Growth Kinetics

1. Phase Transformations Growth Kinetics Byeong-Joo LeePOSTECH-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 • Nucleation vs. Growth • Crystal Growth vs. Grain Growth vs. Precipitate Growth • Driving force • Rate Determining Step • Parallel process vs. Serial Process

4. General Background

5. Classification of Growth Process - 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.

6. Classification of Growth Process - Diffusion 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.   

7. 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 GrowthLateral motion of steps     disordered interface                         ordered/singular interface diffuse interface                             sharp interface         high driving force                               low driving force

8. 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

9. 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

10. 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

11. Interface-Reaction Controlled Growth - Grain growth in polycrystalline solids ▷ Capillary Effect     Consider arbitrarily curved surface element · system condition : Vα=Vβ= V = const.      T = const. · dF   = -S dT - P dV + γdA          = - Pβ dVβ - Pα dVα + γdA          = - (Pβ - Pα) dVβ + γdA @ equilibrium      - (Pβ - Pα) dVβ + γdA = 0 ∴          dA  = (r1 + δr) θ1․(r2 + δr) θ2 - r1 θ1․r2 θ2        = (r1 + r2) δr θ1θ2 + (δr)2θ1 θ2 ≈ (r1 + r2) δr θ1θ2 dVβ 〓 r1r2θ1θ2 δr

12. 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

13. 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).

14. 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.

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 effectat the tip of each layer

22. Diffusion Controlled Growth - Growth of a lamella eutectic/eutectoid

23. Diffusion Controlled Growth - Growth of a lamella eutectic/eutectoid

24. Diffusion Controlled Growth - Growth of a lamella eutectic/eutectoid

25. 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.

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

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

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

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