Thermodynamic and Kinetic Modeling to Predict the Lifetime of Thermal Barrier Coating on Superalloys

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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

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High Temperature Thermochemistry Laboratory

I. Introduction about Thermal Barrier Coatings II. Kinetic Modeling

III. Thermodynamic Modeling

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3

I. Introduction about Thermal Barrier Coatings

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Introduction

4

- 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 (γand γ` phase)

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Introduction

5

- Failure of Thermal Barrier Coatings

Thermal strain caused by CTE mismatch 𝜀𝜀 = − 𝛼𝛼_{𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐} − 𝛼𝛼_{𝑠𝑠𝑠𝑠𝑠𝑠} 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 coating

Failure

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Introduction

6

- Thermodynamics and Kinetics in Thermal Barrier Coatings

Concentration

Top coat TGO Bond coat Superalloy

O O

Al Al

Ni, Cr, Co Al Al

Al Al

O Cr, Co

Al Al

Other Elements

Ni Ni

Al

Al Al

Co, Cr

Distance

Outer Beta Depletion Zone

Inner Beta Depletion Zone

Secondary Reaction Zone

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7

II. Kinetic Modeling

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Diffusion Equation

8

𝐽𝐽_𝑖𝑖 = −𝐷𝐷_𝑖𝑖 𝑑𝑑𝐶𝐶_𝑖𝑖 𝑑𝑑𝑑𝑑

- Fick’s first law

- Fick’s second law

𝑑𝑑𝐶𝐶_𝑖𝑖

𝑑𝑑𝑑𝑑 = 𝐷𝐷_𝑖𝑖 𝑑𝑑²𝐶𝐶_𝑖𝑖 𝑑𝑑𝑑𝑑²

𝐶𝐶_𝐻𝐻 𝐶𝐶_𝐿𝐿

𝐶𝐶_𝐻𝐻

𝐶𝐶_𝐿𝐿 𝐶𝐶

For multi-components,

𝜕𝜕𝐶𝐶_𝑖𝑖

𝜕𝜕𝑑𝑑

= 𝐷𝐷_{𝑖𝑖,𝑖𝑖}𝜕𝜕²𝐶𝐶_𝑖𝑖

𝜕𝜕𝑑𝑑² + 𝜕𝜕𝐷𝐷_{𝑖𝑖,𝑖𝑖}

𝜕𝜕𝑑𝑑 +𝜕𝜕𝐷𝐷_{𝑖𝑖,𝑖𝑖}

𝜕𝜕𝐶𝐶_𝑗𝑗

𝜕𝜕𝑑𝑑 +𝜕𝜕𝐷𝐷_{𝑖𝑖,𝑖𝑖}

𝜕𝜕𝐶𝐶_𝑘𝑘

𝜕𝜕𝑑𝑑

𝜕𝜕𝑑𝑑 + 𝐷𝐷_{𝑖𝑖,𝑗𝑗} 𝜕𝜕²𝐶𝐶_𝑗𝑗

𝜕𝜕𝑑𝑑² + 𝜕𝜕𝐷𝐷_{𝑖𝑖,𝑗𝑗}

𝜕𝜕𝑑𝑑 +𝜕𝜕𝐷𝐷_{𝑖𝑖,𝑗𝑗}

𝜕𝜕𝑑𝑑 + 𝜕𝜕𝐷𝐷_{𝑖𝑖,𝑗𝑗}

𝜕𝜕𝑑𝑑

𝜕𝜕𝑑𝑑 +𝐷𝐷_{𝑖𝑖,𝑘𝑘}𝜕𝜕²𝐶𝐶_𝑘𝑘

𝜕𝜕𝑑𝑑² + 𝜕𝜕𝐷𝐷_{𝑖𝑖,𝑘𝑘}

𝜕𝜕𝑑𝑑 +𝜕𝜕𝐷𝐷_{𝑖𝑖,𝑘𝑘}

𝜕𝜕𝑑𝑑

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Finite Difference Method

9 - Finite Difference Method

∆𝑋𝑋 𝐹𝐹_𝑛𝑛 𝐹𝐹_𝑛𝑛+1 𝐹𝐹₀ 𝐹𝐹₁

𝜕𝜕𝐹𝐹

𝜕𝜕𝑋𝑋 = 𝐹𝐹_𝑛𝑛+1 − 𝐹𝐹_𝑛𝑛−1 2∆𝑋𝑋

𝐹𝐹_𝑛𝑛−1

𝜕𝜕𝐹𝐹

𝜕𝜕𝑋𝑋 = 𝐹𝐹_𝑛𝑛+1− 𝐹𝐹_𝑛𝑛

∆𝑋𝑋

𝜕𝜕𝐹𝐹

𝜕𝜕𝑋𝑋 = 𝐹𝐹_𝑛𝑛 − 𝐹𝐹_𝑛𝑛−1

∆𝑋𝑋

: Forward

𝜕𝜕²𝐹𝐹

𝜕𝜕𝑋𝑋² = 𝐹𝐹_𝑛𝑛+2 −2𝐹𝐹_𝑛𝑛+1 +𝐹𝐹_𝑛𝑛 (∆𝑋𝑋)²

𝜕𝜕²𝐹𝐹

𝜕𝜕𝑋𝑋² = 𝐹𝐹_𝑛𝑛 −2𝐹𝐹_𝑛𝑛−1+𝐹𝐹_𝑛𝑛−2 (∆𝑋𝑋)²

𝜕𝜕²𝐹𝐹

𝜕𝜕𝑋𝑋² = 𝐹𝐹_𝑛𝑛+1 −2𝐹𝐹_𝑛𝑛 +𝐹𝐹_𝑛𝑛−1 (∆𝑋𝑋)²

: Backward

: Central

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III. Thermodynamic Modeling

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Gibb’s Free Energy & Phase Diagram

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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)

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Gibb’s Free Energy of Solution

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- Gibb’s free energy of solution

𝐺𝐺_{𝑠𝑠𝑐𝑐𝑠𝑠𝑛𝑛} = 𝑋𝑋_𝐴𝐴𝐺𝐺_𝐴𝐴 +𝑋𝑋_𝐵𝐵𝐺𝐺_𝐵𝐵 +𝑅𝑅𝑇𝑇 𝑋𝑋_𝐴𝐴ln𝑋𝑋_𝐴𝐴 +𝑋𝑋_𝐵𝐵 ln𝑋𝑋_𝐵𝐵

𝐺𝐺_{𝑠𝑠𝑐𝑐𝑠𝑠𝑛𝑛} = 𝑋𝑋_𝐴𝐴𝐺𝐺_𝐴𝐴 +𝑋𝑋_𝐵𝐵𝐺𝐺_𝐵𝐵 + Ω𝑋𝑋_𝐴𝐴𝑋𝑋_𝐵𝐵 +𝑅𝑅𝑇𝑇 𝑋𝑋_𝐴𝐴ln𝑋𝑋_𝐴𝐴 +𝑋𝑋_𝐵𝐵ln𝑋𝑋_𝐵𝐵

𝐺𝐺_{𝑠𝑠𝑐𝑐𝑠𝑠𝑛𝑛} = 𝑋𝑋_𝐴𝐴𝐺𝐺_𝐴𝐴 + 𝑋𝑋_𝐵𝐵𝐺𝐺_𝐵𝐵 + �

𝑖𝑖,𝑗𝑗≥1

𝜔𝜔_{𝐴𝐴𝐵𝐵}^{𝑖𝑖𝑗𝑗} 𝑋𝑋_𝐴𝐴^𝑖𝑖𝑋𝑋_𝐵𝐵^𝑗𝑗 + 𝑅𝑅𝑇𝑇 𝑋𝑋_𝐴𝐴ln𝑋𝑋_𝐴𝐴 + 𝑋𝑋_𝐵𝐵 ln𝑋𝑋_𝐵𝐵

: Ideal solution

: Regular solution

: General solution

∆𝐻𝐻_{𝑚𝑚𝑖𝑖𝑚𝑚} = 0

∆𝐻𝐻_{𝑚𝑚𝑖𝑖𝑚𝑚} =Ω𝑋𝑋_𝐴𝐴𝑋𝑋_𝐵𝐵

∆𝑆𝑆_{𝑚𝑚𝑖𝑖𝑚𝑚} = 𝑅𝑅(𝑋𝑋_𝐴𝐴ln𝑋𝑋_𝐴𝐴 +𝑋𝑋_𝐵𝐵ln𝑋𝑋_𝐵𝐵)

∆𝑆𝑆_{𝑚𝑚𝑖𝑖𝑚𝑚} =𝑅𝑅(𝑋𝑋_𝐴𝐴ln𝑋𝑋_𝐴𝐴 +𝑋𝑋_𝐵𝐵ln𝑋𝑋_𝐵𝐵)

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Solution Mixing Model

13 - Random Mixing Model

𝐺𝐺_{𝑠𝑠𝑐𝑐𝑠𝑠𝑛𝑛} = 𝑋𝑋_𝐴𝐴𝐺𝐺_𝐴𝐴 +𝑋𝑋_𝐵𝐵𝐺𝐺_𝐵𝐵 +𝑅𝑅𝑇𝑇 𝑋𝑋_𝐴𝐴ln𝑋𝑋_𝐴𝐴 +𝑋𝑋_𝐵𝐵ln𝑋𝑋_𝐵𝐵 + 𝑍𝑍 � 𝑔𝑔_{𝐴𝐴𝐵𝐵}^{𝑖𝑖𝑗𝑗} 𝑋𝑋_𝐴𝐴^𝑖𝑖𝑋𝑋_𝐵𝐵^𝑗𝑗

𝐺𝐺_{𝑠𝑠𝑐𝑐𝑠𝑠𝑛𝑛} = 𝑋𝑋_𝐴𝐴𝐺𝐺_𝐴𝐴 +𝑋𝑋_𝐵𝐵𝐺𝐺_𝐵𝐵 − 𝑇𝑇∆𝑆𝑆^{𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐} +𝑛𝑛_{𝐴𝐴𝐵𝐵}(∆𝑔𝑔_{𝐴𝐴𝐵𝐵}/2)

∆𝑆𝑆^{𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐} = −𝑅𝑅 𝑛𝑛_𝐴𝐴ln𝑋𝑋_𝐴𝐴 +𝑛𝑛_𝐵𝐵ln𝑋𝑋_𝐵𝐵 − 𝑅𝑅 𝑛𝑛_{𝐴𝐴𝐴𝐴} ln(𝑋𝑋_{𝐴𝐴𝐴𝐴}

𝑌𝑌_𝐴𝐴² ) + 𝑛𝑛_{𝐵𝐵𝐵𝐵}ln(𝑋𝑋_{𝐵𝐵𝐵𝐵}

𝑌𝑌_𝐵𝐵² ) + 𝑛𝑛_{𝐴𝐴𝐵𝐵}ln( 𝑋𝑋_{𝐴𝐴𝐵𝐵} 2𝑌𝑌_𝐴𝐴𝑌𝑌_𝐵𝐵) 𝑌𝑌_𝑖𝑖 = 𝑍𝑍_𝑖𝑖𝑋𝑋_𝑖𝑖

𝑍𝑍_𝑖𝑖𝑋𝑋_𝑖𝑖 +𝑍𝑍_𝑗𝑗𝑋𝑋_𝑗𝑗

�

𝑋𝑋_{𝐴𝐴𝐵𝐵}² 𝑋𝑋_{𝐴𝐴𝐴𝐴}𝑋𝑋_{𝐵𝐵𝐵𝐵} = 4 exp(−Δ𝑔𝑔_{𝐴𝐴𝐵𝐵}⁄𝑅𝑅𝑇𝑇)

∆𝑔𝑔_{𝐴𝐴𝐵𝐵} = 𝑓𝑓 𝑑𝑑,𝑇𝑇 = 𝜔𝜔_{𝐴𝐴𝐵𝐵}^° − 𝜂𝜂_{𝐴𝐴𝐵𝐵}^° 𝑇𝑇 + �

(𝑖𝑖+𝑗𝑗≥1)

(𝜔𝜔_{𝐴𝐴𝐵𝐵}^{𝑖𝑖𝑗𝑗} − 𝜂𝜂_{𝐴𝐴𝐵𝐵}^{𝑖𝑖𝑗𝑗} 𝑇𝑇)𝑌𝑌_𝐴𝐴^𝑖𝑖𝑌𝑌_𝐵𝐵^𝑗𝑗

- Modified Quasichemical Model(MQM)

- Random mixing model : ∆𝑆𝑆_{𝑠𝑠𝑐𝑐𝑠𝑠𝑛𝑛} =Δ𝑆𝑆^{𝑖𝑖𝑖𝑖𝑖𝑖𝑐𝑐𝑠𝑠}

- Quasichemical model : ∆𝑆𝑆_{𝑠𝑠𝑐𝑐𝑠𝑠𝑛𝑛} ≠ Δ𝑆𝑆^{𝑖𝑖𝑖𝑖𝑖𝑖𝑐𝑐𝑠𝑠}, varied with A-B interaction energy

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Thermodynamic Modeling

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Thermodynamic modeling is optimization of parameters related to all solutions

I.H. Jung, et al, CALPHAD, 2007, vol. 31 (2), pp. 192-200

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Application of Thermodynamic Calculation

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FCC#1

BCC#1 BCC2#1 L12#1

HCP#1

Liquid

Co + Ni + Cr + Al + Y

Temperature [ ^oC ]

Weight percent [ % ]

600 700 800 900 1000 1100 1200 1300 1400 1500 0

10 20 30 40 50 60 70 80 90 100

1500

Hf2Ni7

Liquid

FCC#1

BCC#1 SIGMA

BCC2#1

BCC2#1 L12#1

L12#1

Ni + Co + Cr + Al + Y + Hf + Si

Temperature [ ^oC ]

600 700 800 900 1000 1100 1200 1300 1400 0

10 20 30 40 50 60 70 80 90 100

FCC#1

BCC#1 SIGMA

BCC2#1

BCC2#1 L12#1

Liquid

Ni + Co + Cr + Al + Y

Temperature [ ^oC ]

600 700 800 900 1000 1100 1200 1300 1400 1500 0

10 20 30 40 50 60 70 80 90 100

FCC#1

BCC#1 BCC2#1

L12#1

IN792 - NiCoCrAlY 1000 ^oC

IN792 NiCoCrAlY

0 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 - CoNiCrAlY 1000 ^oC

IN792 CoNiCrAlY

0 10 20 30 40 50 60 70 80 90 100

• Secondary reaction expectation in interface between MCrAlY bond coats and Ni superalloys

Substrate SRZ Bond coat

Ni, Ta, Re, etc.

Al, Cr, Co, Y

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Summary

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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.

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Thermodynamic and Kinetic Modeling to Predict the Lifetime of Thermal Barrier Coating on Superalloys