1. 3.4.3. Second-Phase Shape: Misfit Strain Effects
lattice parameter의 차이가 적은 경우
→ fully coherent interface →γ-plot에 의해 sphere
→ no strain energy, 그러나…
Elastic strain energy
A. Fully Coherent Precipitates
석출물 격자상수 다름
Fig The origin of coherency strains. The number of lattice points in the hole is conserved.
불일치가 작을 경우
평형모양은
γ-plot으로 예측가능
변형장에 의해 변화
replaced with 2nd phase with small lattice parameter
→ fully coherent interface를 가지면
→ uniform negative dilatational strain

2. if thin disc, (If ν=1/3) δ > 5% δ < 5%
석출물이 얇은 원판인 경우
constrained misfit:
no longer equal. in all direction
Fig For a coherent thin disc there is little misfit parallel to the plane of the disc.
Maximum misfit is perpendicular to the disc.
* 총 탄성 에너지는 석출물의 모양과 기지와 석출물의 탄성 특성에 따라 변화
Elastically Isotropic Materials
& Eβ = Eα
(If ν=1/3)
단, μ=기지의 전단계수, V= 석출물의 부피
→ ΔGs = 석출물 모양에 무관
Influence of strain energy on the equil. shape
δ > 5%
δ < 5%
Interfacial E effect
Dominant: γ-plot
strain E effect
dominant
δ 값에 따라 평형모양 변화

3. B. Incoherent Inclusions
No coherency strain, still misfit strain can arise: vol. misfit
δ (격자 불일치도) → Δ (체적 불일치도)

4. C. Plate-like precipitates
넓은 정합변
부정합 또는 반정합 모서리
두께 증가시 넓은 coherent 면 가로질러 존재하는 constrained misfit 증가
greater strains in the matrix, high shear stress at the corner
넓은 면이 semicoherent로 되는 것이 에너지적으로 바람직함
misfit strain energy가 거의 없는 incoherent inclusion 처럼 행동
Ex) Al-Cu 합금의 θ’ 상

5. 1) Glissile Interfaces Interphase Interfaces in Solid (α/β) (평활 이동 계면)
(interface//burgers vector)
Non-glissile interface
: 전위가 이동해도 계면은 전진하지 않음
: 전위의 이동에 의해 전진할 수 있는 반정합계면
Slip planes must be continuous across the interface.
But, necessarily parallel
→α to be sheared into β : 의미 다음 페이지에서…
전위의 버거스벡터 만큼
Dislocations sliding
Shear deformation
Fig The nature of a glissile interface.

6. 여기서…Low-Angle tilt Boundaries?
Bugers vector → edge dislocation
But, this is not interphase interface.
∵ crystal structure is same,
only lattice rotation
그러므로 위의 에에 해당되지 않는다…
* 전위들이 이동하게 되면 한 결정립은
다른 결정립쪽으로 회전
: 결정구조 변화없고 격자 회전만 있음

7. Glissile Interfaces between two lattices
Shockley partial dislocation
HCP: ABABABAB…
close packed plane: (0001)
close packed directions:
FCC: ABCABCAB…
close packed planes: {111}
close packed directions:
FCC

8. 2) Shockley partial dislocation
Perfect dislocation
C layer에서 보면
C’ → C”
: perfect crystal을 그대로 유지
< Fcc → Fcc >
2) Shockley partial dislocation
C’ → C”로 갈 때
C’ → A → C” 일수 있음 C’
이 burgers vector는
lattice point에 있지 않다.
(격자점과 격자점을 연결하지 못함) A C”
< Fcc → Hcp >

9. Shockley partial dislocation sliding Stacking fault region
적층결함
Dislocation sliding
FCC
Fig (a) An edge dislocation with a Burgers vector b = [112] on (111). (shockley partial dislocation.)
(b) The same dislocation locally changes the stacking sequence from fcc to hcp. a 6
1) 안정한 FCC의 경우, Stacking fault 영역은 높은 자유에너지를 가짐-부분전위 이동 어려움
2) Fcc 격자가 HCP 구조에 대해 준안정상태 (HCP가 더 안정)라면 부분전위 이동 용이/적층결함 영역 증가

10. Glissile Interfaces between two lattices
두개의 (111) 면마다
하나씩 Shorkley partial
dislocation 있음.
Fig Two Shockley partial dislocation on alternate (111) planes create six layers of hcp stacking.
FCC {111} 면
HCP
(0001) 면
Fig An array of Shockley partial dislocations forming a glissile interface between fcc and hcp crystals.

11. C’ A C” 평활 이동 전위가 있는 계면의 중요한 특징: 그 면들이 결정의 모양을 변화시킬 수 있다는 것.
두 개의 (111) 면마다
동일한 Shorkley partial
dislocation 있음.
Pure shear deformation
같은 수를 갖는 3종류의
부분전위에 의한 변태 C’ A C”
모양 변화 없음
Fig Schematic representation of the different ways of shearing cubic close-packed planes into hexagonal close-packed
(a) Using only one Shockley partial, (b) using equal numbers of all three Shockley partials.
조성 변화없이 fcc – hcp 상전이
martensitic transformation
Ex. steel
By glissile dislocation interface

12. 2) Solid / Liquid Interfaces
Interphase Interfaces in Solid (α/β)
2) Solid / Liquid Interfaces
*Essentially same as S/V, *1 원자층 두께의 매우 좁은 천이대,
*Smooth, Faceted, Sharp
Faceted interface
: 일부 금속간 화합물, 대부분의 비금속의 고상/액상, Si, Ge, Sb 같은 원소
Lf/Tm≈ 4R
Lf: heat of fusion
>
>
Fig Solid/liquid interfaces: (a) atomically smooth, (b) and (c) atomically rough, or diffuse interfaces.
*몇 개의 원자층에 걸쳐서 천이가 일어남
*Gradual weakening of interatomic bonds in sol.
& increasing disorder across the interface into liq.
*H and S gradually change across the interface
Diffusion interface (non-faceted)
: 대부분의 금속 ~ R (기체상수)

13. Non-faceted Faceted Cu-Ag 공정 기지내 Ag 수지상 Sn(Sb) 고용체 내에서 생성된 β’-SnSb 화합물
자유에너지 결정학적 방위관계에 무관
Faceted
강한 결정학적 이방성 효과
낮은 지수를 갖는 조밀면으로 둘러싸인 형상

14. γSV > γSL + γLV Solid / Liquid Interfaces at melting point
0.45Lf/Na
γSV > γSL + γLV
γSL ≈ 0.45 γb (결정립계) (≈0.15γSV)
고상/액상 계면 양쪽에서 거리에 따른 H, -TmS 및
G의 변화로서 계면에너지의 생성

15. Interface Migration~모상과 생성상 사이의 원자들의 이동
Heterogeneous Transformation (대부분):
Nucleation (계면 형성) & Growth (계면 이동)
Nucleation barrier Eg. Precipitation
Homogeneous Transformation :
Growth-interface control
No Nucleation barrier Eg. Spinodal decomposition (Chapter 5)
Types of Interface
- Glissile Interface: Athermal, Shape change Military transformation
- Non-Glissile Interface: Thermal, Civilian transformation
원자들의 불규칙 도약
예외) bainite 변태: 열활성화에 의해 일어남/ 평활 이동 계면의 이동에 의한 것과 비슷한 모양 변화 등

16. Classification of Nucleation and Growth Transformation

17. Civilian Transformation
Interface Migration
Civilian Transformation
Interface control
모상과 생성상이 조성변화 없는 경우
(Ex. 순철에서의 α→γ 변태)
계면은 원자가 계면을 가로질러 가는 속도로 성장함
계면의 반응이 느려서 계면 반응 속도가 dominant 함
Composition
Diffusion control
모상과 생성상이 다른 조성을 가지는 경우
(생성상의 성장을 위해 장범위 확산이 필요)
생성상의 성장속도는 격자내 확산에 의해 계면 전방의
과잉 원자들이 제거되는 속도에 의해 제어
Distance β α
Mixed control

18. Diffusion-Controlled and Interface-Controlled Growth
Fig Interface migration with long-range diffusion
(a) 계면 양쪽의 농도 분포
(c) 각 상의 조성에 따른 자유에너지 변화
α/β 계면에 ΔμBi 가 생기는 원인
1) 초기 α상의 조성 X0
2) α상 계면에 B 성분 고갈: B 농도 X0 이하로 감소 Xi
3) 석출물 성장을 위해선 α상→β상으로 B 원자 이동 필요 ΔμBi
4) 석출물 성장을 위해선 계면조성이 평형농도 Xe보다 커야함.
(b) α 상으로 계면이 이동할 때의 구동력

19. 고경각 입계의 이동(3.3.4절)과 유사하게 계면을 가로지르는 B의 실유속으로 인하여 계면 속도 v 생김.
M= 계면 이동도, Vm = β상의 몰당 체적
계면에서 정상상태인 경우,
1) 부정합 계면처럼 계면 이동도가 매우 높다면,
2) 계면 이동도가 낮은 경우,
3) 계면 이동도가 매우 낮은 경우,
ΔμBi 는 최대

20. ΔμBi
는 최대
ΔμBi ~ 0

21. Solidification: Liquid Solid
4 Fold Anisotropic Surface Energy/2 Fold Kinetics, Many Seeds

22. Solidification: Liquid Solid
casting & welding
single crystal growth
directional solidification
rapid solidification
4.1. Nucleation in Pure Metals
Tm : GL = GS
- Undercooling (supercooling) for nucleation: 250 K ~ 1 K
<Types of nucleation>
- Homogeneous nucleation Heterogeneous nucleation

23. ΔG =L-T(L/Tm)≈(LΔT)/Tm
Homogeneous Nucleation
Driving force for solidification
GL = HL – TSL
GS = HS – TSS
ΔG = Δ H -T ΔS
ΔG =0= Δ H-TmΔS
L : ΔH = HL – HS
(Latent heat)
T = Tm - ΔT
ΔS=Δ H/Tm=L/Tm
ΔG =L-T(L/Tm)≈(LΔT)/Tm

24. 4.1.1. Homogeneous Nucleation
: free energies per unit volume
for spherical nuclei (isotropic) of radius : r

25. r < r* : unstable (lower free E by reduce size)
Homogeneous Nucleation
Why r* is not defined by Gr = 0?
r < r* : unstable (lower free E by reduce size)
r > r* : stable (lower free E by increase size)
r* : critical nucleus size
r* dG=0
Unstable equilibrium
Fig The free energy change associated with homogeneous nucleation of a sphere of radius r.

26. 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

27. The creation of a critical nucleus ~ thermally activated process
T↑ → r* ↓
→ rmax↑
of atomic cluster
Fig. 4.5 The variation of r* and rmax with undercooling T

28. Formation of Atomic Cluster
At the Tm, the liquid phase has a volume 2-4% greater than the solid.
Fig. A two-dimensional representation of an instantaneous picture of the liquid structure. Many close-packed crystal-like clusters (shaded) are instantaneously formed.

29. When the free energy of the atomic cluster with radius r is by
Formation of Atomic Cluster
When the free energy of the atomic cluster with radius r is by
how many atomic clusters of radius r would exist in the presence
of the total number of atoms, n0?
# of cluster
of radius r

30. Gibbs Free Energy radius small driving force T ↓ → ΔGr* ↑ → nr ↓
Formation of Atomic Cluster
Compare the nucleation curves
between small and large driving forces.
Gibbs Free Energy
small driving force
T ↓ → ΔGr* ↑ → nr ↓
large driving force
T↑ → ΔGr* ↓ → nr ↑ r* r*
radius

31. ΔGr : excess free energy associated with the cluster
Formation of Atomic Cluster
no : total # of atoms.
ΔGr : excess free energy associated with the cluster
# of cluster of radius r
- holds for T > Tm or T < Tm and r ≤ r*
- nr exponentially decreases with ΔGr
Ex. 1 mm3 of copper at its melting point (no: 1020 atoms)
→ ~1014 clusters of 0.3 nm radius (i.e. ~ 10 atoms)
→ ~10 clusters of 0.6 nm radius (i.e. ~ 60 atoms)
→ effectively a maximum cluster size, ~ 100 atoms
~ 10-8 clusters mm-3 or 1 cluster in ~ 107 mm3

32. The creation of a critical nucleus ~ thermally activated process
Fig. 4.5 The variation of r* and rmax with undercooling T
The number of clusters with r* at T < TN is negligible.

33. 4.1.2. The homogeneous nucleation rate - kinetics
How fast solid nuclei will appear in the liquid at a given undercooling?
C0 : atoms/unit volume
C* : # of clusters with size of C* ( critical size )
clusters / m3
The addition of one more atom to each of these clusters will convert them into stable nuclei.
nuclei / m3∙s
fo ~ 1011 s-1: frequency ∝ vibration frequency energy
of diffusion in liquid surface area
Co ~ 1029 atoms/m3

34. 4.1.2. The homogeneous nucleation rate - kinetics
where
: insensitive to Temp
How do we define TN?
→ critical value for detectable nucleation
- critical supersaturation ratio
- critical driving force
- critical supercooling
→ for most metals, ΔTN~0.2 Tm (i.e. ~200K)
Fig. 4.6 The homogeneous nucleation rate as a function of undercooling ∆T. ∆TN is the critical undercooling for homogeneous nucleation.

35. Real behavior of nucleation
Under suitable conditions, liquid nickel can be undercooled (or supercooled) to 250 K below Tm (1453oC) and held there indefinitely without any transformation occurring.
In the refrigerator, however, water freezes even ~ 1 K below zero.
In winter, we observe that water freezes ~ a few degrees below zero.
Why this happens?
What is the underlying physics?
Which equation should we examine?
Normally undercooling as large as 250 K are not observed.
The nucleation of solid at undercooling of only ~ 1 K is common.

36. 4.1.3. Heterogeneous nucleation
From
Nucleation becomes easy if ↓ by forming nucleus from mould wall.
Fig. 4.7 Heterogeneous nucleation of
spherical cap on a flat mould wall.
In terms of the wetting angle ( ) and the cap radius (r) (Exercies 4.6)
where

37. = 10 → S() ~ 10-4  = 30 → S() ~ 0.02  = 90 → S() ~ 0.5
S(θ) has a numerical value ≤ 1 dependent only on θ (the shape of the nucleus)
= 10 → S() ~ 10-4
 = 30 → S() ~ 0.02
 = 90 → S() ~ 0.5

38. S(θ) has a numerical value ≤ 1 dependent only on θ (the shape of the nucleus)
Fig. 4.8 The excess free energy of solid clusters for homogeneous and heterogeneous nucleation. Note r* is independent of the nucleation site.

39. The Effect of ΔT on ∆G*het & ∆G*hom?
Plot ∆G*het & ∆G* hom vs ΔT
and N vs ΔT.
Fig. 4.9 (a) Variation of ∆G* with undercooling (∆T) for homogeneous and heterogeneous nucleation.
(b) The corresponding nucleation rates assuming the same critical value of ∆G*

40. Barrier of Heterogeneous Nucleation
How about the nucleation at the crevice or at the edge?

41. Nucleation inside the crevice
In both of the nucleation types considered so far it can be shown that
V* : volume of the critical nucleus (cap or sphere)
Nucleation from cracks or crevices should be able to occur at very small undercoolings
even when the wetting angle θ is relatively large. However, that for the crack to be
effective the crack opening must be large enough to allow the solid to grow out without the radius of the solid/liquid inteface decreasing below r*.
Inoculants ~ low values of θ → low energy interface, fine grain size