Computational Theory and Modeling Relevant to

Computational Theory and Modeling Relevant to Fusion Reactor Materials

Roger E. Stoller

in collaboration with: Yuri Osetskiy, Haixuan Xu, Stanislav Golubov, Alexander Barashev

Materials Science and Technology Division Oak Ridge National Laboratory

ICFRM-17 Tutorial Session: Fusion Reactor Materials

Aachen, Germany

11 October 2015

Why do we care about modeling radiation damage in structural materials?

• Although irradiation experiments cannot be replaced by modeling alone, a purely experimental approach to understanding the

effects of irradiation is also not practicable

 costs for design and execution of reactor irradiations

 costs of post-irradiation examination of radioactive materials

 declining facilities for both irradiation and examination

 combinatorial problem: broad range of materials, phenomena, and irradiation conditions - coolants, temperature, loading conditions, dose rate, i.e. because we have to …

 dose

• Recent advances in computational hardware, computational materials science, ... make it more feasible than ever to

aggressively pursue the contribution of modeling

see relevant chapters in: Comprehensive Nuclear Materials, R. J. M. Konings,

because we have to

Components of Primary Radiation Damage Source Term

• incident particles of different types and

energies produce different types of primary damage

– neutrons, heavy charged particles, electrons,

photons

• produce differences in secondary damage accumulation

• need to know both energy spectrum and absolute flux level

nuclear transmutation from thermal neutrons,

E<1 eV

displacement production from elastic collisions

nuclear transmutation from high-energy neutrons, threshold reactions, E>2 MeV

What is unique about modeling the fusion environment?

• the DT fusion reaction source term, unique aspect is 14.1 MeV

neutrons higher energy atomic recoils for

displacement damage production

• higher levels of He and H from transmutation via threshold reactions, e.g. ~12 appm He/dpa and 50 appm H/dpa in AISI-316 for typical first wall spectrum

note: many lower energy neutrons as well

Early mechanistic models used to anticipate role of transmutant helium on swelling:

potential fission-fusion difference

• Helium production higher in fusion than LMFBR (FFTF) environment

– experiments with isotopic tailoring of Ni in austentitic steel used to obtain

intermediate and high helium in water moderated reactors (ORR and HFIR)

– experiments confirmed model predictions of possible non- monotonic behavior

Stoller and Odette, 1982

Relevant phenomena with related computational

and experimental methods

Provide brief examples from various aspects of the multiscale modeling scheme

• ab initio (VASP), development of interatomic potentials, He-Fe potential to account for He defects in iron

• primary damage formation, molecular dynamics and atomistic kinetic Monte Carlo

• mesoscale (reaction rate theory) model of microstructural evolution

– illustrate loosely coupled multiscale modeling (parameter passing) – 2D cluster dynamics

– compare alternate kinetic models

• molecular dynamics (atomistic) simulation of dislocation- defect interactions

• displacement rate effects and ion vs. neutron irradiation

presentation focuses on bulk radiation effects,

Application of ab initio calculations:

Development of He-Fe potential

• Wilson’s He-Fe pair potential developed in 1960s, still extensively used

• A new He-Fe inter-atomic potential has been developed at ORNL

• Extensive fitting to first-principles calculations of point defects and clusters

• First-principles calculations in VASP

• Fitted to both relaxed and unrelaxed defects

– Both forces and energies fitted

• Must predict tetrahedral site as most stable site for He

– Not possible with pair or EAM-style functional forms – Hence 3-body potential needed

• Large clusters e.g. He₃V

– Up to 128 atoms in VASP

Defects Fitted

Unrelaxed He_int oct Unrelaxed He_int tet Unrelaxed He_sub He_int octahedral He_int tetrahedral

He_int midpoint oct-tet He_sub

He-He-V cluster He-He-He-V cluster He_int-He_int pair

ORNL three-body He-Fe potential

• Considerable effort invested to properly understand and implement 3-body

potential: complex dependence

Energy of Fe-He-Fe triplet as a function of angle Q_jik and r_ik where r_ij is equal to 1.8 Angstrom

     



 Q

 Q

Fe k j

ik cut ij

cut ijk

ijk ik

ij r f r f r

r Y

,

2 0.44

cos )

, , (

Interstitial He diffusion

• Fe Potentials:

– AM: Ackland-Mendelev – A97: Ackland-Bacon – FS: Finnis-Sinclair

• He-Fe Potentials

– OR: ORNL 3-body – JN: Juslin-Nordlund – W: Wilson

• Results with ORNL Fe-He

potential lies between those of other Fe-He potentials

• ORNL potential shows

consistent results with different Fe potentials

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 5x10^-10

10^-9 10^-8 6x10^-8

He diffusion coeficient (m2 /s)

Tm/T (K^-1)

Fe Fe-He E_m(eV) D₀(10^-8 m²/s) AM JN 0.062 9.64

FS S 0.065 3.15 A97 S 0.073 3.27 AM S 0.043 2.23 AM W 0.107 4.69

OR OR OR

Formation of Interstitial clusters

• Yellow atoms are He, BCC Fe lattice not shown.

• He-He in Fe has strong binding

– This leads to formation of interstitial clusters

• Mobility decreases with size

• When cluster is large enough, it creates Frenkel pair, emitting SIA

– new He-vacancy cluster immobile – He and He-v clusters are new

source of Frenkel pair

– SIA formation leads to formation of small loops

Comparison of potentials: He clustering

• He diffusion and clustering properties

• Behavior of different He-Fe potentials

– Wilson potential creates FP (SIA emission) more quickly than ORNL potential, higher He binding energy.

– Juslin-Nordlund potential does not form He clusters or create FP above

~400°C.

• Provide parameters for mesoscale microstructural models

He transport with ORNL potential

• Single interstitial helium atoms are very mobile

– but easily captured by vacancies

• SIA can recombine with He_xV if x small

– creates mobile interstitial helium

• Interstitial helium tends to form clusters

• Small He clusters are mobile

– The smaller, the more mobile

• Large He clusters create Frenkel pair to increase local volume, reduce pressure

– transformation to He-vacancy cluster, nascent bubble – He-vacancy cluster is immobile

– ejected interstitials trapped by the He-V cluster and combine to form dislocation loops

Molecular dynamics simulation of primary damage

• MD simulations provide opportunity to investigate displacement cascade evolution, e.g. effects of lattice, PKA energy, T, etc.

• Classical molecular dynamics, typical implementations:

– many millions of atoms, solve Newton’s equation of motion – constant pressure or volume, periodic boundary condition – system may or may not be thermostated to prevent PKA from

heating system

– no electronic losses or electron-phonon coupling, energy of cascade simulation, hence for:,

• to compare with standard NRT displacement model:

n_NRT = 0.8T_dam/(2 E_d), T_dam= kinetic energy lost in elastic collisions – E_MD ~ T_dam (NRT) < E_PKA

– e. g. for T_dam = 100 keV, E_PKA = 175.8 keV and n_NRT = 1000 (see appendix on use of SRIM for dpa calculations)

Comparison of representative neutron and corresponding PKA energy spectra

• differences in neutron flux level lead to different atomic displacement rates

• neutron energy spectrum differences lead to different PKA energy spectra

– different coolants, water for HFIR and PWR vs. sodium for FFTF alter neutron energy spectrum, primarily influence lower energy

– high energy influenced by neutron source, c.f. all fission with ITER fusion

MD cascade database for iron

• an extensive cascade database has been developed that covers a broad range of cascade energy and temperature

– up to 200 keV at 100, 600, and 100 keV at 900K

• this database includes a sufficient number of simulations at each condition to provide a good statistical measure of average cascade behavior

– total number of point defects produced

– in-cascade clustering fractions for both interstitials and vacancies – in-cascade cluster size distributions

– Note: nature of cascade event leads to better statistics for defect production than for clustering

• common use of Finnis-Sinclair potential provides basis for comparison other investigations:

– pre-existing damage (relevant to cascade overlap) using 10 keV cascades

– free surfaces (relevant to in situ experiments) using 10 and 20 keV cascades

– nanograined iron using 10 and 20 keV cascades

Time dependence of defect evolution in atomic displacement cascades

Movies 10keV

50keV

Cascade-induced shock wave

• Three distinct phases in cascade evolution:

– Supersonic – Transonic

– Sonic Calder, et al. (Phil. Mag. 90, 2010)

February 27-March 3, 2011 – San Diego,

The role of shock waves in defect cluster formation

ab initio MD, P. Olsson, C. Domain, EDF R&D, unpublished

Angular dependence of displacement threshold energy:

effect of crystalline lattice on defect formation

Zepeda-Ruiz, PRB 67, 2003 Fe V

Bacon, et al., JNM 205, 1993

classical MD

If normalized to NRT displacements, MD results show reduced defect survival as cascade energy increases

- some of curve structure is significant, related to cascade morphology and subcascade formation

- note small standard errors, measure of mean behavior

- effect of temperature, 100 to 900K, is systematic but not strong

Many of surviving defects are in clusters formed during the cascade event

• significant in nucleation of extended defects

• not accounted for by dpa (or f >x MeV)

Note: poorer statistics, larger standard errors, than for total defect survival

20 keV, 600K

Illustration of subcascade structure at peak damage condition for cascades at 100K

• high energy cascades look like multiple lower energy events, leads to asymptotic behavior with energy

• low energy events between subcascades have higher efficiency

10 keV

5 keV 100 keV

Influence of energy and temperature on SIA clustering

Stoller, JNM 276, 2000

Vacancy clustering in iron

• MD simulations reveal little vacancy clustering in the nearest-neighbor sense - but, vacancies are spatially correlated

arrows indicate 4nn

Typical uncollapsed vacancy cluster, 50 keV cascade at 100 K

• kMC aging of residual defects

indicate that such loose or nascent clusters do tend to collapse into void-like configurations

Influence of energy and temperature on vacancy clustering

Stoller, JNM 276, 2000

MD Results: Application to Neutron Spectrum Effects

• weighting of PKA spectrum using MD defect production

• little effect of neutron energy spectrum observed in spectrum- averaged defect production

(similar for other primary damage parameters)

• yields effective source term for mesoscale models

Point defect survival, per NRT

MD results: averaged over neutron (pka) energy spectrum ( from SPECTER/SPECOMP)

T=100K Finnis-Sinclair

potential ~proportional to

absorbed energy only

Significance to

damage accumulation?

see Stoller and Greenwood, JNM 271 & 272 (1999) and ASTM STP 1366, 2000.

Overall Summary of MD Cascade Simulations

• Cascade energy or PKA energy dependence of stable displacement production is more complex than standard NRT model

• Subcascade formation plays a dominant role in controlling cascade morphology and stable damage production

• Clusters formed directly in the cascade account for a substantial fraction of interstitials, many of which are in glissile configurations

– interstitial clustering fraction and cluster sizes increase with both cascade energy and temperature

• Significantly less in-cascade vacancy clustering appears, nascent clusters coalesce at longer times, e.g. in KMC simulations of

cascade debris

– vacancy clustering increases with cascade energy and decreases with temperature

• Nearby free surfaces (relevant to in situ experiments), pre-existing damage (cascade overlap) and nanograined microstructure alter defect survival and clustering behavior

Stable defect formation is the result of a complex process

involving energetic and coupled many-body reactions

See slides added at the end that include additional information on damage formation

• more details on defect clustering and cascade statistics

• comparison of Fe with other metals

• effect of pre-existing damage on cascade defect production

• effect of free surfaces on cascade defect production

• differences in cascade defect production in nanograined material

• Appendix on recommended approach for using SRIM to

compute dpa

Self-Evolving Atomistic Kinetic Monte Carlo (SEAKMC)

SEAKMC is particularly powerful for large systems with complex defects;

H.X. Xu, Y.N. Osetsky, R.E. Stoller, Physical Review B, Brief Reports, 84, 132103 (2011)

Vacancy Dumbbell

Active Volumes (AVs) – Spatial Localization

Small defects are localized, this can be exploited to speed calculations

Saddle point searches are carried out only within the AVs, different defects have different AV size and shape

The previously (for ~50 years) unexplained

mechanism of <100> interstitial loop formation in iron has been determined for the first time using the SEAKMC method

Results provide a direct link between experiments and atomistic simulations and new insights into defect evolution.

− Formation process of <100> loops involves a distinctly atomistic interaction between two ½

<111> loops.

− Reaction does not follow the conventional assumption of Burgers vector conservation

between the reactants and the reaction product;

process is different from all previously proposed mechanisms.

− Results observed in SEAKMC using multiple interatomic potentials and simulation cell sizes;

predictions at higher temperatures could be confirmed using molecular dynamics with much greater computational effort.

Short-term defect evolution using new self-evolving

atomistic kinetic Monte Carlo (SEAKMC) method*

Application of mesoscale models

• mesoscale models are relevant to many phenomena in materials science and radiation effects

– grain growth

– dislocation evolution, by thermo-mechanical or radiation-induced processes

– void swelling

– precipitation of additional phases, and solute segregation – stress corrosion cracking, and irradiation-assisted SCC

• size scale permits direct comparison with experiments such as TEM and mechanical property measurements

• dependent on fundamental atomistic processes, and controls macroscopic observables such as strength, ductility, creep, ...

• a primary application is the investigation of point defect and solute kinetics and microstructural evolution, includes both mean field

models and Monte Carlo simulations based on reaction rate theory

Example from reaction rate theory modeling

• starting point is continuity equations describing point defect (vacancy, C_v and interstitial, C_i) populations (analogous equations for solutes):

where the  denote spatial derivatives.

• first term on the LHS describes point defect drift to discrete sinks, the U_i,v are interaction energies between the point defects and discrete sinks

• is the total point defect generation rate, including thermal emission from sinks, and the are the point defect diffusivities

• are the total sink strengths for continuum sinks (e.g. cavities,

dislocation, grain boundaries, etc.), recombination rate coefficient is given in terms of an effective recombination radius, r_r:

Dv C

 v D vC

v --- Uk T

 v

 + 

 

 G

v C iC

v D

vC vS

v – T

–

+ t

Cv

=

Di C

 i D iC

i --- UkT

 i

 + 

 

 G

i C iC

v D iC

iS i – T

–

+ t

Ci

=

Gi v ⁼ _{Gdpa Gi v}^{+ em}

 4r r D

v D

+ i

 

= Si vT

Di v

• typical assumptions/simplifications in mean-field models

– the material is treated as a spatially-homogeneous effective medium with embedded effective sinks and sources for point defects

– spatially-averaged point defect generation rates

– these assumptions have been relaxed in particular cases, e.g. to

investigate cascade-induced fluctuations in point defect concentrations – the models are formulated as a series of differential equations describing

the production and fate of point defects and the corresponding evolution of the microstructure

• With these approximations, and assuming that the irradiation

produces only monomers, the time-dependent or steady state point defect concentrations can be obtained as a solution to the following equations:

dC_{i v}

---dt = G_dpa +G_{i v}^em_  – C_iC_v– D_{i v} C_{i v} S_{i v}^T_

• analogous equations can be written to describe an evolving point defect cluster population, for helium generation and distribution, and for the other microstructural components, e.g.:

– if only the monomers are assumed to be mobile, an equation describing the di-interstitial population, C2i, can be written:

– this equation can be generalized to formulate a master equation for interstitial (and vacancy) clusters of arbitrary size. However, this also generates an arbitrarily large number of equations - smallest visible defect clusters ~100 point defects

– various grouping schemes have been devised to minimize number of equations, introduce potential source of error, most accurate is by Golubov, et al., Phil. Mag. A, 2001.

– modern computers reduce driving force for limiting number of equations d C2i

---dt  i iC

i

2  v 3iC

vC

3i C

2i  i 2iC

i  v 2iC + v

 

 

– +

=

• rate equations can be written to describe the evolution of extended defects, e.g. cavity sink strength and growth rate:

• analogous expressions available for grain boundaries, dislocations, dislocation loops, etc.

• greater or lesser detail can be built in as needed to simulate given phenomena e.g. defect nucleation vs. growth regimes, dislocation evolution, effects of solute segregation, etc.

S

= Z

4  rN

 1. + r

S



drv

---dt 1 rv--- Z

v

vDv Cv Cv – v

 

  Zi

vDiCi

 – 

 

=

• solutions obtained from simultaneous integration of equations included in a given model and, when well calibrated with

experimental data, such models have some predictive

capability

However, the success of these models in fitting data can be deceiving (the devil is in the details ...)

• data fitting with incomplete models leads to use of “effective”

parameter values, use of parameter-rich models limits confidence in model extrapolation

• for example, if point defect absorption dominated by dislocations with sink strength S

, swelling rate is proportional to product of  and net dislocation bias (Z

-1):

• In this case, data fitting can not be used to obtain unique set of model parameters. MD cascade simulations provide

independent estimate of  and thereby permit better estimate of Z

dV

---dt

Gdpa

S_v^d

---_^Z_i^{d –}1_^

=

MD cascade simulations provide independent estimate of  and thereby permit better estimate of Z

• more complex models may be ‘stiffer’ with respect to

arbitrary parameter choices, but more complex models

introduce additional parameters

Cluster dynamics simulation of bubble evolution under irradiation: two-dimensional master equation

m

x+1 x-1 x

m+1

m-1

P_v(x-1) P_He(x)

Q_i(x) + Q_v(x,m)

Q_i(x+1) + Q_v(x+1,m)

P_v(x)

P_He(x) Q_He(m) Q_He(m+1) f(x, m+1)

f(x, m-1)

f(x+1, m) f(x-1, m)

Mechanisms include:

   

( , , )

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

x x m m

df x m t

J x m J x m J x m J x m

dt      

• Migration, clustering, and absorption of point defects and He atoms

• Radiation-induced resolutioning of He atoms

• Evaporation of He atoms and vacancies

• Discrete equations up to x=m=10

• Grouping scheme of Golubov, et al.

(Phil. Mag. A81 (2001) 643) for larger sizes

• Radiation-induced generation of point defects and He atoms

• Frenkel pair creation by over- pressurized He-vacancy clusters

Movie to illustrate 2D calculations

Bubble size distributions for nominal input parameters

He bubble size distributions obtained at 100 dpa for irradiation temperatures of 200°C and 330°C

Comparison of kinetic models

• Mean field reaction rate theory (MFRT) and object kinetic Monte Carlo (OkMC) have a common basis in reaction rate theory

• MFRT has a long history in radiation effects modeling; OkMC has seen increasing use since recent advances in

computational power have expanded the accessible domain for these models

• Since MFRT and OkMC models are similar kinetic models

– many details are handled differently in the two approaches

• A direct comparison of the MFRT and OkMC models

– point defect cluster dynamics modeling, relevant to nucleation and evolution of radiation-induced defect structures

– illustrated relative strengths and weaknesses of the two approaches

(see Stoller, et al. J. Nucl. Mater. 382, 2008)

MFRT-OkMC: Inherent differences

parameter or

mechanism MFRT OkMC

solution method deterministic stochastic

time explicit variable inferred from processes and reaction rates

space

smeared, effective medium, possible multi-region MFRT

full spatial dependence

defect production

time and space- averaged, but c.f.

Mansur’s cascade diffusion model

discrete in time and space sink strength, e.g.

dislocations

explicit mathematical expression

inferred from fate of point defects

defect or sink

density essentially unlimited

limited (computationally) by simulation cell size, i.e. N ≥ 1/(x·y·z)

Implications for comparison of simulation results:

Making sure you are simulating the same problem

• deterministic vs. stochastic: not possible to do exact point-to- point comparisons

• spatial correlations when simulating cascade defect production: lost in MFRT, how to handle in OkMC

– electron irradiation with only FP production most straight forward – spatial relationship of point defects and clusters if use cascade

results as input in OkMC

• minimum achievable defect/sink density in OkMC

– e.g. for bcc iron, x=y=z=300a_o,

V=6.42x10^-22 m^-3, 1/V=1.57x10²¹ m³

– c.f. typical defect densities in irradiated materials=10¹⁸ to 10²⁴ m^-3 – imposes limitation on range of irradiation temperature (low) and

displacement rate (high)

Summary: MFRT-OkMC Comparison

• Common issues, primarily related to parameterization and basic model formulation

– e.g. binding and migration energies for small point defect clusters, nature of defect migration (1-D vs 3D)

– are all (or all relevant) mechanisms and reactions accounted for?

• Both approaches produce similar results on well-posed problems, similar dependence on key irradiation and material variables

• Issues for MFRT

– account for effects of discrete time and spatial events, e.g. primary damage

• Issues for OkMC: primarily computational – limited to relatively low doses

– system size limitatation constrains available regime of irradiation temperature and dose rate

• MFRT-OkMC differences remaining (see Stoller, et al. JNM 382, 2008) – cluster densities when statistical nucleation difficult

– damage accumulation under cascade damage conditions – uncorrelated vs correlated cascade debris

- Understand atomic-scale details of dislocation

interactions in complicated environment (point defects, impurities, large obstacles, temperature, variable loading conditions, etc.)

MD-Based Dislocation Dynamics: Objectives

Particular issues :

- mechanical property changes (strengthening, embrittlement, etc.)

- mechanisms of hardening due to localized obstacles

- mechanisms of microstructure modification by moving

dislocations, e.g. cleared

channels

MD-based dislocation dynamics

X - [111]

Y - [110]

Z - [112]

X Y

Z

Direct comparison of MD dislocation dynamics simulation with in situ TEM experiment

73nm

Simulation and experiment reveal dynamic interaction of a gliding dislocation with stacking fault tetrahedron: critical role of free surface in experiment

Importance of simulation accurately simulating the experiment

0.0 0.5 1.0 1.5 2.0 80

100 120 140 160

 MAX (MPA)

He/Vac

2nm at T=300K

Strong

Intermediate

Weak

5 10 15 20 25 30 35 40

0 4 8 12 16 20 24

He/Vac=2.0 He/Vac=1.0 He/Vac=0.75 He/Vac=0.5 He/Vac=0.25

along <110> (a <110> ) void

Results for dislocation – bubble interactions

• obstacle strength is a non-

monontonic function of radius

(Partial) list of issues in understanding how to use charged particle irradiations

• charged particle vs. neutron

– a few orders of magnitude in G_dpa available

– effect of surface, greater for in situ electron irradiation – effect of gradients

– PKA spectrum

– implanted ions, chemistry and interstitials

– lack of transmutatation products, can be “simulated”

• heavy ion vs. electron

– PKA spectrum, lack of cascades (clusters) in electron irradiation – surface influence in electron irradiation

• neutron vs. neutron

– less range in G_dpa available

– PKA spectrum effects, more modest that in ion/neutron comparison

• neutron-neutron comparisons are least ambiguous

56

Comparing ion and neutron irradiation: Packan and Farrell, JNM 78, 1978, pure nickel

57

Simple model-based comparison

• Mansur’s equation for temperature shift (JNM 78,1978):

• Old Stoller-Odette

model, ASTM STP 955

• displacement rate varied in model

• Mansur Eqn. gives 73K

• shift of peak vs shift of cut-off

• shift will be sensitive to E

• "shift" may be different at different doses

75K

~175K

58

Comparing ion and neutron irradiations

• ... its complicated

• when microstructure is sensitive to dose rate, it is reasonable to expect a temperature-shift in radiation-induced phenomena

• relative complexity and stability of unirradiated microstructure is likely to influence T-shift, e.g. FM vs austenitic stainless steels

• new models need to account for influence of mobile interstitial clusters under cascade-producing irradiation

• point defect partitioning is the name of the game, whether dislocation bias, production bias, high sink densities, ...

• when assessing models and their predictions vis a vis

experimental results, need to think carefully about interpretation

– model limitations, missing mechanisms – “effective” parameters

– the art is in knowing what matters

59

Overall Summary

• Substantial progress in understanding and predicting the behavior of materials has been provided by theory and computational modeling

• Further progress requires additional research:

– Advances in electronic structure theory to obtain intrinsic and defect properties in iron and its alloys, including the effects of He and H

– Computationally-efficient, physically-robust interatomic potentials for

multicomponent alloys, including effects of directional bonding and magnetism – Advanced atomistic and mesoscale models describing the defects and

processes that interact in complex ways in multicomponent, multiphase alloys – “Properly” linked and multiscale (atomistic, mesoscale, and continuum)

deformation and fracture models for predicting hardening, plastic instability, changes in ductile-to-brittle fracture, dimensional instability, and creep/creep rupture behavior under realistic loading conditions

• It is critical to validate models* using data from special-purpose and engineering experiments, objective is a predictive capability to avoid technical surprises, e.g. new damage mechanisms at high doses and long times, how to use ion irradiation to understand neutron irradiation

* need to think about model assumptions and what the simulation

Additional slides on:

Cascade statistics, morphology, and other materials

Appendix on calculating dpa:

Comparison of MD defect formation in a range of bcc, fcc, and hcp metals

• different values, but similar power-law depending in defect survival

• energy dependence of SIA clustering similar, varies by a factor <2

Bacon, et al., JNM 276, 2000

• Similar embedded atom type interatomic potentials, edge length of simulation cells is 50a_o.

• Note higher level of in-cascade clustering in more-compact copper cascade.

• Defect survival (relative to NRT) is lower in copper than in iron.

Fe Cu

Comparison of 10 keV cascades in iron and copper

based on molecular dynamics simulations

Fe Cu

• Similar embedded atom type interatomic potentials, edge length of simulation cells are 50a_o.

• Note higher level of in-cascade clustering in more-compact copper cascade.

• Defect survival (relative to NRT) is lower in copper than in iron.

Comparison of 20 keV cascades in iron and copper

based on molecular dynamics simulations

Influence of cascade structure on time signature

• systematic differences in the time dependence of defect

formation are observed around the time of peak damage

Influence of cascade structure on time signature

• subcascade formation reduces influence of pressure wave that creates many small-range, transient displacements

• note that stable defect formation is reduced for structures with higher peak

Formation of point defect clusters

Calder, et al. (Phil. Mag. 90, 2010) have carried out detailed

analysis of high-energy cascades. By tracking individual atoms, and local temperature and atom density, they show:

• the formation of large SIA clusters in Fe is related to the formation of

hypersonic recoils (>10 speed of sound) near the sonic front of the primary cascade

• site of the SIA cluster is determined within ~0.1 ps

• vacancy cluster collapse is a result of prior SIA clustering

Major difference between high-energy cascades:

interstitial cluster size distribution

Pre-existing Damage: Effect on Defect Survival

R. E. Stoller, “The Effect of Free Surfaces on Cascade Damage Production in Iron,”

Journal of Nuclear Materials 307-311 (2002) 935-940.

R. E. Stoller and S. G. Guiriec, “Secondary Factors Influencing Cascade Damage Formation,” Journal of Nuclear Materials 329-333 (2004) 1228-1232.

For extended study: see Y.N. Osetsky, A.F. Calder and R.E. Stoller, "How do Energetic Ions Damage Metallic Surfaces?," Current Opinion in Solid State & Materials

Science 19 (2015) 277-286.

Investigation of potential length-scale effects in primary damage formation

• The MD database provides a good description of primary radiation damage formation from atomic displacement cascades, but most is in perfect, single crystals.

• Large number of mobile defects produced in a displacement cascade, nearby grain boundaries could potentially reduce the residual damage from any given cascade.

• Previous work (e.g. Samaras, et al. in Ni) indicated that the high volume fraction of grain boundaries in bulk

nanocrystalline materials could provide efficient point defect

sinks and/or recombination sites.

Approach

• Parallel version of MOLDY MD code using OpenMP on shared memory platform

• Voronoi technique used to create nanocrystalline system

– “Nucleation site” distribution based on an fcc lattice, melted through Monte Carlo using a Lennard-Jones potential

• Choose nanograin “nucleation sites” → Fill in grains → Remove overlapping atoms → Equilibrate structure

– Nanocrystalline system equilibrated ~200 ps

• Simulate cascade event → Perform nearest neighbor defect analysis → Differentiate grain boundary

reconstruction from in-grain defects

– Cascade simulation was run for ~15 ps, well into the region where the defect count stabilize

Nanocrystalline System: investigate length scale effects on primary damage

• MD simulation cell, 100 lattice parameters (~28.6 nm) on edge, periodic

boundaries

• System shown contains 32 grains

• ~10 nm grain size

• Nanocrystalline system equilibrated ~200 ps

• Representative cascade size:

5 keV 20 keV

Defect Visualization

• Grain boundaries (GBs) distinguished using spherical approximation – 1st and 2nd nearest neighbor analysis

– Before and after visualization can show GB movement or reconstruction

• Possible defects flagged when more than 0.3 of a lattice parameter from an original atom site

• Possible in-grain defects flagged when more than 0.5 a

lattice parameter from original GBs

Cascade Simulations

• Small study of cascade energy and temperature, 8 simulations at each condition:

– 10 keV, 100K – 20 keV, 100K – 20 keV, 600K

• Results compared with “normal” cascade events to

determine difference in residual damage between single crystal and nanocrystalline iron cascades

• Previous cascade database for comparison:

– 10 keV, 100K: 15 events – 20 keV, 100K: 10 events – 20 keV, 600K: 8 events

Comparison of Final Defect State:10 keV, 100K

Single Crystal Iron Nanocrystalline Iron

Residual defects - Interstitial atoms are green, vacant sites are red, grain boundary atoms within 3 lattice parameters of any defect are black.

Comparison with 100 K database

Mean values are shown with the standard error.

• Higher vacancy survival in nanograined material

• Much lower interstitial survival in nanograined material

Results: Stable defect production

Mean values are shown with the standard error.

• Higher vacancy survival in nanograined material

• Much lower interstitial survival in nanograined material

Note:

Ni≠Nv

Results: In-cascade clustering

Much less in-cascade interstitial clustering in nanograined material

Defect production in nanograined Fe

• Strong influence of microstructural length scale (grain size) on primary damage production

• Reduced interstitial survival and clustering will reduce formation of radiation-induced microstructural

components

• Excess vacancy production may lead to higher

supersaturation, greater propensity for cavity formation

• Impact of altered primary damage behavior needs to be evaluated over longer time scale, e.g. using mean field rate theory or Monte Carlo models

• Further analysis of vacancy clustering and grain

boundary motion is underway

Appendix on calculating dpa:

meaning and limitations, and relationship

with damage correlation

• observed radiation damage depends on some measure of radiation exposure,

“damage flux,” f>[0.1, 0.5, 1.0] MeV, but ions, electrons, photons??

• need to account for differences in PKA energy spectra to be able to correlate data from different types of irradiation

• secondary displacement model by Norgett, Robinson and Torrens, Nucl. Engr.

and Des. 33 (1975); based on earlier work by Kinchin and Pease (1955), damage partitioning model by Lindhard, et al. (1963) [see ASTM standards E521 and E693]

• number of displacements, n_NRT, proportional to fraction of PKA energy \ deposited in elastic collisions, T_dam:

• The best way to think of the NRT dpa is to focus on T_dam as an exposure parameter – a measure of absorbed energy

– does not account for anything other than the potential total number of atomic

displacements, e.g. no information on in-cascade formation of point defect clusters – does not account for transmutation production

– does not account for any effects due to ionization

number of dpa may or may not correlate with any particular observed effect of radiation exposure

Ed Tdam

NRT 

  2 8 . n 0

Compare NRT, TRIM, and MD displacements

500 800 1000

see Stoller, et al., NIMB 310, 2013 for recommendations on the use of SRIM for computing dpa

Number of displacements from TRIM

• varies with method chosen:

– use integral of output file: vacancy.txt or “Total target vacancies”

– compute ~damage energy by integrating output files – Kinchin-Pease vs. “full cascade” mode

• if reporting dpa for comparison with neutron irradiation

– use Kinchin-Pease

– use standard displacement threshold energy, i.e. 40 eV for iron (see ASTM E521)

– set lattice and surface binding energy to 0.0 PKA energy

(keV)

NRT damage energy (keV)

NRT

displacements Average MD, 100K

TRIM K-P TRIM Full Cascade

1.0 0.81 8 --- 7.7 to 9.4 ---

78.7 50 500 168 533 to 540

529

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566 to 572 1052-1075

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*