spanning time-scales in simulations of irradiated materials · 2009-07-15 · testing of candidate...

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Lawrence Livermore National Laboratory

Spanning time-scales in simulations

of irradiated materials

Vasily Bulatov


Livermore, California

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Outline

  Materials in nuclear reactors

  Physics of material damage under irradiation: time-scales

  Modeling and simulations in nuclear materials R&D

  Spectrum of simulation approaches Atomistic simulations Lattice Monte Carlo Object Monte Carlo Mean-field Rate Theory (Cluster Dynamics)

  Summary

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Outline





  Summary

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Uses of materials in nuclear power generation

Materials: fuel, reactor core, pressure vessel, control rods,

coolant, moderator, waste forms, …

Extreme conditions: intense irradiation, high temperature,

high pressure, mechanical stress, aggressive

chemistry, long replacement cycle

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Material properties place constraints on reactor design

High radiation tolerance - aggressive (streamlined) design

High operation temperature - efficient energy conversion

High mechanical strength - lower cost

High corrosion resistance - longer replacement cycle

…

Important: need to reduce and/or quantify variations in materials properties

Improved materials reduce cost and environmental impact of nuclear energy generation

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Materials challenge for future nuclear energy

Fusion energy at LLNL

•  NIF inspires research in ICF and new materials •  LIFE-engine concept for fusion/fission energy

•  Applications for dozens of new nuclear plants in NRC •  ANES initiatives: Gen-IV, GNEP, ITER

Critical need for new materials to resist very large doses of irradiation, high temperature and aggressive chemistry

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Testing of candidate materials

To be useful material testing must be accelerated

IFMIF - International Fusion Materials Irradiation Facility

Accelerator-based source of high-energy neutrons

Estimated construction cost ~ €1 billion (Japan, EU, US, Russia)

Estimated completion ~ 2017

Host country - Japan

Catch-22 in materials R&D for nuclear energy

The only fully reliable means to assess a candidate material is …

to build the reactor and run it over the material lifetime

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Ion accelerators for accelerated materials testing

Test Reactors •  ATR ~ 10 dpa/year •  HIFR ~ 20 dpa/year

Ion Beam Systems •  IBA ~ 50 dpa/day •  CAMS ~ 50 dpa/day

JANNUS facility at CEA France

Three accelerators (1) heavy ions (2) helium (3) hydrogen

Advantages

Estimated construction cost ~ €10-30M

High testing throughput

Materials are not activated (safe to handle)

Big question: Can material performance be extrapolated from 6 hours of violent irradiation to 10-50 years in the reactor?

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Computer simulations can play an important role

10-12 10-6 1 106 1012 seconds

Accelerated irradiation tests

Material lifetime in a reactor

Cooling off of hot collision cascades

To bridge the time scale gap simulations must be efficient

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To be useful material simulations should be accurate

(model accuracy) x (computational efficiency) = simulation utility

(acc

urac

y) =

(err

or) -

1

(efficiency) = (computational cost) -1

Often accuracy can be traded for efficiency

Method 1

Method 2

Method 3

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Outline





  Summary

Radiation damage 101

PRA - T PRA - T’ PRA - T’’

SRA - T’ SRA - T’’

TRA - T’’

etc.

PRA - T γ, α (Ηε), p (H) n E’

n - E

Neutrons propagate over 1-10 cm

between scattering and reaction events

n-scattering and reactions → primary recoil atoms (PRA)

PRA average T ≈ 10 keV fission

PRA average T ≈ 50 keV fusion

E E’

E’’ E’’’ n

Displacement cascades

5 nm

Primary damage in�iron at 100K

~ 50 keV PKA

10 keV PKA

* Stoller,JNM 276 (2000) 22.

Each PKA inflicts further damage

First 10 picoseconds: hot stage of damage evolution

7 keV Cascade in Ni (fcc)

•  V and V-clusters

•  I- and I-clusters

•  Replaced atoms or ballistic jumps

self-interstitial atom

vacancy

solute atom (e.g., Cr, Cu,

Mn, Ni)

Diffusion-reaction processes

Vacancy clustering Collapse into vacancy loops

Interstitial clustering Form interstitial loops

Annihilation

10 picoseconds to years

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• Primary damage creates enormous SIA and vacancy super-saturations

• Some vacancies and SIA recombine

• Annihilation at sinks (dislocations, grain boundaries,…)

• Sink bias driven clustering of vacancies (cavities) and SIA (dislocation loops) as well as dislocation climb (by excess SIA flux)

Long range defect migration and interactions

Ei

Dislocations actively remove interstitials

I

V Ev

I

V

Dislocation bias

Co-evolution of irradiation defects and material microstructure

Ji Jv

growing SIA loop

Jv

Ji

growing void absorbs I

jog moves right

absorbs V

jog moves right

extra half-plane of atoms

jogs dislocation

Dislocation climb

Helium accumulation and storage

PRA - T γ, α (Ηε), p (H) n E’

n - E

He

n

n

i v

n

v

B

L

FMS Dislocation

GB

SIA vacancy

voids

GB He bubbles

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Complex microstructural evolution in multicomponent-multiphase alloys

  Ballistic mixing and amorphization

  Vacancy/SIA supersaturations cause radiation enhanced diffusion (RED)

  Coupled solute-defect fluxes to sinks cause radiation induced

segregation (RIS) and precipitation (RIP)

  Destabilization of Fe-Cr-Ni austenite

  Kinetically modified phase boundaries

  Grain boundary segregation

  …

Radiation assisted (induced, driven) alloy kinetics

GB solute segregation in SS

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Exposure to neutrons degrades mechanical properties: - Volumetric swelling (0.3 - 0.6 Tm) - Irradiation hardening and embrittlement/decreased uniform elongation (< 0.4 Tm) - Irradiation (<0.45 Tm) and thermal (>~0.45 Tm) creep - High temperature He embrittlement (> 0.5 Tm)

Irradiation effects on materials properties

0.01

0.1

1

10

100

1000

104

0.1 1 10 100 1000 104

Creep Rupture Life of 20% Cold-workedType 316 Stainless Steel at 550˚C, 310 MPa

Cre

ep

ru

ptu

re life

(h

)

Helium concentration (appm)

(2.0 dpa)

(3.4 dpa)

(43 dpa)

0

Severe embrittlement due to He bubbles

Austenitic steels irradiated infast breeder reactors

Before After

Radiation-inducedswelling Void formation

Complex synergistic interactions with He and H

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Atomistic Monte Carlo simulations of collision cascades

Brute force (honest) atomistic simulations are limited to the first 10-1000 ps: further cascade annealing by defect diffusion takes place on longer time-scales

85 keV displacement cascade in U-235

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How to extend the time horizon of atomistic simulations

The notorious rare-event problem

To access the more interesting infrequent events taking place over longer time scales, the atomistic simulation must be somehow accelerated

A

B

S E

x

kT

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To access the more interesting infrequent events taking place over Longer time-scales, atomistic simulations must be somehow accelerated

A

B

S E

x

kT

How to extend the time horizon of atomistic simulations

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Atomistic Monte Carlo simulations of cascade annealing

Further cascade annealing by defect diffusion takes place on time scales not accessible to brute force atomistic simulations


To access the more interesting infrequent events taking place over longer time scales, the atomistic simulation must be somehow accelerated

A

B

S E

x

kT

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Atomistic simulations of infrequent events is a vast and striving area of current research Distort the system’s dynamics in such a way as to enhance

the probability of sampling infrequent events of interest

An ideal accelerated method should

•  sample rare events much more efficiently than the brute-force (honest) atomistic simulation

•  be unbiased, i.e. should sample the same rare events and with the same probabilities as the brute-force simulations

•  supply an estimate of sampling errors

•  contain information on the true rates of sampled rare events

Slow dynamics of cascade annealing presents a relevant and challenging case for method development

No existing method satisfies all the above requirements

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Building an optimal bias potential on the fly

•  The evolution of the atomistic system is guided via a (external) bias potential defined on a low dimensional subspace of steering coordinates. •  The choice of the steering subspace is not unique but a good choice will make simulations more efficient.

Begin the simulation with zero bias potential

Find the number of times the simulation trajectory passed through each state in the steering subspace

Update the bias potential to steer away from the frequently visited states in the steering subspace

Continue the simulation with the new bias potential

A puppet master

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Two important potential advantages of the low dimensional steering method

Adaptively constructed on-the-fly, the bias potential is provably optimal, steering the atomistic

model most efficiently towards the unexplored states and providing the best statistics on the

weights of the previously explored states.

The method is firmly rooted in the rigorous mathematical formalism of Importance Function

that contains information on the rates at which the observed transitions would have taken

place in an honest simulation as well as the variance between the “smart” and the “honest”

atomistic simulations.

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Kinetic Monte Carlo simulations of damage accumulation kinetics

KMC uses inputs from MD simulations and/or experiment to model kinetics of damage accumulation over longer time scales

Detailed analysis of defect mobility

Cu atoms

Vacancies

MD simulations Individual defects

MD simulations Displacement cascade

Point defect production

10-11sec 10-8m

Kinetic Monte Carlo Damage accumulation

Diffusion, annihilation, clustering

? sec 10-7m Damage source

Adapted from N. Soneda

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2. Lattice Monte Carlo simulations

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B B

B B

B B

B B

B B

B B

B B

B

B B

B

B B

B

B B

B B

B B

B B

B B

B B

B B

B

B B

B

B B

B

The lattice gas models of alloys

*V = vacancy

A sequence of vacancy-atom exchanges

B B

B B

B B

B B

B B

B B

B B

B

B B

B

B B

B


B B

B B

B B

B B

B B

B B

B B

B

B B

B

B B

B


B B

B B

B B

B B

B B

B B

B B

B

B B

B

B B

B

Has been extended to binary alloys under irradiation

The ABV model describes kinetics of alloy

microstructure evolution

•  phase nucleation, growth and coarsening

•  spinodal decomposition

A simple and accurate model of binary alloys, e.g.

Fe-Cu and Fe-Cr.

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Two sources of inefficiency in lattice KMC simulations

1. Low density of mobile species (vacancies, interstitials)

Remedy: use spatial protection and asynchronous propagation as in FPKMC

2. Trapping of mobile species

Trapping of mobile species

Vacancy bound to a B7 cluster

B B

B B

B B

B B

B

B B

B

B

B

B

B

B B

B B

B

B

B

This ubiquitous problem is known as stiffness

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Local traps: clusters of micro-states connected by fast transitions

Potential for efficiency •  Exact and fast first-passage exploration of traps - no repeat transitions! •  Size of the growing web can be effectively controlled by pruning •  Locally adaptive self-tuning algorithm - the web grows larger in stronger traps •  Efficient even when there is no clear time scale separation between fast and slow transitions

First-passage propagation on Markov webs

This method is likely to increase computational efficiency of lattice Monte Carlo simulations by orders of magnitude

System’s micro-states

Current state

Any other state

Transition between two states

Possible next states

Previously visited states

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  Direct coarse-graining of the microscopic Master Equation Master Equation:

€

˙ P (x, t) = P(x',t) ⋅ wx'∑ (x'→ x) − P(x, t) ⋅ w(x → x')

•  Microstate x = (σ1, σ2, …, σN) •  P(x,t) is the probability of microstate x at time t

•  w(x→x’) is the rate of transitions from microstate x to microstate x’ 8x8 cells

Microscopic lattice sites

Define coarse variables:

€

Si = σ jcell∑

€

P(X,t) = P(x, t)x /X∑Coarse state X = (S1, S2, …, SM) and its probability

€

˙ P (X,t) = P(X ',t)w(X '→ X)(t) − P(X,t) w(X → X ')(t)X '∑

X '∑Projected ME:

€

w(X → X ')(t) =def P(x, t)

P(X,t)x /X∑ w(x→ x') =

x ' /X '∑ P(x | X)(t) w(x→ x')

x ' /X '∑

x /X∑where:

Adiabatic elimination closure:

€

P(x | X)(t)≈ Peq (x | X)

time-independent coarse rates

€

w(X → X ')

Solve the coarse-grained Master Equation by KMC on the cells

AB model

+ Coarse-graining

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  Multi-resolution adaptive Monte Carlo

Nested coarse-graining can be repeated as many times as necessary

Local refinement: initialize microscopic simulations where and when needed

(to resolve interesting sub-scale events) Adaptivity: On the finest scale reduces to the (exact) microscopic Monte Carlo

On the coarsest scale treats the largest elements of material microstructure

The Master Equation has the same structure on all scales

AMR software manages refinement and coarsening

AMR research issues Criteria for refinement and coarsening in a fully stochastic simulation

Kolmogorov distance, other measures?

A posteriori error estimates (by concurrent simulations on sub-scales) A priori error bounds?

Further topics What can justify adiabatic elimination in the absence of naturally disparate time scales?

Non-Markov Monte Carlo?

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Computational challenge of low density

  Great many diffusion hops necessary to bring particles to collisions at low density

  Time to collision ~ (inter-particle spacing)3

KMC is a powerful and robust method but grossly inefficient when the density of diffusing particles is low

A kinetic Monte Carlo simulation

Diffusion-controlled annihilation A + A → 0

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0.81 0.43 0.19 0.01

1.21 0.03 0.58

τ1

τ2

τ4 τ3

τ5

τ6

τ7

The new method of first-passage Monte Carlo

For each walker randomly sample first passage time from PFP(t<τ) and order them in a time queue

τ3 < τ4 < τ1 < τ2 < τ7 < τ6 < τ5

Construct disjoint protective regions (cubes, spheres) centered on the particles at t = 0

new τ3

Repeat:

•  Find the earliest time in the queue

•  Propagate the particle to boundary

•  Construct a new protective region

•  Pick a new event time, insert into the queue

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  The algorithm is exact for N-body diffusion-reaction models

  Efficiency: on each step only one particle propagates over a long distance

  Asynchronous algorithm: every particle lives on its own time clock

Diffusion-controlled annihilation A + A → 0

FPKMC simulation

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A test of algorithm efficiency

Kinetics of A + A → 0 annihilation reaction in 1d

Our exact method: 220 decades of time evolution simulated in 15 minutes Previous record: 8 orders of magnitude in time over a month of number-crunching

For a wide class of reaction-diffusion processes the new algorithm is exact and super-efficient

Age of the universe One particle in the universe

5.0−t

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FPKMC has been adapted to simulations of irradiated materials

Needed to introduce additional kinetic mechanisms

Particle insertion: Frenkel pairs, cascades

Clustering: An + Am = An+m

Annihilation: An + Bm = An-m

Emission: An = An-1 + A1

Others

We developed a general protocol that allows to define material models of arbitrary complexity

single particle events

two particle events

Others

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A model of α-Fe under electron irradiation

A. Barbu et al., Philosophical Magazine 85, 541-547 (2005)

Periodic in X and Y

0.287µ

e-

SIA V

• Only single vacancies and single SIA can diffuse.

• All other defects are immobile.

This is not the most realistic but a well-studied model

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FPKMC pushes the simulation limits: α-Fe under electron irradiation

T=200oC and dose rate G = 1.5 x 10-4 dpa/s

10 min: 0.018 dpa 10 days: 18 dpa

Exceeds the previous simulated dose limit by 1000 times

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JERK versus FPKMC : computational efficiency

Cube of α-Fe ( L = 1000 a0 ) 70 K, 300 K, 500 K

From T. Luypaert and C. Marinica

JERK

FPKMC

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Simulations of resistivity recovery

T

electron irradiation isochronal annealing

e- e- e- e- e-

T

Res

istiv

ity (ρ

) T

-dρ/

dT

recovery stages

  recovery stages correspond to annihilation, clustering or dissociation of defects   experiments allow to calibrate the rates of various recovery processes   which mechanisms ?

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Low dose: 2.10-6 dpa

JERK versus FPKMC : cross-validation

JERK: 540 CPU hours

FPKMC: 0.25 CPU hour

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High doses and low dose rates

Such simulations have not been done with any other KMC algorithm

Dose rate dpa/s

Total dose dpa

Simulated time

Performance s/cpus*

Performance dpa/cpuday**

1.5 x 10-4 18 33 hours 0.14 1.80

1.5 x 10-5 2.9 54 hours 1.3 1.67

1.5 x 10-6 4.1 31 days 12.7 1.63

1.5 x 10-7 1.6 125 days 154 1.98

1.5 x 10-8 10 21 years 2,080 2.67

1.5 x 10-9 8.3 175 years 23,200 2.96

*one second of CPU time **one day of CPU time

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Accelerated test (10-4dpa/s) versus reactor (10-8dpa/s) irradiations to 10 dpa

More sophisticated models are being developed for FPKMC simulations of a-Fe JANNUS experiments are planned to test the new models

The new method is efficient at high and low dose rates

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1. FPKMC + Rate Theory

Here we propose:

To extend the FPKMC method to include elastic interactions among the radiation-induced defects

To use FPKMC simulations to examine and improve the accuracy of the Rate Theory models

A. Donev (LLNL), C. Marinica, T. Luypaert (Saclay) and S. Le Bourdiec (EDF)

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Rate Theory

Rate Theory is a mean-field method - ignores spatial correlations

€

˙ V n = kv (n −1)Vn−1V1− ki(n)VnI1+ kdis(n +1)Vn +1 − kdis(n)Vn

€

˙ I n = kv (n −1)In−1I1− ki(n)InV1+ kdis(n +1)In +1 − kdis(n)In

€

˙ V 1 = Γirr− ˙ V 1(sinks) + kdis(n)Vnn= 2

∞

∑ − kv (n)Vnn=1

∞

∑ V1

€

˙ I 1 = Γirr− ˙ I 1(sinks) + kdis(n)Inn= 2

∞

∑ − ki(n)Inn=1

∞

∑ I1

Solve the set of ODE for In and Vn (n = 1, 2, …

For the same simple model of α-Fe under electron irradiation

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The RT method is very efficient but …

1. Mean-field assumption intrinsic to RT can be suspect

Remedy:

Use FPKMC to identify conditions where MF assumption breaks down

Go beyond MF approximation?

2. Difficult to treat complex defect populations, e.g. VnHem

n vacancies

VnHem cluster concentration

Too many equations to solve

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Eureka: Gillespie’s finite volume stochastic method

Solve the master equation directly by kinetic Monte Carlo in a finite volume Ω

Stochastic Rate Theory

Finite (and integer) number of clusters in the finite volume Ω

Monte Carlo rates taken directly from the RT equations

One defect reaction is selected per Monte Carlo cycle with correct probability

€

˙ N i = γ i −α iNi + kijNiN jj∑

Advantages

Extension to complex clusters, e.g. C-V-He, is very simple

Computational cost is controlled by the selection of volume Ω

Finite volume fluctuations are retained

Exact conservation of monomers and clusters species

Gillespie’s algorithm prevailed over ODE in bio-chemistry and cell-biology

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Summary

  Accurate and efficient simulations of irradiation damage and its effects on material properties will be a critical component of materials development for future nuclear energy applications.

  Novel method of First-Passage Kinetic Monte Carlo enables fully resolved simulations of damage accumulation on the reactor time scale.

  Ongoing work on high-performance Monte Carlo methods focuses on three main approaches to microstructure simulations actively employed in the nuclear materials community:

(1) Stochastic Monte Carlo implementation of the Rate Theory (2) Markov-web acceleration for lattice kinetic Monte Carlo (3) Guided atomistic Monte Carlo simulations of cascade annealing

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Collaborators

  LLNL A. Donev, J. Marian, B. Sadigh, A, Arsenlis, G. Gilmer

  CEA Saclay M.-J. Marinica, T. Luypaert, M. Nastar, M. Athenes

  KTH Stockholm T. Oppelstrup

  EDF S. Le Bourdiec

  Stanford W. Cai

  U. Campinas, Brazil M. de Koning

spanning time-scales in simulations of irradiated materials · 2009-07-15 · testing of candidate...

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