Dresden Talk, March 2012 Uniform-acceptance force-biased Monte Carlo: A cheap way to boost MD Barend Thijsse Department of Materials Science and Engineering Delft University of Technology, The Netherlands Erik Neyts Department of Chemistry University of Antwerp, Belgium Maarten Mees Department of Physics Katholieke Universiteit Leuven, Belgium IMEC, Heverlee, Belgium VMM
Dresden Talk, March 2012 Uniform-acceptance force-biased Monte Carlo: A cheap way to boost MD Barend Thijsse Department of Materials Science and Engineering Delft University of Technology, The Netherlands Erik Neyts Department of Chemistry University of Antwerp, Belgium Maarten Mees Department of Physics Katholieke Universiteit Leuven, Belgium IMEC, Heverlee, Belgium VMM
MD: Good for fast mechanics 300 K 10.7 m/s VMM
MD: Slow for thermal activation ν 0 Transition State Theory ! = 1 e Q / kT Time to first transition: Q " 0 10 13 s –1 Surface Bulk Simulation time System time for 5000 atoms (one cpu) 15 min 1 s 30 y 1 ms ↑ relative time 80000 10 d 1 µ s 15 min 1 ns ↑ relative time 30 → difference 0.3 eV 1 s 1 ps 1 fs VMM
Boosting MD by MC: the simple way Force-biased Monte Carlo • No need to detect “events” or “crossings” • Works for small and big activation barriers • Simple algorithm, 5 lines of code, no overhead • No catalogue of transitions needed • Detailed balance satisfied • Can be combined with MD, taking turns or in parallel • Time progress can be measured NEW Mees, Pourtois, Neyts, Thijsse, Stesmans, Phys. Rev. B (2012), accepted VMM
Force-biased Monte Carlo: history • C. Pangali, M. Rao, B.J. Berne, Chem. Phys. Lett. 55 (1978) 413 – Theory only • S. Goldman, J. Comput. Phys. 62 (1986) 464 – H 2 O • G. Dereli, Mol. Simul. 8 (1992) 351 – Amorphous Si • C.H. Grein, R. Benedek, and T. de la Rubia, Comput. Mater. Sci. 6 (1996) 123 – Growth of Ge on Si(100) • M. Timonova, J. Groenewegen, and B.J. Thijsse, Phys. Rev. B 81 (2010) 144107 – Cu surface diffusion, Si phase transitions • E.C. Neyts, Y. Shibuta, A.C.T. van Duin, A. Bogaerts, ACS Nano 4 (2010) 6665 – C nanotube growth VMM
MD and force-biased Monte Carlo Choose a reasonable (first) timestep Δ t (0) Choose a maximum atomic displacement Δ /2 for the problem, then: for the problem, then: In x -direction ( y and z analogously), for each atom i : ( n ) # / 2 ! x , i " F x , i (“effective force”) 2 kT Choose uniform random number R x,i on [0,1] # 1 ln R x , i (e $ # " x , i ) + e ' " x , i # e # " x , i ! x , i = " x , i 2 ( n ) ! t ( n ) & ) ( ) % ( F i ( n ) ! t ( n ) + 1 ( n + 1) = r i ( n ) + v i r i ( n + 1) = r x , i ( n ) + ! x , i " / 2 (always accept) r x , i 2 m i ( n +1) from all r i ( n +1) ( n +1) from all r i ( n +1) Compute F i Compute F i δ x – Δ /2 Δ /2 0 Compute Δ t ( n +1) (optional) Big F , cool ( n ) ! t ( n ) + F i ( n + 1) ! t ( n + 1) F i ( n ) + 1 ( n + 1) = v i v i 2 m i Small F , hot VMM
Early success Recrystallization of ion-beam bombarded Si(100) Start Molecular Force-biased Dynamics Monte Carlo Monte Carlo Δ /2 = 0.075 Å is more than 100 times faster VMM
Uphill motion, detailed balance Fx ! x " ! U 2 kT # K " 1 e p ( ! x ) = K " 1 e 2 kT Probability density of a displacement Why this factor 2 here? F x " / 2 P (uphill) = 1 ! 1 Probability of an uphill move 2 4 2 kT ) 2 , # & F x " / 2 ! x RMS = " / 2 1 + 1 (more agitation with + . RMS displacement in a move % ( greater Δ /2 and smaller T ) + . 15 2 kT 3 $ ' * - Erik Neyts W ( ! r r ) P ( r ) = W ( r ! r ) P ( ! r ) Detailed balance r r )e " U ( r )/ kT = W ( r ! r )e " U ( ! r )/ kT Canonical W ( ! Transition probability ( W ) = Displacement ( D ) × acceptance ( A ) probabilities: r r )e " U ( r )/ kT = D ( r ! r )e " U ( ! r )/ kT D ( ! r r ) A ( ! r ) A ( r ! # & # & r r ) = min 1, D ( r ! r ) = min 1, D ( r ! r ) e " ( U ( ! r ) " U ( r ))/ kT e " ) U / kT A ( ! % ( % ( D ( ! r r ) D ( ! r r ) % ( % ( $ ' $ ' VMM
Detailed balance, uniform acceptance $ ' r r ) = min 1, D ( r ! r ) e " # U / kT A ( ! & ) D ( ! r r ) & ) % ( Metropolis: Trial move is uniformly sampled in its domain: D ( ! r r ) = D ( r ! r ) $ & r r ) = min 1, e " # U / kT Therefore acceptance is A ( ! % ' # $ U x x ) " K # 1 e 2 kT UFMC: D ( ! + $ U This UFMC is not unique. K # 1 e 2 kT D ( x x ) " ! ! D ( r ! r ) If x and x’ are not too far apart: K = K’ and F = F’ = e " U / kT If D ( ! r r ) r r ) = min 1, e + " U /2 kT $ ' e # " U /2 kT e # " U / kT A ( ! ) = 1 always uniform acceptance. & % ( This explains the factor 2. Therefore: always acceptance and detailed balance VMM
Metropolis vs UFMC 6 6 6 6 7 7 8 8 5 5 4 4 3 3 2 2 1 2 1 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 6 6 Metropolis UFMC VMM
Time Maarten Mees Which time interval Δ t can be associated with iteration step n ? Define as follows: # x ( n ) ! / 6 ≈ Δ /6, a very slow function ! t ( n ) " ! x ( n ) ! t " of the effective force 2 kT / # m v ( n ) Next, Δ should be made mass-dependent to allow for several atomic masses being present and have the same time interval for each species, ! i # ! m min / m i ! " Δ t in fs for several m min and Δ /2 = 0.1 R nnb 300 K 800 K 1300 K 1800 K ! / 6 ! t " m min 2 kT / # m min H 2.0 1.2 1.0 0.8 Larger Δ → more boost Si 33 20 16 13 but more deviation from F = F ’ Cu 54 33 26 22 W 98 60 47 40 VMM
One particle in cosine potential U = Q 0 # 1 ! cos 2 " x & Q 0 = 0.25 eV, L = 1 Å % ( 2 $ L ' n j = ! 0 t e " Q / kT TST: Number of jumps in time t : UFMC Metropolis Somewhat different UFMC version Δ /2 = 0.10 Å Q from straight line = 0.247 eV ν 0 = 0.9e13 Hz Counting arrivals in 5% region around a new minimum Counting crossings (incl recrossings) VMM
Quasivacancies Counting “vacancies” in crystals, amorphous, liquids in a consistent way i – Δ R i Quasivacancy (QV) “Missing neighbor” (MN) of atom i: r MN = r i – Δ R i QV concentration = (MN concentration)/ Z Δ R i = ∑ j ( r j – r i ), should be > 0.8 R nnb VMM
Test: When UFMC should do nothing Silicon crystal, MEAM-potential (M. Timonova, B.J. Lee, BJT). Quite good, but T m = 2990 K (too high) Δ /2 values just MD just MD Potential energy during UFMC Quasivacancy concentration during UFMC As expected: more agitation with greater Δ /2 and smaller (!) T. “Effective forces” are larger. Robust: All UFMC results, followed by MD, return to a perfect crystal (except o o o o → l l l l = violent UFMC, with Δ /2 = 0.17 R eq ). More robust: each atom returns to its own position when UFMC in green area is followed by MD. So: Δ /2 = 0.15 R eq is safe. Also at surface (100), including dimerization. VMM
Si phase transformation where MD fails Δ /2 = 0.06 R eq Δ /2 = 0.11 R eq C Polycrystal formed at C Potential energy during cooling+heating MD: 1 ps/K (lines) -- much slower than UFMC MD: Cooling to amorphous (relaxed), heating to glass transition, xtallization, melting (liq: Z = 5.6). UFMC with Δ /2 = 0.06 R eq : Cooling to amorphous (relaxed), heating to liquid (Z = 5.5) . UFMC with Δ /2 = 0.11 R eq : Cooling to polycrystal, heating to liquid. The amorphous phase also crystallizes to a polycrystal at 300 K VMM
Si recrystallization: UFMC faster than MD MD UFMC++ ∆ /2 = 0.06 R eq ∆ /2 = 0.11 R eq ∆ /2 = 0.14 R eq 303 K 0 % Ar evapor. 0 % Ar evapor. 17 % Ar evapor. 2.5 % Ar evapor. 17 % QV 17 % QV 2.5 % QV 2.5 % QV 1518 K /10 5 8.7 % Ar evapor. 3.6 % Ar evapor. 15 % Ar evapor. 33 % Ar evapor. 11 % QV 4 % QV 16 % QV 2 % QV 2024 K 12 % Ar evapor. 8,7 % Ar evapor. 71 % Ar evapor. 19 % Ar evapor. 8 % QV 8 % QV 1.8 % QV 2.5 % QV 2530 K 60 % Ar evapor. 88 % Ar evapor. 78 % Ar evapor. 89 % Ar evapor. 1 % QV 2 % QV 1 % QV /10 5 1 % QV VMM
Other successes E.C. Neyts, Y. Shibuta, A.C.T. van Duin, A. Bogaerts, ACS Nano 4 (2010) 6665 UFMC, well defined chirality MD Also: Ni nanocluster melting, Neyts/Bogaerts JCP 2009 VMM
Surprises Fe: fcc → bcc transformation UFMC: A cheap method to construct a polycrystal? Here bcc Fe bcc MD fcc bcc 2 neighbors ( l ) ( i ) = ( l ) ( ! ij , " ij ) # w ( m ) Y ( m ) j Rotationally invariant: l w ( l ) ( i ) = 4 ! # ( l ) w ( m ) UFMC 2 l + 1 UFMC 1000 K → MD 1000 K → Quench m = " l Number of fcc atoms after 3 × 10 5 steps UFMC 1000 K → Quench UFMC goes the wrong way! VMM
Conclusions • UFMC has much potential as MD booster • Very easy to handle • Only 5 lines of program code • No thermostat needed, T is built into the method • Solid statistical basis • Time can be implemented sensibly • Mixture of atomic masses can be handled consistently • Dynamic creation and annihilation of QV appears essential • For further study • Boost is not always spectacular • Alternative displacement statistics may be better • Does not work in close packed systems? • Convenient method to generate polycrystals? VMM
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