Molecular Dynamics Simulations of Fusion Materials: Challenges and Opportunities (Recent Developments)

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 m 3 10 6 mols 10 30 atoms Person time scales Lifetime 100 years 10 9 s Time scale for atomic motions is 10-12 s, so time step is 10-13 s Total number of time steps 10 22 Person simulation At 10 calculations per each atom per each time step we get 10 x 10 30 x 10 22 = 10 53 calculations Present computing capability is teraflop 10 12 FLOPS (floating point operations per second) Person simulation takes 10 41 s 10 34 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 cm 2 /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 10 10 atoms, and increasing, but still limited.

Need for Accelerated Molecular Dynamics Methods Many important problems lie beyond the timescales accessible to conventional MD, due to the infrequent surface or bulk diffusion mechanisms involved, including: 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

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 T high will lead to a corruption of the dynamics.

Example of parallel replica hyperdynamics Ag 10 /Ag(111) diffusion Cu/Cu(100) epitaxial growth A. F. Voter, Phys. Rev. Lett. 78, 3908 (1997). 10 monolayers (ML) per second

Example of parallel replica hyperdynamics Competing Kinetics and He Bubble Morphology in W 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 L. Sandoval et al. Phys. Rev. Lett. 114, 105502 (2015).

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

Accelerated Molecular Dynamics 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

Steady-State Accelerated MD q SSAMD method (rough potential surface and slow dynamics) Ø Add boost potentials slowly S 6 S 5 A B S 1 S2 S 1 S 2 S 4 S 1 S 3 A B 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 A B

Basic Ideas q Vibration magnitude of atoms at a given temperature A i =! R i t! R i 0 λ = n i=1 A i λ (nm) λ of 2000 atoms at 400 K in Fe ª The system reaches an equilibrium state λ becomes a constant

Basic Ideas q Vibration magnitude of atoms at a given temperature A i =! R i t! R i 0 λ = n i=1 A i λ (nm) 2000 atoms 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 (temperatureinduced states).

Ø Form boost potential ΔV = F(λ S ) Basic Ideas λ S = A i,s 1) A i, S - the vibration magnitude of ith atom in S state 2) Activated volume (only boost atoms in the activated volume) AV

Boost Potential Ø Two conditions: 1. both the original and biased systems obey transition state theory ΔV Ab (r) = 0 k TST A = v A δ A (r)e βδv b (r) e βδv b (r) A b A b The bias potential vanishes at any dividing surface ΔV(r) Ab = 0 where δ Ab (r) 0 2. Concave boost potential!!"#$ (!!,,!!!" ;!) =!! (!) 1!!!,..,!!!"!!!!(!(!)!!!,..,!!!" )! E b is constant parameter related to state s and we slowly increase E b (t) and q(t)

General Picture Ø System evolves from one equilibrium state to another: λ (Å) ª The boost potential allows the system to jump from S 0 S 1 S 2. ª When the system reaches state S 1, the vibration λ reaches to q 1. ª The system can move along the corresponding constant energy contour. q k = q k 1 + dq E b,k = E b,k 1 + de

Transition State Ø The time processing involving the boost potential at step k " t k = t MD exp$ ΔVk # k B T % ' & t MD : normal MD time step Ø Determine initial q!!"#$ (!!,,!!!" ;!) =!! (!) 1!!!,..,!!!"!!!!(!(!)!!!,..,!!!" )! At each state, there exists an average value of total vibration magnitude of n atoms: q s0 -> initial λ at a given temperature q S1 -> state S 1 q- k = q k 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

Vacancy Migration One of the 1 st nearest neighbor jumps to vacancy site Fe atom vacancy site Ø At 300 K, a jump ~ 2.8 x 10-5 s Ø 100 jumps ~ 2.8 x 10-3 s E m ~0.6 ev E m (NEB) ~ 0.63 ev (Ackland JPCM)

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)

Material Issues in Fission and Fusion Reactors Radiation damage defects He accumulation alters microstructures and properties q Annual defect production in fission and fusion reactor environments Defects Fission (LWR) Fusion (3.5 MW/m 2 ) Displacement per atom/year 5 40 Helium (appm/year) 25 504 Hydrogen (appm/year) 250 1865 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

HeV 2 Cluster at 400 K q Two paths Path I Fe atom He atom vacancy Ø Both paths are consistent with ab initio calculations (CC Fu, PRB) Ø Vacancy driven migration of the H n V m clusters (m>n) Path II

Migration of He n V m Cluster (n>m) q He 6 V 3 (He 9 V 6 ) (a) (b) v2 v1 v3 v4 v1 Vacancy He atom v2 v4 v4 v1 v3 v1 v2 v2 (c) (d) (1) Creation of SIA (2) Orientation change of the dumbbell interstitial to crowdion (3)The crowdion diffuses along the cluster surface Inters88al v3 (4) Combination with original vacancy: V +SIA->0 Ø SIA driven migration of He n V m (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 6 V 4 T = 500 K He atom Vacancy Interstitial Mass center of He atoms Original mass center of He atoms Ø An self-interstitial atom driven migration

Migration of He n V m Cluster (n>m) q Migration energies of He n V m clusters (200, 300, 400 and 500 K) D = D 0 exp( E a k B T ) Ø D 0 and E a for the He 6 V 4 are 1.0756e+14 and 0.88 ev, respectively. Ø D 0 and E a for the He 9 V 6 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 He n V m cluster from Small Clusters q He 6 V 4 and He 9 V 6 He 13 V 10 (600K) He atom Vacancy Interstitial (a) He 9 V 6 He 6 V 4 (b) He 9 V 5 (c) He 10 V 7 He 6 V 5 (d) (e) He 13 V 10 (f) He 5 V 3 6.79 µs He 2 The temporal vacancy and interstitial are formed: He 6 V 4 - >He 6 V 5 +SIA Mass transfer occurs between them.

SSAMD Growth of He-V clusters Total time is around 6.79µs. Interaction between He 9 V 7 and He 6 V 3 T = 600 K He atom Vacancy Interstitial Ø The small He-V disappears and transfers to the larger one Ø Ostwald Ripening (mass transportation) an important mechanism He 13 V 10 38

Electronic Effects: Two Temperature Model 39

Electronic Effects: Two Temperature Model 40