1. Nucleation Kinetics Byeong-Joo Lee Phase Transformations POSTECH - MSE

2. Motivation

3. General Background

4. 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. 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

6. 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. 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           

8. 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

9. Homogeneous Nucleation
Nucleation of liquid from vapor
assume spherical nucleus

10. Homogeneous Nucleation

11. 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. Nucleation in Solids

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

14. Nucleation in Solids

15. 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

16. Nucleation in Solids
Substituting into general expression for I :

17. Nucleation in Solids

18. 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