Adam F. Wallace (afw@udel.edu) & Kendra J. Lynn (kjlynn@udel.edu)Depar tment of Geological Sciences

University of Delaware

Applications of High-Performance Computing in Geochemistry

Kendra J. LynnPostdoc with Dr. Jessica Warren (left)

LabMembersandAffiliates

AdamWallace

Yifei MaPh.D.student

JuneHazewskiM.S.student

ChunleiWangPh.D.student(Sturchio Lab)

BriannaMcEvoyM.S.2017

Geochemistsareinterestedinprocessesthatoccuroverawidevarietyoftimeandlengthscales

AtomicScale MineralSurfaces GrainBoundaryProcesses

WholeRockRegional

Global

2.5 x 2.5 µm

QC(ab initio/DFT)

ReactiveMD(i.e.ReaxFF)

MD

CoarseGrained

Continuum

Distance

Time

ion-ionion-water

gas-water

ion-hydrolysis/polymerizationIonsorption/desorptionDiffusioninsolution

nucleation/growthrecrystallizationmineral-waterequilibria

pore-scalewholemineral/rockregionalscale

Modelingiscriticalbecausemanylocationsandconditionsofinterestarephysicallyinaccessibleinthefieldandinthelab

Ourlabusesexperimentalandcomputationaltoolstoinvestigatemineral-waterinteractions

AFM-basedstudiesofmineralnucleation&growth

7.5 x 7.5 µm

2.5 x 2.5 µm

Σ3 i

cos θr ir

iξ a

ngle

=

Σ3

i

ri rir0

2ξbond =

Theoreticalmodelsofmineralstabilityandreactivity

• DensityFunctionalTheory(DFT),abinitiomethods

• BasedonapproximatesolutionstoSchrodinger’s equation:

• Physicsmodeledwith“high”accuracy.

• Requiresalotofcomputerpower.

• 10-20ps trajectoriestypical

Overviewofatomisticsimulationmethods

HΨ =EΨ

ElectronicStructureMethods

• BasedonNewton’sequationsofmotion.

• Physicsmodeledwithwithlessprecision.

• Requiresalotofcomputerpowerbutlessthanelectronicstructuremethods.

• 10-100nstrajectoriestypical

ClassicalMolecularDynamics

F=ma

Reactionsofgeochemicalinterestareoftentooslowtobeobservedwithdirectsimulationmethods

• Earth’scrustisprimarilycomposedofsilicateminerals

• Atsurfaceconditionsmostsilicatesdissolvevery

slowly(10-6 to10-14 mol /m2 sec).

• Evenat200°Cquartzdissolvesat~10-6 mol /m2 secin

saltwatersolutions.

Dove and Nix (2002) GCA, 61:3329-3340.

Thisisequivalenttothereleaseof~10-4 moleculesof

H4SiO4° per100nm2 ofsurfacepermicrosecond.

Specializedmethodsareneededtoovercometimescalelimitationsandenhancesampling

• MinimaHopping(LJ38)• ReplicaExchangeMolecularDynamics(CaCO3.nH2Oclusters)

Globalenergyminimizationstrategies

• UseoftheColvars LibraryinLAMMPS• UmbrellaSampling(waterexchangeonCa2+)• Metadynamics (Si-Obondhydrolysis)

Explorationoffreeenergylandscapeswithbiaseddynamics

• Directmethods• ReactiveFlux• ForwardFluxSampling

Calculationofreactionrates(waterexchangereactions)

• 2PT(substitutionofCO2 forH2Oinsepiolite)AbsoluteFreeEnergymethods

GlobalEnergy

MinimizationStrategies

MinimaHopping (Goedecker (2004) J. Chem. Phys., 120:9911)

• Hoppingisperformedbyactivation-relaxationsteps.

• Duringanactivationstep,amoleculardynamicstrajectoryisinitiatedwithagivenkineticenergy(Ekin)andfolloweduntilthepotentialenergydecreases.

• Duringarelaxationstepthesystemenergyisminimized tothenearestlocalminimum.

• Themovetothenewminimumisacceptediftheenergydifferencebetweenthenewandoldminimaislessthanaparameter(Ediff)thatisdynamicallyadjustedsothat~50%ofthemovesareaccepted.

• IfthesystemreturnstoaminimumithasalreadyvisitedEkin isincreased.

PotentialEne

rgy

ReactionProgress

FailedescapeEkin increases

SuccessfulescapeEkin decreases

MinimaHopping (Goedecker (2004) J. Chem. Phys., 120:9911)

• ActivationstepsareperformedwithLAMMPS• RelaxationstepsareperformedwithLAMMPSorGULP.• Apythonscriptcontrolsthesetupandexecutionof

LAMMPS/GULP.

ImplementationDetails

Energy

MinimaVisited

Exchangebetweenreplicasoccurssubjecttoaconditionalprobabilityrule.

ParallelTempering/ReplicaExchangeMolecularDynamics

Potential Energy Landscape

ReplicaExchangeSamplingCircumventsHighEnergyBarriers

Potential Energy Landscape

ReplicaExchangeSamplingCanLocateGlobalMinima

EnhancedsamplingofhydratedCaCO3 clusterswithReplicaExchangeMolecularDynamics().Wallace et al. (2013) Science, 341:885-889

ExplorationofFreeEnergy

LandscapeswithBiasedDynamics

UseoftheColvars LibraryinLAMMPS

• TheColvars libraryisincludedinLAMMPSdistributionsasanoptionalpackage.

• Thelibraryenablesanumberofbiasedsamplingmethods,including:umbrellasampling,metadynamicsandadaptivebiasforce.

• Thelibraryisinvokedasa“fix”intheLAMMPSinputscript.

AnatomyofasimpleCOLVARSinputfile

Blockthatdefinesthecollectivevariabletoapplythebiasto.Inthiscasethedistancebetweentwogroupscontaining1atomeach.

Outputandrestartfrequency.

Thisblockappliesaharmonicrestrainingpotentialtomaintainthevalueofthecollectivevariable“DIST”centeredaround3.1.Theenergyunitsarethoseusedbylammps.InthiscasetheforceconstantisineV.

SampleCOLVARSOutput

Instantaneousvalueof“DIST”MDStep

Obtainingthefreeenergybarrier(UmbrellaSampling)

Definereactionprogresscoordinate(oftenmetal-oxygendistanceforwaterexchangereactions)

1 2 Applyabiasalongthereactioncoordinate

3 Obtainbiasedprobabilitydistributions

4 Convertprobabilities tofreeenergiesforeachwindowandsubtractthebiaspotential.

5 Combine freeenergysegments

UsingWHAMutilitytoProcessCOLVARSOutput

WeuseautilitymaintainedbyAlanGrossfield atU.RochestertoprocessumbrellasamplingoutputfromCOLVARS.

http://membrane.urmc.rochester.edu/content/wham

Theutilitytakesafile”metadata”asinputthatspecifiesthelocationsoftheCOLVARSoutputforallthesamplingwindows,andwritesanoutputfile”freefile”thatcontainsthefreeenergywithrespecttothebiasescoordinate.

UsingWHAMutilitytoProcessCOLVARSOutput

PathtoCOLVARSoutputfileswindowcenter

Restraintforce

constant(kcal/mol)

Samplemetadatafile

RunningWHAM

Van Speybroeck et al. (2014) Chem. Soc. Rev., 43:7326-7357.

Metadynamics Basics

Themetadynamics methodacceleratestheexplorationofenergylandscapesbytheslowbuildupofahistorydependentbias.

Thebiasencouragesthesimulationtoexplorenewstatesandultimatelygrowstoformthenegativeimageoftheenergylandscape.

min(Si-OW)

max(Si-O

br)

Metadynamics SimulationofSi-OBondHydrolysis

SampleLAMMPSandCOLVARSinputfilesfor2DMetadynamics

LAMMPSInput COLVARSInput

LAMMPSgroupswrittentofiletobereadbyCOLVARS

Filetoreadgroupdefinitions from

Definitionofcollectivevariable“one”astheminimumSi-OW distance.

Definitionofcollectivevariable“two”asthemaximumSi-Obr distance.ThisusesacustomfunctionthatdependsontheLeptonlibrary.

Settingsthatcontroltheaccumulationofthemetadynamicsbiasfunction.

Calculation

WaterExchangeRates

!I# H%O '] + H%O∗+,- [I# H%O '/0 H%O∗]+ H%O

Richens (2005) Chem. Rev., 105(6):1961-2002.

𝜏234 =1𝑘38

CorrelationofWaterExchangeRates&BulkReactivityTrends

Casey (2017) Adv. Inorg. Chem., 69:91-115.

Comp.by:

DElaya

raja

Stage:Proof

ChapterNo.:27

Title

Name:ADIO

CH

Date:12/1/17

Tim

e:15:36:32

PageNumber:5

Fig. 2f0010 Dissolution rates at pH 2 of simple oxide (A) and orthosilicate (B and C) minerals containing divalent metals. The abscissa is the rate ofexchange of water from the corresponding metal ion (e.g., Mg2+(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:M2SiO4(s) and isolated SiO4

4! tetrahedra. The end-member compositions of orthosilicates are shown in red squares in (B), whereas themixed-metal compositions are shown as blue dots. For the mixed-metal compositions, dissolution rates are plotted against the weightedsum of the logarithm of rates of water exchange for the component ions. Note that the dissolution rates for mixed-metal compositions (bluedots in (B)) fall intermediate between the end-member compositions. The rates of ligand exchange of the aquated ions ( ) and the dissolutionrates 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-fieldstabilization 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.

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(OxidesandOrthosilicates)

Dove and Nix (2002) GCA, 61:3329-3340.

(SiO2)3338 P .M. Dove and C.J. Nix

(a)

o

= ©

_= @

~3 o ©

©

(b)

0

O

©

@

o

-6.0

-6.5

-7.0

-7.5

-8.0 0.00

[] water, electrolytes not present • 0.01 molal M e C 1 2 ~ 2 © 0.05 molal MeC12 o

q

2+

[]

Si 4+ 200oC i I r I i I I I r i i [ i ~ i i i i r

2.00 4.00 6.00 8.00 10.00 log kex of aqueous ions (s -1)

-6.0

-6.5

-7.0

-7.5

-8.0 0.00

[] water, electrolytes not present

[] []

200°C i K i I i i I I i i i I i i i I i r i

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 suggests that these relative rates hold over the temperature range of 20-200°C. In the broader perspective of Earth envi- ronments, any of these IA or IIA cations have a greater rate enhancing effect than organic compounds. For example, Ben- nett et al. (1988) and Bennett (1991) showed that organic acids, citrate, and oxalate, enhanced quartz dissolution rates by

a factor of two compared to pure water. In contrast, solutions containing potassium or barium ion can enhance rates as much as forty times at 200°C.

5.1. Relationship to Previous Studies The relationship between quartz dissolution rate and the

kex of solute ions shown in Fig. 5a,b is consistent with previ-

CorrelationofWaterExchangeRates&BulkReactivityTrends

Pokrovsky and Schott (2002) Environ. Sci. Technol., 36:426-432.

(Carbonates)CorrelationofWaterExchangeRates&BulkReactivityTrends

5

7

6

1

3

4

2

DirectRateMethodsforLigandSubstitution

𝜏234 =1𝑘38

= 9 𝑅(𝑡) 𝑑𝑡?

@

(Impey’s Method)

1

2

3

45

67 𝑅(𝑡) =

1𝑛(0)

C𝜃E(0)𝜃E(𝑡)'(F)

EG0

Ca2+

Impey ResidenceTime:Ca2+inSPC/Fw waterτ res(ps) k ex(ps

-1)227.3 0.0044200.5 0.0050186.0 ± 0.0054 0.0049 ± 0.0003204.5 0.0049207.7 0.0048274.4 0.0036274.3 0.0036254.4 ± 0.0039 0.0038 ± 0.0002234.7 0.0043268.8 0.0037279.0 0.0036197.0 0.0051220.3 ± 0.0045 0.0045 ± 0.0005242.1 0.0041202.9 0.0049

Impey

⟨k ex⟩(ps-1)⟨τ res⟩(ps)

NVT 205.2 13.3

DPD 15.2

CSVR 228.3 29.8

261.3

Wallace & Ma, submitted.

Rateri et al. (2015) J. Phys. Chem. C, 119(43):24447–24458

𝜏234 = 230psec

DirectRateMethodsforLigandSubstitution(Hofer’s “Direct” Method)

RESERVOIR𝑟E' 𝑟PQF

𝑟E' = 𝑟PQF = massexchangerate

𝜏234 =ReservoirMass

MassExchangeRate

DirectRateMethodsforLigandSubstitution(Hofer’s “Direct” Method)

𝜏234 =ReservoirMass

MassExchangeRate

ReservoirMass = 𝑛(ioncoordinationnumber)

MassExchangeRate = 𝑟38 =numberofexchangesperion

lengthofsimulation

𝜏234 =1𝑘38

=𝑛

𝑟38 Ca%f⁄

Indirectratemethods(ReactiveFlux)

𝑘hi = 𝜅𝑘klk

FreeEne

rgy

ReactionProgress

ΔG*Transitionstatetheoryrateconstant(determinedbytheheightoftheenergybarrier)

Transmissioncoefficient(dynamicalcorrectionto𝑘klk)

TheTSTrateisbasedentirelyonthebarrierheight

Wallace & Ma, submitted.

Transition State Theory Residence Time

𝜏klk~36psec

TSTassumeseveryvisittothetransitionstateissuccessful(thisisn’tture)

𝜅 𝑡 =�̇� 0 𝜃[𝑟 𝑡 − 𝑟‡r s�̇� 0 𝜃[�̇�(0)] s

TocorrecttheTSTratewecalculatethetransmissioncoefficient:

NumberofsuccessfultrajectoriesTotalnumberofattempts 0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

k(t)

Time(psec)

k TST(ps-1)

NVT 0.0322 0.1748 ± 0.0062 0.0056 ± 0.0002 178.1 ± 7.0DPD 0.0273 0.1678 ± 0.0211 0.0046 ± 0.0006 221.4 ± 31.2CSVR 0.0257 0.1707 ± 0.0213 0.0044 ± 0.0005 231.4 ± 34.3

ReactiveFlux

k k RF(ps-1) t RF(ps)

ApplyingthedynamicalcorrectionyieldsτRF≈ τres

Wallace & Ma, submitted.

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

K(t)

Time(psec) Time(psec) Time(psec)

CSVRDPDNVT

𝑘hi =1𝜏hi

= 𝜅𝑘klk

Indirectratemethods(ForwardFluxSampling)AttemptRateCalculation DeterminationofInterfaceFluxes

𝑃 𝜆w 𝜆@) =x𝑃 𝜆Ef0 𝜆E)'/0

EG@𝑘iil =

1𝜏iil

= Φz{,w = Φz{,}~𝑃 𝜆w 𝜆@)

Elapsedtimeisusedintheplaceofatraditionalorderparameter

𝑛 = CC𝑠E� 𝑟�E

• Variablepathlength• Influenceoforderparameterminimized• Definingproduct stateisnotnecessary

(enablesunguided searchesformetastablestates)

Thetransitioninterfacesaredefinedwithrespecttotheamountoftimeelapsedoutsidethereactantbasin.

PotentialAdvantages:

Reactant/Productbasinsaredefinedbythresholdvaluesofn:

TypicalConvergenceBehavioroftheInterfaceCrossingProbabilities

τFFS isComparabletoτrxn fromHofer’sDirectMethod

DPD 0.3915 ± 0.0059 0.5648 ± 0.0055 0.5611 ± 0.0162 0.2357 ± 0.0086 0.0292 ± 0.0014 34.3 ± 1.6CSVR 0.4447 ± 0.0232 0.0806 ± 0.0027 - - 0.0358 ± 0.0022 28.0 ± 1.6

τ FFS(ps)

ForwardFluxSampling

(ps-1) P (λ 1,λ 0) P (λ 2,λ 1) P (λ 3,λ 2) k FFS(ps-1)Φ"#,%0

NVT 0.0344 ± 0.0023 29.2 ± 1.9 0.0049 ± 0.0003 204.6 ± 13.5 0.0049 ± 0.0003 205.2 ± 13.3DPD 0.0273 ± 0.0010 36.7 ± 1.4 0.0039 ± 0.0001 256.8 ± 9.8 0.0038 ± 0.0002 261.3 ± 15.2CSVR 0.0352 ± 0.0022 28.5 ± 1.6 0.0050 ± 0.0003 199.3 ± 11.5 0.0045 ± 0.0005 228.3 ± 29.8

τ res(ps)Impey

k rxn=r ex/[Ca2+](ps-1) τ rxn(ps) k ex(ps

-1) τ res(ps)Hofer

k ex(ps-1)

𝑟38 = 𝑛𝑘234[Ca%f] = 𝑘28'[Ca%f]ForwardFluxSamplingresidencetimes

DPD≈ 240psCSVR≈ 196ps

LookingTowardstheFuture

Student/Postdoc OpportunitiesContact: afw@udel.edu

ξbond dominated

ξbond andξangle

MODELING THE COMPOSITIONAL ZONING OF MINERALS

A GEOCHEMICAL TOOL FOR INVESTIGATING THE

TIMESCALES OF EARTH PROCESSES

Kendra J. Lynn

DIFFUSION CHRONOMETRY

• Minerals in rocks can be used to calculate the timescales of Earth processes

Rosen (2016)

CHEMICAL ZONING: OLIVINE

Rosen (2016)

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

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.

!"!# =

!!% & !"!%

Crank (1975)

• Finite difference, forward time, centered space discretization

• Fick’s second Law: Continuity equation

DIFFUSION ANISOTROPY

• Diffusion is anisotropic

• Da=Db=1/6Dc

Shea et al. 2015

DIFFUSION ANISOTROPY

• Diffusion is anisotropic

• Da=Db=1/6Dc

Dtrav = Da(cosα)2 + Db(cosβ)2 + Dc(cosγ)2

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 same models are run in < 8 hours on Farber

• Future plans to model 3 different minerals with many elements each at the same time

PleasecontactKendraJ.Lynn(kjlynn@udel.edu)