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

2. Byeong-Joo Lee www.postech.ac.kr/~calphad 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. Byeong-Joo Lee www.postech.ac.kr/~calphad General Background Wikipedia Jeroen R. Mesters, Univ. of Lübeck

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

5. Byeong-Joo Lee www.postech.ac.kr/~calphad 1.Crystal Growth vs. Grain Growth vs. Precipitate Growth Driving force & Rate Determining Step 2.Parallel process vs. Serial Process 3.Interface Reaction vs. Diffusion Controlled Process 4. Interface: Continuous Growth vs. Lateral Growth Objective

6. Byeong-Joo Lee www.postech.ac.kr/~calphad Classification of Growth Process - Interface-Reaction Controlled Growth Interface-Reaction Controlled Growth □ 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.

7. Byeong-Joo Lee www.postech.ac.kr/~calphad Classification of Growth Process - Diffusion Controlled Growth Controlled Growth □ 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.

8. Byeong-Joo Lee www.postech.ac.kr/~calphad 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. Byeong-Joo Lee www.postech.ac.kr/~calphad 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)·ΔG df a thermally activated migration of grain boundaries ⇒ v = M·ΔG df for example, for solidification ⇒ v = k 1 ․ ΔT i

10. Byeong-Joo Lee www.postech.ac.kr/~calphad Interface-Reaction Controlled Growth - Crystal Growth Mechanism

11. Byeong-Joo Lee www.postech.ac.kr/~calphad 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 ( - k 2 /ΔT i ) (ii) spiral growth : ⇒ v = k 3 ·(ΔT i ) 2 (iii) twin boundary : similar to the spiral growth mechanism

12. Byeong-Joo Lee www.postech.ac.kr/~calphad 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 = K L T' L + v L v 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 ΔT c /r

13. Byeong-Joo Lee www.postech.ac.kr/~calphad Interface-Reaction Controlled Growth - Grain growth in polycrystalline solids ▷ Capillary Effect Consider arbitrarily curved surface element · system condition : V α = V β = V = const. T = const. · dF = -S dT - P dV + γdA = - P β dV β - P α dV α + γdA = - (P β - P α ) dV β + γdA @ equilibrium - (P β - P α ) dV β + γdA = 0 ∴ dA = (r 1 + δr) θ 1 ․ (r 2 + δr) θ 2 - r 1 θ 1 ․ r 2 θ 2 = (r 1 + r 2 ) δr θ 1 θ 2 + (δr) 2 θ 1 θ 2 ≈ (r 1 + r 2 ) δr θ 1 θ 2 dV β 〓 r 1 r 2 θ 1 θ 2 δr

14. Byeong-Joo Lee www.postech.ac.kr/~calphad Interface-Reaction Controlled Growth - Grain growth in polycrystalline solids ▷ Reaction rate · jump frequency ν βα = ν o exp(-ΔG*/RT) ν αβ = ν o exp(-[ΔG*+ΔG df ]/RT) ⇒ ν net = ν = ν o exp(-ΔG*/RT) (1 - exp(-ΔG df /RT)) if ΔG df << RT ∴ ν 〓 ν o exp(-ΔG*/RT)·ΔG df / RT ▷ Growth rate, u u = λν ; λ - jump distance

15. Byeong-Joo Lee www.postech.ac.kr/~calphad Interface-Reaction Controlled Growth - Grain growth in polycrystalline solids ▶ Grain Growth - 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 ※ Role of Mobility / Role of Anisotropy in Grain boundary Energy 1. "Grain Growth Behavior in the System of Anisotropic Grain Boundary Mobility," Nong Moon Hwang, Scripta Materialia 37, 1637-1642 (1997). 2. "Texture Evolution by Grain Growth in the System of Anisotropic Grain Boundary Energy," Nong Moon Hwang, B.-J. Lee and C.H. Han, Scripta Materialia 37, 1761-1767 (1997).

16. Byeong-Joo Lee www.postech.ac.kr/~calphad Interface-Reaction Controlled Growth - Grain growth in polycrystalline solids ▶ 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) ※ Correlation between Deformation Texture and Recrystallization Texture 1. "The evolution of recrystallization textures from deformation textures," Dong Nyung Lee Scripta Metallurgica et Materialia, 32(10), 1689-1694, 1995 2. "Maximum energy release theory for recrystallization textures," Dong Nyung Lee Metals and Materials 2(3), 121-131, 1996.

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

18. Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusion Controlled Growth - Precipitate Growth

19. Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusion Controlled Growth - Precipitate Growth ※ As a thermally activated process with a parabolic growth law · v ∝ ΔX o · x ∝ t 1/2

20. Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusion Controlled Growth - Precipitate Growth

21. Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusion Controlled Growth - Effect of interfacial energy

22. Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusion Controlled Growth - Lengthening of Needles (spherical tip)

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

24. Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusion Controlled Growth - Growth of a lamella eutectic/eutectoid

25. Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusion Controlled Growth - Growth of a lamella eutectic/eutectoid

26. Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusion Controlled Growth - Growth of a lamella eutectic/eutectoid

27. Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusion Controlled Growth - Growth of a lamella eutectic/eutectoid ∴ by examining the dependence of growth rate on S, one can see which one of the two diffusion mechanisms is more important.

28. Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusion Controlled Growth - Coarsening of Precipitates (Ostwald ripening)

29. Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusion Controlled Growth - Coarsening of Precipitates (Ostwald ripening)

30. Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusion Controlled Growth - Coarsening of Precipitates (Ostwald ripening)

31. Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusion Controlled Growth - Coarsening of Precipitates (Ostwald ripening)