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Molecular Dynamics Simulations of Fusion Materials: Challenges and
Opportunities (Recent Developments) Fei Gao
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
40
Electronic Effects: Two Temperature Model
41