Molecular Dynamics Simulations of Fusion Materials: Challenges and

Opportunities (Recent Developments) Fei Gao

gaofeium@umich.edu

Limitations of MD

•  Time scales

•  Length scales (PBC help a lot)

•  Accuracy of forces

•  Classical nuclei

Computer Simulation – Multi-Scale Ø  Time scale for radiation damage evolution and the corresponding simulation

methods

C+-Si<100> Defect properties

BCA – Binary Collision Approximation MD – Molecular Dynamics KMC – Kinetic Monte Carlo

Computer Simulation – Multi-Scale

Ø  Time and length scales are always challenging for materials science

Can’t Just Simulate Everything at Atomic Level: The Human Simulator?

•  Person length scales –  Size ≈ 1 m3 ≈ 106 mols ≈ 1030

atoms •  Person time scales

–  Lifetime ≈ 100 years ≈ 109 s –  Time scale for atomic motions is ≈ 10-12 s, so time step is ≈

10-13 s –  Total number of time steps ≈ 1022

•  Person simulation –  At 10 calculations per each atom per each time step we get ≈ 10 x 1030 x 1022 = 1053 calculations

–  Present computing capability is teraflop ≈ 1012 FLOPS (floating point operations per second) ⇒ Person simulation takes ≈ 1041s ≈ 1034 years.

Limitations of MD – Rare Events, System Size

•  Rare Events –  Rare events occur very infrequently, so do not happen on MD

time scales. –  E.g., diffusion (D ≈ 10-12 cm2/s) has atom hops only every ≈ 10-3

s. Cannot see them with normal MD simulations (can use temperature accelerated MD, but that is an advanced topic).

•  System size –  Time for a simulation scales with number of atoms, i.e, O(N). –  Programs can be parallelized to work by spatial decomposition of

different regions on different processors –  Can treat up to 1010 atoms, and increasing, but still limited.

Need for Accelerated Molecular Dynamics Methods

•  Vapor-deposited thin film growth (metals and semiconductors) especially heteroepitaxial layers

•  Stress-assisted diffusion of point defects, defect clusters, and dislocations

•  Ion implantation or radiation damage annealing

Many important problems lie beyond the timescales accessible to conventional MD, due to the infrequent surface or bulk diffusion mechanisms involved, including:

Infrequent Event Systems

Transition State Theory (TST)

Three Recent Methods for Infrequent Events

Hyperdynamics: Basic Approach

Parallel Replica Dynamics: Basic Concept

Temperature Accelerated Dynamics

Temperature Accelerated Dynamics

¨  Key issue: As TAD relies upon harmonic TST for validity, any anharmonicity error at Thigh will lead to a corruption of the dynamics.

A. F. Voter, Phys. Rev. Lett. 78, 3908 (1997).

Ag10/Ag(111) diffusion Cu/Cu(100) epitaxial growth

10 monolayers (ML) per second

Example of parallel replica hyperdynamics

L. Sandoval et al. Phys. Rev. Lett. 114, 105502 (2015).

Competing Kinetics and He Bubble Morphology in W

Example of parallel replica hyperdynamics

W vacancies

W interstitial Additional vacancies

q  Fast and slow growth regimes are defined relative to typical diffusion hopping times of W interstitials around the He bubble

TAD Example: 1/4 ML Cu/Cu(111) at 150 K

M.R. Sørensen and A.F. Voter, J. Chem. Phys. 112, 9599 (2000).

Comparison of Different Acceleration Methods

q Accelerated Dynamics v  Hyperdynamics – boost potential ΔV(r) v  Parallel replica method for dynamics v  Temperature accelerated dynamics v  Long-time dynamics based on the dimer method v  ABC – slowly add penalty functions + static relaxation

q  Issues "   Solve the rough potential surface

problem for dynamics simulation "   Coupling motion of fast and slow

dynamics

Energy Landscape - Rough

Initial Final

Accelerated Molecular Dynamics

Steady-State Accelerated MD

B A S1

S2

S1

A B

q SSAMD method (rough potential surface and slow dynamics) Ø  Add boost potentials slowly

Ø  How to determine boost potential during MD simulation

Ø  In normal MD, increase T - the total vibration magnitude of atoms, driving the system to change state smoothly from A to B, and overcoming the energy barrier to state B

S2

S1

S3

S4

S5

S6

A B

Basic Ideas

q Vibration magnitude of atoms at a given temperature

Ai =!Rit −!Ri0

ª  The system reaches an equilibrium state – λ becomes a constant

λ = Aii=1

n

∑

λ (n

m)

λ of 2000 atoms at 400 K in Fe

Basic Ideas

q Vibration magnitude of atoms at a given temperature

Ai =!Rit −!Ri0

ª  The system occupies the different steady states at different temperatures.

ª  The states approach a

continuum function as the increase in temperature is infinitesimal (temperature-induced states).

2000 atoms

λ = Aii=1

n

∑

λ (n

m)

Basic Ideas

q Vibration magnitude of atoms at a given temperature

Ai =!Rit −!Ri0

ª  The system occupies the different steady states at different temperatures.

ª  The states approach a

continuum function as the increase in temperature is infinitesimal (temperature-induced states).

2000 atoms

λ = Aii=1

n

∑

λ (n

m)

Basic Ideas

Ø  Form boost potential

1)  Ai, S - the vibration magnitude of ith atom in S state 2) Activated volume (only boost atoms in the activated volume)

λS = Ai,S∑ΔV = F(λS )

AV

Boost Potential

Eb is constant parameter related to state s and we slowly increase Eb (t) and q(t)

kA→TST =

vA δA(r)eβΔVb (r )

Ab

eβΔVb (r )Ab

ΔV(r)Ab= 0 where δAb

(r) ≠ 0

ΔVAb(r) = 0 Ø  Two conditions:

1. both the original and biased systems obey transition state theory

The bias potential vanishes at any dividing surface 2. Concave boost potential

!!"#$( !!,… , !!!" ; !) = !!(!) 1− ! !!,… . . , !!!"! !

!!(!(!)− ! !!,… . . , !!!" )!

General Picture

Ø System evolves from one equilibrium state to another:

ª  The boost potential allows the system to jump from S0 S1 S2. ª When the system reaches state S1, the vibration λ reaches to q1. ª  The system can move along the corresponding constant energy contour.

qk = qk−1 +dq Eb,k = Eb,k−1 +dE

λ (Å)

Transition State

Ø  The time processing involving the boost potential at step k

tk = tMD expΔVk

kBT

"

#$$

%

&'' tMD: normal MD time step

At each state, there exists an average value of total vibration magnitude of n atoms: qs0 -> initial λ at a given temperature qS1 -> state S1 -

Ø Determine initial q

!!"#$( !!,… , !!!" ; !) = !!(!) 1− ! !!,… . . , !!!"! !

!!(!(!)− ! !!,… . . , !!!" )!

qk = qk−1 +dq

Validate SSAMD

q Migration of a mono vacancy in Fe and Surface Diffusion of Clusters in Cu

Ø  Ackland’s potential for Fe and Cu, and Fe-He potential by PNNL

Ø  Time step – 1 fs, Temperature – 200 ~ 500 K.

Ø  MD simulation in the canonical ensemble of NVT is performed to get the initial equilibrium state and determine the initial q (initial λ at a given temperature).

Ø  Set appropriate dq and dE, and add boost potential to system. At each state, the system is relaxed with boost potential and then evolved to next state (dq ~0.1-0.2 A and dE ~ 0.3 eV).

Ø  When a transition occurs, the simulation returns to normal MD which will bring the system to another equilibrium state.

Ø  Repeat the above steps, and explore the potential surface

•  One of the 1st nearest neighbor jumps to vacancy site

Fe atom

vacancy site

Vacancy Migration

Em~0.6 eV Em(NEB) ~ 0.63 eV (Ackland JPCM)

Ø  At 300 K, a jump ~ 2.8 x 10-5 s

Ø  100 jumps ~ 2.8 x 10-3 s

Surface Diffusion of Cu Atom

Ø Surface diffusion – atoms go underground

Nature Materials 2(2003)211

Surface Diffusion of Cu Atom

Ø Surface diffusion – hop-exchange of a dimer

T = 300 K Simulation time ~ 5.6 x10-3 s (5.6 ms)

Radiation damage – defects He accumulation – alters microstructures and properties

Material Issues in Fission and Fusion Reactors

Defects Fission (LWR) Fusion (3.5 MW/m2)

Displacement per atom/year 5 40

Helium (appm/year) 25 504

Hydrogen (appm/year) 250 1865

q  Annual defect production in fission and fusion reactor environments

Ø  Helium has significant effects in fusion structure materials, affecting the global evolution of microstructures and degrading the mechanical properties of materials.

Ø  One of the most important issues is helium bubble formation and growth

HeV2 Cluster at 400 K

q  Two paths

Fe atom

vacancy He atom Path I

Ø  Both paths are consistent with ab initio calculations (CC Fu, PRB)

Ø  Vacancy driven migration of the HnVm clusters (m>n)

Path II

Migration of HenVm Cluster (n>m)

(a)  

(b)  

(c)  

(d)  

v1  

v2  

v3  

v1  

v2  

v3  

v4  

v1  v2  

v3  

v4  

v1  v2  

v4  

He  atom  Vacancy  

Inters88al  

q  He6V3 (He9V6) (1) Creation of SIA (2) Orientation change of the dumbbell interstitial to crowdion (3)The crowdion diffuses along the cluster surface (4) Combination with original vacancy: V+SIA->0

Ø  SIA driven migration of HenVm (n>m) clusters （a two-step mechanism － creation of an SIA and recombination of the SIA with a vacancy)

SSAMD – Migration of He-V clusters

Total diffusion time is around 0.22s.

He atom

Vacancy

He6V4 T = 500 K

Mass center of He atoms Original mass center of He atoms

Ø  An self-interstitial atom driven migration

Interstitial

Migration of HenVm Cluster (n>m)

q  Migration energies of HenVm clusters (200, 300, 400 and 500 K)

D = D0 exp(−EakBT

)

Ø  D0 and Ea for the He6V4 are 1.0756e+14 and 0.88 eV, respectively.

Ø  D0 and Ea for the He9V6 are 2.1445e+13 and 0.87 eV

ª  The migration energy is close. ª  The mechanisms are same for these two He-V clusters.

Growth of A HenVm cluster from Small Clusters

q  He6V4 and He9V6 He13V10 (600K)

(a)   (c)  

(d)   (e)   (f)  

(b)  He9V6  

He6V4  

He9V5  

He6V5  

He13V10  

He2  

He10V7  

He5V3  

Interstitial He atom Vacancy

The temporal vacancy and interstitial are formed: He6V4->He6V5+SIA Mass transfer occurs between them.

6.79 µs

38

SSAMD – Growth of He-V clusters

Total time is around 6.79µs. Interaction between He9V7 and He6V3 T = 600 K

Ø  The small He-V disappears and transfers to the larger one

Ø  Ostwald Ripening (mass transportation) – an important mechanism

He atom

Vacancy Interstitial

He13V10

39

Electronic Effects: Two Temperature Model

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Electronic Effects: Two Temperature Model

41