UNCERTAINTY QUANTIFICATION OF PHASE EQUILIBRIA AND THERMODYNAMICS - NOAH PAULSON

1. PHASE FIELD WORKSHOP IX
EVANSTON, IL
NOVEMBER 5, 2019
UNCERTAINTY
QUANTIFICATION OF
PHASE EQUILIBRIA AND
THERMODYNAMICS
NOAH PAULSON
Computational Materials Scientist
Applied Materials Division
Argonne National Laboratory
Lemont, IL

2. CHIMAD UQPET
2
• Uncertainty Quantification of Phase Equilibria and Thermodynamics (UQPET)
working group
• Researchers from Argonne, NIST, U Chicago, QuesTek, etc.
• Focused on UQ for thermodynamics and CALPHAD, in addition to other
applications
• Discussion of grand challenges
• Collaboration development
• Organization of workshops, etc.

3. CONTEXT
3
Cu-Mg phase diagram
Mole fraction Mg
Temperature(K)
Gibbs energies (650K)
Paulson, N.H. et al., Acta Mat.,
174, pp.9-15. (2019)

4. BACKGROUND
BAYESIAN STATISTICS
4
Pr 𝛉 𝐃, 𝑀 =
Pr 𝐃 𝛉, 𝑀 Pr 𝛉 𝑀
Pr 𝐃 𝑀
posterior
parameter
distribution
data
Likelihood
prior
parameter
distribution
marginal Likelihood
• Bayesian statistics provides a framework for
model selection, calibration and UQ
• for some model, 𝑀, Bayes’ theorem states:
𝑦=𝑤0+𝑥𝑤1
Paulson, N.H., et al., IJES, 142
pg.74-93 (2019)

5. BAYESIAN FRAMEWORK
THERMODYNAMIC CONSISTENCY
5
𝐶 𝑝 𝑇 =
𝑑
𝑑𝑇
𝐻 𝑇 =
3𝑅 ൗ𝜃
𝑇
2 𝑒 ൗ𝜃
𝑇
ൗ𝜃
𝑇 − 1
2 + 𝑎𝑇 + 𝑏𝑇2
Einstein Model
Paulson, N.H., et al., IJES, 142
pg.74-93 (2019)

6. BAYESIAN FRAMEWORK
DATA WEIGHTING
6
Pr 𝐃 𝚯, 𝜶, 𝑀 = ෑ
i
ෑ
j
N yj
i
𝑀 xj
i
, 𝚯 , Τεj
i
αiLikelihood:
dataset index
data point index
Bayesian
hyperparameter
reported
error
Paulson, N.H., et al., IJES, 142
pg.74-93 (2019)

7. BAYESIAN FRAMEWORK FINAL MODELS
7 Paulson, N.H., et al., IJES, 142
pg.74-93 (2019)

8. Investigate effect of removing individual datasets in Bayesian analysis:
8
BAYESIAN FRAMEWORK HF FINAL MODELS
Paulson, N.H., et al., IJES, 142
pg.74-93 (2019)

9. Parameter distribution
samples from MCMC
FRAMEWORK
9
...
PARAMETER G(FCC_A1,CU;0) 1 GFCCCU#; 10000 N !
PARAMETER G(FCC_A1,MG;0) 1 GFCCMG#; 10000 N !
PARAMETER L(FCC_A1,CU,MG;0) 1 VV0002#; 10000 N !
PARAMETER L(FCC_A1,CU,MG;1) 1 VV0001#; 10000 N !
...
CALPHAD models
Superimposed
CALPHAD
predictions
Paulson, N.H. et al., Acta Mat.,
174, pp.9-15. (2019)

10. UQ ANALYSIS
INVARIANT FEATURE
10
Paulson, N.H. et al., Acta Mat.,
174, pp.9-15. (2019)

11. 11
XMg: 0.214
T: 1003K
UQ-ANALYSIS
X-T-P POINT
Phase Phase Probability
liquid 0.24
FCC + liquid 0.21
Laves + liquid 0.03
FCC + Laves 0.51
PHASE TOTALS 1.00000
Paulson, N.H. et al., Acta Mat., 174, pp.9-15. (2019)

12. UQ-ANALYSIS
X-T-P REGION
12
FCC Laves
liquid FCC + liquid
Laves + liquid
FCC + Laves
Paulson, N.H. et al., Acta Mat.,
174, pp.9-15. (2019)

13. UQ ANALYSIS METASTABLE PHASE DIAGRAM
13
Paulson, N.H. et al., Acta Mat.,
174, pp.9-15. (2019)

14. UQ ANALYSIS METASTABLE PHASE DIAGRAM
14
Paulson, N.H. et al., Acta Mat.,
174, pp.9-15. (2019)

15. 15
UQ FRAMEWORK PDUQ
Documentation: https://pduq.readthedocs.io
Code Repository: https://github.com/npaulson/pduq

16. FRAMEWORK
16
This research was supported by:

17. ABSTRACT
Uncertainty Quantification (UQ) for the CALculation of PHAse Diagrams (CALPHAD)
method has been investigated by various researchers over the past several decades;
however, interest in this topic has exploded with the widespread recognition of the impact
that CALPHAD modeling has made on the design of complex materials, including high
entropy and additively manufactured alloys. The open-source Python package, ESPEI,
employs Markov chain Monte Carlo (MCMC) sampling to perform Bayesian inference on
CALPHAD parameters and to propagate the uncertainty to the prediction of thermodynamic
properties. While some studies demonstrate the propagation of parameter uncertainty to
phase boundaries and invariant temperatures, no software package has been made
available. In this presentation, we introduce the open-source Phase Diagram Uncertainty
Quantification (PDUQ) Python package, which directly leverages ESPEI outputs to perform
UQ for a wide range of predictions, including overall phase diagram morphologies, invariant
positions, probabilities of phase stability and distributions of phase fraction/composition.
17