1. Growth Kinetics Byeong-Joo Lee Microstructure Evolution POSTECH - MSE

2. General Background ※ References:
1. W.D. Kingery, H.K. Bowen and D.R. Uhlmann,
"Introduction to Ceramics", John Wiley & Sons.  Chap. 8.
     2. Christian, section 56 & 54.
     3. J. Burke, "The Kinetics of Phase Transformations in Metals,"
         Pergamon Press. Chap. 6.

3. General Background
Jeroen R. Mesters, Univ. of Lübeck
Wikipedia

4. General Background

5. Objective Crystal Growth vs. Grain Growth vs. Precipitate Growth
Driving force & Rate Determining Step
Parallel process vs. Serial Process
Interface Reaction vs. Diffusion Controlled Process
4. Interface: Continuous Growth vs. Lateral Growth

6. Classification of Growth Process - Diffusion Controlled Growth
▷ Changes which involve long-range diffusional transport
▷ Assumptions
       
local equilibrium at the interface : the concentration on either
side of the interface is given by the phase diagram
   ※ for conditions under which this assumption might break down,
        see: Langer & Sekerka, Acta Metall. 23, 1225 (1975).
capillarity effects are ignored.
the diffusion coefficient is frequently assumed to be
independent from concentration.   

7. Classification of Growth Process - Interface-Reaction Controlled Growth
    ▷ Changes which do not involve long-range diffusional transport
          ex) growth of a pure solid
             grain growth - curvature driven kinetics
              recrystallization
              massive transformation
              martensitic transformation
              antiphase domain coarsening
              order-disorder transformation
    ※ Even phase transformations that involve composition changes
may be interface-reaction limited.
- local equilibrium is not applied at the interface.

8. Continuous Growth Lateral motion of steps
Interface-Reaction Controlled Growth - Mechanism
□ Two types of IRC growth mechanism
    - Continuous growth and growth by a lateral migration of steps
      Continuous growth can only occur when the boundary is unstable
with respect to motion normal to itself.
    - It can add material across the interface at all points with
equal ease.
    - Comparison of the two mechanisms
Continuous Growth                             Lateral motion of steps
   disordered interface                        ordered/singular interface
diffuse interface                             sharp interface
    high driving force                             low driving force

9. Interface-Reaction Controlled Growth - Crystal Growth Mechanism

10. Interface-Reaction Controlled Growth - Growth of a pure Solid
▷ Lateral growth
     ex) solidification of materials with a high entropy of melting
         minimum free energy ⇔ minimum number of broken bond
     source of ledge of jog :  (i)    surface nucleation
                               (ii)   spiral growth
                              (iii)  twin boundary
                               
    (i) surface nucleation :  
two-dimensional homogeneous nucleation problem
          existence of critical nucleus size, r*
          the growth rate normal to the interface ∝ nucleation rate
                             ⇒   v ∝ exp ( - k2 /ΔTi )
    (ii) spiral growth :       ⇒   v = k3·(ΔTi)2
    (iii) twin boundary :      similar to the spiral growth mechanism

11. Interface-Reaction Controlled Growth - Growth of a pure Solid
ex) single crystal growth during solidification or deposition
▷ Continuous growth
     reaction rate in a thermally activated process 
(in Chemical Reaction Kinetics)
          ⇒   (ν/RT)·exp (-ΔG*/RT)·ΔGdf
       a thermally activated migration of grain boundaries
          ⇒   v = M·ΔGdf
       for example, for solidification        
          ⇒   v = k1․ΔTi

12. Interface-Reaction Controlled Growth - Growth of a pure Solid
▷ Heat Flow and Interface Stability (for pure metal)
     In pure metals solidification is controlled by the conduction rate of the latent heat.
     Consider solid growing at a velocity v with a planar interface into a superheated liquid. 
     Heat flux balance equation  KsT's = KLT'L + v Lv
      when T'L < 0, planar interface becomes unstable and dendrite forms.
     Consider the tip of growing dendrite and assume the solid is isothermal (T's = 0).  
T'L is approximately given by ΔTc/r

13. Interface-Reaction Controlled Growth - Grain growth in polycrystalline solids
▷ Reaction rate
· jump frequency
  νβα = νo exp(-ΔG*/RT)
  ναβ = νo exp(-[ΔG*+ΔGdf]/RT)
⇒ νnet = ν = νo exp(-ΔG*/RT) (1 - exp(-ΔGdf/RT))
 if ΔGdf << RT
 ∴ ν 〓 νo exp(-ΔG*/RT)·ΔGdf / RT
▷ Growth rate, u            
u = λν   ; λ - jump distance

14. Interface-Reaction Controlled Growth - Grain growth in polycrystalline solids
      - no composition change & no phase (crystal structure) change
      - capillary pressure is the only source of driving force
         · α and β is the same phase
         ·
        
       ∴             : normal growth equation
▶ Recrystallization (primary)
     - no composition change & no phase (crystal structure) change
      - stored strain energy is the main source of driving force
        · α and β is the same phase, but α has higher energy (strain energy)

15. Interface-Reaction Controlled Growth - Grain growth in polycrystalline solids
▶ Phase Transformations
     - no composition change & phase (crystal structure) change
     - Gibbs energy difference is the main source of driving force
     - ex) Massive transformation in alloys, Polymorphism
※ Linear relationship between interfacial velocity and driving force are common but not the rule.

16. Diffusion Controlled Growth - Precipitate Growth

17. Diffusion Controlled Growth - Precipitate Growth
※  As a thermally activated process with
a parabolic growth law
· v ∝ ΔXo · x ∝ t 1/2

18. Diffusion Controlled Growth - Precipitate Growth

19. Diffusion Controlled Growth - Effect of interfacial energy

20. Diffusion Controlled Growth - Lengthening of Needles (spherical tip)

21. Diffusion Controlled Growth - Growth of a lamella eutectic/eutectoid
※ Exactly the same results can be obtained when
considering capillarity effect at the tip of each layer

22. Diffusion Controlled Growth - Growth of a lamella eutectic/eutectoid
The interfacial energy serves as an energy barrier, and there exists a critical size in the interlamella spacing S.

23. Diffusion Controlled Growth - Coarsening of Precipitates (Ostwald ripening)

24. Diffusion Controlled Growth - Coarsening of Precipitates (Ostwald ripening)

25. Diffusion Controlled Growth - Coarsening of Precipitates (Ostwald ripening)