UNCERTAINTY OF THERMODYNAMIC DATA: HUMANS AND MACHINES

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MARIUS STANSenior Scientist and Group LeaderComplex Physical SystemsSystems Security CenterANL

ATPESC 2017St, Charles, ILAug. 5, 2017

THE PROBLEM

Temperature-composition phase diagrams

3

Humans can draw simple diagrams

4

Ternary diagram

Binary diagram

HUMANS ONLY

The Gibbs rule

J. W. Gibbs, On the Equilibrium of Heterogeneous Substances, in Transactions of the Connecticut Academy of Arts and Sciences (1875-1878).

6

Josiah Willard Gibbs (1839-1903)

),,(),,(

),,(),,(

xPTxPT

xPTxPT

BB

AA

At thermodynamic equilibrium, the chemical potential of

each component A, B, C … must be the same in all

phases α, β, γ …

Calculating equilibrium binary phase diagrams

M. I. Baskes, K. Muralidharan, M. Stan, S. M. Valone, and F. J. Cherne, JOM, 55 (2003) 41-50. 7

0 1

T = T1

,

A

,

B

)(xG

)(xG

A B

x

PT

Ax

xPTGxxPTGxPT

,

),,(),,(),,(

Phase diagram

Free energy of components A, B, C …

in phases α, β, γ …

Chemical potentials

PT

Bx

xPTGxxPTGxPT

,

),,()1(),,(),,(

Thermodynamic equilibrium

(Gibbs’ rule)

),,(),,(

),,(),,(

xPTxPT

xPTxPT

BB

AA

),,(...,,

...,, xPTG CBA

The Hf-O Phase Diagram

8

HUMANS AND MACHINES

Solid (A11)

M.I. Baskes et al., Phys. Rev. B, 66 (2002) 104

Liquid Liquid

Melting via Molecular Dynamics

10

),,(

),,(

xPTfH

xPTfH

S

L

Enthalpy

Predicting the equilibrium phase diagrams from atomistic simulations

1M. I. Baskes, K. Muralidharan, M. Stan, S. M. Valone, and F. J. Cherne, JOM, 55 (2003) 41-50.2M. I. Baskes and M. Stan, Metall. Mater. Trans. 34A (2003) 435-39

11

PT

Ax

xPTGxxPTGxPT

,

),,(),,(),,(

Phase diagram

Free energy of components A, B, C …

in phases α, β, γ …by thermal integration1,2

Chemical potentials by

composition derivation

PT

Bx

xPTGxxPTGxPT

,

),,()1(),,(),,(

Thermodynamic equilibrium

(Gibbs’ rule)

),,(),,(

),,(),,(

xPTxPT

xPTxPT

BB

AA

''

),,'(),,(),,(2

T

T

ref

refref

ref

dTT

xPTH

T

xPTG

T

xPTG

LEARNING MACHINES

The Bayesian Method1

1Philosophical Transactions of London, 53 (1763) 370 (free on Google Books).13

Take a guess Acquire info Improve model

N

i

ii

jj

j

MPMDP

MPMDPDM

1

|

||

Prior

Posterior

Reverend Thomas Bayes (1702-1761)That is how Humans and Machines learn

Bayesian Polynomial Regression

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Bayes Rule:

𝑝 𝐰 𝐗, 𝐭 ∝ 𝑝 𝐭 𝐗,𝐰 𝑝 𝐰

predictive distribution:

𝑝 t 𝐱, 𝐗, 𝐭 = න𝑝 t 𝐱,𝐰 𝑝 𝐰 𝐗, 𝐭 𝑑𝐰

posterior likelihood prior

likelihood posterior

deg: 1

deg: 2

N. Paulson et al.

𝐰 𝑙 samples using MCMC (Metropolis-Hastings algorithm)

Uncertainty Quantification of Phase Diagrams

REFERENCES

3200

3100

3000

2900

2800

2700

Liquidus [2]

Liquidus [1]

Solidus [1]

Solidus [2]

3200

UO2 PuO2

3100

3000

2900

2800

2700

Te

mp

era

ture

(K

)

Mol % PuO2

20 40 60 80

UO2 + PuO2

Liquid

* M. Stan and B. J. Reardon, CALPHAD, 27 (2003) 319-323.

[1] M. G. Adamson, E. A. Aitken, and R. W. Caputi, J. Nucl. Mater., 130 (1985) 349-365.

[2] T. D. Chikalla, J. Am. Ceram. Soc., 47 (1964) 309-309.

• Uncertainty in fuel thermo-mechanical properties is often >10%

• Uncertainty of chemical properties (free energy) can be 10-15 %

Example:

• Uncertainty quantification the UO2-PuO2phase diagram*. ΔT = ±50K, δc = 5%

• Bayesian analysis of 15 data sets (melting temperatures , transformation enthalpies, …).

• Optimization via a genetic algorithm.

15

Bayesian Analysis

M. Stan and B. Reardon, CALPHAD 27 (2003) 319

TM

B

M

A

M

B

M

A TTHHM ,,,

TNDDDD ,...,, 21

Model parameter vector

Data vector

Diagram line model m

B

m

A

S

B

S

A

S

m

B

m

A

L

B

L

A

L

TTHHTfx

TTHHTfx

,,,,

,,,,

N

i

ii

jj

j

MPMDP

MPMDPDM

1

|

||Bayes’ Theorem

The mean model j

jj

i

i DMMM |

Standard deviation l

llCSTD

Correlation matrix mmll

lmlm

CC

CR

Posterior covariance matrix T

i

T

i

i

i MMDMMMC ||

16

Evolutionary (Genetic) Algorithms

17

TM

B

M

A

M

B

M

A TTHHM ,,,

TNDDDD ,...,, 21

TM

B

M

A

M

B

M

A TTHHM ,,,

m

B

m

A

L

B

L

A

S

m

B

m

A

L

B

L

A

L

TTHHTfx

TTHHTfx

,,,,

,,,,

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Uncertainty of Hf-O Lattice Parameters

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Thermodynamics of Hf-O1

1D. Shin, R. Arroyave, Z.K. Liu, CALPHAD, 30 (2006) 375-386

Uncertainty of the Hf-O Phase Diagram

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THERMODYNAMIC DATA

NIST ThermoData Engine (TDE)

• Requires

– Knowledge Base: A trusted data archive with full, machine-interpretable metadata

– Inference Engine: software developed via systematic, test-driven analysis of real data systems with a team of subject experts

• Delivers

– A virtual data expert backed by a well-curated library for engineers

CMOS Gate Stack: it is very smallNIST/TRC SOURCE

Data Archival System

as of 2017-01-05

System Type

Chemical Systems

Data Points

Pure 24,000 1,600,000

Binary 52,000 3,300,000

Ternary 15,000 1,100,000

Reactions 7,200 17,000

TDE - NIST SRD 103a/b

Uncertainty Evaluation

Simulation software

Direct data dissemination

Property Prediction

Critically Evaluated Data

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OUTLOOK

The role of computation is changing

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19901950 1970 2010

com

ple

xity

calculations

document

processing

intelligent

assistance

simulations

Impact on Science and Technology

25

MATERIAL GENOME INITIATIVECreate a multi-dimensional Periodic Table of

multi-component, multi-phase materials

Personal assistance

Immersive visualization

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