thermodynamic and kinetic modeling to predict the lifetime

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


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)


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


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


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


+𝜕𝜕𝐷𝐷𝑖𝑖,𝑖𝑖𝜕𝜕𝐶𝐶𝑗𝑗

𝜕𝜕𝐶𝐶𝑗𝑗𝜕𝜕𝑑𝑑

+𝜕𝜕𝐷𝐷𝑖𝑖,𝑖𝑖𝜕𝜕𝐶𝐶𝑘𝑘

𝜕𝜕𝐶𝐶𝑘𝑘𝜕𝜕𝑑𝑑


+ 𝐷𝐷𝑖𝑖,𝑗𝑗𝜕𝜕2𝐶𝐶𝑗𝑗𝜕𝜕𝑑𝑑2

+𝜕𝜕𝐷𝐷𝑖𝑖,𝑗𝑗𝜕𝜕𝐶𝐶𝑖𝑖


+𝜕𝜕𝐷𝐷𝑖𝑖,𝑗𝑗𝜕𝜕𝐶𝐶𝑗𝑗


+𝜕𝜕𝐷𝐷𝑖𝑖,𝑗𝑗𝜕𝜕𝐶𝐶𝑘𝑘



+𝐷𝐷𝑖𝑖,𝑘𝑘𝜕𝜕2𝐶𝐶𝑘𝑘𝜕𝜕𝑑𝑑2

+𝜕𝜕𝐷𝐷𝑖𝑖,𝑘𝑘𝜕𝜕𝐶𝐶𝑖𝑖


+𝜕𝜕𝐷𝐷𝑖𝑖,𝑘𝑘𝜕𝜕𝐶𝐶𝑗𝑗


+𝜕𝜕𝐷𝐷𝑖𝑖,𝑘𝑘𝜕𝜕𝐶𝐶𝑘𝑘




9Finite Difference Method

- Finite Difference Method

∆𝑋𝑋

𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛+1𝐹𝐹0 𝐹𝐹1

𝜕𝜕𝐹𝐹𝜕𝜕𝑋𝑋

=𝐹𝐹𝑛𝑛+1 − 𝐹𝐹𝑛𝑛−1

2∆𝑋𝑋

𝐹𝐹𝑛𝑛−1


=𝐹𝐹𝑛𝑛+1 − 𝐹𝐹𝑛𝑛

∆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


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)


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)


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


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


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


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!

thermodynamic and kinetic modeling to predict the lifetime

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