Grain Coarsening at High Temp. annealing: 줄어드는 결정립에 있는 원자들 입계 에너지에 의해 높은 압력을 받음. 격자로부터 떨어져 나가서 성장하는 결정립의 격자 자리에 다시 붙음. Fir. 3. 23 (a) The atomic mechanism of boundary migration. The boundary migrates to the left if the jump rate from grain 1 → 2 is greater than 2 → 1. Note that the free volume within the boundary has been exaggerated for clarity. (b) Step-like structure where close-packed planes protrude into the boundary.

Grain coarsening at high T, annealing → metastable equil. state : # ↓ , size ↑ curvature ~ ΔP ~Δμ High energy Low energy ← ΔG = 2Vm/r ~ Δμ : effect of pressure difference by curved boundary Work : F dx = (2Vm/r) (dx/Vm) → F = 2/r = ΔG/Vm (by curvature) Gibbs-Thomson Eq. Driving force for grain growth : F 결정립 B로 들어가는 물질의 몰수: 1 (dx/Vm) Fig. 3.25 A boundary separating grains with different free energies is subjected to a pulling force F. Pulling force per unit area of boundary : 물질의 단위체적당 자유에너지 차이

Gibbs-Thompson Equation G of a spherical particle of radius, r G of a supersaturated solute in liquid in equilibrium with a particle of radius, r Equil. condition for open system  should be the same. /mole or / per unit volume r r*: in (unstable) equilibrium with surrounding liquid

* How fast boundary moves ? : Grain Growth Kinetics Grain boundary migration by thermally activated atomic jump * (1) → (2) : Flux (1) atoms in probable site : n1 Vibration frequency : ν1 A2 : probability of being accommodated in grain (2) → A2 n1 ν1 exp(-ΔGa/RT) atom/m2s = J1→2 * (2) →(1) : Flux → A1 n2 ν2 exp[-(ΔGa +ΔG) /RT] = J2→1 (1) (2) When ΔG=0, there is no net boundary movement. A2 n1 ν1≈A1 n2 ν2 = Anν When ΔG >0, there will be a net flux from grain 1 to 2. (A2 n1 ν1≈A1 n2 ν2 = Anν) J1→2 - J2→1 = An νexp(-ΔGa/RT) [1- exp(-ΔG/RT)] (고경각 경계 A1≈A2≈1)

If the boundary is moving with a velocity v, the above flux must also be equal to ? J=c•v → v/( Vm/Na ) (Vm/Na : atomic volume) If ΔG is small [ΔG <<RT] exp (- ΔG/RT)항을 Tayler 전개 Jnet = A2n1 ν1exp(-ΔGa/RT) [ΔG/RT] (atom/m2s)= v(Vm/Na) v ~ ΔG/Vm driving force 입계 이동 속도 → F = ΔG/Vm M: 입계 이동도 단위 구동력하에서의 속도 M : mobility = velocity under unit driving force ~ exp (-1/T) The boundary migration is a thermally activated process.

αM(2/D) = dD/dt (α = proportionality const ~1) Kinetic of grain growth * driving force F = ΔG/Vm → v = M (ΔG/Vm) M : exponentially increase with temp. v : relation to grain coarsening 입계 성장속도 D Mean grain size : D Mean radius of curvature of boundary : r if D ∝ r, Mean velocity : v = αM (ΔG/Vm) = dD/dt (ΔG = 2γVm/r) αM(2/D) = dD/dt (α = proportionality const ~1) dD/dt (rate of grain growth) ~ 1/D , exponentially increase with T

→ D2 = Do2 + kt → ∫D dD = ∫ 2αMdt → ½ (D2 - Do2) = 2αMγt → (D2 - Do2) = 4αMγt = kt → D2 = Do2 + kt Do D r = average radius of particles fv = volume fraction of particles if Do ≈ 0 → D = k’t1/2 → D = k’tn (experimental : n << ½, ½ at 순금속 or only high temp.)

Schematic diagram illustrating the possible interactions of second phase particles and migrating grain boundaries.

Pinning by particle

Effect of Second-Phase Particles Interaction with particles Zener Pinning Derive the expression for the pinning effect of grain boundary migration by precipitates. = AB 원의 둘레 단일 입자로 인해 생긴 힘의 최대값

Interaction with particles Zener Pinning fv = volume fraction of randomly distributed particles of radius r Ntotal = number of particles per unit volume If the boundary is essentially planar, Ninteract = 2rNtotal = 3fv/2r2 Given the assumption that all particles apply the maximum pinning force, the total pinning pressure This force will oppose the driving force for grain growth, Only particles within one radius (solid circles) can intersect a planar boundary

Interaction with particles Zener Pinning This force will oppose the driving force for grain growth, For fine grain size → a large volume fraction of very small particles → F = 2/r = ΔG/Vm (by curvature) * Effect of second-phase particles on grain growth → D = k’tn : 미세한 결정립이 안정화되려면 매우 작은 입자 (r )가 많아야 한다 (f ).

Whose mobility would be high between special and random boundaries? 입계구조의 고찰을 통해 High energy G.B. => Open G.B. structure => High mobility Low energy G.B. => closed (or dense) G.B. structure => Low mobility 같은 구동력 Ex) 정합 쌍정립계 But, Ideal Real 특정한 경우 특수 입계가 무질서한 고경각 입계보다 이동도가 크다. Why? 불순물의 양과 거의 무관 합금원소의 양이 증가함에 따라 급격히 감소 불순물의 양에 따라 민감하게 변함 불순물과 입계의 상호작용이 입계의 종류에 따라 변화 Migration rate of special and random boundaries at 300 oC in zone-refined lead alloyed with tin under equal driving forces

Solute drag effect In general, Gb and mobility of Pure metal decreases on alloying. 편석 경향을 나타내는 Gb 는 기지 고용도 감소할 수록 증가 ~Impurities tend to stay at the GB. Generally, Gb, tendency of seg- regation, increases as the matrix solubility decreases. Xb/X0: GB enrichment ratio - 온도 증가시 입계 편석 감소 Gb Xb Mobility of G.B. Alloying elements affects mobility of G.B. X0 : matrix solute concentration/ Xb : boundary solute concentration Gb : free energy reduced when a solute is moved to GB from matrix. 양수, 용질원자와 기지원자와의 원자크기차 클수록 증가, 용질원자간 결합력 감소하면 증가

Normal Grain Growth Grain boundary moves to reduce area and total energy Large grain grow, small grains shrink Average grain size increases Little change of size distribution

Abnormal Grain Growth - Local breaking of pinning by precipitates Discontinuous grain growth of a few selected grains - Local breaking of pinning by precipitates - Anisotropy of grain boundary mobility - Anisotropy of surface & grain boundary energy - Selective segregation of impurity atoms - Inhomogeneity of strain energy Bimodal Size distribution

Abnormal Grain Growth ex) Si steel = discontinuous grain growth or secondary recrystallization

3.4 Interphase Interfaces in Solids coherent, semicoherent incoherent Interphase boundary - different two phases : different crystal structure different composition 3.4.1 Coherent interfaces Perfect atomic matching at interface 화학적인 것은 무시한 채 계면 양쪽 상이 같은 원자배열을 갖고, 두 결정이 특정한 방위를 이루고 있는 경우 α β Fig. 3.32 Strain-free coherent interfaces. (a) Each crystal has a different chemical composition but the same crystal structure. (b) The two phases have different lattices

Which plane and direction will be coherent between FCC and HCP? 3.4.1 Coherent interfaces Which plane and direction will be coherent between FCC and HCP? : Interphase interface will make lowest energy and thereby the lowest nucleation barrier ex) hcp silicon-rich  phase in fcc copper-rich  matrix of Cu-Si alloy → the same atomic configuration → Orientation relation 원자간 거리도 동일 Cu Si - of Cu-Si ~ 1 mJm-2 In general,  (coherent) ~ 200 mJm-2 γcoherent = γstructure + γchemical = γchemical Fig. 3.33 The close-packed plane and directions in fcc and hcp structures.  (coherent) = ch hcp/ fcc 계면의 경우: 정합을 이루는 면은 하나만 존재

When the atomic spacing in the interface is not identical between the adjacent phase, what would happen? Lattice가 같지 않아도 Coherent interface를 만들 수 있다. → lattice distortion → Coherency strain → strain energy Fig. 3.34 A coherent interface with slight mismatch leads to coherency strains in the adjoining lattices. 정합 계면에서 형성된 strain은 계의 총 에너지를 증가시킴. How can this coherent strain can be reduced?

→ “misfit dislocations” If coherency strain energy is sufficiently large, → “misfit dislocations” → semi-coherent interface δ가 작다면, b: Burgers vector of disl. [b=(dα + dβ )/2] Fig. 3.35 A semicoherent interface. The misfit parallel to the interface is accommodated by a series of edge dislocations.

(2) Semicoherent interfaces 전위의 burgers vector (2) Semicoherent interfaces st → due to structural distortions caused by the misfit dislocations dα < dβ In general,  (semicoherent) ~ 200~500 mJm-2 semi 불일치 전위 간격이 감소함에 따라 변형지역이 중복되어 서로 상쇄됨. γ δ = (dβ - dα)/ dα : misfit → D vs. δ vs. n (n+1) dα = n dβ = D δ = (dβ/ dα) – 1, (dβ/ dα) = 1 + 1/n = 1 + δ → δ = 1/n D = dβ / δ ≈ b / δ [b=(dα + dβ )/2] 0.25 δ 1 dislocation per 4 lattices δ=4 δ가 작다면, 전위의 burgers vector

3) Incoherent Interfaces ~ high angle grain boudnary 1) δ > 0.25 2) different crystal structure (in general) 격자가 잘 일치하는 것이 불가능해짐 In general,  (incoherent) ~ 500~1000 mJm-2 incoherent Fig. 3.37 An incoherent interface.

4) Complex Semicoherent Interfaces If bcc  is precipitated from fcc , which interface is expected? Which orientation would make the lowest interface energy? Nishiyama-Wasserman (N-W) Relationship Kurdjumov-Sachs (K-S) Relationships (두 방위관계의 유일한 차이점은 조밀면에서 5.26°만큼 회전시킨 것임)

Complex Semicoherent Interfaces Semicoherent interface observed at boundaries formed by low-index planes. (atom pattern and spacing are almost equal.) N-W relationship 격자가 잘 일치하는 부분: 점선 영역으로 제한됨. 이러한 넓은 계면은 부정합 임. K-S 방위관계에서도 유사한 거동 나타남. Fig. 3.38 Atomic matching across a (111)fcc/(110)bcc interface bearing the NW orientation relationship for lattice parameters closely corresponding to the case of fcc and bcc iron.

Complex Semicoherent Interfaces The degree of coherency can, however, be greatly increased if a macroscopically irrational interface is formed. The detailed structure of such interfaces is , however, uncertain due to their complex nature.