1. Algorithms and Software for Large-Scale Simulation of Reactive Systems _______________________________ Metin Aktulga, Sagar Pandit, Alejandro Strachan, and Ananth Grama PRISM Purdue University

2. Molecular Simulation Methods Ab-initio methods (few approximations but slow) Density Functional Theory (DFT) Car-Parrinello Molecular Dynamics (CPMD) Electron and nuclei treated explicitly Classical atomistic methods (coarser approximations) Classical molecular dynamics Monte Carlo Brownian dynamics No electronic degrees of freedom (approximated through fixed partial charges on atoms). Continuum methods (no atomistic details)

3. Molecular Simulation Methods Statistical and continuum methods ps ns ss ms nm mm mm Ab-initio methods Atomistic methods

4. V = V bond + V angle + V dihedral + V LJ + V Elecrostatics Interactions in Classical MD

5. Implementation of Classical Interactions Molecular topologies are fixed Bonded interactions are implemented as static neighbor lists Non-bonded interactions are implemented using dynamic neighbor lists Usually not updated at every time step Only two body interactions, relatively easy to implement.

6. Reactive Systems Chemical reactions correspond to association and dissociation of chemical bonds Classical simulations cannot simulate reactions ab-initio methods calculate orbital overlap to model chemical reactions Reax force field postulates a classical bond order interaction to mimic the association and dissociation of chemical bonds 1 1 van Duin et al, J. Phys. Chem. A, 105, 9396 (2001)

7. Bond Order Interaction 1 van Duin et al, J. Phys. Chem. A, 105, 9396 (2001) Bond order for C-C bond Uncorrected bond order: where  is for  and  bonds  The total uncorrected bond order is sum of three types of bonds Bond order requires correction to account for the correct valency

8. Upon correction, the bond order between a pair of atoms depends on the uncorrected bond orders of the neighbors of each atoms Bond orders rapidly decay to zero as a function of distance so it is reasonable to construct a neighbor list for efficient computation of bond orders Bond Order Interaction

9. Neighbor Lists for Bond Order Efficient implementation critical to performance Implementation based on a dynamic oct-tree decomposition of the domain Has implications for parallelism (issues identical to parallelizing multipole methods)

10. Bond Order : Choline

11. Bond Order : Benzene

12. Other Local Energy Terms Other interaction terms common to classical MD (bond energy, valence angle, torsion) are modified, and contribute to non-zero bond order pairs of atoms These terms become many body interactions as bond order itself depends on the neighbors and neighbor’s neighbors Due to variable bond structure there are other interaction terms, such as over/under coordination energy, lone pair interaction, 3- and 4- body conjugation, and 3-body penalty energy

13. Non Bonded van der Waals Interaction van der Waals interactions are modeled using distance corrected Morse potential where R(r ij ) is the shielded distance, given by

14. Electrostatics Shielded electrostatic interaction is used to model orbital overlap of electrons at closer distances Long range electrostatics interactions are handled through cut-offs

15. Charge Equilibration (QEq) Method The fixed partial charge model used in classical simulations is inadequate for reacting systems. One must compute the partial charges on atoms at each time step using an ab-initio method. We compute the partial charges on atoms at each time step using a simplified approach call the QEq method.

16. Charge Equilibration (QEq) Method Expand electrostatic energy as a Taylor series in charge around neutral charge. Equate the term linear in charge to electronegativity of the atom and the quadratic term to electrostatic potential and self energy. Using these, solve for self-term of partial derivative of electrostatic energy.

17. QEq Method We need to minimize: subject to: where

18. Qeq Method

19. QEq Method From charge neutrality, we get:

20. QEq Method Let where or

21. QEq Method Substituting back, we get: We need to solve 2n equations with kernel H for s i and t i.

22. QEq Method Observations: H is dense. The diagonal term is J i The shielding term is short-range Long range behavior of the kernel is 1/r

23. Computational Units Bonded Forces: Bond order computation Bond forces Lone-pair forces, Over/under-coordination forces Valence angle forces Torsion angle forces Hydrogen bond forces (if necessary) Derivative of bond order computation is nicely coupled with total bonded-force computation reducing memory and processing overhead

24. Computational Units Non-bonded Forces: charge equilibration (QEq) electrostatic forces van der Waals forces GMRES(30) with a diagonal preconditioner and quadratic interpolation for initialization shielding and tabulation with interpolation yield significant performance improvement while retaining accuracy

25. Implementation, Performance, and Validation

26. Serial Performance Profiling 6540-atom water system using a relative residual norm of 10 -6 for GMRES(30) and tabulated long range interactions. Neighbor Generation (23%) Force Calculation (76%) Bonded Forces (15%):  Hydrogen bond forces (10%)  Others (5%) Non-Bonded Forces (60%):  QEq (42%)  Electrostatics and van der Waals forces (18%)  Evolving the System(1%)

27. Serial Performance: Scaling

28. Serial Performance: Memory Scaling

29. Parallel Performance Total execution times per MD timestep for the ReaxFF MD with scaled workloads—12,192 x p atom water systems (p = 1,..,1024).

30. Current Development Efforts Integration into LAMMPS. Qeq optimization. Force field development.

31. Develop first principles-based constitutive relationships and provide atomic level insight for coarse grain models Atomistic Materials Simulations in PRISM Identify and quantify the molecular level mechanisms that govern performance, reliability and failure of PRISM device using: Ab initio simulations Large-scale MD simulations

32. Atomistic Modeling of Contact Physics How: classical MD with ab initio-based potentials Size: 200 M to 1.5 B atoms Time scales: nanoseconds Mechanical response: Force-separation relationships (history dependent) Generation of defects in metal & roughness evolution Generation of defects in dielectric (dielectric charging) Electronic properties: Thermal role of electrons in metals Current crowding and Joule Heating Chemistry: Surface chemical reactions Predictions: Role of initial microstructure & surface roughness, moisture and impact velocity on: Main Challenges Interatomic potentials Implicit description of electrons

33. Atomistic Modeling of Contact Physics: II Mobility of dislocations in metal, Interactions with other defects (e.g. GBs) Link to phase fields Defects in semiconductor Mobility and recombination Role of electric charging Surface chemical reactions Reactive MD using ReaxFF Fluid-solid interaction: Interaction of single gas molecule with surface (accommodation coefficients) for rarefied gas regime Smaller scale (0.5 – 2 M atom) and longer time (100 ns) simulations to uncover specific physics:

34. Planned Development Efforts Upscaling MD to: Fluid Dynamics Electronic Processes Micromechanics Thermal Effects Atomistic Simulations: Potential development Interface to MD Interface to Continuum