Applications of High-Performance Computing in Geochemistry Adam F. Wallace ([email protected]) & Kendra J. Lynn ([email protected]) Department of Geological Sciences University of Delaware
Kendra J. Lynn Postdoc with Dr. Jessica Warren (left) Lab Members and Affiliates
Adam Yifei Ma June Hazewski Chunlei Wang Brianna McEvoy Wallace Ph.D. student M.S. student Ph.D. student M.S. 2017 (Sturchio Lab) Geochemists are interested in processes that occur over a wide variety of time and length scales
Atomic Scale Mineral Surfaces Grain Boundary Processes
2.5 x 2.5 µm
Whole Rock Global Regional Modeling is critical because many locations and conditions of interest are physically inaccessible in the field and in the lab
pore-scale whole mineral/rock regional scale
nucleation / growth recrystallization Time mineral-water equilibria ion-hydrolysis / polymerization Continuum Ion sorption / desorption Diffusion in solution Coarse Grained gas-water MD
ion-ion ion-water Reactive MD (i.e. ReaxFF) QC (ab initio / DFT) Distance Our lab uses experimental and computational tools to investigate mineral-water interactions AFM-based studies of mineral nucleation & growth Theoretical models of mineral stability and reactivity i r i r cos θ cos 3 i Σ = angle angle ξ
3 2.5 x 2.5 µm ri ri ξbond = r 2 Σ 0 i
7.5 x 7.5 µm Overview of atomistic simulation methods
Electronic Structure Methods Classical Molecular Dynamics
• Density Functional Theory (DFT), • Based on Newton’s equations of ab initio methods motion.
• Based on approximate solutions to Schrodinger’s equation: F = ma
HΨ = EΨ • Physics modeled with with less precision.
• Physics modeled with “high” • Requires a lot of computer accuracy. power but less than electronic structure methods. • Requires a lot of computer power. • 10-100 ns trajectories typical
• 10-20 ps trajectories typical Reactions of geochemical interest are often too slow to be observed with direct simulation methods
• Earth’s crust is primarily composed of silicate minerals • At surface conditions most silicates dissolve very slowly (10-6 to 10-14 mol / m2 sec). • Even at 200°C quartz dissolves at ~10-6 mol / m2 sec in salt water solutions.
This is equivalent to the release of ~10-4 molecules of
2 H4SiO4° per 100 nm of surface per microsecond.
Dove and Nix (2002) GCA, 61:3329-3340. Specialized methods are needed to overcome timescale limitations and enhance sampling
Global energy minimization strategies
• Minima Hopping (LJ38) • Replica Exchange Molecular Dynamics (CaCO3.nH2O clusters) Exploration of free energy landscapes with biased dynamics • Use of the Colvars Library in LAMMPS • Umbrella Sampling (water exchange on Ca2+) • Metadynamics (Si-O bond hydrolysis) Calculation of reaction rates (water exchange reactions) • Direct methods • Reactive Flux • Forward Flux Sampling Absolute Free Energy methods
• 2PT (substitution of CO2 for H2O in sepiolite) Global Energy Minimization Strategies Minima Hopping (Goedecker (2004) J. Chem. Phys., 120:9911) • Hopping is performed by activation-relaxation steps.
• During an activation step, a molecular dynamics trajectory is initiated with a given kinetic energy (Ekin) and followed until the potential energy decreases.
• During a relaxation step the system energy is minimized to the nearest local minimum. Successful escape Ekin decreases Failed escape • The move to the new minimum is accepted if the Ekin increases energy difference between the new and old minima Potential Energy is less than a parameter (Ediff) that is dynamically adjusted so that ~50% of the moves are accepted.
• If the system returns to a minimum it has already Reaction Progress visited Ekin is increased. Minima Hopping (Goedecker (2004) J. Chem. Phys., 120:9911) Implementation Details • Activation steps are performed with LAMMPS • Relaxation steps are performed with LAMMPS or GULP. • A python script controls the setup and execution of LAMMPS/GULP. Energy
Minima Visited Parallel Tempering / Replica Exchange Molecular Dynamics
Exchange between replicas occurs subject to a conditional probability rule. Replica Exchange Sampling Circumvents High Energy Barriers
Potential Energy Landscape Replica Exchange Sampling Can Locate Global Minima
Potential Energy Landscape Enhanced sampling of hydrated CaCO3 clusters with Replica Exchange Molecular Dynamics ( ).Wallace et al. (2013) Science, 341:885-889 Exploration of Free Energy Landscapes with Biased Dynamics Use of the Colvars Library in LAMMPS
• The Colvars library is included in LAMMPS distributions as an optional package.
• The library enables a number of biased sampling methods, including: umbrella sampling, metadynamics and adaptive bias force.
• The library is invoked as a “fix” in the LAMMPS input script. Anatomy of a simple COLVARS input file
Output and restart frequency.
Block that defines the collective variable to apply the bias to. In this case the distance between two groups containing 1 atom each.
This block applies a harmonic restraining potential to maintain the value of the collective variable “DIST” centered around 3.1. The energy units are those used by lammps. In this case the force constant is in eV. Sample COLVARS Output
MD Step Instantaneous value of “DIST” Obtaining the free energy barrier (Umbrella Sampling)
Define reaction progress coordinate Apply a bias along the 1 2 (often metal-oxygen distance for water reaction coordinate exchange reactions) Obtain biased probability 3 distributions
Convert probabilities to free 4 energies for each window and subtract the bias potential.
Combine free energy 5 segments Using WHAM utility to Process COLVARS Output
We use a utility maintained by Alan Grossfield at U. Rochester to process umbrella sampling output from COLVARS.
http://membrane.urmc.rochester.edu/content/wham
The utility takes a file ”metadata” as input that specifies the locations of the COLVARS output for all the sampling windows, and writes an output file ”freefile” that contains the free energy with respect to the biases coordinate. Using WHAM utility to Process COLVARS Output
Sample metadata file window Path to COLVARS output files center
Restraint force constant (kcal/mol)
Running WHAM Metadynamics Basics
The metadynamics method accelerates the exploration of energy landscapes by the slow buildup of a history dependent bias.
The bias encourages the simulation to explore new states and ultimately grows to form the negative image of the energy landscape.
Van Speybroeck et al. (2014) Chem. Soc. Rev., 43:7326-7357. Metadynamics Simulation of Si-O Bond Hydrolysis ) br O - max(Si
min(Si-OW) Sample LAMMPS and COLVARS input files for 2D Metadynamics
File to read group definitions from
Definition of collective variable “one”
as the minimum Si-OW distance.
Definition of collective variable “two” LAMMPS groups written to file as the maximum Si-Obr distance. This to be read by COLVARS uses a custom function that depends on the Lepton library.
Settings that control the accumulation of the metadynamics bias function.
LAMMPS Input COLVARS Input Calculation Water Exchange Rates ∗ ∗ I H O ] + H O [I H O H O ] + H O
1 � = �
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Correlation of Water Exchange Rates & Bulk Reactivity Trends (Oxides and Orthosilicates)
f0010 Fig. 2 Dissolution rates at pH 2 ofCasey simple (2017) oxide Adv. (A) Inorg and. Chem., orthosilicate 69:91-115. (B and C) minerals containing divalent metals. The abscissa is the rate of 2+ exchange of water from the corresponding metal ion (e.g., Mg (aq)), and the ordinate is the dissolution rate of the mineral (e.g., Mg2SiO4, forsterite) normalized to area. The oxide minerals have the rocksalt structure. The orthosilicate minerals (5) have the stoichiometry: 4 M2SiO4(s) and isolated SiO4 À tetrahedra. The end-member compositions of orthosilicates are shown in red squares in (B), whereas the mixed-metal compositions are shown as blue dots.Forthemixed-metalcompositions,dissolutionratesareplottedagainsttheweighted sum of the logarithm of rates of water exchange for the component ions. Note that the dissolution rates for mixed-metal compositions (blue dots in (B)) fall intermediate between the end-member compositions. The rates of ligand exchange of the aquated ions ( ) and the dissolution rates of the orthosilicate minerals ( ) at pH 2 are plotted as a function of the number of d electrons in (C), illustrating the role of ligand-field stabilization in the rates. Panels (B) and (C) are adapted from Ref. Casey, W.H.; Swaddle, T.W. Reviews of Geophysics 2003, 41 (2), 4/1–4/20. Correlation of Water Exchange Rates & Bulk Reactivity Trends3338 P.M. Dove and C.J. Nix (SiO ) (a) 2 -6.0 [] water, electrolytes not present • 0.01 molal MeC12 ~2 © 0.05 molal MeC12 o -6.5 q o
= © _= -7.0 @ 2+ ~3 o © -7.5 [] © Si4+ 200oC -8.0 i I r I i I I I r i i [ i ~ i i i i r 0.00 2.00 4.00 6.00 8.00 10.00 log kex of aqueous ions (s -1) (b) -6.0 [] water,Dove andelectrolytes Nix (2002) not present GCA, 61:3329-3340.
-6.5 0
O -7.0 ©
@ -7.5 [] [] o 200°C -8.0 i K i I i i I I i i i I i i i I i r i 0.00 2.00 4.00 6.00 8.00 10.00 log kex of aqueous ions (s -1)
Fig. 5. Dissolution rate of quartz at 200°C vs. log rate of solvent exchange, kox, for the predominant cation at the silica-solution interface. Slowest rates are measured in salt-free solutions where only low concentrations H4SiO4, is present. Rates increase with the introduction of IIA cations in a predictable trend that is linearly related to their kex value. (a) A comparison of rates measured in 0.01 and 0.05 molal solutions shows that the slopes of these linear relationships increase with the total MeClz concentration. Trends have r 2 of 0.996 and 0.983, respectively. (b) Quartz dissolution rates in solutions containing one of six different monovalent and divalent cations having comparable ionic strengths of 0.15 show an empirical correlation given by Eqn. 10.
studies of amorphous and crystalline silica polymorphs which a factor of two compared to pure water. In contrast, solutions suggests that these relative rates hold over the temperature containing potassium or barium ion can enhance rates as much range of 20-200°C. In the broader perspective of Earth envi- as forty times at 200°C. ronments, any of these IA or IIA cations have a greater rate enhancing effect than organic compounds. For example, Ben- 5.1. Relationship to Previous Studies nett et al. (1988) and Bennett (1991) showed that organic The relationship between quartz dissolution rate and the acids, citrate, and oxalate, enhanced quartz dissolution rates by kex of solute ions shown in Fig. 5a,b is consistent with previ- Correlation of Water Exchange Rates & Bulk Reactivity Trends (Carbonates)
Pokrovsky and Schott (2002) Environ. Sci. Technol., 36:426-432. Direct Rate Methods for Ligand Substitution (Impey’s Method) 1 � = = �(�) �� � 5 1 7 ( ) 1 7 6 �(�) = � (0)� (�) �(0) 6 2 1 Ca2+ 3 3 4 4 5 2 Impey Residence Time: Ca2+ in SPC/Fw water
Impey -1 -1 τ res (ps) ⟨τ res⟩ (ps) k ex (ps ) ⟨k ex⟩ (ps ) 227.3 0.0044 200.5 0.0050 NVT 186.0 205.2 ± 13.3 0.0054 0.0049 ± 0.0003 204.5 0.0049 207.7 0.0048 274.4 0.0036 274.3 0.0036 DPD 254.4 261.3 ± 15.2 0.0039 0.0038 ± 0.0002 234.7 0.0043 268.8 0.0037 279.0 0.0036 197.0 0.0051 CSVR 220.3 228.3 ± 29.8 0.0045 0.0045 ± 0.0005 242.1 0.0041 202.9 0.0049
Wallace & Ma, submitted.
� = 230 psec Rateri et al. (2015) J. Phys. Chem. C, 119(43):24447–24458 Direct Rate Methods for Ligand Substitution (Hofer’s “Direct” Method)
� � RESERVOIR
� = � = mass exchange rate
Reservoir Mass � = Mass Exchange Rate Direct Rate Methods for Ligand Substitution (Hofer’s “Direct” Method)
Reservoir Mass � = Mass Exchange Rate
Reservoir Mass = � (ion coordination number)
number of exchanges per ion Mass Exchange Rate = � = length of simulation
1 � � = = � � ⁄ Ca Indirect rate methods (Reactive Flux)
� = ��
Transition state theory rate constant ΔG* (determined by the height of the energy barrier)
Transmission coefficient Free Energy (dynamical correction to � )
Reaction Progress The TST rate is based entirely on the barrier height
Transition State Theory Residence Time
� ~ 36 psec
Wallace & Ma, submitted. TST assumes every visit to the transition state is successful (this isn’t ture)
To correct the TST rate we calculate 1 the transmission coefficient: 0.8 �̇ 0 �[� � − �‡ � � = 0.6 �̇ 0 �[�̇(0)] ) k(t 0.4
0.2 Number of successful trajectories 0 Total number of attempts 0 0.5 1 1.5 2 Time (psec) Applying the dynamical correction yields τRF ≈ τres 1 � = = �� �
1 1 1
0.8 0.8 0.8 NVT DPD CSVR 0.6 0.6 0.6 K(t)
0.4 0.4 0.4
0.2 0.2 0.2
0 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Time (psec) Time (psec) Time (psec)
Reactive Flux TST -1 RF -1 k (ps ) k k (ps ) t RF (ps) NVT 0.0322 0.1748 ± 0.0062 0.0056 ± 0.0002 178.1 ± 7.0 DPD 0.0273 0.1678 ± 0.0211 0.0046 ± 0.0006 221.4 ± 31.2 CSVR 0.0257 0.1707 ± 0.0213 0.0044 ± 0.0005 231.4 ± 34.3 Wallace & Ma, submitted. Indirect rate methods (Forward Flux Sampling)
Attempt Rate Calculation Determination of Interface Fluxes 1 � = = Φ = Φ � � � ) � � � ) = � � � ) � , , Elapsed time is used in the place of a traditional order parameter
Reactant / Product basins are defined by threshold values of n:
� = � �
The transition interfaces are defined with respect to the amount of time elapsed outside the reactant basin.
Potential Advantages: • Variable path length • Influence of order parameter minimized • Defining product state is not necessary (enables unguided searches for metastable states) Typical Convergence Behavior of the Interface Crossing Probabilities τFFS is Comparable to τrxn from Hofer’s Direct Method
Forward Flux Sampling Φ" -1 P ( , ) P ( , ) P ( , ) FFS -1 (ps) (ps#,%0 ) λ 1 λ 0 λ 2 λ 1 λ 3 λ 2 k (ps ) τ FFS DPD 0.3915 ± 0.0059 0.5648 ± 0.0055 0.5611 ± 0.0162 0.2357 ± 0.0086 0.0292 ± 0.0014 34.3 ± 1.6 CSVR 0.4447 ± 0.0232 0.0806 ± 0.0027 - - 0.0358 ± 0.0022 28.0 ± 1.6
Hofer Impey 2+ -1 -1 -1 k rxn = r ex /[Ca ] (ps ) τ rxn (ps) k ex (ps ) τ res (ps) k ex (ps ) τ res (ps) NVT 0.0344 ± 0.0023 29.2 ± 1.9 0.0049 ± 0.0003 204.6 ± 13.5 0.0049 ± 0.0003 205.2 ± 13.3 DPD 0.0273 ± 0.0010 36.7 ± 1.4 0.0039 ± 0.0001 256.8 ± 9.8 0.0038 ± 0.0002 261.3 ± 15.2 CSVR 0.0352 ± 0.0022 28.5 ± 1.6 0.0050 ± 0.0003 199.3 ± 11.5 0.0045 ± 0.0005 228.3 ± 29.8
Forward Flux Sampling residence times � = �� [Ca ] = � [Ca ] DPD ≈ 240 ps CSVR ≈ 196 ps Looking Towards the Future
ξbond dominated Student/Postdoc Opportunities Contact: [email protected]
ξbond and ξangle MODELING THE COMPOSITIONAL ZONING OF MINERALS
A GEOCHEMICAL TOOL FOR INVESTIGATING THE TIMESCALES OF EARTH PROCESSES
Kendra J. Lynn DIFFUSION CHRONOMETRY
Rosen (2016)
• Minerals in rocks can be used to calculate the timescales of Earth processes CHEMICAL ZONING: OLIVINE a) Initially homogeneous core of composition X b) Secondary growth of a crystal rim with composition Y c) Chemical diffusion between core and rim - d) System attempts to reach equilibrium between compositions X and Y
Rosen (2016) MODELING CHEMICAL ZONING
• Goal: Model diffusive re-equilibration in 3D MODELING CHEMICAL ZONING
• Goal: Model diffusive re-equilibration in 3D
Zoning patterns are directly related to storage time.
!" ! !" • Finite difference, forward time, centered space = & discretization !# !% !% • Fick’s second Law: Continuity equation
Crank (1975) DIFFUSION ANISOTROPY
• Diffusion is anisotropic
• Da=Db=1/6Dc
Shea et al. 2015 DIFFUSION ANISOTROPY
• Diffusion is anisotropic
• Da=Db=1/6Dc
2 2 2 Dtrav = Da(cosα) + Db(cosβ) + Dc(cosγ)
Shea et al. 2015 MODELING MULTIPLE ELEMENTS
best-fit
Modeling multiple elements in parallel is computationally expensive ZONING IN THREE DIMENSIONS ZONING IN THREE DIMENSIONS
Stability Criterion – where R < 0.166 • To maintain a high spatial resolution (small ∆x), each timestep iteration (∆t) must also be small • High resolution diffusion models are time intensive, requiring high performance computing SUMMARY
• Current 3D models require 1 month to simulate 3 elements on desktop computer with 32 GB RAM • The Please contact Kendra J. Lynn same models are run in < 8 hours on Farber ([email protected]) • Future plans to model 3 different minerals with many elements each at the same time