NIST Diffusion Workshop Gaithersburg, MD 5/3-5/4/2012

1. NIST Diffusion WorkshopGaithersburg, 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 WorkshopGaithersburg, 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. Alloy 1 Alloy 2 Background 1997 Experiment by Hopfe Alloy 1 2.0 at%Cr 34.2 at%Al Alloy 2 7.0 at%Cr 32.0 at%Al Cr Composition vector Ni Al

6. Comparison of 1-DC with 2-DC Diffusivities and Predictions [D] at 1200C for Ni-4.5at%Cr-33.1at%Al -phase from W.D. Hopfe (PhD thesis, University of Connecticut, 1997) 88% error 512% error -125% error -97% error DCrCr DCrAl DAlCr DAlAl 1DC 11.42 13.41 -1.40 0.54 2DC 6.08 2.19 5.57 15.44 units 10-9cm2/sec 2-DC diffusivity predictions 1-DC diffusivity predictions

7. Comparison of 1-DC with 2-DC Diffusivities and Predictions [D]1-DCand [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) DCrCr DCrAl DAlCr DAlAl 1DC 11.42 13.41 -1.40 0.54 2DC 6.08 2.19 5.57 15.44 units 10-9cm2/sec Data from the second diffusion couple, at 2 2-DC diffusivity predictions 1-DC diffusivity predictions

8. Predicted rms error as a function of composition vector angle = 12 = 118 rms error in D11 vs  Eigenvector directions of [D] 1% error in measurables Composition vector direction

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

10. 114 Predicted rms error as a function of composition vector angle = 12 = 118 rms error in D11 vs  67% predicted error Note that the error scales with the % error Composition vector used to calculate [D]1-DC 1% error in measurables

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

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

13. Predicted Errors for 1-DC Diffusivitiesfrom 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 r11 versus composition vector angle = 30° = 119°

15. Comparison of rms error of r11 and r22 vs composition vector angle = 30° = 119°

16. rms error of r12 versus composition vector angle = 30° = 119°

17. Comparison of rms error of r12 and r21 vs composition vector angle = 30° = 119°

18. Comparison of rms error of D11 and r11 vs composition vector angle = 30° = 119° Note that D error is ~ twice the r error and

19. Comparison of rms error of D12 and r12 vs composition vector angle = 30° = 119° and

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

21. Discussion Questions 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)? 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° r11

23. What is the probability that a randomly selected composition vector will give an acceptable [D]? = 30° = 119° r12

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

27. Reference to Diffusivity measurements by Hopfe 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.

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

29. 114 Predicted rms error as a function of composition vector angle = 12 = 118 rms error in D12 vs 

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 Diffusivitiesfrom 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.