# Presentation: 771546

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

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

**Jeffrey C. LaCombe, Alonso V. Jaques**
**University of Nevada, Reno, USA**

**Notes:**
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## 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 .  Below ~850 C, it is joined by equilibrium , P, and , phases and oP6 LRO (undesirable).
- Precipitation and growth kinetics are slow.
- But are they “slow enough”?

**Notes:**
The ternary model is used here, recognizing that the other alloying elements affect the stability of the precipitate phases (Turchi et al. 2006&2007).  These affect corrosion properties and can cause hydrogen embrittlement
The Long Range Ordering is of the Pt2Mo type (oP6 or C11b).

## 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**.

**Notes:**
Turchi et al. validated their DICTRA model with TTT diagrams of NiCr.  They replicated the oP6 transformation curve pretty well, which necessitated using the kinetic data at temperatures of 450, 500, and 530 C.  The time for the transformation was very accurately modeled, although the critical temperature asymptote is off a tad more.

## “High” Temperature [D] Measurements in Alloy-22

**High Temp Repository Model**

**Low Temp Repository Model**

***T******melt***

**900 ********C**

**Repository Temperatures**

- Jaques, A.V. and J.C. LaCombe, Defect and Diffusion Forum, 2007. **266**: p. 181-190.

**500 ********C**

**Notes:**
We were able to perform some experimental measurements using ternary diffusion couples (published), but extrapolation of the Arrhenius function to the repository operating conditions is not trustworthy.  Even if we had 900 C data, it would still be a very long extrapolation, with a good chance that GB effects would not be captured.

Data
Melts between 1357 and 1399 C (Haynes Intl.).  (1000/T = 0.6)
HTRM: 450 K = 175 C
LTRM: 360 K = 90 C
Look for diffusion mode transition below about 0.79 homologous temperature (after the Ag data).  This corresponds to 1045 C, or 1000/T = 0.957

## Open Questions

- Is it feasible to experimentally characterize [*D*] in ** 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?

**For Reference:**
**Age of Universe =**
- ~5  1012 days

**Notes:**
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## Experiment
Sample preparation
Initial Composition, 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)

- Sample preparation
- Initial Composition, 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)

**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

**Sources of Error**

## Types of Instrument Errors in Diffusion

- We seek to connect measurable uncertainties with the uncertainty in [D].

**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)

- These sources of error are normally superimposed onto errors resulting from phenomenological assumptions and data reduction...
- Concentration Dependence
- Integration/Differentiation
- Smoothing
- Regression
- Etc.

**Notes:**
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## Analysis Approach

- Start with exact expressions for the “true” concentration profiles, using specified *D**ij* values.

**Composition**

**Position**

- Discretize the “true” profiles to simulate analytical instrument sampling (averaging) effects.  Produces a data set analogous to microprobe, etc., varying...

- Use established methods to determine the *measured* diffusivity matrix elements, *D**ij*.

- Compare these measured *D**ij* values with the original values.

- End-point compositions
- Diffusion time/distance
**Added Errors**: spot size, position, etc. (quantified).

**Notes:**
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## Monte Carlo Simulations (Ternary)

**Monte Carlo Simulations (Ternary)**

_**Measured**_** Diffusion****Coefficients, *****D******ij***

**Perform Diffusion Couple**
**Analysis on “Observed”**
**Concentration Profile**

**Error Estimates, *********D******ij***
**(Measured vs. “True” *****D******ij*****)**

**Select “True”**
**Diffusivity**
**Values, *****D******ij******,***** ****and**
**Compositions**

**Simulation Results**
**(Diffusivity Msmts.)**

**Create ‘True”**
**Concentration Profile**

**Simulate**
**“****Experimentally Observed”**
**Concentration Profile**
**(****Instrument Sampling****)**

**Notes:**
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## 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**

**Sample Surface**

**Analytical Volume**
**(X-Ray Source)**

***z***

- _Where,_
*C**i**Obs*	=	Experimentally observed composition (averaged)
*C**i**Orig*	=	Original (true) composition
*V* 	=	Analytical Volume
**	=	Spatial variation of X-ray emissions

- We simplify, by considering a spherical analytical volume, with homogeneous & isotropic emission of x-rays... i.e., **(*r***) = const.

***C***

***x***

***r******Spot***

**Notes:**
Note that analytical expressions for the sampling volume size are available in the literature.

## Error in Dii

**Error in *****D******ii***
**Parabolic dependence of error, ***********D******ii*****, on Instrument Spot Size**

**Notes:**
The errors (uncertainties) in the diagonal terms were found to be always positive (i.e., biased or not symmetric).  Off-diagonal terms of the errors were much smaller, but had much more scatter, and WERE symmetric.
A follow-on to this is that because dDii are always positive, and because the diffusion distance derives mostly from the diagonal terms (through Emax), predictions of the diffusion distance that are based on [D] tend strongly to be under-estimations.

## Type I Error in Dii (Spot Size)

**Type I Error in *****D******ii****** *****(****Spot Size****)**
**Normalized to the Major Eigenvalue (Dimensionless)**

**Notes:**
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## Type II Error: Spatial Positioning

- The spatial positioning resolution of the measurement points, as well as the spacing between points introduce error.

**Sample Surface**

***C***

***x***

- *x*

- *x*

- *x*

- *x*

- *x*

**Notes:**
Note that analytical expressions for the sampling volume size are available in the literature.

## Type II Error: Spatial Positioning

**Notes:**
Study of the precise scaling behaviors (i.e., the constants) is still a work in progress.
Note that sigmaij is the Coefficient of Variation (Stdev/Value) now.  I.e., it is the standard deviation in the dDij measurements divided by the Dij values themselves.

## Type III Error: Composition

- The compositional resolution (scatter) in the measurement profile.

**Sample Surface**

***C***

***x***

**Notes:**
Note that analytical expressions for the sampling volume size are available in the literature.

## Type III Msmt Error: Composition

- Relative error in composition

- Uniform noise was added to the concentration data...

- This work is still very-much in progress...

**Notes:**
Study of the precise scaling behaviors (i.e., the constants) is still a work in progress.
Note that sigmaij is the Coefficient of Variation (Stdev/Value).  I.e., it is the standard deviation in the dDij measurements divided by the Dij values themselves.

## 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, *D**ii*, can be estimated.

**Notes:**
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## Summary (II)Valid for Linear Ternary Diffusion Couples

**Type II Errors** (positioning) Relative errors in [*D*] have power law scaling with positioning error, ***x*, and *n**points*.

- These error sources both contribute *stochastically*, rather than deterministically (compared with Type I errors).

**Type III Errors** (composition).  Relative errors in  [*D*] scale linearly with the %error in the concentration ...

- Note that ***C*  *t*1/2 on the instrument

## 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.

**Notes:**
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## 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)

**Notes:**
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