Uncertainties in Multicomponent Diffusivities and the Determination of Long-Term Diffusivities at Low Temperatures

1. Uncertainties in Multicomponent Diffusivities and theDetermination of Long-Term Diffusivities at Low Temperatures Jeffrey C. LaCombe, Alonso V. Jaques University of Nevada, Reno, USA

2. Motivation for Study of Alloy-22 • Alloy-22 (Ni-Cr-Mo-W-Fe-Co) used as a corrosion barrier on waste package outer surface. 10,000+ yr design life. • Long-term phase stability in Alloy-22 (Modeled as Ni-Cr-Mo). • Nominally metastable single phase fcc g. Below ~850 C, it is joined by equilibrium m, P, and s, phases and oP6 LRO (undesirable). • Precipitation and growth kinetics are slow. • But are they “slow enough”?

3. Diffusion in Alloy-22 at Repository Temperatures • Selected work in this area • Campbell, C.E., W.J. Boettinger, and U.R. Kattner, Development of a diffusion mobility database for Ni-base superalloys. Acta Materialia, 2002. 50(4): p. 775-792. • Turchi, P.E.A., L. Kaufman, and Z.-K. Liu, Modeling of Ni-Cr-Mo based alloys: Part I– phase stability. Calphad, 2006. 30(1): p. 70-87. • Turchi, P.E.A., L. Kaufman, and Z.-K. Liu, Modeling of Ni-Cr-Mo based alloys: Part II– Kinetics. Calphad, 2007. 31(2): p. 237-248.1 • Kinetic Data Used in Thermocalc/DICTRA Models: • Best (and only?) kinetic data available for this alloy system. • Derived from experiments above 900 C. • Grain boundary effects observed below 900 C, but not accounted for in model.

4. “High” Temperature [D] Measurements in Alloy-22 Experimentally-Measured Data (2 temperatures) Repository Temperatures Low Temp Repository Model High Temp Repository Model Tmelt 900 C 500 C Jaques, A.V. and J.C. LaCombe, Defect and Diffusion Forum, 2007. 266: p. 181-190.

5. Open Questions For Reference: Age of Universe = ~5  1012 days • Is it feasible to experimentally characterize [D] in g phase Alloy 22, at lower temperatures (long duration)? • Grain boundary contributions expected to appear below ~900 C. • Repository conditions are likely unreachable, but can we characterize [D] under grain-boundary-affected conditions?

6. Sources of Error Phenomenological Linear vs. Non-Linear Assumptions (Invariant Density, …) Thermodynamic Factor Fast diffusion paths (GBs, other defects) Other Arrhenius deviations (divacancies) Onsanger Redlich Kister Dimensionality Data Reduction Numerical smoothing, filtering Outliers Numerical Integration/ Derivation. Truncation of C(x,t) profile Data weighting Experiment Sample preparation InitialComposition, segregation, lack of homogenization. Temperature Calibration / Control Impurities, precipitates, secondary phase nucleation, reactions, porosity Grain size, dislocation density Minimal replication of experiments Sampling (non independence, split plot) Interface location (Kirkendall Markers) Residual or imposed stresses Interfacial bonding Instrument: Positioning Analytical spot size (precision/accuracy) Composition (precision/accuracy) Detection Limit Probe volume (shape as well as size)

7. Types of Instrument Errors in Diffusion • Sampling (Volume Averaging) • Probe spot size • Spatial Positioning Resolution • Uncertainty in probe position • Number of measurement points (spacing) • Concentration (instrument sensitivity/accuracy) Most measurements of [D] in the literature report semi-quantitative estimates of the uncertainties. We seek to connect measurable uncertainties with the uncertainty in [D]. • These sources of error are normally superimposed onto errors resulting from phenomenological assumptions and data reduction… • Concentration Dependence • Integration/Differentiation • Smoothing • Regression • Etc.

8. Penetration Distance 70 60 50 40 30 20 10 0 -25 -20 -15 -10 -5 0 5 10 15 20 25 Start with exact expressions for the “true” concentration profiles, using specified Dij values. Analysis Approach Discretize the “true” profiles to simulate analytical instrument sampling (averaging) effects. Produces a data set analogous to microprobe, etc., varying… Composition • End-point compositions • Diffusion time/distance • Added Errors: spot size, position, etc. (quantified). Use established methods to determine the measured diffusivity matrix elements, Dij. Position Compare these measured Dij values with the original values.

9. All 14 parameters randomly chosen in each simulation ~2200 Simulations Monte Carlo Simulations (Ternary) Select “True” Diffusivity Values, Dij,and Compositions Create ‘True” Concentration Profile Simulate “Experimentally Observed” Concentration Profile (Instrument Sampling) Measured DiffusionCoefficients, Dij Simulation Results (Diffusivity Msmts.) Perform Diffusion Couple Analysis on “Observed” Concentration Profile Error Estimates, dDij (Measured vs. “True” Dij)

10. Type I Errors: Spot Size The observed concentration profile in a diffusion couple derives from sampling measurements in a finite volume beneath the probe spot. Probe Beam Where, CiObs = Experimentally observed composition (averaged) CiOrig = Original (true) composition V = Analytical Volume f = Spatial variation of X-ray emissions Sample Surface z rSpot Analytical Volume (X-Ray Source) C x We simplify, by considering a spherical analytical volume, with homogeneous & isotropic emission of x-rays… i.e., f (r, q) = const.

11. ! dDii [ ] cm2 s Error in Dii Parabolic dependence of error, dDii, on Instrument Spot Size

12. ! Type I Error in Dii(Spot Size) Normalized to the Major Eigenvalue (Dimensionless)

13. Probe Beam sx Type II Error: Spatial Positioning The spatial positioning resolution of the measurement points, as well as the spacing between points introduce error. Sample Surface sx sx sx sx sx C x

14. Type II Error: Spatial Positioning

15. Probe Beam Type III Error: Composition The compositional resolution (scatter) in the measurement profile. Sample Surface C x

16. Type III Msmt Error: Composition Uniform noise was added to the concentration data… This work is still very-much in progress… Relative error in composition

17. Summary (I)Valid for Linear Ternary Diffusion Couples Type I Errors (due to spot size averaging of concentration), scale with the spot size (area) and the diffusion time: • Knowing your instrument’s spot size, the necessary diffusion time for a desired uncertainty, dDii, can be estimated.

18. Summary (II)Valid for Linear Ternary Diffusion Couples Type II Errors (positioning) Relative errors in [D] have power law scaling with positioning error, dx, and npoints. • Type III Errors (composition). Relative errors in [D] scale linearly with the %error in the concentration … Note that dCt1/2 on the instrument These error sources both contribute stochastically, rather than deterministically (compared with Type I errors).

19. Current & Future Work Short Term… Attempt to identify more clear scaling for Type II & III errors. Superposition of error contributions Nonlinear diffusion models (variable [D]). Higher order systems (4+ components). Working Towards…? Develop design of experiment (DOE) methodology for diffusion measurements. Incorporate Uncertainty Quantification (UQ) into transport property databases to permit calculation of uncertainties in [D] based on how the properties in the database were measured.

20. Acknowledgements Financial Support: DOE: DE-FG02-04ER 63819 DOE:ORD-FY04-015 NSF: DMR-0349300 Additional Assistance:A. Manavbasi (UNR) S. Vadwalas (UNR) G. Larios (UNR)