nist diffusion working group, may 12-13 2008 1 uncertainties in multicomponent diffusivities and the...

NIST Diffusion Working Group, May 12-13 2008 1

Uncertainties in Multicomponent Diffusivities and the

Determination of Long-Term Diffusivities at Low Temperatures

Jeffrey C. LaCombe, Alonso V. JaquesUniversity of Nevada, Reno, USA


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”?

Cr, wt%

Mo, wt%

10 20 30 40 50 60

10

20

30

40

50

60

Ni

850 C

Alloy-22

P


Diffusion in Alloy-22 at Repository Temperatures

Selected work in this area1. 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.

2. 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.

3. 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.


“High” Temperature [D] Measurements in Alloy-22Arrhenius plot multicomponent diffusion

-180.0

-160.0

-140.0

-120.0

-100.0

-80.0

-60.0

-40.0

-20.0

0.000 0.500 1.000 1.500 2.000 2.500 3.000

1000/(T[K])

ln |D

ij[cm

2 /s]|

ln D11

ln D12

ln D21

ln D22

Hig

h T

emp

R

epo

sito

ry M

od

el

Lo

w T

emp

R

epo

sito

ry M

od

el

Tmelt

Experimentally-Measured Data (2 temperatures)

900 C

Repository Temperatures

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

500 C


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?

100 nm 1 m 100 m 500 m

1200 5.9E-08 5.9E-06 5.9E-02 1.5E+00900 6.5E-05 6.5E-03 65 1,636600 9 906 9.1E+06 2.3E+08300 3.0E+11 3.0E+13 3.0E+17 7.6E+18175 1.1E+20 1.1E+22 1.1E+26 2.7E+2790 1.6E+29 1.6E+31 1.6E+35 4.0E+36

Diffusion DistanceTemp (C)

Est'd Time (days) for Diffusion Couple Expt.

For Reference:

Age of Universe =~5 1012 days


ExperimentSample 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)

PhenomenologicalLinear vs. Non-Linear

Assumptions (Invariant Density, …)

Thermodynamic Factor

Fast diffusion paths (GBs, other defects)

Other Arrhenius deviations (divacancies)

Onsanger

Redlich Kister

Dimensionality

Data ReductionNumerical smoothing, filtering

Outliers

Numerical Integration/ Derivation.

Truncation of C(x,t) profile

Data weighting

Sources of Error


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].

Types of Instrument Errors in Diffusion

I. Sampling (Volume Averaging)

• Probe spot size

II. Spatial Positioning Resolution

• Uncertainty in probe position

• Number of measurement points (spacing)

III. 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.


Originalij

Calculatedijij DDD

Analysis ApproachStart with exact expressions for the “true” concentration profiles, using specified Dij values.

Co

mp

osi

tio

n

Position

0 5 10 15 20 25-25 -20 -15 -10 -5

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, Dij.

Compare these measured Dij values with the original values.

• End-point compositions• Diffusion time/distance• Added Errors: spot size,

position, etc. (quantified).0

10

20

30

40

50

60

70

te

xerfcA

te

xerfcA

CCCCCtxC

ii

ii

22

11

2

22

2

11

22

,

2221

1211

reference)later (for Values True""

DD

DD

2221

1211

Values Measured

DD

DD

Penetration Distance

tEmax4 DistanceDiffusion


All 14 parameters randomly chosen in each simulation

t

C

C

C

C

C

C

C

C

DD

DD

B

B

B

B

A

A

A

A

2

1

2

1

2

1

2

1

2221

1211

Monte Carlo Simulations (Ternary)

Measured DiffusionCoefficients, Dij

Perform Diffusion CoupleAnalysis on “Observed”

Concentration Profile

Error Estimates, Dij

(Measured vs. “True” Dij)

Select “True”Diffusivity

Values, Dij, andCompositions

Simulation Results(Diffusivity Msmts.)

Create ‘True”Concentration Profile

Simulate“Experimentally Observed”

Concentration Profile(Instrument Sampling)

spotr

~2200 Simulations


Type I Errors: Spot SizeThe 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

V

Origi

Obsi dVVCV

VtxC 1

,

Where,

CiObs = Experimentally

observed composition (averaged)

CiOrig = 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

rSpot


0.E+00

2.E-11

4.E-11

6.E-11

8.E-11

1.E-10

1.E-10

1.E-10

2.E-10

0.00 0.10 0.20 0.30 0.40 0.50 0.60

(r interaction/ Depth)2

Dii

tE

rr

Max

Spot

Depth

Spot

16

22

scmE

210max 1016.3

scmE

210max 1021.4

scmE

210max 1026.5

Error in DiiParabolic dependence of error, Dii, on Instrument Spot Size

Dii [

]

cm2

s

0

:

,

ii

iijiij

Originalij

Calculatedijij

D

DD

DDD

Note

!


0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.00 0.05 0.10 0.15 0.20 0.25 0.30

(r Spot / Depth)2

Dii/E

max

tE

rr

E

D

Max

Spot

Depth

Spotii

16

22

max

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

2

Depth

Spotr

0.08

0.09

0.10

0.11

0.12

0.08 0.09 0.10 0.11 0.12

t

rD Spot

ii

2

222

22Spotr

tD

211

11Spotr

tD

0

:

,

ii

iijiij

Originalij

Calculatedijij

D

DD

DDD

Note

!


Type II Error: Spatial Positioning

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

Sample Surface

Probe Beam

C

x

x x x x x x


Type II Error: Spatial Positioning

0.001

0.01

0.1

1

10

100

0.01 0.1 1 10 100h

Per

cent

Err

or in

Dij

D

ij/D

ij

d D11/D11

d D12/D12

d D21/D21

d D22/D22

31

1

b

a

n

xb

points

ah

bpoints

a

ij

D

n

x

Dij


Type III Error: Composition

The compositional resolution (scatter) in the measurement profile.

Sample Surface

Probe Beam

C

x


Type III Msmt Error: Composition

0%

5%

10%

15%

20%

25%

30%

35%

40%

0.00% 0.20% 0.40% 0.60% 0.80% 1.00% 1.20%

E (Uniform relative error in the concentration)

ij/D

ij

D11

D12

D21

D22

ED

ExCxC

ij

D

nSimin

noisei

ij

1,1rand1.

C

C

Dij

Dij

C

CE

Relative error in

composition

Uniform noise was added to the concentration data…

This work is still very-much in progress…


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:

t

rD

r

E

D

Spotii

Depth

Spotii

2

2

max

1.0

• Knowing your instrument’s spot size, the necessary diffusion time for a desired uncertainty, Dii, can be estimated.


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 npoints.

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

31

1

b

a

n

x

D bpoints

a

ij

Dij

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

C

Dij

Dij

Note that C t1/2 on the instrument


Current & Future WorkShort 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.


Acknowledgements

Financial Support: DOE: DE-FG02-04ER 63819DOE: ORD-FY04-015NSF: DMR-0349300

Additional Assistance:

A. Manavbasi (UNR)S. Vadwalas (UNR)G. Larios (UNR)

nist diffusion working group, may 12-13 2008 1 uncertainties in multicomponent diffusivities and the...

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