Ab Initio Calculation of Equilibrium Isotopic Fractionations ofPotassium and Rubidium in Minerals and WaterHao Zeng,†,‡ Viktor F. Rozsa,‡ Nicole Xike Nie,† Zhe Zhang,† Tuan Anh Pham,§ Giulia Galli,‡,∥,⊥

and Nicolas Dauphas*,†

†Origins Laboratory, Department of the Geophysical Sciences and Enrico Fermi Institute, and ‡Pritzker School of MolecularEngineering, The University of Chicago, Chicago, Illinois 60637, United States§Quantum Simulations Group, Lawrence Livermore National Laboratory, Livermore, California 94551, United States∥Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, United States⊥Department of Chemistry, University of Chicago, Chicago, Illinois 60637, United States

*S Supporting Information

ABSTRACT: We used first-principle approaches to calculate theequilibrium isotopic fractionation factors of potassium (K) and rubidium(Rb) in a variety of minerals of geological relevance (orthoclase, albite,muscovite, illite, sylvite, and phlogopite). We also used moleculardynamics simulations to calculate the equilibrium isotopic fractionationfactors of K in water. Our results indicate that K and Rb form bonds ofsimilar strengths and that the ratio between the equilibriumfractionations of K and Rb is approximately 3−4. Under low-temperature conditions relevant to weathering of continents or alteration of seafloor basalts (∼25 °C), the K isotopicfractionation between solvated K+ and illite (a proxy for K-bearing clays) is +0.24‰, exceeding the current analytical precision,so equilibrium isotopic fractionation can induce measurable isotopic fractionations for this system at low temperature. Thesefindings, however, cannot easily explain why the δ41K value of seawater is shifted by +0.6‰ relative to igneous rocks. Ourresults indicate that part of the observed fractionation is most likely due to kinetic effects. The narrow range of mean forceconstants for K and Rb in silicate minerals suggests that phase equilibrium is unlikely to create large K and Rb isotopicfractionations at magmatic temperatures (at least in silicate systems). Kinetic effects associated with diffusion can, however,produce large K and Rb isotopic fractionations in igneous rocks.KEYWORDS: isotopes, equilibrium fractionation, concentration effect, potassium, rubidium

1. INTRODUCTION

Potassium is a moderately volatile, lithophile element that ispresent in relatively high abundance in the ocean (eighth mostconcentrated element),1 Earth’s crust (eighth),1−3 bulk silicateEarth (BSE; seventeenth),4,5 and in the solar system(twentieth).6−8 The limited precision achievable by massspectrometry in measuring the ratios of the isotopicabundances of 39K, 40K, and 41K has long limited theapplication of K isotope systematics to cosmochemistry,where large isotopic variations have been found.9−16 Thanksto improvements in purification protocols and multicollectorinductively coupled plasma mass spectrometry (MC-ICPMS),the isotopic composition of K (expressed using the 41K/39Kratio) can now be measured with a precision of ∼0.1‰,17−24

spurring wide interest in K isotopes, which have now beenused to study weathering,21,25 seafloor alteration,23 and volatileelement depletion in planetary materials.10,12,14,17 Potassiumisotopic compositions are reported as the per mil (‰)deviation (δ41K) of the ratio of two potassium stable isotopes41K (6.730%) and 39K (93.258%), relative to the inferredcomposition of the mantle or the NIST standard referencematerial 3141a. Because of direct isobaric interferences from

40Ar+ and 40Ca+ on 40K+ in MC-ICPMS and the low abundanceof 40K (0.012%), the latter isotope is often not reported. Thisis inconsequential because K isotopic variations are thought tofollow the laws of mass-dependent fractionation, meaning thatthe relative deviations in the 40K/39K ratio are approximatelyhalf those of the 41K/39K ratio (δ40K = δ41K/2).26,27

Also belonging to the alkali group, Rb shares similarchemical and physical properties with K and substitutes for Kin minerals such as feldspar. As a result, the K/Rb weight (g/g)ratio varies little both among meteorites28 and within variousgeochemical reservoirs in the Earth (400 for the BSE; 232 forthe crust).1−5 Notable exceptions to the constancy in the K/Rbweight ratio are rivers (∼1000)29 and seawater (∼3765).30The elevated K/Rb ratio of seawater compared to that of theBSE could be due to the preferential mobilization of K relativeto Rb during continental weathering,29 as well as thepreferential uptake of Rb relative to K during seafloor

Received: June 21, 2019Revised: October 4, 2019Accepted: October 16, 2019Published: October 16, 2019

Article

http://pubs.acs.org/journal/aesccqCite This: ACS Earth Space Chem. 2019, 3, 2601−2612

© 2019 American Chemical Society 2601 DOI: 10.1021/acsearthspacechem.9b00180ACS Earth Space Chem. 2019, 3, 2601−2612

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alteration.1,4,31,32 Rubidium isotopic ratios can be measuredwith great precision,33−36 but the similarity in the geochemicalbehaviors of K and Rb raises the question of whethermeasuring the isotopic compositions of K and Rb on thesame samples provides additional insight compared tomeasuring K or Rb alone.Understanding how K and Rb isotopes are fractionated at

equilibrium between vapor and condensed phases is importantto use these systems for understanding why planetary bodiesare depleted in these moderately volatile elements. It has beenshown that lunar rocks are enriched in the heavy isotopes of Kby ∼0.4‰, which was interpreted by Wang and Jacobsen toreflect equilibrium isotopic fractionation between liquid andvapor under conditions relevant to the Moon-forming giantimpact event.17 Nie and Dauphas et al. contended that at thetemperature of ∼3500 K, most likely relevant to Moonformation, the equilibrium fractionation between vapor andcondensate would be too small, implying the involvement ofkinetic isotopic fractionation presumably associated withevaporation.36 Similarly, lunar rocks are enriched in heavyRb isotope relative to terrestrial rocks33,36 and those data areconsistent with a scenario that involves loss of volatile elementsfrom the protolunar disk by accretion onto the Earth, leavingbehind a Moon that is depleted in moderately volatileelements.36

Water−mineral interactions play a significant role inmaintaining habitable conditions at the surface of the Earththrough silicate weathering and carbonate precipitation. Theisotopic compositions of K and Rb could help trace continentalweathering and reverse weathering in marine sediments, whichboth influence climate through their controls of CO2 partialpressure in the atmosphere (and associate greenhouse effect).Li et al.,37 Wang and Jacobsen,18 and Morgan et al.38 foundthat the ocean is significantly enriched in the heavy isotopes ofK relative to the BSE by δ41Kocean−BSE ≈ +0.6‰ and the causeof this enrichment remains unclear. As summarized bySantiago-Ramos et al.,21 this enrichment could arise from theremoval of K from the oceans involving authigenic Al-silicateformation in marine sediments and/or low-temperaturealteration of the oceanic crust.39−42 Alternatively, the enrich-ment could originate from isotopic fractionation duringmobilization of K associated with continental silicate weath-ering or high-temperature basalt alteration.41,43,44 Santiago-Ramos et al. studied K isotopic fractionation in sediment pore-fluids and concluded that K removal in authigenic Al-silicatesin sediments could be responsible for such fractionation aspore-fluids tend to be enriched in the light isotopes of K. Thedriver for this fractionation could be diffusion through theporewater, which is kinetic in origin. Equilibrium fractionationbetween K+ dissolved in porewater and K incorporated inminerals could also play a role but the extent of thisfractionation is unknown. Characterizing equilibrium isotopicfractionation between aqueous fluids and K-bearing silicates isalso important for understanding how K isotopes arefractionated during weathering.Despite progress made in measuring the isotopic composi-

tions of K and Rb, understanding the cause of the isotopicvariations for those elements in natural systems can bechallenging, partly due to an insufficient theoretical under-standing of how equilibrium and kinetic processes control thefractionations of K and Rb isotopes. To address this, Li, Y. etal. recently calculated the reduced partition function ratio(rpfr) of K in alkali feldspar and studied the concentration

effect on K fractionation using density functional theory(DFT).45 Additionally, Li, W. et al. studied experimentallyequilibrium fractionation between K dissolved in water and K-salts such as halides, sulfates, and carbonates.22 They alsoperformed ab initio calculations for the salts that they studied.Clearly, more work is needed to understand how the isotopiccompositions of Rb and K are fractionated at equilibriumbetween different minerals, gaseous species, and water. Thesefractionations are critical for interpreting the origin of Kisotopic variations in low-temperature aqueous systems andplanetary/nebular processes that involved the volatilization ofK and Rb.Understanding the isotopic fractionation of K between water

and K-bearing minerals requires knowledge of the speciation ofK in water. As ∼99% of aqueous K in the oceans is present ashydrated K+, we chose to focus on hydrated K+ in thisstudy.46−48 In contrast to other cations, in particular divalentones (e.g., Zn2+),49 the hydration shell of K+ is not as welldefined, with water molecules rapidly exchanging within thefirst solvation shell.50 Because equilibrium fractionationproperties are highly sensitive to coordination numbers andto nearest neighbor distances, we used first-principle moleculardynamics (FPMD) to study hydrated K+, capturing manyinstantaneous configurations of the fully hydrated K+, incontrast to a cluster approach where static hydratedconfigurations are considered.Our results suggest that (a) equilibrium fractionation

between K-bearing clays (illite) and seawater could be asignificant source of oceanic 41K enrichment and (b) the bondstrengths of Rb and K are similar, so equilibrium processes areexpected to impart correlated isotopic fractionations to Rb andK that differ from kinetic processes, providing a means ofdistinguishing between equilibrium and kinetic processes innature.

2. METHOD2.1. Equilibrium Fractionation. Equilibrium isotope

fractionation for an element X between two phases A and B(αA−B) can be calculated using the rpfr (or β-factor) of eachphase26,27,51

X XX X

( / )( / )A B

A

B

A

B

αββ

=′′

=−(1)

where X, X′ refer to the abundances of two isotopes of anelement. The rpfr can be calculated from51

i

k

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h kT

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h kT

n N

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3 /2

/

/

/2

1/i

i

i

i

q

q

q

q

q,

,

,

,∏ ∏β

νν

=′

×−

× −ν

ν

ν

ν=

−

−

− ′

− ′

·

(2)

where N is the total number of atoms in the unit cell, νq,i andνq,i′ are the frequencies of vibrational mode i for two isotopes ata given wavevector q, Nq is the total number of q-vectors, andn is the number of isotopic sites in the unit cell. Whenmeasured experimentally, equilibrium isotopic fractionation (in‰) between two phases is expressed as

(‰) 1000 ln 1000(ln ln )A B A B A Bα β βΔ = = −− − (3)

To a good approximation, one can write 1000 ln β as apolynomial expansion in even powers of the inverse of thetemperature52,53

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AT

AT

AT

1000 ln 12

24

36β = + +

(4)

where A1, A2, and A3 are constants that depend on themineral/phase considered. The coefficient of the first-orderterm, which always dominates the rpfr, especially at hightemperature (>200 K), can be rewritten as

ikjjj

y{zzzA

M M kF1000

1 181

2

2=′

− ℏ(5)

where ℏ is the reduced Planck constant, k is Boltzmann’sconstant,M′ andM are the masses of the two isotopes, and ⟨F⟩is the mean force constant of the K or Rb bonds (equivalent tothe spring constant of a harmonic spring), which can becalculated from the partial phonon density of states (PDOS,g(E))52

FM

E g E E( )d2 0

2∫=ℏ

+∞

(6)

At high temperature for 39/41K and 85/87Rb, eq 4 can bewritten as

FT

1000 ln 5500K 2β ≃ ⟨ ⟩(7)

FT

1000 ln 1189Rb 2β ≃ ⟨ ⟩(8)

At low temperature relevant to weathering, for example, thefull expansion in eq 4 should be used.2.2. FPMD of the Potassium Ion in Water. We carried

out FPMD of aqueous K+ using the Perdew, Burke, andErnzerhof (PBE) exchange−correlation functional.54 Thisrelatively simple functional was chosen as it was reportedthat there is no significant difference in rpfr calculated fromPBE and van der Waals functionals.49 We used the samefunctional for minerals and hydrated potassium, to facilitatecomparing fractionation factors for different systems and totake advantage of cancellation of errors.We performed Born−Oppenheimer molecular dynamics

(BOMD) simulations with the Qbox code.55 We used a cubiccell containing 63 water molecules and one K+ and a volumechosen so as to obtain the measured density of water atambient conditions (edge length equal to 12.4 Å). Acompensating uniform background charge was used to ensurecharge neutrality. All hydrogen atoms were replaced withdeuterium to allow for a larger timestep, chosen to be 10atomic units. The 3s and 3p semicore states of K were treatedas valence states, and the interaction between valence and coreelectrons was represented by an optimized norm-conservingpseudopotential (ONCV)56 for K and a norm-conservingHamann−Schluter−Chiang−Vanderbilt57 pseudopotentials forhydrogen and oxygen. The electronic wavefunctions wererepresented with a plane-wave basis set and a kinetic energycutoff of 85 Ry. We generated trajectories for ∼50 ps in theNVT ensemble, with a temperature of 400 K set by a velocityscaling thermostat. Using 400 K instead of room temperatureis a commonly used approximation to correct the well-knownproblem of an overstructured liquid water using PBE at roomtemperature.58−60 We considered configurations of solvated K+

from 20 uncorrelated snapshots over the entire trajectory(taken from the postequilibrated heavy water trajectory with100 steps between each snapshot corresponding to a totalduration of 24 fs).

2.3. DFT Calculations for Solid Phases. Unlessotherwise specified, structural relaxations and phononcalculations (including all minerals and aqueous K+ snapshots)were carried out with the Quantum ESPRESSO code,61,62 withplane-wave basis sets, the PBE functional,54 and ONCVpseudopotentials.63 The kinetic energy and charge-densitycutoff were set to 80 and 320 Ry, respectively. Our choices ofMonkhorst k-point grids64 and q-point grids are given in TableS1. To compare our results with those of Li, Y. et al.45 forfeldspar, we also used the local density approximation (LDA)with LDA pseudopotentials from the GBRV library65 andQuantum ESPRESSO PSlibrary (PSL).66 Pseudopotentialsfrom PSL used in this study include Na.pz-n-vbc.UPF, K.pz-n-vbc.UPF (1-valence), K.pz-spn-kjpaw_psl.1.0.0.UPF (9-va-lence), Al.pz-n-kjpaw_psl.0.1.UPF, Si.pz-n-kjpaw_psl.0.1.UPF,and O.pz-n-kjpaw_psl.0.1.UPF. For GBRV and PSL pseudo-potentials, the kinetic energy was set at 60 and 75 Ry, andcharge-density cutoff was set at a multiple of 10 and 4 ofkinetic energy cutoff, respectively.The minerals (and their formulas) investigated here are

orthoclase (KAlSi3O8), microcline (KAlSi3O8), albite (NaAl-Si3O8), anorthite (CaAl2Si2O8), muscovite (KAl3Si3O12H2),illite (KAl2Si4O11F), phlogopite (KAlSi3Mg3O12H2), andsylvite (NaCl). Their crystal structures were taken fromexperiments67−74 and the lattice and atomic positions wererelaxed until the total force and stress were smaller than 10−4

atomic units and 0.1 kbar, respectively. A phonon calculationwas then performed for each structure using DFPT (densityfunctional perturbation theory).62,75 The same calculationswere performed for K and Rb by substituting the second forthe first in the mineral structures. For albite−microcline (Naand K feldspars) solid solution and K-substituted anorthite,phonon frequencies were only calculated at the Γ point. Formicrocline−anorthite (K and Ca feldspars) solid solution, forevery Ca-substituted by K, one Al was replaced with Si tobalance charge. For aqueous K+, with configurations frommolecular dynamics (MD) trajectory, only the atomicpositions were relaxed.To model infinite dilution we used two strategies. The first is

what we call the constrained cell method (Rb in muscovite,orthoclase, phlogopite, and illite): (1) we fully relaxed thelattice and atomic positions of the host phase; (2) we madeone substitution (i.e., replaced one K with Rb); (3) we heldthe lattice constant fixed but relaxed the atomic positions ofthe structure after substitution; and (4) we calculated thephonon frequencies and rpfr. The second is the supercellapproach (Rb in sylvite combines the supercell and con-strained cell approaches): (1) we obtained a primitive cell; (2)we made one substitution (i.e., replaced one K with Rb); (3)we calculated the phonon frequencies and rpfr; (4) we made asupercell that was twice as large as the primitive cell andrepeated steps 2 and 3; (5) we made another supercell that wastwice as large as the cell in the previous step and repeated steps2 and 3; and (6) we repeated step 5 until 1000 ln β converged.One caveat when using the constrained cell approach to

study infinite dilution is that the primitive cell must berelatively large so there is minimal interaction betweensubstituents. In cases like KCl, where there are 2 atoms inthe primitive cell, one still needs to build a supercell. For Rb-sylvite, we used tetragonal K8Cl8 with one Rb atomsubstituting for K.

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3. RESULTSThe structures of minerals investigated here are provided inFigure 1, and the mineral lattice parameters calculated from

first-principles are given in Table S1. The PDOS of theminerals are given in the Supporting Information (Figure S1)We also report the calculated frequencies as compared toexperimentally measured infrared and Raman frequencies(Figure S6), showing good agreement. Similar to the resultsreported by Ducher et al.,76 we found that the presence ofhydrogen atoms leads to slightly distorted lattice parameters.In general, the computed parameters are 1−2% larger than thecorresponding measured values, consistent with previousobservations that the PBE functional tends to overestimatelattice parameters of semiconductors and insulators.77

The coefficients in the expansion describing the temperaturedependence of 1000 ln β, the force constant of the chemicalbonds, and the 1000 ln β values at 300, 600, and 1200 K aregiven in Table 1 for K, Rb, and aqueous K+. The temperaturedependence of rpfr is shown in Figure 2, where the rpfr scaleslinearly with 1/T2. The ⟨F⟩ values for Rb are ∼20−50% largerthan those for K in the same minerals and given the massdifference (eqs 7 and 8), we expect the ratio Δ41/39KA−B/Δ87/85RbA−B to be ∼3.5 for equilibrium fractionation betweentwo systems A and B if K and Rb are fractionated together byan equilibrium process.We calculated the rpfr for aqueous K+ by extracting 20

snapshots (see Figure S5 for rpfr convergence with respect tonumber of snapshots) from MD trajectories and performedphonon calculations on the relaxed snapshot structures. In thePDOS calculated for aqueous K+, we can clearly see couplingsto the H−O−H bending and O−H stretching modes at higherfrequency (Figure 3).The calculated rpfr of aqueous K+ shows some variability

(see Figure 2) across different snapshots due to dynamicalchanges in the coordination environment of K+. In our FPMDsimulation, we found that the coordination number of K+

varies from 4 to 10 between snapshots, with an averagecoordination number of 6.7 (Figure S2). We averaged the rpfrobtained from 20 snapshots uniformly spaced in time, whoseaverage coordination number (7.2, Figure S2) is comparable tothat obtained over the whole trajectory. Similar to the resultsof Ducher et al. for solvated Zn, we found little correlationbetween rpfr and coordination numbers (R2 = 0.05), so thesmall difference between the average over the selected samplesand whole trajectory should have an insignificant impact on thecomputed rpfr.49 We also assume that the dependence ofphonon frequencies on temperature is weak and use thecomputed phonon frequencies across all temperatures.

Figure 1. Structures (rendered by VESTA)78 of minerals and aqueousK+ investigated in this paper. Albite, anorthite, microcline, andorthoclase share the feldspar structure. Red represents O, whiterepresents H, green represents Cl, dark blue represents Si, light bluerepresents Al, purple represents K, brown represents Mg, and greyrepresents F.

Table 1. Force Constants for K and Rb (Infinite Dilution in K-Bearing Minerals), Expansion Coefficients for 1000 ln β = A1x +A2x

2 + A3x3 With x = 106/T2, and 1000 ln β Values at Selected Temperatures

expansion coefficient 1000 ln β

⟨F⟩ (N/m) A1 A2 A3 300 K 600 K 1200 K

muscovite K 30 0.169 −3.25 × 10−4 3.84 × 10−6 1.875 0.469 0.117Rb 35 0.043 −6.54 × 10−5 8.11 × 10−7 0.474 0.119 0.030

orthoclase K 27 0.152 −3.07 × 10−4 4.16 × 10−6 1.690 0.423 0.106Rb 38 0.046 −6.58 × 10−5 8.87 × 10−7 0.508 0.127 0.032

sylvite K 27 0.148 −8.40 × 10−5 7.47 × 10−8 1.643 0.411 0.103Rb 39 0.047 −2.26 × 10−5 2.02 × 10−8 0.517 0.129 0.032

phlogopite K 26 0.144 −2.97 × 10−4 3.77 × 10−6 1.605 0.401 0.100Rb 32 0.038 −6.27 × 10−5 8.04 × 10−7 0.427 0.107 0.027

illite K 22 0.121 −2.74 × 10−4 4.11 × 10−6 1.350 0.337 0.084Rb 35 0.042 −6.44 × 10−5 9.33 × 10−7 0.465 0.116 0.029

[K(H2O)63]+ K 25 0.145 −4.61 × 10−4 6.24 × 10−6 1.610 N/A N/A

microcline K 27 0.153 −3.79 × 10−4 6.36 × 10−6 1.662 0.422 0.106microcline−albite 50% K K 34 0.192 −3.49 × 10−4 3.12 × 10−6 2.093 0.530 0.133microcline−albite 25% K K 46 0.263 −9.47 × 10−4 1.34 × 10−5 2.829 0.725 0.183microcline−albite 12.5% K K 52 0.291 −7.55 × 10−4 7.06 × 10−6 3.150 0.803 0.202albite 0% K K 53 0.295 −4.32 × 10−4 1.32 × 10−6 3.225 0.816 0.205microcline−anorthite 12.5% K K 59 0.329 −5.98 × 10−4 2.48 × 10−6 3.584 0.909 0.228anorthite 0% K K 69 0.388 −9.61 × 10−4 5.34 × 10−6 4.203 1.071 0.269

Albite 0% K, anorthite 0% K, Rb-muscovite, orthoclase, phlogopite, and illite were calculated using the constrained cell approach. Rb-sylvite wascalculated using a combination of the supercell and constrained cell approaches.

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4. DISCUSSION

To the best of our knowledge, there is no calculation of rpfr forRb with which we could compare our results, nor could we findany experimentally measured, direct determination of equili-brium intermineral fractionation data for Rb. The mostrelevant comparisons are a theoretical study on alkali feldsparby Li, Y. et al.45 and the work of Li, W. et al. on K fractionationbetween salts and aqueous solution.22 We will first discuss andcompare our results to both studies and then present theimplications of our results.4.1. Mineral Fractionation: Comparison with a

Previous Theoretical Study. Li, Y. et al. examined theinfluence of the Na−K solid solution in alkali feldspar bycalculating the rpfr of K in K-feldspars using DFT and the LDAfunctional and pseudopotentials. To make the discussion morestraightforward, we converted the expansion coefficientsreported by Li, Y. et al. to force constants (see eqs 4 and 5).For K in microcline, they obtained a force constant of 48 N/m,while for 50−50 microcline−albite solid solution, the value is64 N/m. These values are significantly larger than the values of28 and 35 N/m that we obtained for the same minerals usingPBE. Li, Y. et al. claimed that the LDA approximation is moreappropriate because it gave better agreement betweencomputed and experimental lattice parameters, and the rpfris inversely proportional to the unit cell volume.45

One potential concern in Li, Y. et al.’s study is the use of 1-valence pseudopotentials for the alkali atoms (Na and K), as itis well known that considering only the outermost valenceelectrons of these elements may cause significant errors in mostcomputed properties.75,79,80 To evaluate the influence of suchpseudopotentials, we tested 3 sets of LDA pseudopotentialsfrom different libraries for microcline. The three sets of LDA

pseudopotentials are (1) a set from the Quantum ESPRESSOPSlibrary (PSL1) that uses 1-valence alkali pseudopotentials asused by Li, Y. et al.; (2) a set from Quantum ESPRESSOPSlibrary (PSL9), in which we used 9-valence alkalipseudopotentialsthe only difference between PSL9 andPSL1 is the alkali pseudopotential; and (3) one set from theGBRV pseudopotential library that has 9-valence alkalipseudopotentials.The resulting cell parameters, unit cell volume, and the force

constant for microcline are shown in Table 2 and compared

with the experimental results. Temperature-dependent 1000 lnβ values are given in Figure 4. First, we note that the resultreported by Li, Y. et al. was reproduced when using thepseudopotential set PSL1 (Table 2). We found that twodifferent sets of pseudopotentials with 9-valence alkalipseudopotentials gave similar lattice parameters and rpfr

Figure 2. Temperature dependence of 1000 ln β for K (panel a) and Rb (panel b) in various minerals investigated here. The error bars are thestandard errors of the mean value for aqueous K+.

Figure 3. PDOS for aqueous K+. The full spectrum in logarithm scale shows higher frequency couplings to the H−O−H bending and O−Hstretching modes.

Table 2. Experimental and Calculated Lattice Parameterswith Different Pseudopotentials, Unit Cell Volume, andForce Constant of K-Bonds (Which Controls EquilibriumIsotopic Fractionation through Eq 7; gamma-point onlycalculation) for Microcline

a (Å) b (Å) c (Å) V (Å3)Vdiff(%)

⟨F⟩(N/m)

exp.measured

8.571 12.964 7.221 720.99 0.0 N/A

Li et al.45 8.600 12.942 7.199 717.79 −0.4 48LDA-PSL1 8.633 12.928 7.208 722.17 0.2 45LDA-PSL9 8.417 12.901 7.180 699.61 −3.0 38LDA-GBRV 8.425 12.917 7.186 702.12 −2.6 38PBE 8.698 13.105 7.313 752.03 4.3 28

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(Figure 4). These are both significantly smaller than thosereported by Li, Y. et al. who had argued that our lower estimateof the force constant was due to a larger calculated unit cellvolume but as shown here, our calculated cell volume isactually smaller. In addition, PSL9 and GBRV PP gave similarrelative errors in unit cell volumes, compared with the resultsobtained with PBE. Using LDA pseudopotentials from GBRVlibraries, we conducted another test with a 50−50 microcline−albite structure as reported by Li, Y. et al., and we againobtained a smaller force constant (49 N/m) than that reportedby Li, Y. et al. (64 N/m). Finally, we tested the PSL1 PP forthe NaCl structure of sylvite and found that total energyconvergence could not be achieved even with kinetic energycutoff set at 120 Ry (Figure S3).We further calculated rpfr for K in the microcline−albite,

and microcline−anorthite solid solution (Figure 5) with PBEpseudopotentials. We consistently found a ∼40% difference inabsolute 1000 ln β values calculated with LDA and PBE, whichcould be explained by the difference in predicted unit cellvolume as suggested by Li, Y. et al.45 Although this differencemight appear concerning, when the intermineral fractionation(ΔA−B) is considered, a much smaller difference is obtained(Figure S4) if the same functional is used in the calculation of

rpfr because LDA/PBE consistently underestimate/overesti-mate lattice parameters.77

The results presented above suggest that the differencebetween our findings and those of Li, Y. et al. is due to the useof LDA or PBE approximation, as well as the use ofpseudopotentials for the metal ions. We suggest that the 1-valence pseudopotential used by Li, Y. et al. is ill-suited and werecommend using state-of-the-art pseudopotential libraries incalculations of rpfr.63,81,82

4.2. Mineral/Water Fractionation: Comparison withExperiments. While the database of K isotope measurementsof sediments and products of weathering is rapidlyexpanding,19−21 the rpfr of K+ in aqueous media has notbeen studied. The most relevant data that we can compare ourcalculations with are the experimental results from Li, W. et al.,who studied equilibrium isotopic fractionation of K betweensoluble K-salts and their saturated solutions, a proxy foraqueous K+.22

Li, W. et al. found indistinguishable K fractionation betweensylvite and aqueous K+. One concern with such experiments isthat achieving equilibrium between phases at room temper-ature can be difficult. Our calculated rpfr indicates that sylviteand aqueous K+ have indistinguishable fractionation at roomtemperature, given the current analytical precision on δ41Kmeasurements (∼0.1‰). Our results thus agree well with theexperimental results of Li, W. et al.One potential concern is that Li, W. et al. used saturated

sylvite solution while in our calculation K+ is not saturated.However, we do not expect the concentration to have asignificant impact based on the study by Wang et al.,83 whichshows that an increase in the simulation concentration of Mg2+

from 1 Mg in 50 H2O to 1 Mg in 30 H2O only increases therpfr of Mg by ∼3%.

4.3. Mineral/Water Fractionation: Implications. Wanget al. reported a K isotopic fractionation between seawater andthe BSE of δ41Kocean−BSE = ∼+0.6‰.18 The residence time of Kis on the order of 10 Myr,84 indicating that although only ahandful of measurements were reported, they are likelyrepresentative of Earth’s oceans as a whole because theocean mixing timescale is only around 1 kyr.85 The heavyisotope enrichment in seawater most likely involve K isotopicfractionation (kinetic or equilibrium) between solvated K+ andminerals, either during terrestrial weathering41,43,44 or due to

Figure 4. Temperature-dependent 1000 ln β values calculated fromthree different sets of pseudopotentials for K in microcline. PSL1 isthe result calculated with 1-valence alkali pseudopotentials whilePSL9 and GBRV are results calculated with 9-valence alkalipseudopotentials. The PDOS of all atoms in these minerals areprovided in a supplementary online file.

Figure 5. (a) Temperature-dependent 1000 ln β values for K in microcline−albite solid solution from microcline (microcline 100% K) to infinitelydiluted K in albite (albite 0% K), microcline−anorthite solution from 12.5% K (microcline−anorthite 12.5% K) to infinitely diluted K in anorthite(anorthite 0% K)infinite dilution (0% K) is modeled with a constrained cell approach (see Method section). (b) Linear fit of K force constant inmicrocline−albite solid solution with respect to Na concentration.

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uptake in silicate minerals in sediments and hydrothermalsystems.39−42

Li, S. et al. evaluated the global mass-balance of K in theoceans using δ41K values of the sources and sinks as inputs.25

They found that potassium dissolved in river waters seems tobe shifted in its δ41K value by +0.55 ± 0.29‰ relative to theclay fraction in the same rivers, and globally, the δ41K value ofrivers is −0.22 ± 0.04‰, which is shifted by ∼+0.3‰ relativeto the BSE, which has a δ41K value of ∼ -0.52.18,25,38 Theriverine δ41K value is low relative to the seawater value of∼+0.06 ± 0.10‰,37 which corresponds to a seawater-riverinerun off difference in δ41K values of +0.28 ± 0.11‰. There isinsufficient data available at present to evaluate what processesaffect K isotopes on a global scale, which is important if onewants to transfer knowledge of the present terrestrial cycle tothe geological record.Our calculation of the K isotopic fractionation between

solvated K+ and illite allows us to partially address thisquestion. At a temperature of 25 °C relevant to weathering atthe surface of continents, we calculate an equilibrium δ41Kfractionation between water and illite of +0.24‰. At a highertemperature of 100 °C relevant to hydrothermal systems orsmectite-illite conversion in sediments (∼50−100 °C), wecalculate a fractionation of +0.16‰. The K inputs in theoceans comprise (1) continental weathering and (2) midocean ridge hydrothermal fluxes. The sinks that remove K fromthe oceans are (1) the formation of K-bearing authigenic claysand ion exchange during sediment diagenesis and (2) lowtemperature basalt alteration. Given the elevated temperaturesinvolved in hydrothermal systems, it is unlikely that thehydrothermal flux has a δ41K value higher than seawater tobalance the low δ41K value of the rivers. To explain theelevated δ41K value of seawater relative to the sources (riversand mid ocean ridges), there must therefore be a negativefractionation between the K sinks and seawater (δ41Ksinks −δ41Kseawater < 0). Li, S. et al. estimated that the isotopicfractionation between sediments (formation of authigenic claysand ion exchange) and seawater must be −0.6 to −0.3‰.25

The equilibrium fractionation that we calculate between illiteand solvated K+ is −0.24‰, which could explain the seawatervalue if the −0.3‰ shift between sediment and seawater wasthe correct number or would be insufficient if the shift wasactually −0.6‰ (but it could still contribute to ∼ half of theoverall shift). Further work is clearly needed to better constrainthe global geochemical cycle of K. In particular, combining Kand Rb isotopic analyses could provide new insights into theseprocesses.4.4. Teasing Apart Equilibrium and Kinetic Processes

from Combined K and Rb Fractionations. The moststraightforward manner to compare K and Rb equilibriumisotopic fractionations is to compare the strength of the bondsthat they form. To the best of our knowledge, the only nominalRb-bearing natural mineral is rubicline (Rb-rich microcline).The reason for this scarcity is the low abundance of Rb, whichalways substitutes for K in K-bearing minerals. To model thelow Rb concentration in natural samples (the K/Rb weightratio on Earth is 400),4 we studied Rb in K-bearing minerallattices (see Method section for details). As seen in Figure 6,the force constants of Rb in a variety of minerals are relativelyconstant and are ∼20−50% larger than those of K. Given thestrength of K and Rb chemical bonds, and the relative massdifferences between 87/85Rb and 41/39K, we would thus expect

the isotopic compositions of two phases A and B in equilibriumto be related through the relationship86,87

cK RbA B41/39 87/35

A BΔ ≈ ·Δ− − (9)

where c is ∼3.5 depending on the host phase.Although K and Rb are chemically similar, they do not

necessarily behave in the same way during geochemicalprocesses. For instance, plagioclase is known to preferentiallypartition K relative to Rb,88 and Rb is preferentiallyincorporated in alteration phases (e.g., palagonite, smectite)during alteration on the seafloor.31,32,40,89 This indicates thatthe K and Rb isotopic compositions of a single phase may notnecessarily follow eq 9, even if the phase is formed underequilibrium conditions. However, we know that for two phasesin equilibrium, eq 9 should hold, and we can use this relationto assess whether a system is at equilibrium. Most kineticprocesses would impart isotopic fractionations for K and Rbthat would differ from the equilibrium fractionation given byeq 9. The word “kinetic” covers a multitude of processesincluding (1) unidirectional chemical reactions in which oneisotope can react faster than the other, (2) evaporation/condensation, and (3) diffusion.90,91 These kinetic processeshave been discussed extensively in the context of fractionationbetween isotopes (e.g. the relationship between δ17O andδ18O)87,92,93 and the same reasoning can be applied to K andRb.By combining K and Rb isotopic analyses, it is potentially

possible to determine whether K and Rb isotopic fractionationsare controlled by kinetic or equilibrium processes. To illustratethis, we have performed a simple calculation of K and Rbfractionation due to diffusion in water, which could berepresentative of diffusion through porewater in sediments,which Santiago-Ramos et al.21 argued could be a cause of theelevated δ41K value of seawater if light K isotopes diffuse fasterthan heavy ones. The ratio of the diffusion coefficients Di andDj of two isotopes i and j depend on their masses mi and mjthrough94

ikjjjjj

y{zzzzz

DD

m

mi

j

j

i

b

=(10)

where b is a coefficient that is derived from fitting results fromexperiments or MD simulations (b = 0.049 for K diffusion inaqueous medium, from MD simulation).95 The diffusivity of Kin water is 3.85 × 10−9 m2/s (MD simulation, fromliterature).95 Due to lack of data, we further model Rb as aheavier potassium isotope given that K and Rb have similar

Figure 6. Rb (infinite dilution in K-bearing minerals) and K meanforce constants in various K-bearing minerals.

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chemical properties and K and Cs have similar diffusioncoefficients,95 which leads to a diffusivity of Rb in water at 25°C of 3.70 × 10−9 m2/s. We calculate K and Rb isotopicfractionation assuming 1D diffusion between two semi-infinitemedia with initial concentrations C0 = 0 (x < 0) and C1 = 0.2M (for x > 0). In these conditions, the isotopic fractionation isgiven by the following equation96

i

kjjjjjj

y

{zzzzzz

Ä

Ç

ÅÅÅÅÅÅÅÅÅ

É

Ö

ÑÑÑÑÑÑÑÑÑ

mm

u

buu

(‰) 250 1 exp( /4)

/0.5 erfc( /2) 1

i j i

j

/ 2δ

π

= − − −

− (11)

where u x Dt/= is dimensionless time (the solution is self-similar, meaning that the isotopic profile stretches in spacewith the square root of time).As shown in Figure 7, the kinetic isotopic fractionation

produced by diffusion of K and Rb yields δ41K/δ87Rb = 2.2,

which differs from our computed value of ∼3−4 forequilibrium isotopic fractionation. This suggests that kineticand equilibrium fractionation would lead to distinct δ41K/δ87Rb ratios, but we acknowledge that other processes such asmixing could complicate interpretations of coupled K and Rbisotopic variations in geological samples.97 While the databaseof K isotopic analyses has grown tremendously over the pastseveral years,18−22,37,38 only a few natural rocks have beenstudied for their Rb isotopic compositions33−36 and thereforesuch comparison between K and Rb is not yet possible.However, our calculation illustrates the virtue of combiningthese two systems.4.5. Applications in High-Temperature Isotope Geo-

chemistry. The equilibrium isotopic fractionation between Kand Rb in silicate melt and minerals cannot be reliablycalculated at present as it is difficult to model silicate melts byMD. Several studies examining the properties of silicate meltshave refined our understanding of the speciation of alkalielements in silicate melts.98−101 In particular, their structuralposition seems to be influenced by the proportion of alkalielements and aluminum. In model silicate melt composition,when Na/Al < 1, all Na is bound to Al with tetrahedralcoordination as it locally charge-compensates Al3+. When Na/Al > 1, Na might be present in network-forming triclusters orat network-modifying octahedral sites.98,99,102 The equilibriumisotopic fractionations for K and Rb could thus be influencedby the nature of the silicate melts. Nevertheless, we expect

relatively small fractionation between silicate melts andcrystallized minerals. Assuming that the force constants of Kand Rb between silicate melts and crystallized minerals differby 35 and 7 n/m respectively, based on the values calculatedfor silicate minerals, we estimate that at 1100 °C, theequilibrium fractionation between silicate melt and mineralshould be lower than 0.10‰ for δ41K and 0.006‰ for δ87Rb.Using a lower temperature of 900 °C, one would expect theequilibrium isotopic fractionations between silicate melt andmineral to be lower than 0.14 and 0.009‰ for δ41K and δ87Rb,respectively. For comparison, K and Rb isotopic compositionscan be measured with precisions of ∼0.1 and 0.03‰,respectively.17−23,34,37,38 Partial melting and magmatic differ-entiation processes are thus unlikely to be associated withsignificant equilibrium isotopic fractionations for K and Rb. Ifany large isotopic fractionation is found for these systems atmagmatic temperatures, it is therefore likely to be the productof kinetic isotope effects associated with diffusive transport.103

For example, laboratory experiments have shown that Kisotopes could be readily fractionated by Soret diffusion insilicate melts at high temperature, with a fractionation of1.06‰/amu per 100 °C gradient.103,104 While equilibriumisotopic fractionation decreases as the inverse of the square ofthe temperature, diffusion in silicate can impart large isotopicfractionation even at magmatic temperatures. Such fractiona-tion has been extensively documented in olivine for Mg andFe.105−109 The geological settings where K and Rb diffusioncould happen at high temperature involve (1) large-scalediffusive transport in the mantle where solids and meltsinteract, as documented for Li and Fe (e.g., throughmetasomatism),110−113 (2) transport of K-rich aqueous fluidsfrom the subducting slab to the mantle wedge where thosefluids can induce flux melting, and (3) late diffusive re-equilibration of xenoliths114 or zoned minerals with their hostmelts.105−109 Again, the correlation between K and Rb isotopicfractionations in magmatic systems could provide insights intowhether these fractionations are driven by equilibrium orkinetic processes.

5. CONCLUSIONWe report a thorough first-principles study of equilibriumfractionation properties of aqueous K+, K, and Rb in commonK-bearing minerals using DFT and FPMD. The motivation forthis work is to provide a framework for interpreting theisotopic variations documented in natural systems for K andRb.In water, we do not find a clear correlation between the

strength of the K bonds and the coordination number. Themean force constant of K-bonds in water is ∼25 N/m. Thisfalls within the range of force constant values calculated by usand others for K-bearing minerals (between 22 and 30 N/m,not including solid solution, Table 1).The value of rpfr computed for sylvite is in good agreement

with a previous study; however, the rpfr value of feldspar differssignificantly from that reported in the literature. We ascribe thedifference to the choice of the pseudopotential. Ourdescription of the valence-core partition is more accurate butthe overall accuracy of the theory, in particular the functionalused, remains to be fully tested. We emphasize, however, thatany error present in DFT calculation of the absolute value ofthe rpfr is substantially reduced when calculating equilibriumisotopic fractionation factors between phases (Figure S4), asthis results in a partial cancellation of systematic errors.

Figure 7. Isotopic fractionations of K and Rb in a model (t = 5s) of1D diffusion between 2 semi-infinite slabs.

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We also calculate the force constants of Rb in nominal K-bearing minerals. The force constants of Rb in the mineralsinvestigated here are slightly larger than those for K, indicatingthat in most systems, equilibrium isotopic fractionationbetween two phases A and B should be characterized by aratio (δ41KB − δ41KA)/(δ

87RbB − δ87RbA) of approximately 3−4. We show through a diffusion calculation relevant to K andRb diffusion in sediment porewater that kinetic isotopicfractionation would depart from the equilibrium value; hencemeasuring K and Rb isotopic ratios together with K/Rb ratioscould provide new insights into the drivers of K and Rbfractionations in natural systems.Based on the computed rpfr, we obtained the equilibrium K

isotopic fractionation between aqueous K+ and illite, taken as aproxy mineral for clays. At 25 °C, the fractionation is∼+0.24‰, suggesting that equilibrium fractionation betweenaqueous K+ and K-bearing clay minerals is insufficient toexplain the observed K heavy isotope enrichment in seawateror river waters. Instead, this heavy K isotope enrichment couldpartly reflect kinetic isotopic fractionation associated withdiffusive transport or unidirectional chemical reactions.

■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acsearthspace-chem.9b00180.

(1) PDOS for minerals investigated; (2) calculated andexperimental lattice parameters for minerals; (3)coordination number of aqueous K+; (4) test of totalenergy convergence for the NaCl structure with respectto kinetic energy cutoff; (5) 1000 ln β calculated byLDA and PBE for K in albite−microcline solid solution;(6) convergence of 1000 ln β with respect to number ofsnapshots used for calculation; and (7) calculated versusmeasured infrared and Raman frequencies of K-bearingminerals (PDF)Partial phonon density of states of all atoms in theminerals and aqueous species investigated (XLSX)

■ AUTHOR INFORMATIONCorresponding Author*E-mail: dauphas@uchicago.edu.ORCIDTuan Anh Pham: 0000-0003-0025-7263FundingThis work was supported by NASA grants NNX17AE86G(LARS), NNX17AE87G (Emerging Worlds), andNSSC17K0744 (Habitable Worlds) to N.D., and NASANESSF grant NNX15AQ97H to N.X.N. V.F.R. acknowledgesthe Department of Energy National Nuclear SecurityAdministration Stewardship Science Graduate Fellowship.Part of this work was performed under the auspices of theU.S. Department of Energy by Lawrence Livermore NationalLaboratory under contract no. DE-AC52-07NA27344. T.A.P.(performed simulations of solvated ions) was supported as partof the Center for Enhanced Nanofluidic Transport, an EnergyFrontier Research Center funded by the U.S. Department ofEnergy, Office of Science, Basic Energy Sciences under awardno. DE-SC0019112. G.G. acknowledges support fromAMEWS (Advanced Materials for Energy-Water SystemCenter funded by the U.S. Department of Energy, Office of

Science, Basic Energy Sciences. V.F.R. acknowledges supportfrom the Department of Energy National Nuclear SecurityAdministration Stewardship Science Graduate Fellowshipunder award no. DE-NA0003864. This work was completedin part with resources provided by the University of Chicago’sResearch Computing Center.

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

Discussions with Marc Blanchard, Merlin Meheut, FrankRichter, and Fangzhen Teng and Matteo Gerosa were greatlyappreciated.

■ ABBREVIATIONS

rpfr, reduced partition function ratio; LDA, localized densityapproximation; BOMD, Born−Oppenheimer molecular dy-namics; MD, molecular dynamics; DFT, density functionaltheory; DFPT, density-functional perturbation theory

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