH - [email protected]@postech.ac.kr

Nucleation Nucleation KineticsKinetics


MotivationMotivation


General BackgroundGeneral Background


Classical theory of nucleationClassical theory of nucleationReferences : 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


□ 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 p⇆ 2 p⇆ 3 p⇆ 4 p⇆ 5 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 nucleationClassical theory of nucleation


□ Becker and Döring (1935) :

▷ Improved the treatment considering accommodation factor and reverse reaction

0 < α < 1 accommodation factorβ: correction for reverse reaction

I = α βZA*n*

※ "No one can prove"

H.Reiss, J. Chem. Phys. 20, 1216 (1952)

kTnGv v

*2

ZZ '

Classical theory of nucleationClassical theory of nucleation


Nucleation Rate in SolidsNucleation 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


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 Δ․ Gc 70 ≲ kT

Christian :

Nucleation Rate in SolidsNucleation Rate in Solids


Homogeneous NucleationHomogeneous Nucleation

Nucleation of liquid from vaporassume spherical nucleus


Homogeneous NucleationHomogeneous Nucleation


Nucleation of solid between liquid and solid mould

Heterogeneous NucleationHeterogeneous Nucleation

Derive Derive it!it!


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

Heterogeneous NucleationHeterogeneous Nucleation


Nucleation in SolidsNucleation in Solids


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

with elastic isotropywith elastic isotropy


※ 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



Substituting into general expression for I :


Byeong-Joo Lee Byeong-Joo Lee POSTECH - MSE Nucleation Kinetics.

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