1. NIST Diffusion Workshop Gaithersburg, MD 5/3-5/4/2012 The RMS Error of Ternary Diffusivities Measured from One Diffusion Couple John Morral and Laura Turcer The Ohio State University Columbus, OH 43210-1178 William Hopfe Global Materials Engineering & Joining Van Buren Township, MI 48111-5711

2. Concentration profiles below were predicted by two different [D]s, Which [D] was the most accurate?

3. NIST Diffusion Workshop Gaithersburg, MD 5/3-5/4/2012 The RMS Error of Ternary Diffusivities Measured from One Diffusion Couple John Morral and Laura Turcer The Ohio State University Columbus, OH 43210-1178 William Hopfe Global Materials Engineering & Joining Van Buren Township, MI 48111-5711

4. 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

5. Background Alloy 1 2.0 at%Cr 34.2 at%Al Alloy 2 7.0 at%Cr 32.0 at%Al 1997 Experiment by Hopfe NiAl Cr Alloy 2 Composition vector Alloy 1

6. [D] at 1200  C for Ni-4.5at%Cr-33.1at%Al  -phase from W.D. Hopfe (PhD thesis, University of Connecticut, 1997) D CrCr D CrAl D AlCr D AlAl 1DC 11.4213.41-1.40 0.54 2DC 6.08 2.19 5.5715.44 units 10 -9 cm 2 /sec 1-DC diffusivity predictions 2-DC diffusivity predictions 88% error 512% error -125% error -97% error Comparison of 1-DC with 2-DC Diffusivities and Predictions

7. D CrCr D CrAl D AlCr D AlAl 1DC 11.4213.41-1.40 0.54 2DC 6.08 2.19 5.5715.44 units 10 -9 cm 2 /sec 1-DC diffusivity predictions 2-DC diffusivity predictions Data from the second diffusion couple, at  2 [D] 1-DC and [D] 2-DC at 1200  C 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

8. Predicted rms error as a function of composition vector angle 1% error in measurables rms error in D 11 vs  = 12  = 118  Eigenvector directions of [D] Composition vector direction

9. Explanation of why the rms error goes to infinity at the eigenvector directions Cannot recover [D] from one eigenvalue and one eigenvector direction when

10. Predicted rms error as a function of composition vector angle 1% error in measurables rms error in D 11 vs  = 12  = 118  67% predicted error Note that the error scales with the % error 114  Composition vector used to calculate [D] 1-DC

11. Measurable Quantities for a Constant D Analysis SiSi x = 0 Distance Concentration

12. Calculation of the rms error 1. Calculate the rms error of the square root diffusivity 2. Equations for calculating [r] from one diffusion couple 4 equations → 4 r ij 3. The error for each r ij is: 4. The rms error for each r ij is: Calculate the partial derivatives from the above equations All these terms contain in the denominator

13. Predicted Errors for 1-DC Diffusivities from in  phase diffusion couples at 1100  C with average compositions of Ni-9.5at%Cr-7.5at%Al MatLab Program Inputs 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.

14. rms error of r 11 versus composition vector angle = 30°= 119°

15. = 30°= 119° Comparison of rms error of r 11 and r 22 vs composition vector angle

16. = 30°= 119° rms error of r 12 versus composition vector angle

17. = 30°= 119° Comparison of rms error of r 12 and r 21 vs composition vector angle

18. and = 30°= 119° Comparison of rms error of D 11 and r 11 vs composition vector angle Note that D error is ~ twice the r error

19. and = 30°= 119° Comparison of rms error of D 12 and r 12 vs composition vector angle

20. Conclusions 1.Measuring [D] with 1-DC is an ill-posed problem for n  3 2.Expected error = f(  ) not  C 3.Expected error  in D ij is proportional to the error in the measurables.

21. 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?

22. What is the probability that a randomly selected composition vector will give an acceptable [D]? = 30°= 119° r 11

23. = 30°= 119° What is the probability that a randomly selected composition vector will give an acceptable [D]? r 12

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

27. 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

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

29. Predicted rms error as a function of composition vector angle rms error in D 12 vs  114  = 12  = 118 

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

31. Predicted Errors for 1-DC Square root Diffusivities from in  phase diffusion couples at 1100  C with average compositions of Ni-9.5at%Cr-7.5at%Al MatLab Program Inputs 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.