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Byeong-Joo Leewww.postech.ac.kr/~calphad

ByeongByeong--Joo LeeJoo Lee

POSTECH POSTECH -- MSEMSE

[email protected]@postech.ac.kr

Pure W W + 0.4wt% Ni

Surface Transition and Alloying Effect Surface Transition and Alloying Effect –– N.M. Hwang et al., 2000.N.M. Hwang et al., 2000.


Pure W W + 0.4wt% Ni

Vaccum Annealing

Abnormal Grain Growth Abnormal Grain Growth –– from N.M. Hwang from N.M. Hwang


ScopeScope

Fundamentals

1. Concept of Surface Energy2. A Rough Estimation of Solid Surface Energy3. Anisotropy of Surface Energy4. Curvature Effect

• Capillary Pressure• Vapor Pressure• Solubility


• Solubility• Melting Point

5. Measurement of Surface/Interface Energy

Interface Phenomena

1. Adsorption2. Segregation3. Crystal Shape 4. Grain Growth

Fdxdxl =γ2

PTA

G

,

∂∂

=γdAVdPSdTdG γ++−=

Concept of Surface Energy and Surface TensionConcept of Surface Energy and Surface Tension


l

F

2=γ

※ Surface Energy and Tension

for Liquids and Solids

A

s

N

Hw

42

3 ∆== ε

εAs ZNH 5.0=∆Pair approximation

Necessary Work for Creationof (111) surface in fcc (/atom)

For fcc (111): N/A = 4/(31/2a2)fcc (100): N/A = 2/a2

Rough Estimation of Solid Surface EnergyRough Estimation of Solid Surface Energy

∆=

A

N

N

H

A

s

4γ


15.13

2

)111(

)100( ==γ

γ

For Cu: a = 3.615 Å △Hs =337.7J/molγ(111) = 2460 erg/cm

2 (1700 by expt.)

For fcc ※ Origin of Anisotropy

fcc (100): N/A = 2/a

Approximation for Estimation of Grain Boundary EnergyApproximation for Estimation of Grain Boundary Energy


Anisotropy of Grain Boundary Energy Anisotropy of Grain Boundary Energy –– [100] and [110] tilt Bd. of Al [100] and [110] tilt Bd. of Al


Why Interface/Grain Boundary Properties?Why Interface/Grain Boundary Properties?


Wetting angle : 36o Wetting angle : 120o

Fe - 0.5% Mn – 0.1% C, dT/dt = 1 oC/s

from SG Kim, Kunsan University

Grain Boundary Identification SchemeGrain Boundary Identification Scheme

How to uniquely define misorientation and inclination between two neighboring grains

(RD) 1x

TD)(2x

ND)(3x

cx2

cx1

cx1

cx1

cx1

cx1

cx1

cx2cx2

cx2cx2

cx2

cx3

cx3

cx3

cx3

cx3

cx3


Grain Boundary Energy of BCC FeGrain Boundary Energy of BCC Fe


Sigma (Σ) Theta (θ) (hkl) plane Sigma (Σ) Theta (θ) (hkl) plane

5 36.87 100 11 144.9 3103 70.53 110 5 180 31011 50.48 110 7 115.38 3109 38.94 110 3 146.44 3113 60 111 9 67.11 3117 38.21 111 11 180 3113 131.81 210 5 95.74 3119 96.38 210 11 100.48 3207 73.4 210 7 149 3205 180 210 7 180 3213 180 211 9 123.75 3215 101.54 211 9 152.73 32211 62.96 211 11 82.16 3317 135.58 211 7 110.92 3319 90 221 5 154.16 3315 143.13 221 11 180 332

Curvature Effect Curvature Effect –– Capillary PressureCapillary Pressure

dAdVPP

dAdVPdVP

dAPdVSdTdF

γ

γ

γ

βαβ

ααββ

+−−=

+−−=

+−−=

)(

System condition

T = constantVα = Vβ = V = constant

@ equilibrium


2121 θθδβ rrrdV =

2121212

212122112211 )()()()()( θθδθθδθθδθθθδθδ rrrrrrrrrrrrrdA +≅++=⋅−+⋅+=

@ equilibrium

βαβ γ

dV

dAPPP =−=∆ )(

+=−=∆

21

11)(

rrPPP γαβ

αβαβ µµγ −=

+=∆=−=∆

21

11

rrVPVGGG mmmm

Curvature Effect Curvature Effect –– Capillary Pressure Effect on Vapor Pressure and SolubilityCapillary Pressure Effect on Vapor Pressure and Solubility

Solubility of pure B phase in a dilute solution

o

o

rvrvmcrc

P

PRTV

r ∞∞∞ =−==− ln

2,,,, µµγµµ

Vapor Pressure

rRT

V

P

P m

o

o

r γ2ln ⋅=

∞


1, =⋅ ∞Bxk

Solubility of pure B phase in a dilute solution

rRT

V

x

xBAm

B

rB −

∞

⋅=γ2

ln,

,

rBBmBA

BrB kxRTaRTVr

,,, lnln2

===− −∞

γµµ

∞∞ =−

,

,,, ln

B

rB

BrBx

xRTµµ

Curvature Effect Curvature Effect –– Capillary Pressure Effect on Melting PointCapillary Pressure Effect on Melting Point

22

r

drVdTS mLSm −=∆ γ

02)()(2=−−−− −

r

drVdPVVdTSS SSLLSLSL γ

rPP LS

LS

−+=γ2

LLLL dPVdTSd +−=µ

++−=+−= −

rPdVdTSdPVdTSd SL

LSSSSS

γµ

2


22

rVdTS mLSm −=∆ γ

rH

TV

rS

VT

m

mmLS

m

mLS

m

122⋅

∆−=

∆−=∆ −− γγ

22

r

drVdTS

rr T

TmLS

T

Tm ∫∫

∞∞−=∆ γ

r

VTSTTS mLSmmrm

−∞ −=∆∆=−∆

γ2)(

Thermodynamics of Nano Particles or Thermodynamics of Nano Particles or NanowiresNanowires


Size dependence of melting points of Au nano particles.

Au

Melting points of Nano ParticlesMelting points of Nano Particles

PtNi


Pt

WMg

PtNi

Melting points of Nano WiresMelting points of Nano Wires


Pt

WMg

Measurement of Surface Energy Measurement of Surface Energy –– Surface Energy of LiquidSurface Energy of Liquid


Rgh

γθρ

⋅=

cos2

θcos/Rr =

rPPP oi

γ2=−=∆

r

mg

πγ =

Measurement of Surface Energy Measurement of Surface Energy –– Surface Energy of SolidSurface Energy of Solid

dAdlmgwrev ⋅=⋅= γδ

dll

rdr

2

1−=

rlA π2=

.2 constlrV == π

)(2 ldrrdldA += π

02 2 =+= dlrldrdV ππ


rπγ =

( )2rnddAgb π=

gbl

nrrmg γπγπ 2−=

gbgbdAdAdlmg γγ +⋅=⋅

Measurement of Surface Energy Measurement of Surface Energy –– Relative Interface Energy of SolidRelative Interface Energy of Solid

2

31

3

12

1

23

sinsinsin θγ

θγ

θγ

==

=2

cos2 1211

θγγ


SVγLSγθ

LVγθγγγ cosLVLSSV +=

2

Interfacial Phenomena Interfacial Phenomena –– AdsorptionAdsorption

θ

θaa

KR

PKR

=

−= )1(

Physisorption : Van der Waals

Chemisorption : Chemical Bonding

Langmuir adsorption : monolayer chemisorptionθ : fraction of surface site covered


KP

KP

+=1

θ

)()1( 2

1 TKK

K

P==

−θθ

θθ da KPK =− )1(

θdd KR =

@ equilibrium

)()1(

TKP

AA =−− θθ

θ

Interfacial Phenomena Interfacial Phenomena –– Competitive AdsorptionCompetitive Adsorption

BAAaa

KR

PKR

θ

θθ

=

−−= )1(

θA : fraction of coverage of AθB : fraction of coverage of B

• A(g) + S ↔ AS• B(g) + S ↔ BS

For A :


)()1(

TKP

A

BAA

=−− θθ

BB

AA

B

A

PTK

PTK

)(

)(=

θθ

Bdd

BABaa

KR

PKR

θ

θθ

=

−−= )1(

Add KR θ=

For B : )()1(

TKP

B

BAB

B =−− θθ

θ

Interfacial Phenomena Interfacial Phenomena –– Multilayer Adsorption (BET isotherm)Multilayer Adsorption (BET isotherm)

1. A(g) + S ↔ AS θ1 = θ0 K1P2. A(g) + AS ↔ A2S θ2 = θ1 K2P3. ….n. A(g) + An-1S ↔ AnS θn = θn-1 KnP

K1 : chemisorption


11 )( −= n

n KPθθ

K1 : chemisorptionK2, …, Kn : the same physisorption (interface)

assume K2 = K3 = … = Kn = K

Interfacial Phenomena Interfacial Phenomena –– Segregation (Guttmann)Segregation (Guttmann)

iiiii

o XRTG σωµγ φφφφ +=+ ln

Assume a one atomic layer surface phase and consider equilibrium

between bulk and surface

where ωi is the molar surface areaAssume ωi = ωj = … = ω

B

i

B

i

B

i

B

i

o XRTG µγ =+ ln

Bfrom µµ φ = iiBoo XRTRTGG

φφφ γ

σω lnln][ ++−=


RTG

B

n

B

i

n

iseg

eX

X

X

X /∆−=φ

φ

B

iifrom µµ φ =

∑−

=

∆−

∆−

−+=

1

1

/

/

)1(1n

j

RTGB

j

RTGB

ii

segj

segi

eX

eXXnentsmulticompofor φ

B

i

i

B

i

iB

i

o

i

o

X

XRTRTGGφ

γγ

σω lnln][ ++−=

B

in

B

niB

n

o

n

oB

i

o

i

oseg RTGGGGGγγγγ

φ

φφφ ln][][ +−−−=∆

Interfacial Phenomena Interfacial Phenomena –– Segregation (Physical Meaning of Quantities)Segregation (Physical Meaning of Quantities)

Bfrom µµ φ =

][][

ln][][

B

n

xsB

i

xs

n

xs

i

xs

n

o

i

o

B

in

B

niB

n

o

n

oB

i

o

i

oseg

GGGG

RTGGGGG

∆−∆−∆−∆+−=

+−−−=∆

φφ

φ

φφφ

σωσω

γγγγ


B

iifrom µµ φ =

B

i

i

B

i

iB

i

o

i

o

iX

XRTRTGG

φφφ

γγ

σω lnln][ ++−=

B

i

i

i

B

i

xs

i

xs

i

B

i

o

i

o

i X

XRTGGGG

φφφ

ωωωσ ln][

1][

1+∆−∆+−=

Interfacial Phenomena Interfacial Phenomena –– Segregation (Butler/Tanaka)Segregation (Butler/Tanaka)

L=≠• ji ωω

)/ln(][1

][1

)/ln(][1

][1

B

nn

n

B

n

xs

n

xs

n

B

n

o

n

o

n

B

ii

i

B

i

xs

i

xs

i

B

i

o

i

o

i

XXRT

GGGG

XXRT

GGGG

φφφ

φφφ

ωωω

ωωωσ

+∆−∆+−=

=

+∆−∆+−=

L

)/ln()/ln(][1

][1

][1

][1 B

nn

n

B

ii

i

B

n

xs

n

xs

n

B

i

xs

i

xs

i

B

n

o

n

o

n

B

i

o

i

o

i

XXRT

XXRT

GGGGGGGG φφφφφφ

ωωωωωω−=

∆−∆−∆−∆+−−−−


( ) ( ) RTGXXXX seg

i

B

nn

B

iini ////ln '11

∆−=

ωφωφ

( ) RTGB

nn

B

ii

segi

n

i

eXXXX/// ∆−⋅= ω

ωφφ

][][][][B

n

xs

n

xs

n

iB

i

xs

i

xsB

n

o

n

o

n

iB

i

o

i

oseg

i GGGGGGGGG ∆−∆−∆−∆+−−−=∆φφφφ

ωω

ωω

Interfacial Phenomena Interfacial Phenomena –– Segregation (Example)Segregation (Example)


Interfacial Phenomena Interfacial Phenomena –– Equilibrium Shape of A CrystalEquilibrium Shape of A Crystal


102

22 γγ c

caES +

−=

−=

2

122γγ

a

c

.4

22 const

caA =−=

cc

afor =

−=

2

2210 γγ

Interfacial Phenomena Interfacial Phenomena –– Abnormal Grain Growth (N.M. Hwang)Abnormal Grain Growth (N.M. Hwang)


Download - ByeongByeong--Joo Lee Joo Lee - Postech CMSE Lab. · ByeongByeong--Joo Lee Joo Lee POSTECH POSTECH ... Rough Estimation of Solid Surface Energy ... ∆Gi = G − G − G − G + ∆

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