Article

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

Ab Initio Calculation of Equilibrium Isotopic Fractionations of Potassium and Rubidium in Minerals and Water † ‡ ‡ † † § ‡ ∥ ⊥ Hao 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 Institute, and Pritzker School of Molecular Engineering, 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 the equilibrium isotopic fractionation factors of potassium (K) and rubidium (Rb) in a variety of minerals of geological relevance (, , muscovite, illite, sylvite, and phlogopite). We also used molecular dynamics simulations to calculate the equilibrium isotopic fractionation factors of K in water. Our results indicate that K and Rb form bonds of similar strengths and that the ratio between the equilibrium fractionations 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 isotopic fractionation 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. These findings, however, cannot easily explain why the δ41K value of seawater is shifted by +0.6‰ relative to igneous rocks. Our results indicate that part of the observed fractionation is most likely due to kinetic effects. The narrow range of mean force constants for K and Rb in silicate minerals suggests that phase equilibrium is unlikely to create large K and Rb isotopic fractionations 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 40Ar+ and 40Ca+ on 40K+ in MC-ICPMS and the low abundance 40 Potassium is a moderately volatile, lithophile element that is of K (0.012%), the latter isotope is often not reported. This

Downloaded via UNIV OF CHICAGO on November 25, 2019 at 15:53:03 (UTC). present in relatively high abundance in the ocean (eighth most is inconsequential because K isotopic variations are thought to − concentrated element),1 Earth’s crust (eighth),1 3 bulk silicate follow the laws of -dependent fractionation, meaning that 4,5 the relative deviations in the 40K/39K ratio are approximately Earth (BSE; seventeenth), and in the solar system 41 39 40 41 26,27 See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. − δ δ (twentieth).6 8 The limited precision achievable by mass half those of the K/ K ratio ( K= K/2). spectrometry in measuring the ratios of the isotopic Also belonging to the alkali group, Rb shares similar abundances of 39K, 40K, and 41K has long limited the chemical and physical properties with K and substitutes for K application of K isotope systematics to cosmochemistry, in minerals such as . As a result, the K/Rb weight (g/g) − 28 where large isotopic variations have been found.9 16 Thanks ratio varies little both among meteorites and within various fi geochemical reservoirs in the Earth (400 for the BSE; 232 for to improvements in puri cation protocols and multicollector 1−5 inductively coupled plasma mass spectrometry (MC-ICPMS), the crust). Notable exceptions to the constancy in the K/Rb ∼ 29 ∼ 30 the isotopic composition of K (expressed using the 41K/39K weight ratio are rivers ( 1000) and seawater ( 3765). − ratio) can now be measured with a precision of ∼0.1‰,17 24 The elevated K/Rb ratio of seawater compared to that of the spurring wide interest in K isotopes, which have now been BSE could be due to the preferential mobilization of K relative 29 used to study weathering,21,25 seafloor alteration,23 and volatile to Rb during continental weathering, as well as the element depletion in planetary materials.10,12,14,17 Potassium preferential uptake of Rb relative to K during seafloor isotopic compositions are reported as the per mil (‰) 41 deviation (δ K) of the ratio of two potassium stable isotopes Received: June 21, 2019 41K (6.730%) and 39K (93.258%), relative to the inferred Revised: October 4, 2019 composition of the mantle or the NIST standard reference Accepted: October 16, 2019 material 3141a. Because of direct isobaric interferences from Published: October 16, 2019

© 2019 American Chemical Society 2601 DOI: 10.1021/acsearthspacechem.9b00180 ACS Earth Space Chem. 2019, 3, 2601−2612 ACS Earth and Space Chemistry Article alteration.1,4,31,32 Rubidium isotopic ratios can be measured effect on K fractionation using density functional theory − with great precision,33 36 but the similarity in the geochemical (DFT).45 Additionally, Li, W. et al. studied experimentally behaviors of K and Rb raises the question of whether equilibrium fractionation between K dissolved in water and K- measuring the isotopic compositions of K and Rb on the salts such as halides, sulfates, and carbonates.22 They also same samples provides additional insight compared to performed ab initio calculations for the salts that they studied. measuring K or Rb alone. Clearly, more work is needed to understand how the isotopic Understanding how K and Rb isotopes are fractionated at compositions of Rb and K are fractionated at equilibrium equilibrium between vapor and condensed phases is important between different minerals, gaseous species, and water. These to use these systems for understanding why planetary bodies fractionations are critical for interpreting the origin of K are depleted in these moderately volatile elements. It has been isotopic variations in low-temperature aqueous systems and shown that lunar rocks are enriched in the heavy isotopes of K planetary/nebular processes that involved the volatilization of by ∼0.4‰, which was interpreted by Wang and Jacobsen to K and Rb. reflect equilibrium isotopic fractionation between liquid and Understanding the isotopic fractionation of K between water vapor under conditions relevant to the Moon-forming giant and K-bearing minerals requires knowledge of the speciation of impact event.17 Nie and Dauphas et al. contended that at the K in water. As ∼99% of aqueous K in the oceans is present as ∼ hydrated K+, we chose to focus on hydrated K+ in this temperature of 3500 K, most likely relevant to Moon − formation, the equilibrium fractionation between vapor and study.46 48 In contrast to other cations, in particular divalent condensate would be too small, implying the involvement of ones (e.g., Zn2+),49 the hydration shell of K+ is not as well kinetic isotopic fractionation presumably associated with defined, with water molecules rapidly exchanging within the evaporation.36 Similarly, lunar rocks are enriched in heavy first solvation shell.50 Because equilibrium fractionation Rb isotope relative to terrestrial rocks33,36 and those data are properties are highly sensitive to coordination numbers and consistent with a scenario that involves loss of volatile elements to nearest neighbor distances, we used first-principle molecular from the protolunar disk by accretion onto the Earth, leaving dynamics (FPMD) to study hydrated K+, capturing many behind a Moon that is depleted in moderately volatile instantaneous configurations of the fully hydrated K+,in elements.36 contrast to a cluster approach where static hydrated Water−mineral interactions play a significant role in configurations are considered. maintaining habitable conditions at the surface of the Earth Our results suggest that (a) equilibrium fractionation through silicate weathering and carbonate precipitation. The between K-bearing clays (illite) and seawater could be a isotopic compositions of K and Rb could help trace continental significant source of oceanic 41K enrichment and (b) the bond weathering and reverse weathering in marine sediments, which strengths of Rb and K are similar, so equilibrium processes are fl both in uence climate through their controls of CO2 partial expected to impart correlated isotopic fractionations to Rb and pressure in the atmosphere (and associate greenhouse effect). K that differ from kinetic processes, providing a means of Li et al.,37 Wang and Jacobsen,18 and Morgan et al.38 found distinguishing between equilibrium and kinetic processes in that the ocean is significantly enriched in the heavy isotopes of nature. δ41 ≈ ‰ K relative to the BSE by Kocean−BSE +0.6 and the cause of this enrichment remains unclear. As summarized by 2. METHOD 21 Santiago-Ramos et al., this enrichment could arise from the 2.1. Equilibrium Fractionation. Equilibrium isotope removal of K from the oceans involving authigenic Al-silicate fractionation for an element X between two phases A and B formationinmarinesediments and/or low-temperature (α − ) can be calculated using the rpfr (or β-factor) of each 39−42 A B alteration of the oceanic crust. Alternatively, the enrich- phase26,27,51 ment could originate from isotopic fractionation during (/)XX′ β mobilization of K associated with continental silicate weath- α = A = A ering or high-temperature basalt alteration.41,43,44 Santiago- AB− (/)XX′ B β (1) Ramos et al. studied K isotopic fractionation in sediment pore- B fluids and concluded that K removal in authigenic Al-silicates where X, X′ refer to the abundances of two isotopes of an in sediments could be responsible for such fractionation as element. The rpfr can be calculated from51 pore-fluids tend to be enriched in the light isotopes of K. The 1/nN· q ff 3N − ν − ν′ driver for this fractionation could be di usion through the ν e hkTq,i/2 1e− hkTq,i/ β = i × × porewater, which is kinetic in origin. Equilibrium fractionation ∏∏ −hkTν / −hkTν′ /2 + ν′ q,i q,i between K dissolved in porewater and K incorporated in i=1 q i 1e− e minerals could also play a role but the extent of this i y (2) j z fractionation is unknown. Characterizing equilibrium isotopic j zν fractionation between aqueous fluids and K-bearing silicates is where Nj is the total number of in the unit cell,z q,i and ν′ are thej frequencies of vibrational mode i for two isotopesz at also important for understanding how K isotopes are q,i k { a given wavevector q, N is the total number of q-vectors, and fractionated during weathering. q n is the number of isotopic sites in the unit cell. When Despite progress made in measuring the isotopic composi- measured experimentally, equilibrium isotopic fractionation (in tions of K and Rb, understanding the cause of the isotopic ‰) between two phases is expressed as variations for those elements in natural systems can be challenging, partly due to an insufficient theoretical under- ΔAB−−(‰)==− 1000 lnαββ AB 1000(ln ln ) (3) standing of how equilibrium and kinetic processes control the AB fractionations of K and Rb isotopes. To address this, Li, Y. et To a good approximation, one can write 1000 ln β as a al. recently calculated the reduced partition function ratio polynomial expansion in even powers of the inverse of the (rpfr) of K in alkali feldspar and studied the concentration temperature52,53

2602 DOI: 10.1021/acsearthspacechem.9b00180 ACS Earth Space Chem. 2019, 3, 2601−2612 ACS Earth and Space Chemistry Article

A1 A2 A3 2.3. DFT Calculations for Solid Phases. Unless 1000 ln β =++ fi T2 T 4 T 6 (4) otherwise speci ed, structural relaxations and phonon calculations (including all minerals and aqueous K+ snapshots) where A1, A2, and A3 are constants that depend on the were carried out with the Quantum ESPRESSO code,61,62 with mineral/phase considered. The coefficient of the first-order 54 plane-wave basis sets, the PBE functional, and ONCV term, which always dominates the rpfr, especially at high pseudopotentials.63 The kinetic energy and charge-density temperature (>200 K), can be rewritten as cutoff were set to 80 and 320 Ry, respectively. Our choices of 11ℏ2 Monkhorst k-point grids64 and q-point grids are given in Table A = 1000 − F 45 1 MM′ 8k2 (5) S1. To compare our results with those of Li, Y. et al. for feldspar, we also used the local density approximation (LDA) ℏ ’ where is the reduced Planck constant, k is Boltzmann s with LDA pseudopotentials from the GBRV library65 and constant, M′ andji M are thezy of the two isotopes, and ⟨F⟩ 66 j z Quantum ESPRESSO PSlibrary (PSL). Pseudopotentials is the mean forcek constant{ of the K or Rb bonds (equivalent to from PSL used in this study include Na.pz-n-vbc.UPF, K.pz-n- the spring constant of a harmonic spring), which can be vbc.UPF (1-valence), K.pz-spn-kjpaw_psl.1.0.0.UPF (9-va- calculated from the partial phonon density of states (PDOS, lence), Al.pz-n-kjpaw_psl.0.1.UPF, Si.pz-n-kjpaw_psl.0.1.UPF, g(E))52 and O.pz-n-kjpaw_psl.0.1.UPF. For GBRV and PSL pseudo- +∞ M 2 potentials, the kinetic energy was set at 60 and 75 Ry, and F = 2 ∫ EgE()dE ℏ 0 (6) charge-density cutoff was set at a multiple of 10 and 4 of kinetic energy cutoff, respectively. At high temperature for 39/41K and 85/87Rb, eq 4 can be The minerals (and their formulas) investigated here are written as orthoclase (KAlSi3O8), (KAlSi3O8), albite (NaAl- ⟨⟩F Si3O8), anorthite (CaAl2Si2O8), muscovite (KAl3Si3O12H2), 1000 lnβK ≃ 5500 2 T (7) illite (KAl2Si4O11F), phlogopite (KAlSi3Mg3O12H2), and sylvite (NaCl). Their crystal structures were taken from ⟨⟩F 67−74 1000 lnβ ≃ 1189 experiments and the lattice and atomic positions were Rb T2 (8) relaxed until the total force and stress were smaller than 10−4 At low temperature relevant to weathering, for example, the atomic units and 0.1 kbar, respectively. A phonon calculation full expansion in eq 4 should be used. was then performed for each structure using DFPT (density 62,75 2.2. FPMD of the Potassium Ion in Water. We carried functional perturbation theory). The same calculations out FPMD of aqueous K+ using the Perdew, Burke, and were performed for K and Rb by substituting the second for Ernzerhof (PBE) exchange−correlation functional.54 This the first in the mineral structures. For albite−microcline (Na relatively simple functional was chosen as it was reported and K ) solid solution and K-substituted anorthite, that there is no significant difference in rpfr calculated from phonon frequencies were only calculated at the Γ point. For PBE and van der Waals functionals.49 We used the same microcline−anorthite (K and Ca feldspars) solid solution, for functional for minerals and hydrated potassium, to facilitate every Ca-substituted by K, one Al was replaced with Si to comparing fractionation factors for different systems and to balance charge. For aqueous K+, with configurations from take advantage of cancellation of errors. molecular dynamics (MD) trajectory, only the atomic We performed Born−Oppenheimer molecular dynamics positions were relaxed. 55 (BOMD) simulations with the code. We used a cubic To model infinite dilution we used two strategies. The first is + cell containing 63 water molecules and one K and a volume what we call the constrained cell method (Rb in muscovite, chosen so as to obtain the measured density of water at orthoclase, phlogopite, and illite): (1) we fully relaxed the ambient conditions (edge length equal to 12.4 Å). A lattice and atomic positions of the host phase; (2) we made compensating uniform background charge was used to ensure one substitution (i.e., replaced one K with Rb); (3) we held charge neutrality. All hydrogen atoms were replaced with the lattice constant fixed but relaxed the atomic positions of deuterium to allow for a larger timestep, chosen to be 10 the structure after substitution; and (4) we calculated the atomic units. The 3s and 3p semicore states of K were treated phonon frequencies and rpfr. The second is the supercell as valence states, and the interaction between valence and core approach (Rb in sylvite combines the supercell and con- was represented by an optimized norm-conserving strained cell approaches): (1) we obtained a primitive cell; (2) pseudopotential (ONCV)56 for K and a norm-conserving Hamann−Schluter−Chiang−Vanderbilt57 pseudopotentials for we made one substitution (i.e., replaced one K with Rb); (3) hydrogen and oxygen. The electronic wavefunctions were we calculated the phonon frequencies and rpfr; (4) we made a represented with a plane-wave and a kinetic energy supercell that was twice as large as the primitive cell and cutoff of 85 Ry. We generated trajectories for ∼50 ps in the repeated steps 2 and 3; (5) we made another supercell that was NVT ensemble, with a temperature of 400 K set by a velocity twice as large as the cell in the previous step and repeated steps β scaling thermostat. Using 400 K instead of room temperature 2 and 3; and (6) we repeated step 5 until 1000 ln converged. is a commonly used approximation to correct the well-known One caveat when using the constrained cell approach to problem of an overstructured liquid water using PBE at room study infinite dilution is that the primitive cell must be − temperature.58 60 We considered configurations of solvated K+ relatively large so there is minimal interaction between from 20 uncorrelated snapshots over the entire trajectory substituents. In cases like KCl, where there are 2 atoms in (taken from the postequilibrated heavy water trajectory with the primitive cell, one still needs to build a supercell. For Rb- 100 steps between each snapshot corresponding to a total sylvite, we used tetragonal K8Cl8 with one Rb duration of 24 fs). substituting for K.

2603 DOI: 10.1021/acsearthspacechem.9b00180 ACS Earth Space Chem. 2019, 3, 2601−2612 ACS Earth and Space Chemistry Article

3. RESULTS The coefficients in the expansion describing the temperature β The structures of minerals investigated here are provided in dependence of 1000 ln , the force constant of the chemical Figure 1, and the mineral lattice parameters calculated from bonds, and the 1000 ln β values at 300, 600, and 1200 K are given in Table 1 for K, Rb, and aqueous K+. The temperature dependence of rpfr is shown in Figure 2, where the rpfr scales linearly with 1/T2. The ⟨F⟩ values for Rb are ∼20−50% larger than those for K in the same minerals and given the mass ff Δ41/39 di erence (eqs 7 and 8), we expect the ratio KA−B/ Δ87/85 ∼ RbA−B to be 3.5 for equilibrium fractionation between two systems A and B if K and Rb are fractionated together by an equilibrium process. We calculated the rpfr for aqueous K+ by extracting 20 snapshots (see Figure S5 for rpfr convergence with respect to number of snapshots) from MD trajectories and performed phonon calculations on the relaxed snapshot structures. In the PDOS calculated for aqueous K+, we can clearly see couplings to the H−O−H bending and O−H stretching modes at higher

78 frequency (Figure 3). Figure 1. Structures (rendered by VESTA) of minerals and aqueous + + The calculated rpfr of aqueous K shows some variability K investigated in this paper. Albite, anorthite, microcline, and ff orthoclase share the feldspar structure. Red represents O, white (see Figure 2) across di erent snapshots due to dynamical represents H, green represents Cl, dark blue represents Si, light blue changes in the coordination environment of K+. In our FPMD represents Al, purple represents K, brown represents Mg, and grey simulation, we found that the coordination number of K+ represents F. varies from 4 to 10 between snapshots, with an average coordination number of 6.7 (Figure S2). We averaged the rpfr fi rst-principles are given in Table S1. The PDOS of the obtained from 20 snapshots uniformly spaced in time, whose minerals are given in the Supporting Information (Figure S1) average coordination number (7.2, Figure S2) is comparable to We also report the calculated frequencies as compared to that obtained over the whole trajectory. Similar to the results experimentally measured infrared and Raman frequencies of Ducher et al. for solvated Zn, we found little correlation (Figure S6), showing good agreement. Similar to the results 2 76 between rpfr and coordination numbers (R = 0.05), so the reported by Ducher et al., we found that the presence of ff hydrogen atoms leads to slightly distorted lattice parameters. small di erence between the average over the selected samples fi In general, the computed parameters are 1−2% larger than the and whole trajectory should have an insigni cant impact on the 49 corresponding measured values, consistent with previous computed rpfr. We also assume that the dependence of observations that the PBE functional tends to overestimate phonon frequencies on temperature is weak and use the lattice parameters of semiconductors and insulators.77 computed phonon frequencies across all temperatures.

fi ffi β A x Table 1. Force Constants for K and Rb (In nite Dilution in K-Bearing Minerals), Expansion Coe cients for 1000 ln = 1 + A x2 A x3 x 6 T2 β 2 + 3 With =10/ , and 1000 ln 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 0.117 Rb 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 0.106 Rb 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 0.103 Rb 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 0.100 Rb 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 0.084 Rb 35 0.042 −6.44 × 10−5 9.33 × 10−7 0.465 0.116 0.029 + − × −4 × −6 [K(H2O)63] K 25 0.145 4.61 10 6.24 10 1.610 N/A N/A microcline K 27 0.153 −3.79 × 10−4 6.36 × 10−6 1.662 0.422 0.106 microcline−albite 50% K K 34 0.192 −3.49 × 10−4 3.12 × 10−6 2.093 0.530 0.133 microcline−albite 25% K K 46 0.263 −9.47 × 10−4 1.34 × 10−5 2.829 0.725 0.183 microcline−albite 12.5% K K 52 0.291 −7.55 × 10−4 7.06 × 10−6 3.150 0.803 0.202 albite 0% K K 53 0.295 −4.32 × 10−4 1.32 × 10−6 3.225 0.816 0.205 microcline−anorthite 12.5% K K 59 0.329 −5.98 × 10−4 2.48 × 10−6 3.584 0.909 0.228 anorthite 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 was calculated using a combination of the supercell and constrained cell approaches.

2604 DOI: 10.1021/acsearthspacechem.9b00180 ACS Earth Space Chem. 2019, 3, 2601−2612 ACS Earth and Space Chemistry Article

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 the standard 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−H stretching modes.

4. DISCUSSION pseudopotentials are (1) a set from the Quantum ESPRESSO PSlibrary (PSL1) that uses 1-valence alkali pseudopotentials as To the best of our knowledge, there is no calculation of rpfr for used by Li, Y. et al.; (2) a set from Quantum ESPRESSO Rb with which we could compare our results, nor could we find PSlibrary (PSL9), in whichweused9-valencealkali any experimentally measured, direct determination of equili- pseudopotentialsthe only difference between PSL9 and brium intermineral fractionation data for Rb. The most PSL1 is the alkali pseudopotential; and (3) one set from the relevant comparisons are a theoretical study on alkali feldspar GBRV pseudopotential library that has 9-valence alkali by Li, Y. et al.45 and the work of Li, W. et al. on K fractionation 22 fi pseudopotentials. between salts and aqueous solution. We will rst discuss and The resulting cell parameters, unit cell volume, and the force compare our results to both studies and then present the constant for microcline are shown in Table 2 and compared implications of our results. 4.1. Mineral Fractionation: Comparison with a Table 2. Experimental and Calculated Lattice Parameters Previous Theoretical Study. Li, Y. et al. examined the ff fl − with Di erent Pseudopotentials, Unit Cell Volume, and in uence of the Na K solid solution in alkali feldspar by Force Constant of K-Bonds (Which Controls Equilibrium calculating the rpfr of K in K-feldspars using DFT and the LDA Isotopic Fractionation through Eq 7; gamma-point only functional and pseudopotentials. To make the discussion more calculation) for Microcline straightforward, we converted the expansion coefficients ⟨ ⟩ reported by Li, Y. et al. to force constants (see eqs 4 and 5). Vdiff F 3 For K in microcline, they obtained a force constant of 48 N/m, a (Å) b (Å) c (Å) V (Å ) (%) (N/m) while for 50−50 microcline−albite solid solution, the value is exp. 8.571 12.964 7.221 720.99 0.0 N/A fi measured 64 N/m. These values are signi cantly larger than the values of 45 − 28 and 35 N/m that we obtained for the same minerals using Li et al. 8.600 12.942 7.199 717.79 0.4 48 LDA-PSL1 8.633 12.928 7.208 722.17 0.2 45 PBE. Li, Y. et al. claimed that the LDA approximation is more LDA-PSL9 8.417 12.901 7.180 699.61 −3.0 38 appropriate because it gave better agreement between LDA-GBRV 8.425 12.917 7.186 702.12 −2.6 38 computed and experimental lattice parameters, and the rpfr PBE 8.698 13.105 7.313 752.03 4.3 28 is 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 it with the experimental results. Temperature-dependent 1000 ln is well known that considering only the outermost valence β values are given in Figure 4. First, we note that the result electrons of these elements may cause significant errors in most reported by Li, Y. et al. was reproduced when using the computed properties.75,79,80 To evaluate the influence of such pseudopotential set PSL1 (Table 2). We found that two pseudopotentials, we tested 3 sets of LDA pseudopotentials differentsetsofpseudopotentialswith9-valencealkali from different libraries for microcline. The three sets of LDA pseudopotentials gave similar lattice parameters and rpfr

2605 DOI: 10.1021/acsearthspacechem.9b00180 ACS Earth Space Chem. 2019, 3, 2601−2612 ACS Earth and Space Chemistry Article

rpfr because LDA/PBE consistently underestimate/overesti- mate lattice parameters.77 The results presented above suggest that the difference between our findings and those of Li, Y. et al. is due to the use of LDA or PBE approximation, as well as the use of pseudopotentials for the metal ions. We suggest that the 1- valence pseudopotential used by Li, Y. et al. is ill-suited and we recommend using state-of-the-art pseudopotential libraries in calculations of rpfr.63,81,82 4.2. Mineral/Water Fractionation: Comparison with Experiments. While the database of K isotope measurements of sediments and products of weathering is rapidly − expanding,19 21 the rpfr of K+ in aqueous media has not β been studied. The most relevant data that we can compare our Figure 4. Temperature-dependent 1000 ln values calculated from calculations with are the experimental results from Li, W. et al., three different sets of pseudopotentials for K in microcline. PSL1 is the result calculated with 1-valence alkali pseudopotentials while who studied equilibrium isotopic fractionation of K between PSL9 and GBRV are results calculated with 9-valence alkali soluble K-salts and their saturated solutions, a proxy for + 22 pseudopotentials. The PDOS of all atoms in these minerals are aqueous K . provided in a supplementary online file. Li, W. et al. found indistinguishable K fractionation between sylvite and aqueous K+. One concern with such experiments is that achieving equilibrium between phases at room temper- (Figure 4). These are both significantly smaller than those ature can be difficult. Our calculated rpfr indicates that sylvite reported by Li, Y. et al. who had argued that our lower estimate and aqueous K+ have indistinguishable fractionation at room of the force constant was due to a larger calculated unit cell temperature, given the current analytical precision on δ41K volume but as shown here, our calculated cell volume is ∼ ‰ actually smaller. In addition, PSL9 and GBRV PP gave similar measurements ( 0.1 ). Our results thus agree well with the relative errors in unit cell volumes, compared with the results experimental results of Li, W. et al. One potential concern is that Li, W. et al. used saturated obtained with PBE. Using LDA pseudopotentials from GBRV + libraries, we conducted another test with a 50−50 microcline− sylvite solution while in our calculation K is not saturated. albite structure as reported by Li, Y. et al., and we again However, we do not expect the concentration to have a significant impact based on the study by Wang et al.,83 which obtained a smaller force constant (49 N/m) than that reported 2+ by Li, Y. et al. (64 N/m). Finally, we tested the PSL1 PP for shows that an increase in the simulation concentration of Mg the NaCl structure of sylvite and found that total energy from 1 Mg in 50 H2O to 1 Mg in 30 H2O only increases the ∼ convergence could not be achieved even with kinetic energy rpfr of Mg by 3%. cutoff set at 120 Ry (Figure S3). 4.3. Mineral/Water Fractionation: Implications. Wang We further calculated rpfr for K in the microcline−albite, et al. reported a K isotopic fractionation between seawater and δ41 ∼ ‰ 18 and microcline−anorthite solid solution (Figure 5) with PBE the BSE of Kocean−BSE = +0.6 . The residence time of K 84 pseudopotentials. We consistently found a ∼40% difference in is on the order of 10 Myr, indicating that although only a absolute 1000 ln β values calculated with LDA and PBE, which handful of measurements were reported, they are likely could be explained by the difference in predicted unit cell representative of Earth’s oceans as a whole because the volume as suggested by Li, Y. et al.45 Although this difference ocean mixing timescale is only around 1 kyr.85 The heavy might appear concerning, when the intermineral fractionation isotope enrichment in seawater most likely involve K isotopic Δ ff + ( A−B) is considered, a much smaller di erence is obtained fractionation (kinetic or equilibrium) between solvated K and (Figure S4) if the same functional is used in the calculation of minerals, either during terrestrial weathering41,43,44 or due to

Figure 5. (a) Temperature-dependent 1000 ln β values for K in microcline−albite solid solution from microcline (microcline 100% K) to infinitely diluted 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 in microcline−albite solid solution with respect to Na concentration.

2606 DOI: 10.1021/acsearthspacechem.9b00180 ACS Earth Space Chem. 2019, 3, 2601−2612 ACS Earth and Space Chemistry Article uptake in silicate minerals in sediments and hydrothermal − systems.39 42 Li, S. et al. evaluated the global mass-balance of K in the oceans using δ41K values of the sources and sinks as inputs.25 They found that potassium dissolved in river waters seems to be shifted in its δ41K value by +0.55 ± 0.29‰ relative to the clay fraction in the same rivers, and globally, the δ41K value of rivers is −0.22 ± 0.04‰, which is shifted by ∼+0.3‰ relative to the BSE, which has a δ41K value of ∼ -0.52.18,25,38 The riverine δ41K value is low relative to the seawater value of ∼+0.06 ± 0.10‰,37 which corresponds to a seawater-riverine run off difference in δ41K values of +0.28 ± 0.11‰. There is ffi insu cient data available at present to evaluate what processes Figure 6. Rb (infinite dilution in K-bearing minerals) and K mean affect K isotopes on a global scale, which is important if one force constants in various K-bearing minerals. wants to transfer knowledge of the present terrestrial cycle to the geological record. the isotopic compositions of two phases A and B in equilibrium Our calculation of the K isotopic fractionation between 86,87 solvated K+ and illite allows us to partially address this to be related through the relationship question. At a temperature of 25 °C relevant to weathering at 41/39 87/35 Δ KRb≈·Δc (9) the surface of continents, we calculate an equilibrium δ41K AB−−AB fractionation between water and illite of +0.24‰. At a higher where c is ∼3.5 depending on the host phase. temperature of 100 °C relevant to hydrothermal systems or Although K and Rb are chemically similar, they do not smectite-illite conversion in sediments (∼50−100 °C), we necessarily behave in the same way during geochemical calculate a fractionation of +0.16‰. The K inputs in the processes. For instance, plagioclase is known to preferentially 88 oceans comprise (1) continental weathering and (2) mid partition K relative to Rb, and Rb is preferentially ocean ridge hydrothermal fluxes. The sinks that remove K from incorporated in alteration phases (e.g., palagonite, smectite) 31,32,40,89 the oceans are (1) the formation of K-bearing authigenic clays during alteration on the seafloor. This indicates that and ion exchange during sediment diagenesis and (2) low the K and Rb isotopic compositions of a single phase may not temperature basalt alteration. Given the elevated temperatures necessarily follow eq 9, even if the phase is formed under involved in hydrothermal systems, it is unlikely that the equilibrium conditions. However, we know that for two phases hydrothermal flux has a δ41K value higher than seawater to in equilibrium, eq 9 should hold, and we can use this relation balance the low δ41K value of the rivers. To explain the to assess whether a system is at equilibrium. Most kinetic δ41 processes would impart isotopic fractionations for K and Rb elevated K value of seawater relative to the sources (rivers ff and mid ocean ridges), there must therefore be a negative that would di er from the equilibrium fractionation given by δ41 − eq 9. The word “kinetic” covers a multitude of processes fractionation between the K sinks and seawater ( Ksinks δ41K < 0). Li, S. et al. estimated that the isotopic including (1) unidirectional chemical reactions in which one seawater isotope can react faster than the other, (2) evaporation/ fractionation between sediments (formation of authigenic clays ff 90,91 and ion exchange) and seawater must be −0.6 to −0.3‰.25 condensation, and (3) di usion. These kinetic processes have been discussed extensively in the context of fractionation The equilibrium fractionation that we calculate between illite between isotopes (e.g. the relationship between δ17O and and solvated K+ is −0.24‰, which could explain the seawater δ18O)87,92,93 and the same reasoning can be applied to K and value if the −0.3‰ shift between sediment and seawater was Rb. the correct number or would be insufficient if the shift was By combining K and Rb isotopic analyses, it is potentially actually −0.6‰ (but it could still contribute to ∼ half of the possible to determine whether K and Rb isotopic fractionations overall shift). Further work is clearly needed to better constrain are controlled by kinetic or equilibrium processes. To illustrate the global geochemical cycle of K. In particular, combining K this, we have performed a simple calculation of K and Rb and Rb isotopic analyses could provide new insights into these fractionation due to diffusion in water, which could be processes. representative of diffusion through porewater in sediments, 4.4. Teasing Apart Equilibrium and Kinetic Processes which Santiago-Ramos et al.21 argued could be a cause of the from Combined K and Rb Fractionations. The most elevated δ41K value of seawater if light K isotopes diffuse faster straightforward manner to compare K and Rb equilibrium than heavy ones. The ratio of the diffusion coefficients D and isotopic fractionations is to compare the strength of the bonds i Dj of two isotopes i and j depend on their masses mi and mj that they form. To the best of our knowledge, the only nominal through94 Rb-bearing natural mineral is rubicline (Rb-rich microcline). b The reason for this scarcity is the low abundance of Rb, which D mj i = always substitutes for K in K-bearing minerals. To model the D m low Rb concentration in natural samples (the K/Rb weight j i (10) ratio on Earth is 400),4 we studied Rb in K-bearing mineral where b is a coefficient that is derived from fitting results from ji zy lattices (see Method section for details). As seen in Figure 6, experimentsj orz MD simulations (b = 0.049 for K diffusion in j z the force constants of Rb in a variety of minerals are relatively aqueous medium,k { from MD simulation).95 The diffusivity of K constant and are ∼20−50% larger than those of K. Given the in water is 3.85 × 10−9 m2/s (MD simulation, from strength of K and Rb chemical bonds, and the relative mass literature).95 Due to lack of data, we further model Rb as a differences between 87/85Rb and 41/39K, we would thus expect heavier potassium isotope given that K and Rb have similar

2607 DOI: 10.1021/acsearthspacechem.9b00180 ACS Earth Space Chem. 2019, 3, 2601−2612 ACS Earth and Space Chemistry Article chemical properties and K and Cs have similar diffusion relatively small fractionation between silicate melts and coefficients,95 which leads to a diffusivity of Rb in water at 25 crystallized minerals. Assuming that the force constants of K °C of 3.70 × 10−9 m2/s. We calculate K and Rb isotopic and Rb between silicate melts and crystallized minerals differ fractionation assuming 1D diffusion between two semi-infinite by 35 and 7 n/m respectively, based on the values calculated ° media with initial concentrations C0 =0(x < 0) and C1 = 0.2 for silicate minerals, we estimate that at 1100 C, the M (for x > 0). In these conditions, the isotopic fractionation is equilibrium fractionation between silicate melt and mineral given by the following equation96 should be lower than 0.10‰ for δ41K and 0.006‰ for δ87Rb. Using a lower temperature of 900 °C, one would expect the ij/2mi equilibrium isotopic fractionations between silicate melt and δ (‰)=− 250 − 1 exp( −u /4) ‰ δ41 δ87 mj mineral to be lower than 0.14 and 0.009 for K and Rb, respectively. For comparison, K and Rb isotopic compositions ibu/ π y can be measured with precisions of ∼0.1 and 0.03‰, j z − j z respectively.17 23,34,37,38 Partial melting and magmatic differ- 0.5 erfc(j u /2)z− 1 (11) k { entiation processes are thus unlikely to be associated with ÅÄ ÑÉ fi where u = xDt/ Å is dimensionlessÑ time (the solution is self- signi cant equilibrium isotopic fractionations for K and Rb. If Å Ñ fi similar, meaningÅ that the isotopicÑ pro le stretches in space any large isotopic fractionation is found for these systems at with the squareÇÅ root of time). ÖÑ magmatic temperatures, it is therefore likely to be the product As shown in Figure 7, the kinetic isotopic fractionation of kinetic isotope effects associated with diffusive transport.103 produced by diffusion of K and Rb yields δ41K/δ87Rb = 2.2, For example, laboratory experiments have shown that K isotopes could be readily fractionated by Soret diffusion in silicate melts at high temperature, with a fractionation of 1.06‰/amu per 100 °C gradient.103,104 While equilibrium isotopic fractionation decreases as the inverse of the square of the temperature, diffusion in silicate can impart large isotopic fractionation even at magmatic temperatures. Such fractiona- tion has been extensively documented in olivine for Mg and − Fe.105 109 The geological settings where K and Rb diffusion could happen at high temperature involve (1) large-scale diffusive transport in the mantle where solids and melts interact, as documented for Li and Fe (e.g., through − metasomatism),110 113 (2) transport of K-rich aqueous fluids from the subducting slab to the mantle wedge where those fluids can induce flux melting, and (3) late diffusive re- Figure 7. Isotopic fractionations of K and Rb in a model (t = 5s) of equilibration of xenoliths114 or zoned minerals with their host 1D diffusion between 2 semi-infinite slabs. − melts.105 109 Again, the correlation between K and Rb isotopic fractionations in magmatic systems could provide insights into which differs from our computed value of ∼3−4for whether these fractionations are driven by equilibrium or equilibrium isotopic fractionation. This suggests that kinetic kinetic processes. and equilibrium fractionation would lead to distinct δ41K/ δ87Rb ratios, but we acknowledge that other processes such as 5. CONCLUSION mixing could complicate interpretations of coupled K and Rb We report a thorough first-principles study of equilibrium isotopic variations in geological samples.97 While the database fractionation properties of aqueous K+, K, and Rb in common of K isotopic analyses has grown tremendously over the past K-bearing minerals using DFT and FPMD. The motivation for − several years,18 22,37,38 only a few natural rocks have been this work is to provide a framework for interpreting the − studied for their Rb isotopic compositions33 36 and therefore isotopic variations documented in natural systems for K and such comparison between K and Rb is not yet possible. Rb. However, our calculation illustrates the virtue of combining In water, we do not find a clear correlation between the these two systems. strength of the K bonds and the coordination number. The 4.5. Applications in High-Temperature Isotope Geo- mean force constant of K-bonds in water is ∼25 N/m. This chemistry. The equilibrium isotopic fractionation between K falls within the range of force constant values calculated by us and Rb in silicate melt and minerals cannot be reliably and others for K-bearing minerals (between 22 and 30 N/m, calculated at present as it is difficult to model silicate melts by not including solid solution, Table 1). MD. Several studies examining the properties of silicate melts The value of rpfr computed for sylvite is in good agreement have refined our understanding of the speciation of alkali with a previous study; however, the rpfr value of feldspar differs − elements in silicate melts.98 101 In particular, their structural significantly from that reported in the literature. We ascribe the position seems to be influenced by the proportion of alkali difference to the choice of the pseudopotential. Our elements and aluminum. In model silicate melt composition, description of the valence-core partition is more accurate but when Na/Al < 1, all Na is bound to Al with tetrahedral the overall accuracy of the theory, in particular the functional coordination as it locally charge-compensates Al3+. When Na/ used, remains to be fully tested. We emphasize, however, that Al > 1, Na might be present in network-forming triclusters or any error present in DFT calculation of the absolute value of at network-modifying octahedral sites.98,99,102 The equilibrium the rpfr is substantially reduced when calculating equilibrium isotopic fractionations for K and Rb could thus be influenced isotopic fractionation factors between phases (Figure S4), as by the nature of the silicate melts. Nevertheless, we expect this results in a partial cancellation of systematic errors.

2608 DOI: 10.1021/acsearthspacechem.9b00180 ACS Earth Space Chem. 2019, 3, 2601−2612 ACS Earth and Space Chemistry Article

We also calculate the force constants of Rb in nominal K- Science, Basic Energy Sciences. V.F.R. acknowledges support bearing minerals. The force constants of Rb in the minerals from the Department of Energy National Nuclear Security investigated here are slightly larger than those for K, indicating Administration Stewardship Science Graduate Fellowship that in most systems, equilibrium isotopic fractionation under award no. DE-NA0003864. This work was completed between two phases A and B should be characterized by a in part with resources provided by the University of Chicago’s δ41 − δ41 δ87 − δ87 − ratio ( KB KA)/( RbB RbA) of approximately 3 Research Computing Center. 4. We show through a diffusion calculation relevant to K and ff Notes Rb di usion in sediment porewater that kinetic isotopic fi fractionation would depart from the equilibrium value; hence The authors declare no competing nancial interest. measuring K and Rb isotopic ratios together with K/Rb ratios could provide new insights into the drivers of K and Rb ■ ACKNOWLEDGMENTS fractionations in natural systems. Discussions with Marc Blanchard, Merlin Meheut, Frank Based on the computed rpfr, we obtained the equilibrium K + Richter, and Fangzhen Teng and Matteo Gerosa were greatly isotopic fractionation between aqueous K and illite, taken as a appreciated. proxy mineral for clays. At 25 °C, the fractionation is ∼+0.24‰, suggesting that equilibrium fractionation between ABBREVIATIONS aqueous K+ and K-bearing clay minerals is insufficient to ■ explain the observed K heavy isotope enrichment in seawater rpfr, reduced partition function ratio; LDA, localized density or river waters. Instead, this heavy K isotope enrichment could approximation; BOMD, Born−Oppenheimer molecular dy- partly reflect kinetic isotopic fractionation associated with namics; MD, molecular dynamics; DFT, density functional diffusive transport or unidirectional chemical reactions. theory; DFPT, density-functional perturbation theory

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2612 DOI: 10.1021/acsearthspacechem.9b00180 ACS Earth Space Chem. 2019, 3, 2601−2612