1. 3. Crystal interfaces and microstructure
Types of Interface
solid
vapor
1. Free surface (solid/vapor interface)
2. Grain boundary (α/ α interfaces)
> same composition, same crystal structure
> different orientation
3. inter-phase boundary (α/β interfaces)
> different composition &
crystal structure β α
defect
energy ↑

2. 3.1. Interfacial free energy
Interfacial energy ( : J/m2)
→ The Gibbs free energy of a system containing an interface of area A
→ Gbulk + Ginterface → G = G0 +  A
vapor
solid
Interfacial energy ( ) vs. surface tension (F: a force per unit length)
1) work done : F dA = dG
2) dG =  dA + A d
→ F =  + A d /dA
In case of a liq. film, d /dA = 0, F =  (N/m = J/m2)
Ex) liq. : d /dA = 0
sol. : d /dA ≠ 0, but, very small value
At near melting temperature d /dA = 0
Why? Rearrangement(재배열)을 통한 일정한 표면구조 유지
Fig. 3.1 A liquid film on a wire frame.

3. 3.2 Solid / Vapor Interfaces
* Hard sphere model
- Fcc : density of atoms in these planes decreases as (h2+k2+l2) increases
Fig. 3.2 Atomic configurations on the three closest-packed planes in fcc crystals; (111), (200), and (220).
For (111) plane CN=12
# of Broken Bonds per atom at surface?

4. For (111) plane
# of broken bond at surface : 3 broken bonds
Bond Strength: ε for each atom : ε/2
Lowering of Internal Energy per atom on surface: 3ε/2 ↓
For (200) plane CN=12
# of Broken Bonds per atom at surface?

5. → LS = 12 Na ε/2 ESV = 3 ε/2 = 0.25 LS /Na ESV vs  ? For (111) plane
# of broken bond at surface : 3 broken bonds
Bond Strength: ε for each atom : ε/2
Lowering of Internal Energy per atom on surface: 3ε/2
Heat of Sublimation (승화) in terms of ε?
→ LS = 12 Na ε/2
Energy per atom of a {111} Surface?
ESV = 3 ε/2 = 0.25 LS /Na
ESV vs  ?
인접원자들 간의 관계만 고려하여 근사치임.
 interfacial energy = free energy (J/m2)
→  = G = H – TS
= E + PV – TS (: PV is ignored)
→  = Esv – TSsv (Ssv thermal entropy, configurational entropy)
→ ∂ /∂T = - S : surface energy decreases with increasing T
표면>내부
표면에 공공 등의 형성으로 extra S 존재
0< S < 3 (mJ/m-2K-1)
due to increased contribution of entropy

6. Average Surface Free Energies of Selected Metals
측정 어려움, near Tm
 of Sn : 680 mJ/m2 (Tm : 232ºC)
 of Cu : 1720 mJ/m2 (Tm : 1083ºC)
Higher Tm,
>> stronger bond (large negative bond energy)
>> larger surface energy
cf) G.B. energy gb is about one third of sv
The measured  values for pure metals near the melting temperature
γSV = 0.15 LS /Na J / surface atom

7. Surface energy for high or irrational {hkl} index
Closer surface packing > smaller number of broken bond > lower surface energy
표면에서 끊어진 결합수 {111} {200} {220} 면을 따라 증가 > γSV 면지수 순으로 증가
A crystal plane at an angle  to the close-packed plane will contain broken bonds in excess of the close-packed plane due to the atoms at the steps.
Surface with high {hkl} index
Low index [ex. (111)]
Fig. 2.2 The ‘broken-bond’ model for surface energy.
(cosθ/a)(1/a) : broken bonds
from the atoms on the steps
(sin|θ|/a)(1/a): additional broken bonds

8. Surface energy for high or irrational {hkl} index
ESV = 3 ε/2 = 0.25 LS /Na
Surface energy for high or irrational {hkl} index
(cosθ/a)(1/a) : broken bonds from the atoms on the steps
(sin|θ|/a)(1/a) : additional broken bonds from the atoms on the steps
Attributing /2 energy to each broken bond,
E- plot
Fig. 3.4 Variation of surface energy as a function of θ
The close-packed orientation ( = 0) lies at a cusped minimum in the E plot.
Similar arguments can be applied to any crystal structure
for rotations about any axis from any reasonably close-packed plane.
All low-index planes should therefore be located at low-energy cusps.
Equilibrium shape of a crystal?

9. Distance from center : γsv
Equilibrium shape: Wulff surface
Distance from center : γsv
Several plane A1, A2 etc. with energy 1 , 2
Total surface energy : A11 + A22 .…
= ∑ Ai i → minimum
→ equilibrium morphology
Analytical solution in 2D is reported
Wulff plane
단결정의 평형모형을 예측하는데 유용
How is the equilibrium shape
determined?
- plot
ESV-θ diagram 보다 엔트로피 효과로
cusped minimum 발견 어려움.

10. Equilibrium shape: Wulff surface
{110} 면 등은
나타나지 않음
FCC 금속의 평형모형
정사각형의 {100} 과
정육각형 {111}로 구성
Equilibrium shape can be determined experimentally by annealing small single crystals at high temperatures in an inert atmosphere, or by annealing small voids inside a crystal.
Of course when γ is isotropic, as for liquid droplets, both the γ-plots and equilibrium shapes are spheres.

11. 3.3 Boundaries in Single-Phase Solids
Grain boundary
(α/α interfaces)
Single phase - Poly grain
(hkl) L . . . → → . . .
> same composition, same crystal structure
> different orientation
(hkl)
G.B.
1) misorientation of lattice in two grains
2) orientation of grain boundary
두 개 인접한 결정립의 방위차이 cf. 두 조밀면 만남
인접 결정립과 입계면의 방위관계

12. 3.3 Boundaries in Single-Phase Solids
두 결정립 격자 단일축을 중심으로 적당한 각으로 회전시 일치됨.
tilt boundary
twist boundary
→ misorientation
→ tilt angle
→ misorientation
→ twist angle
- symmetric tilt or twist boundary
- non-symmetric tilt or twist boundary

13. 3.3.1 Low-Angle and High-Angle Boundaries
Low-Angle Boundaries
Symmetrical low-angle tilt boundary
Symmetrical low-angle twist boundary
Fig. 3.7 (a) Low-angle tilt boundary, (b) low-angle twist boundary: ○ atoms in crystal below, ● atoms in crystal above boundary.
평행한 칼날전위의 배열
서로 직교하는 나선전위들의 배열

14. Dislocations

15. Non-symmetric Tilt Boundary
Fig. 3.8 An unsymmetrical tilt boundary. Dislocations with two different Burgers vectors are present.
If the boundary is unsymmetrical, dislocations with different
Burgers vectors are required to accommodate the misfit.
In general boundaries of a mixture of the tilt and twist type,
→ several sets of different edges and screw dislocations.

16. 3.3.1 Low-Angle and High-Angle Boundaries
Low-Angle Tilt Boundaries
→ around edge dislocation : strain ↑
but, LATB ~ almost perfect matching
→ g.b. energy : g.b. → E /unit area
(energy induced from dis.)
양쪽 결정의 방위차
전위의 버거스 벡터
소각 경계 에너지 ~ 입계의 단위면적 안에 있는 전위의 총 에너지
~ 전위의 간격 (D)에 의존

17. 3.3.1 Low-Angle and High-Angle Boundaries
Low-Angle Tilt Boundaries
→ around edge dislocation : strain ↑
but, LATB ~ almost perfect matching
→ g.b. energy : g.b. → E /unit area
(energy induced from dis.)
Sinθ = b/D , at low angle
→ D=b/θ → g.b. is proportional to 1/D
→ Density of edge dislocation in low
angle tilt boundary
(cf. low angle twist boundary → screw dis.)
* Relation between D and  ?
가 매우 작은 경우
D가 매우 크다.
소각 경계 에너지 ~ 입계의 단위면적 안에 있는 전위의 총 에너지
~ 전위의 간격 (D)에 의존

18. Low-Angle tilt Boundaries
1) As θ increases, g.b. ↑ ㅗ ㅗ ㅗ ㅗ
Strain field overlap
→ cancel out ㅗ ㅗ ㅗ ㅗ ㅗ
→ 2) g.b. increases and the increasing rate of  (=d /d θ) decreases.
→ 3) if θ increases further, it is impossible to physically identify the individual dislocations
→ 4) increasing rate of g.b. ~ 0
5) 전위간격이 너무 작아 전위의 구별이 불가능해지고 결정립계 에너지가 방위차와 무관해짐.

19. Soap Bubble Model
소각경계와 고경각경계의 구조적 차이
전위구별 가능
Fig Rafts of soap bubbles showing several grains of varying misorientation. Note that
the boundary with the smallest misorientation is made up of a row of dislocations, whereas
the high-angle boundaries have a disordered structure in which individual dislocations
cannot be identified.
무질서해서 전위구별 불가능

20. High Angle Grain Boundary
→ Broken Bonds
Fig Disordered grain boundary structure (schematic).
High angle boundaries contain large areas of poor fit and have
a relatively open structure.
→ high energy, high diffusivity, high mobility (cf. gb segregation)

21. High Angle Grain Boundary
Low angle boundary
→ almost perfect matching (except dislocation part)
High angle boundary (almost)
→ open structure, large free volume
* low and high angle boundary
high angle g.b.≈ 1/3 S/V.
→ Broken Bonds
Measured high-angle grain boundary energies
* ESV 처럼 γb 도 온도 증가시 감소하는 온도의존형

22. Chapter 3 Crystal Interfaces and Microstructure
계면의 단순한 형태를 사용하여 계면 자유에너지의 근원을 알아보고 계면에너지를 구할 수 있는 몇 가지 방법을 보여줌.
Boundaries in Single-Phase Solids
(a) Low-Angle and High-Angle Boundaries
- symmetric tilt or twist boundary
- non-symmetric tilt or twist boundary
Θ < 15° : 단위 면적 안에 있는 전위의 총 에너지
- Relation between D and  ?
→ low angle tilt boundary
Sinθ = b/D , at low angle
→ D=b/θ → g.b. is proportional to 1/D
→ Density of edge dislocation
high angle g.b.≈ 1/3 S/V.
Θ > 15° : 전위 간격이 너무 작아 전위중심은 중복되고 각각의 전위 물리적 구별이 어려움.
→ Broken Bonds

23. Special High-Angle Grain Boundaries
Boundaries in Single-Phase Solids
(a) Low-Angle and High-Angle Boundaries
Special High-Angle Grain Boundaries
: high angle boundary but with low g.b. .
쌍정립계 ≠ 쌍정면
Coherent twin boundary
(b)
Incoherent twin boundary
symmetric twin boundary
→ low g.b. .
asymmetric twin boundary
→ low g.b. .
입계의 원자들이 변형되지
않은 위치에 존재
쌍정립계 E 입계면의 방위에 따라 민감하게 변화

24. Special High-Angle Grain Boundaries
(c) Twin boundary energy as a function of the grain boundary orientation
Table 3.3 Measured Boundary Free Energies for Crystals in Twin Relationships
(Units mJ/m2)
<<
<

25. Special High-Angle Grain Boundaries
2개의 결정립이 <100> 축을 중심으로 회전
2개의 결정립이 <110> 축을 중심으로 회전
<
고경각 경계~대략 같은 E 가짐
Fig Measured grain boundary energies for symmetric tilt boundaries in Al (a) When
The rotation axis is parallel to <100>, (b) when the rotation axis is parallel to <110>.
Why are there cusps in Fig (b)?
FCC 금속에서 쌍정립계 양쪽 결정의 <110> 축은 서로 70.5° 이룸: 정합 쌍정립계

26. 대칭적 경각입계
소규모 집단의 반복
: 입계원자들
빈공간 거의 없음.
Fig Special grain boundary.

27. Equilibrium in Polycrystalline Materials
현미경 조직 → 서로 다른 입계들이 공간에서 어떻게 연결되는가에 따라 결정
서로 다른 E 갖고 있는 입계 때문에 다결정체 재료의 미세구조가 어떻게 영향을 받는지 고려
Fig Microstructure of an annealed crystal of austenitic stainless steel.

28. Poly grain material 1) Fx =  2) Fy ?
두 결정립은 면(입계), 세 결정립은 선 (결정 모서리),
네 결정립은 점(결정립 모퉁이) 에서 만남.
Poly grain material
입계 교차점에서 평형조건
1) Fx = 
2) Fy ?
P is moved at a small distance(δy)
A. work done by : Fy δy
B. increase boundary energy caused
by the change in orientation δθ ~ l (d/dθ) δθ
Fy δy = l (d/dθ) δθ
→ Fy = d/dθ torque force
→ segment of g.b. moves to low energy position
G.B.는 고 에너지 영역
Equil. ~ no grain boundary
→ 다결정재료 실제 평형조직 아님
→ 어닐링시 다결정체 내의 입계는
이동하거나 회전하여 입계의 교차점
에서 준안정 평형상태 유지

29. → There is little effect of orientation
(3)
(2)
(1)
(1)
(2)
Fy = d/dθ
입계에너지 최소~torque = 0
회전하지 않고 유지하기 위해 입계에 cusp까지 끌어당기는 힘에 대응하는 힘 작용
(3)
→ There is little effect of orientation
How metastable equil. ? → force (torque)

30. * general high angle boundary : dγ/dθ ≈ 0
동일한 입계에너지/방위와 무관
→ consider more simply
→ 3 grain 사이의 연결점에서 준안정 평형에 필요한 요구조건
→ θ = 120º
결정상 2, 3이 1 과 다른 경우도 성립
If the solid-vapor energy (γS/V) is
the same for both grains,
grain 1
grain 2
grain 3
(단, torque 효과 무시)
입계에너지 측정하는 한 방법:
높은 온도에서 시편 어닐링 후
입계와 표면 교차점의 각도 측정

31. 3.3.4. Thermally Activated Migration of Grain Boundaries
G.B 만날 때 1) 입계 E 균형 + 2) 표면 장력 균형
Thermally Activated Migration of Grain Boundaries
If the boundary is curved in the shape of cylinder, Fig. 3.20a,
it is acted on by a force of magnitude /r towards its center of curvature.
Therefore, the only way the boundary tension forces can balance
in three dimensions is if the boundary is planar (r = ) or if it is
curved with equal radii in opposite directions, Fig. 3.20b and c.
Net Force due to Surface Tension
Balance
in three dimensions
Fig (a) A cylindrical boundary with a radius of curvature r is acted on by a force γ/r.
(b) A planar boundary with no net force. (c) A doubly curved boundary with no net force.
* 무질서한 다결정체, 항상 어느한방향으로 실곡률 가짐 → 어닐링시 입계 이동

32. Direction of Grain Boundary Migration during Grain Growth
For isotropic grain boundary energy in two dimensions,
Equilibrium angle at each boundary junction?
→120o
Equilibrium angle at each boundary junction in 3D?
→109o28’
Morphology of metastable equilibrium state
고온 어닐링시 이동
Boundaries around Grain < 6
; grain shrink, disappear
Boundaries around Grain = 6
; equilibrium
Boundaries around Grain > 6
; grain growth
Reduce the # of grains, increase the mean grain size, reducing the total G.B. energy
called grain growth (or grain coarsening): at high temperature above about 0.5 Tm

33. Grain Growth (Soap Bubble Model)
Fig Two-dimensional cells of a soap solution illustration the process of grain growth. Numbers are time in minutes.