Byeong-Joo Lee

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

Byeong-Joo Lee General Background Wikipedia Jeroen R. Mesters, Univ. of Lübeck

Byeong-Joo Lee General Background

Byeong-Joo Lee 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

Byeong-Joo Lee 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.

Byeong-Joo Lee 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.

Byeong-Joo Lee 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

Byeong-Joo Lee 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

Byeong-Joo Lee Interface-Reaction Controlled Growth - Crystal Growth Mechanism

Byeong-Joo Lee 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

Byeong-Joo Lee 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

Byeong-Joo Lee 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 + 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

Byeong-Joo Lee 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

Byeong-Joo Lee 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, (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, (1997).

Byeong-Joo Lee 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), , "Maximum energy release theory for recrystallization textures," Dong Nyung Lee Metals and Materials 2(3), , 1996.

Byeong-Joo Lee 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.

Byeong-Joo Lee Diffusion Controlled Growth - Precipitate Growth

Byeong-Joo Lee Diffusion Controlled Growth - Precipitate Growth ※ As a thermally activated process with a parabolic growth law · v ∝ ΔX o · x ∝ t 1/2

Byeong-Joo Lee Diffusion Controlled Growth - Precipitate Growth

Byeong-Joo Lee Diffusion Controlled Growth - Effect of interfacial energy

Byeong-Joo Lee Diffusion Controlled Growth - Lengthening of Needles (spherical tip)

Byeong-Joo Lee 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

Byeong-Joo Lee Diffusion Controlled Growth - Growth of a lamella eutectic/eutectoid

Byeong-Joo Lee Diffusion Controlled Growth - Growth of a lamella eutectic/eutectoid

Byeong-Joo Lee Diffusion Controlled Growth - Growth of a lamella eutectic/eutectoid

Byeong-Joo Lee 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.

Byeong-Joo Lee Diffusion Controlled Growth - Coarsening of Precipitates (Ostwald ripening)

Byeong-Joo Lee Diffusion Controlled Growth - Coarsening of Precipitates (Ostwald ripening)

Byeong-Joo Lee Diffusion Controlled Growth - Coarsening of Precipitates (Ostwald ripening)

Byeong-Joo Lee Diffusion Controlled Growth - Coarsening of Precipitates (Ostwald ripening)