NIST Diffusion WorkshopGaithersburg, MD 5/3-5/4/2012

The RMS Error of Ternary DiffusivitiesMeasured from One Diffusion Couple

John Morral and Laura TurcerThe Ohio State University

Columbus, OH 43210-1178

William HopfeGlobal Materials Engineering & JoiningVan Buren Township, MI 48111-5711

Concentration profiles below were predicted by two different [D]s,

Which [D] was the most accurate?

NIST Diffusion WorkshopGaithersburg, MD 5/3-5/4/2012

The RMS Error of Ternary DiffusivitiesMeasured from One Diffusion Couple

John Morral and Laura TurcerThe Ohio State University

Columbus, OH 43210-1178

William HopfeGlobal Materials Engineering & JoiningVan Buren Township, MI 48111-5711

Outline

• Background: Hopfe’s experiments • rms error predictions for Ni-Cr-Al

-phase diffusion couples• Derivation of the rms error • Additional predictions for Ni-Cr-Al

-phase diffusion couples• Conclusions• Questions

Background

Alloy 12.0 at%Cr34.2 at%Al

Alloy 27.0 at%Cr32.0 at%Al

1997 Experiment by Hopfe

Ni Al

Cr

Alloy 2

Composition vector

Alloy 1

.deg661 .deg12

[D] at 1200C for Ni-4.5at%Cr-33.1at%Al -phase from W.D. Hopfe (PhD thesis, University of Connecticut, 1997)

DCrCr DCrAl DAlCr DAlAl

1DC 11.42 13.41 -1.40 0.542DC 6.08 2.19 5.57 15.44

units 10-9cm2/sec

1-DC diffusivity predictions 2-DC diffusivity predictions

88%error

512%error

-125%error

-97%error

100ij

ij

D

D

Comparison of 1-DC with 2-DC Diffusivities and Predictions

DCrCr DCrAl DAlCr DAlAl

1DC 11.42 13.41 -1.40 0.542DC 6.08 2.19 5.57 15.44

units 10-9cm2/sec

1-DC diffusivity predictions 2-DC diffusivity predictions

Data from the second diffusion couple, at 2

[D]1-DCand [D]2-DC at 1200C for Ni-4.5at%Cr-33.1at%Al -phase From W.D. Hopfe (PhD thesis, University of Connecticut, 1997)

Comparison of 1-DC with 2-DC Diffusivities and Predictions

Predicted rms error as a function of composition vector angle

D

D%

1% error in measurables

rms error in D11 vs

e1

e2= 12 = 118

Eigenvector directions of [D]

Composition vector direction

Explanation of why the rms error goes to infinity at the eigenvector directions

tD

xerfA

tD

xerfAtxc

eieii

2

2

1

122

,

2221

1211

DD

DD ee

ee DD

21

21

ee

ee DD

21

21

Cannot recover [D] from one eigenvalue and one

eigenvector direction

when e1

Predicted rms error as a function of composition vector angle

D

D%

1% error in measurables

rms error in D11 vs

e1

e2= 12 = 118

67% predicted

errorNote that the error

scales with the % error

114

Composition vector used to calculate

[D]1-DC

Measurable Quantities for a Constant D Analysis

Si

x = 0

Distance

Con

cent

ratio

n

Calculation of the rms error

1. Calculate the rms error of the square root diffusivity

2. Equations for calculating [r] from one diffusion couple

022

011 CrCr

tS iii

2221

1211

rr

rr4 equations→ 4 rij

3. The error for each rij is:

02

10

00 k

k k

ijk

k

ijk

k

ijij C

C

rC

C

rS

S

rr

4. The rms error for each rij is:

2

1

2

00

2

00

2

kk

k

ijk

k

ijk

k

ijrmsij C

C

rC

C

rS

S

rr

Calculate the partial derivatives from the

above equations

All these terms contain

in the denominator

01

02

02

01 CCCC

022

011

0 2 CrCrtC iii

Predicted Errors for 1-DC Diffusivities from in phase diffusion couples at 1100C

with average compositions of Ni-9.5at%Cr-7.5at%Al

smDC

/1026.178.0

76.020.2 214

1100

smrC

/1008.131.0

30.045.1 27

1100

MatLab Program Inputs

01.2

2

1

1

1

1

1

1

2

2

1

1

S

S

S

S

C

C

C

C

C

C

C

C

Thompson, M. S., J. E. Morral and A. D. Romig, Jr. 1990. Applications of

the square root diffusivity to diffusion in Ni‑Al‑Cr alloys. Metall. Trans. A. 21A:2679‑2685.

rms error of r11 versus composition vector angle

e1

e2= 30° = 119°

e1

e2= 30° = 119°

Comparison of rms error of r11 and r22 vs composition vector angle

e1

e2= 30° = 119°

rms error of r12 versus composition vector angle

e1

e2= 30° = 119°

Comparison of rms error of r12 and r21 vs composition vector angle

and

D

D%

e1

e2= 30° = 119°

Comparison of rms error of D11 and r11 vs composition vector angle

Note that D error is ~ twice the r error

and

D

D%

e1 e

2= 30° = 119°

Comparison of rms error of D12 and r12 vs composition vector angle

Conclusions

1. Measuring [D] with 1-DC is an ill-posed problem for n 32. Expected error = f() not C3. Expected errorin Dij is proportional to the error in the

measurables.

This program is based on using the square diffusivity equations. Will it predict the error if another method is used (e.g Roper and Whittle)?

Discussion Questions

Can these equations be extended to systems in which n>3?

How can this program be used if you need to now the diffusivity before selecting a 1-DC composition vector?

What is the probability that a randomly selected composition vector will give an acceptable [D]?

e1

e2= 30° = 119°

r11

e1

e2= 30° = 119°

What is the probability that a randomly selected composition vector will give an acceptable [D]?

r12

What can you tell by inserting a 1-DC [D] into the Error prediction program?

Hopfe, W. D., Y.-H. Son, J. E. Morral and A. D. Romig, Jr. Measuring the diffusivity of B2 nickel aluminide alloys containing

chromium using the square root diffusivity analysis. Diffusion in Ordered Alloys. ed. by B. Fultz, R. W. Cahn and D. Gupta.

(TMS. Warrendale, PA.1993) pp. 69-76.

Reference to Diffusivity measurements by Hopfe

Diffusion couple Data at 1100C and [D]* measured for the Ni-9.0at%Cr-7.5at%Al phase

smDC

/1026.178.0

76.020.2 214

1100

smrC

/1008.131.0

30.045.1 27

1100

smDC

/10608.219.0

557.0544.1 212

1200

smrC

/107597.1101.0

2799.0230.1 26

1200

Predicted rms error as a function of composition vector angle

D

D%

rms error in D12 vs

114

e1

e2= 12 = 118

Diffusion couple Data at 1100C and [D]* measured for the Ni-9.0at%Cr-7.5at%Al phase

smDC

/1026.178.0

76.020.2 214

1100

smrC

/1008.131.0

30.045.1 27

1100

smDC

/10608.219.0

557.0544.1 212

1200

smrC

/107597.01101.0

2799.0230.1 26

1200

Predicted Errors for 1-DC Square root Diffusivities from in phase diffusion couples at 1100C

with average compositions of Ni-9.5at%Cr-7.5at%Al

smDC

/1026.178.0

76.020.2 214

1100

smrC

/1008.131.0

30.045.1 27

1100

MatLab Program Inputs

01.2

2

1

1

1

1

1

1

2

2

1

1

S

S

S

S

C

C

C

C

C

C

C

C

Thompson, M. S., J. E. Morral and A. D. Romig, Jr. 1990. Applications of

the square root diffusivity to diffusion in Ni‑Al‑Cr alloys. Metall. Trans. A. 21A:2679‑2685.