Crossing the barriers - simulations of activated processes

Transcription

1 Crossing the barriers - simulations of activated processes Mgr. Ján Hreha for 6 th Student Colloquium and School on Mathematical Physics Faculty of Mathematics, Physics and Informatics Comenius University in Bratislava August 21, 2012 Ján Hreha (FMFI UK) Crossing the barriers August 21, / 23

2 Outline 1 Activated process 2 How to deal with them? 3 Metadynamics (MTD) MD 4 Silica as an important material 5 Our simulations 6 Summary Ján Hreha (FMFI UK) Crossing the barriers August 21, / 23

3 Activated process (Free) Energy surface Activated processes involve crossing high energy barriers. Ja n Hreha (FMFI UK) Crossing the barriers August 21, / 23

4 Activated process Activated process complicated free energy landscape many metastable valleys separated by steep barriers too high for thermal fluctuations high dimensionality time scale gap for simulations (even >10 orders of magnitude) Ján Hreha (FMFI UK) Crossing the barriers August 21, / 23

5 Activated process Motivation Activated processes tend to appear in: chemical reactions reconstructive phase transitions formation of nano-materials protein folding proton transport trough biomembranes. Ján Hreha (FMFI UK) Crossing the barriers August 21, / 23

6 How to deal with them? How to deal with activated processes? give more power reduce the barriers avoid returning back Ján Hreha (FMFI UK) Crossing the barriers August 21, / 23

7 Hire parachute troops How to deal with them? The best method to map a high dimensional surface is random sampling. Can be used only with newest (super)computers. Ján Hreha (FMFI UK) Crossing the barriers August 21, / 23

8 How to deal with them? Overheating e.g. simulated annealing sampling unrealistic ensemble- collective modes wrong reaction pathways Ján Hreha (FMFI UK) Crossing the barriers August 21, / 23

9 How to deal with them? Lowering barriers Raise the pressure! sampling different ensemble no chance to extract real transition pressure overpass metastable valleys on the way Ján Hreha (FMFI UK) Crossing the barriers August 21, / 23

10 Lowering barriers How to deal with them? Importance Sampling, Umbrella Sampling, Weighted Histogram Analysis Method sampling different distribution biased by w π( r) = w( r) e U( r)/k BT d r w( r ) e U( r )/K B T A = A/w w 1/w w Ján Hreha (FMFI UK) Crossing the barriers August 21, / 23

11 Metadynamics Metadynamics (MTD) dimensionality reduction by splitting fast and slow DoF steepest descent dynamics in slow parameters s t+1 = s t + δs Φt Φ t, Φt = F t (s) s Gaussian history dependent biasing potential F t (s) = F(s) + ( s t s 2 ) W exp 2δs 2 t <t (1) (2) FES mapping without actually calculating the F Figure: Free energy surface exploring by metadynamics. Ján Hreha (FMFI UK) Crossing the barriers August 21, / 23

12 Metadynamics (MTD) MD Molecular dynamics MD for Fast degrees of freedom: Quantum mechanics for Electron under adiabatic approximation Classical approximation for ions Hellmann-Feynman forces from e.g. DFT RI E({R}) = RI Ψ 0 H Ψ 0 = Ψ 0 RI (H) Ψ 0 (3) Newton s equation integration - MD steps m i R i = E [ ] = F i {R} R i (4) Ján Hreha (FMFI UK) Crossing the barriers August 21, / 23

13 Metadynamics (MTD) MD Parrinello-Rahman scheme Constant pressure scaled coordinates r i = h s i. collective variables h = (a, b, c) for simulation box parametrisation extended Lagrangian for box relaxation L = 1 m i ṡ T i h T hṡ i u(r i,j ) W Tr(ḣT ḣ) P det h (5) i<j i Ján Hreha (FMFI UK) Crossing the barriers August 21, / 23

14 Metadynamics (MTD) Metadynamics application MD supercell parameters as collective variables h t+1 = h t + δh Φt Φ t, Φt = G t (h) h (6) 2 G t (h) = G (h) + h ht W e 2 δh 2 (7) t <t G = V [ h 1 (σ + P) ] h ji, (8) ij dimensionality reduction - upper triangular form of h Ján Hreha (FMFI UK) Crossing the barriers August 21, / 23

15 Silica as an important material Silica as an important material abundant compound with rich polymorphism geological importance use in electronics Amir Chossrow Akhavan. quartzpage.de [2008] Ján Hreha (FMFI UK) Crossing the barriers August 21, / 23

16 Silica polymorphs Silica as an important material quartz cristobalite seifertite Ján Hreha (FMFI UK) Crossing the barriers August 21, / 23

17 Silica as an important material Post-stishovite phases J. P. Driver (2008) rutile-like kinked pyrite-like Ján Hreha (FMFI UK) Crossing the barriers August 21, / 23

18 Simulations Our simulations Table: Simulations with 48 atoms in a supercell label P [kbar] T [K] Gaussian width phase transition total # of meta=steps A yes 791 B yes 96 Table: Simulations with 96 atoms in a supercell label P [kbar] T [K] Gaussian width phase transition total # of meta-steps no no no no no 47 C /100 yes 422 Ján Hreha (FMFI UK) Crossing the barriers August 21, / 23

19 run A: 48 atoms, 2000 kbar, 2500 K monitored parameters: 1 enthalpy 2 volume 3 supercell lengths 4 supercell angles

20 Our simulations Transition from pyrite-like SiO 2 into 4 2 kinked structure Figure: Visualization of the transition process using the polyhedral view. We can see the different orientation of octahedra in the broken plane. Ján Hreha (FMFI UK) Crossing the barriers August 21, / 23

21 Summary Summary Results: Demonstration of the MTD method applicability for phase transitions under extreme pressure observed series of transitions into post-stishovite phases description of (new) transition mechanisms on atomic level Ján Hreha (FMFI UK) Crossing the barriers August 21, / 23

22 Summary References Understanding Molecular Simulation: From Algorithms to Applications, Dean Frenkel, Berend Smit, Academic press (2002) Predicting Crystal Structures: The Parrinello-Rahman Method, Revisited, R. Martonak, A. Laio, and M. Parrinello, Phys. Rev. Lett. 90, (2003) Simulation of structural phase transitions by metadynamics, R. Martonak, A. Laio, M. Bernasconi, C. Ceriani, 220 (2005) Study of structural transitions in SiO 2 by ab initio metadynamics, Hreha, R. Martonak, FMFI UK (2012) Ján Hreha (FMFI UK) Crossing the barriers August 21, / 23

23 Summary Thank you Thank you for your attention. Ján Hreha (FMFI UK) Crossing the barriers August 21, / 23