Thermodynamic and Kinetic Modeling to Predict the Lifetime of Thermal Barrier Coating on Superalloys High Temperature Thermochemistry Laboratory & Korea Institute of Materials Science Date: 13th April 2021 Yeon Woo Yoo
Thermodynamic and Kinetic Modeling to Predict the Lifetime of Thermal Barrier Coating
on Superalloys
High Temperature Thermochemistry Laboratory&
Korea Institute of Materials ScienceDate: 13th April 2021
Yeon Woo Yoo
High Temperature Thermochemistry Laboratory
2Contents
I. Introduction about Thermal Barrier Coatings
II. Kinetic Modeling
III. Thermodynamic Modeling
3
I. Introduction about Thermal Barrier Coatings
High Temperature Thermochemistry Laboratory
4Introduction
- Thermal Barrier Coatings
• Top coating- Yttria stabilized zirconia (8YSZ), GZO(Gd2Zr2O7), LZO(La2Zr2O7)
- Thermal insulation from high temperature environment
- Low thermal conductivity and porous microstructure
• Bond coating- MCrAlX M= Ni and/or Co , X = Y, Ta, Hf, and/or Si, other minor
elements
- Intermediate thermal expansion coefficient between top coating and
bottom Ni based superalloys
- Directly related to the thermal lifetime of thermal barrier coatings
• Ni based superalloys- Maintain excellent mechanical strength at high temperature
(γ and γ` phase)
High Temperature Thermochemistry Laboratory
5Introduction
- Failure of Thermal Barrier Coatings
Thermal strain caused by CTE mismatch
𝜀𝜀 = − 𝛼𝛼𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 − 𝛼𝛼𝑠𝑠𝑠𝑠𝑠𝑠 𝑇𝑇 − 𝑇𝑇0 = −Δ𝛼𝛼Δ𝑇𝑇
Repeating heating and cooling in TBCs as the gas turbine operation
Thermal stress caused by CTE mismatch between bond coating and top coatingFailure
High Temperature Thermochemistry Laboratory
6Introduction
- Thermodynamics and Kinetics in Thermal Barrier Coatings
Con
cent
ratio
nTop coat Bond coat SuperalloyTGO
O
O
Al
Al
Ni, Cr, Co
Al
Al
Al
Al
O Cr, Co
Al
Al
Other Elements
NiNi
Al
Al
AlCo, Cr
Co, Cr
Distance
Outer Beta Depletion Zone
Inner Beta Depletion Zone
SecondaryReaction Zone
7
II. Kinetic Modeling
High Temperature Thermochemistry Laboratory
8Diffusion Equation
𝐽𝐽𝑖𝑖 = −𝐷𝐷𝑖𝑖𝑑𝑑𝐶𝐶𝑖𝑖𝑑𝑑𝑑𝑑
- Fick’s first law
- Fick’s second law
𝑑𝑑𝐶𝐶𝑖𝑖𝑑𝑑𝑑𝑑
= 𝐷𝐷𝑖𝑖𝑑𝑑2𝐶𝐶𝑖𝑖𝑑𝑑𝑑𝑑2
𝐶𝐶𝐻𝐻 𝐶𝐶𝐿𝐿
𝐶𝐶𝐻𝐻
𝐶𝐶𝐿𝐿
𝐶𝐶
For multi-components,
𝜕𝜕𝐶𝐶𝑖𝑖𝜕𝜕𝑑𝑑
= 𝐷𝐷𝑖𝑖,𝑖𝑖𝜕𝜕2𝐶𝐶𝑖𝑖𝜕𝜕𝑑𝑑2
+𝜕𝜕𝐷𝐷𝑖𝑖,𝑖𝑖𝜕𝜕𝐶𝐶𝑖𝑖
𝜕𝜕𝐶𝐶𝑖𝑖𝜕𝜕𝑑𝑑
+𝜕𝜕𝐷𝐷𝑖𝑖,𝑖𝑖𝜕𝜕𝐶𝐶𝑗𝑗
𝜕𝜕𝐶𝐶𝑗𝑗𝜕𝜕𝑑𝑑
+𝜕𝜕𝐷𝐷𝑖𝑖,𝑖𝑖𝜕𝜕𝐶𝐶𝑘𝑘
𝜕𝜕𝐶𝐶𝑘𝑘𝜕𝜕𝑑𝑑
𝜕𝜕𝐶𝐶𝑖𝑖𝜕𝜕𝑑𝑑
+ 𝐷𝐷𝑖𝑖,𝑗𝑗𝜕𝜕2𝐶𝐶𝑗𝑗𝜕𝜕𝑑𝑑2
+𝜕𝜕𝐷𝐷𝑖𝑖,𝑗𝑗𝜕𝜕𝐶𝐶𝑖𝑖
𝜕𝜕𝐶𝐶𝑖𝑖𝜕𝜕𝑑𝑑
+𝜕𝜕𝐷𝐷𝑖𝑖,𝑗𝑗𝜕𝜕𝐶𝐶𝑗𝑗
𝜕𝜕𝐶𝐶𝑗𝑗𝜕𝜕𝑑𝑑
+𝜕𝜕𝐷𝐷𝑖𝑖,𝑗𝑗𝜕𝜕𝐶𝐶𝑘𝑘
𝜕𝜕𝐶𝐶𝑘𝑘𝜕𝜕𝑑𝑑
𝜕𝜕𝐶𝐶𝑗𝑗𝜕𝜕𝑑𝑑
+𝐷𝐷𝑖𝑖,𝑘𝑘𝜕𝜕2𝐶𝐶𝑘𝑘𝜕𝜕𝑑𝑑2
+𝜕𝜕𝐷𝐷𝑖𝑖,𝑘𝑘𝜕𝜕𝐶𝐶𝑖𝑖
𝜕𝜕𝐶𝐶𝑖𝑖𝜕𝜕𝑑𝑑
+𝜕𝜕𝐷𝐷𝑖𝑖,𝑘𝑘𝜕𝜕𝐶𝐶𝑗𝑗
𝜕𝜕𝐶𝐶𝑗𝑗𝜕𝜕𝑑𝑑
+𝜕𝜕𝐷𝐷𝑖𝑖,𝑘𝑘𝜕𝜕𝐶𝐶𝑘𝑘
𝜕𝜕𝐶𝐶𝑘𝑘𝜕𝜕𝑑𝑑
𝜕𝜕𝐶𝐶𝑘𝑘𝜕𝜕𝑑𝑑
High Temperature Thermochemistry Laboratory
9Finite Difference Method
- Finite Difference Method
∆𝑋𝑋
𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛+1𝐹𝐹0 𝐹𝐹1
𝜕𝜕𝐹𝐹𝜕𝜕𝑋𝑋
=𝐹𝐹𝑛𝑛+1 − 𝐹𝐹𝑛𝑛−1
2∆𝑋𝑋
𝐹𝐹𝑛𝑛−1
𝜕𝜕𝐹𝐹𝜕𝜕𝑋𝑋
=𝐹𝐹𝑛𝑛+1 − 𝐹𝐹𝑛𝑛
∆𝑋𝑋
𝜕𝜕𝐹𝐹𝜕𝜕𝑋𝑋
=𝐹𝐹𝑛𝑛 − 𝐹𝐹𝑛𝑛−1
∆𝑋𝑋
: Forward𝜕𝜕2𝐹𝐹𝜕𝜕𝑋𝑋2
=𝐹𝐹𝑛𝑛+2 − 2𝐹𝐹𝑛𝑛+1 + 𝐹𝐹𝑛𝑛
(∆𝑋𝑋)2
𝜕𝜕2𝐹𝐹𝜕𝜕𝑋𝑋2
=𝐹𝐹𝑛𝑛 − 2𝐹𝐹𝑛𝑛−1 + 𝐹𝐹𝑛𝑛−2
(∆𝑋𝑋)2
𝜕𝜕2𝐹𝐹𝜕𝜕𝑋𝑋2
=𝐹𝐹𝑛𝑛+1 − 2𝐹𝐹𝑛𝑛 + 𝐹𝐹𝑛𝑛−1
(∆𝑋𝑋)2
: Backward
: Central
10
III. Thermodynamic Modeling
High Temperature Thermochemistry Laboratory
11Gibb’s Free Energy & Phase Diagram
G = H − TS
- Gibb’s free energy
- Gibb’s free energy and phase diagram
- At temperature T, the phase which has lowest G is the most stable
Porter, D.A., and Easterling, K.E., Phase Transformation in Metals and Alloys, 2nd Ed. CHAMAN & HALL (1992)
High Temperature Thermochemistry Laboratory
12Gibb’s Free Energy of Solution
- Gibb’s free energy of solution
𝐺𝐺𝑠𝑠𝑐𝑐𝑠𝑠𝑛𝑛 = 𝑋𝑋𝐴𝐴𝐺𝐺𝐴𝐴 + 𝑋𝑋𝐵𝐵𝐺𝐺𝐵𝐵 + 𝑅𝑅𝑇𝑇 𝑋𝑋𝐴𝐴 ln𝑋𝑋𝐴𝐴 + 𝑋𝑋𝐵𝐵 ln𝑋𝑋𝐵𝐵
𝐺𝐺𝑠𝑠𝑐𝑐𝑠𝑠𝑛𝑛 = 𝑋𝑋𝐴𝐴𝐺𝐺𝐴𝐴 + 𝑋𝑋𝐵𝐵𝐺𝐺𝐵𝐵 + Ω𝑋𝑋𝐴𝐴𝑋𝑋𝐵𝐵 + 𝑅𝑅𝑇𝑇 𝑋𝑋𝐴𝐴 ln𝑋𝑋𝐴𝐴 + 𝑋𝑋𝐵𝐵 ln𝑋𝑋𝐵𝐵
𝐺𝐺𝑠𝑠𝑐𝑐𝑠𝑠𝑛𝑛 = 𝑋𝑋𝐴𝐴𝐺𝐺𝐴𝐴 + 𝑋𝑋𝐵𝐵𝐺𝐺𝐵𝐵 + �𝑖𝑖,𝑗𝑗≥1
𝜔𝜔𝐴𝐴𝐵𝐵𝑖𝑖𝑗𝑗 𝑋𝑋𝐴𝐴𝑖𝑖𝑋𝑋𝐵𝐵
𝑗𝑗 + 𝑅𝑅𝑇𝑇 𝑋𝑋𝐴𝐴 ln𝑋𝑋𝐴𝐴 + 𝑋𝑋𝐵𝐵 ln𝑋𝑋𝐵𝐵
: Ideal solution
: Regular solution
: General solution
∆𝐻𝐻𝑚𝑚𝑖𝑖𝑚𝑚 = 0
∆𝐻𝐻𝑚𝑚𝑖𝑖𝑚𝑚 = Ω𝑋𝑋𝐴𝐴𝑋𝑋𝐵𝐵
∆𝑆𝑆𝑚𝑚𝑖𝑖𝑚𝑚 = 𝑅𝑅(𝑋𝑋𝐴𝐴 ln𝑋𝑋𝐴𝐴 + 𝑋𝑋𝐵𝐵 ln𝑋𝑋𝐵𝐵)
∆𝑆𝑆𝑚𝑚𝑖𝑖𝑚𝑚 = 𝑅𝑅(𝑋𝑋𝐴𝐴 ln𝑋𝑋𝐴𝐴 + 𝑋𝑋𝐵𝐵 ln𝑋𝑋𝐵𝐵)
High Temperature Thermochemistry Laboratory
13Solution Mixing Model
- Random Mixing Model
𝐺𝐺𝑠𝑠𝑐𝑐𝑠𝑠𝑛𝑛 = 𝑋𝑋𝐴𝐴𝐺𝐺𝐴𝐴 + 𝑋𝑋𝐵𝐵𝐺𝐺𝐵𝐵 + 𝑅𝑅𝑇𝑇 𝑋𝑋𝐴𝐴 ln𝑋𝑋𝐴𝐴 + 𝑋𝑋𝐵𝐵 ln𝑋𝑋𝐵𝐵 + 𝑍𝑍�𝑔𝑔𝐴𝐴𝐵𝐵𝑖𝑖𝑗𝑗 𝑋𝑋𝐴𝐴𝑖𝑖𝑋𝑋𝐵𝐵
𝑗𝑗
𝐺𝐺𝑠𝑠𝑐𝑐𝑠𝑠𝑛𝑛 = 𝑋𝑋𝐴𝐴𝐺𝐺𝐴𝐴 + 𝑋𝑋𝐵𝐵𝐺𝐺𝐵𝐵 − 𝑇𝑇∆𝑆𝑆𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐 + 𝑛𝑛𝐴𝐴𝐵𝐵(∆𝑔𝑔𝐴𝐴𝐵𝐵/2)
∆𝑆𝑆𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐 = −𝑅𝑅 𝑛𝑛𝐴𝐴 ln𝑋𝑋𝐴𝐴 + 𝑛𝑛𝐵𝐵 ln𝑋𝑋𝐵𝐵 − 𝑅𝑅 𝑛𝑛𝐴𝐴𝐴𝐴 ln(𝑋𝑋𝐴𝐴𝐴𝐴𝑌𝑌𝐴𝐴2
) + 𝑛𝑛𝐵𝐵𝐵𝐵 ln(𝑋𝑋𝐵𝐵𝐵𝐵𝑌𝑌𝐵𝐵2
) + 𝑛𝑛𝐴𝐴𝐵𝐵 ln(𝑋𝑋𝐴𝐴𝐵𝐵2𝑌𝑌𝐴𝐴𝑌𝑌𝐵𝐵
)
𝑌𝑌𝑖𝑖 =𝑍𝑍𝑖𝑖𝑋𝑋𝑖𝑖
𝑍𝑍𝑖𝑖𝑋𝑋𝑖𝑖 + 𝑍𝑍𝑗𝑗𝑋𝑋𝑗𝑗�𝑋𝑋𝐴𝐴𝐵𝐵2 𝑋𝑋𝐴𝐴𝐴𝐴𝑋𝑋𝐵𝐵𝐵𝐵 = 4 exp(− ⁄Δ𝑔𝑔𝐴𝐴𝐵𝐵 𝑅𝑅𝑇𝑇)
∆𝑔𝑔𝐴𝐴𝐵𝐵 = 𝑓𝑓 𝑑𝑑,𝑇𝑇 = 𝜔𝜔𝐴𝐴𝐵𝐵° − 𝜂𝜂𝐴𝐴𝐵𝐵° 𝑇𝑇 + �(𝑖𝑖+𝑗𝑗≥1)
(𝜔𝜔𝐴𝐴𝐵𝐵𝑖𝑖𝑗𝑗 − 𝜂𝜂𝐴𝐴𝐵𝐵
𝑖𝑖𝑗𝑗 𝑇𝑇)𝑌𝑌𝐴𝐴𝑖𝑖𝑌𝑌𝐵𝐵𝑗𝑗
- Modified Quasichemical Model(MQM)
- Random mixing model : ∆𝑆𝑆𝑠𝑠𝑐𝑐𝑠𝑠𝑛𝑛 = Δ𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖𝑐𝑐𝑠𝑠
- Quasichemical model : ∆𝑆𝑆𝑠𝑠𝑐𝑐𝑠𝑠𝑛𝑛 ≠ Δ𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖𝑐𝑐𝑠𝑠, varied with A-B interaction energy
High Temperature Thermochemistry Laboratory
14Thermodynamic Modeling
Thermodynamic modeling is optimization of parameters related to all solutions
I.H. Jung, et al, CALPHAD, 2007, vol. 31 (2), pp. 192-200
High Temperature Thermochemistry Laboratory
15Application of Thermodynamic Calculation
FCC#1
FCC#1
BCC#1
BCC2#1
L12#1
HCP#1
Liquid
Co + Ni + Cr + Al + Y
Temperature [ oC ]
Wei
ght p
erce
nt [
% ]
600 700 800 900 1000 1100 1200 1300 1400 15000
10
20
30
40
50
60
70
80
90
100
1500
Hf2Ni7
Liquid
FCC#1
FCC#1
BCC#1
SIGMA
BCC2#1
BCC2#1L12#1
L12#1
Ni + Co + Cr + Al + Y + Hf + Si
Temperature [ oC ]
Wei
ght p
erce
nt [
% ]
600 700 800 900 1000 1100 1200 1300 14000
10
20
30
40
50
60
70
80
90
100
FCC#1
FCC#1
BCC#1
SIGMA
BCC2#1
BCC2#1L12#1
Liquid
Ni + Co + Cr + Al + Y
Temperature [ oC ]
Wei
ght p
erce
nt [
% ]
600 700 800 900 1000 1100 1200 1300 1400 15000
10
20
30
40
50
60
70
80
90
100
FCC#1
BCC#1
BCC2#1
L12#1
IN792 - NiCoCrAlY1000 oC
Wei
ght p
erce
nt [
% ]
IN792 NiCoCrAlY0
10
20
30
40
50
60
70
80
90
100
• Phase fractions of MCrAlY bond coats as function of a temperature
FCC#1
BCC2#1
IN792 - CoNiCrAlY1000 oC
Wei
ght p
erce
nt [
% ]
IN792 CoNiCrAlY0
10
20
30
40
50
60
70
80
90
100
• Secondary reaction expectation in interface between MCrAlY bond coats and Ni superalloys
Substrate SRZ Bondcoat
Ni, Ta, Re, etc.
Al, Cr, Co, Y
High Temperature Thermochemistry Laboratory
16Summary
1. Lifetime prediction of thermal barrier coatings were required due to the difficulty of real parts experiment and long time experiment.
2. Thermodynamics and kinetics should be considered to predict lifetime of thermal barrier coatings.
3. Kinetic modeling of multicomponent diffusion could be solved by finite difference method.
4. Thermodynamic modeling can be used to predict stable phase at high temperature and reaction between bond coat and superalloys.
Thank you for
your attention!