American Mineralogist, Volume 94, pages 600–608, 2009

Effective radii of noble gas atoms in silicates from first-principles molecular simulation

Li q u n Zh a n g ,1,2 Ja m e s A. Va n Or m a n ,1,* a n d Da n i e l J. La c k s 2

1Department of Geological Sciences, Case Western Reserve University, Cleveland, Ohio 44106, U.S.A. 2Department of Chemical Engineering, Case Western Reserve University, Cleveland, Ohio 44106, U.S.A.

Ab s t r a c t An understanding of how noble gas atoms are dissolved in mantle minerals and melts is necessary to infer geological information from the constraints provided by noble gas geochemistry. For this purpose, first-principles molecular simulations are carried out on liquid and crystalline (stishovite) silica systems with dissolved noble gas atoms (He, Ne, Ar, Kr, and Xe). The first principles nature of the simulations, which do not involve empirical force field parameters, enables the determination of the effective radii and structural environments of the noble gas atoms. The noble gas atoms are shown to be highly compressible, so that their effective radii depend strongly on the molar volume of the system (which in turn depends on pressure). Due to the continuous nature of interatomic forces, the effective radii also depend on the extent to which the surrounding atoms can relax in response to the presence of the noble gas atom. In this regard, different definitions of effective radii are relevant in different situations: “equilibrium radii” that correspond to the optimal interatomic distances at the molar volume of the system, and “repulsive wall” radii that correspond to the interatomic distances where the interatomic potentials of mean force change from attractive to repulsive at that molar vol- ume. The equilibrium radii determine the interatomic distances in a melt, and the repulsive wall radii determine the interatomic distances for interstitial sites in a crystal. Based on these effective radii, the structural environment surrounding the noble gas atoms at high pressure is shown to correspond to a close packing of O atoms around the central noble gas atom. Compression of the noble gas atoms is shown to correspond closely to the compression of the porosity within the silicate melt structure. Keywords: Silicate melt, ionic radius, molecular dynamics simulation, density functional theory

In t r o d u c t i o n solubility to be partially retained in the mantle during partial The noble gases and their isotopes are important tracers in melting. In another recent study, Parman et al. (2005) found that mantle geochemistry, providing unique insight into the structure He is less incompatible than U and Th during mantle melting and 3 4 of the deep Earth, its formation and differentiation, and the style that, contrary to the usual interpretation, high He/ He ratios in and efficiency of mantle convection (e.g., Allègre et al. 1987, basalts may be derived from depleted rather than enriched mantle 1996; Ozima and Igarashi 2000; Parman 2007). Noble gases sources. Other recent experimental studies on the solubility of are commonly considered to behave as perfectly incompatible argon in mantle minerals (Watson et al. 2007; Thomas et al. 2008) elements during partial melting in Earth’s upper mantle, enter- indicate that it behaves as a compatible element during mantle ing the melt en masse as soon as melt forms, then degassing melting, with a solid/melt partition coefficient >>1. These results to the atmosphere during ascent and eruption. Melting is thus imply that Ar would be retained in the mantle despite extensive often approximated as a perfectly efficient process for remov- partial melting, and that other mechanisms may be necessary to 40 ing noble gases from the mantle, with noble gases completely explain the accumulation of Ar in Earth’s atmosphere. extracted from regions of the mantle that have undergone partial The change with pressure in the solubility of noble gases in melting. silicate melts is also incompletely understood. It is well estab- This simple picture of noble gas behavior during mantle melt- lished that the solubility of noble gases increases with pressure ing is increasingly difficult to reconcile with experimental data up to ~5 GPa (White et al. 1989; Montana et al. 1993; Chamorro- on the solubility of noble gases in mantle minerals, and on their Pérez et al. 1996, 1998; Schmidt and Keppler 2002; Bouhifd partitioning between minerals and melts. The experimental data and Jephcoat 2006). At higher pressures the noble gas solubility themselves vary over a wide range among different studies and may plateau (Schmidt and Keppler 2002; Chamorro-Perez et al. have profoundly different implications for the extraction of noble 1996) and/or may drop abruptly at pressures between 5 and 18 gases from the mantle. Some recent studies (e.g., Brooker et al. GPa (Chamorro-Pérez et al. 1996, 1998; Bouhifd and Jephcoat 2003; Heber et al. 2007) have found that noble gases are indeed 2006). The discontinuous drop in solubility at high pressures is highly incompatible in mantle minerals, but still have sufficient not well understood from a mechanical perspective. Sarda and Guillot (2005) were able to reproduce the observed variation in Ar solubility with pressure up to ~5 GPa, in a range of melts, * E-mail: [email protected] using an empirical hard sphere model for the melt. Their model

0003-004X/09/0004–600$05.00/DOI: 10.2138/am.2009.3086 600 Zhang et al.: Effective radii of noble gas atoms 601 was unable to explain a precipitous decline in solubility with Si m u l at i o n d e ta i l s pressure, and these authors suggested that an abrupt change in First-principles molecular dynamics (FPMD) simulations the melt structure, not described by the hard sphere model, may are used to study the structures of liquid silica systems with dis- be necessary to explain such behavior. solved noble gas atoms. The simulations are carried out with the It is clear that to interpret accurately the constraints provided VASP computer program (Kresse and Furthmüller 1996a, 1996b, by noble gas geochemistry, it is necessary to have a better under- 2008; Kresse and Hafner 1993, 1994a). In these simulations, standing of how noble gas atoms are dissolved in mantle minerals the classical dynamics of the nuclei are determined, based on and melts. The sites that noble gas atoms occupy in minerals are forces obtained with a density functional theory solution of the very poorly known, and there is a lack of reliable information on electronic structure. We emphasize that the density functional basic properties of the noble gas atoms themselves, especially calculations of the interatomic forces are not exact, due to the their radii and how they change with pressure and the local struc- limited basis set, the use of approximate pseudopotentials to tural environment. The solubility of an element in a mineral is represent the core electrons, and the approximate nature of the a strong function of the radius of the atom or ion relative to the density functional. ideal radius of the crystallographic site it occupies. Blundy and To connect our results with the recent FPMD results for pure Wood (1994) presented a model that quantifies this dependence, silica liquids of Karki et al. (2007), the same methodology for and the model has been widely applied to the partitioning of trace the electronic structure calculation is used. In particular, the local elements between minerals and silicate melts, including the noble density approximation (LDA) functional (Perdew and Zunger gases (Wood and Blundy 2001; Brooker et al. 2003). The solubil- 1981) is used with a plane wave cutoff of 400 eV and gamma ity of noble gases in melts is also considered to depend primarily point sampling, and ultra-soft pseudopotentials (Vanderbilt 1990; on the sizes of the atoms and on the availability of spaces in the Kresse and Hafner 1994b) are used where available. While these melt structure large enough to accommodate them (Lux 1987; pseudopotentials are available for Si, O, Ne, Kr, and Xe, they Carroll and Stolper 1993; Guillot and Guissani 1996; Sarda and are not available for He and Ar, and so the projector-augmented Guillot 2005). While the radii of most trace elements are well wave pseudopotentials (Kresse and Joubert 1999; Blöchl 1994) known based on crystallographic data (Shannon 1976), the radii are used for these atoms; tests on Ne with the two types of pseu- of neutral noble gas atoms are much less well established. To our dopotentials showed that the results are insensitive to the type knowledge there exists only one experimental measurement of of pseudopotential used. the radius of a noble gas atom dissolved in a condensed phase, The simulations are performed in the canonical ensemble, based on EXAFS data for Kr dissolved in vitreous silica (Wulf such that the number of atoms in the system, the temperature et al. 1999). Zhang and Xu (1995) provided estimates of neutral (T), and the volume (V) are kept constant. The systems studied noble gas radii by interpolating (or extrapolating) to zero charge include 32 Si atoms, 64 O atoms, and one noble gas atom; these the charge vs. radius data for cations and anions with identical compositions correspond to mass percentages of the noble gas electronic structures. The basis for these estimates is reasonable, species ranging from 0.2% for He to 6% for Xe. A time step of 1 but data on the radii of noble gas atoms, and on their unknown fs is used in the simulation, and the results reported here are based dependences on pressure and local structural environment, are on simulations with durations of 24–28 ps. Most simulations are needed. carried out at T = 4000 K, which is near the lowest temperature Here, we present noble gas radii based on ab initio molecular for which the systems reliably equilibrate in the duration of the dynamics simulations of noble gases dissolved in liquid and simulation. The simulations are carried out in cubic simulation crystalline silica (SiO2). Previous molecular simulations of noble cells, with periodic boundary conditions used to remove surface gases dissolved in silicate melts and minerals (Guillot and Guis- effects. The volumes range from V = 27.57 to 13.79 cm3/mol; the sani 1996; Du et al. 2008) have relied on empirical interatomic FPMD results of Karki et al. (2007) show that this volume range potentials. These simulations provide essential insights, but are corresponds to pressures ranging from 0 to 50 GPa (note that the limited by the sparse data available to calibrate interatomic po- pressures in the present systems will be slightly higher due to the tentials involving noble gas atoms. This limitation makes them addition of the rare gas atom to the same volume). unsuitable, in particular, for determining the radii of noble gas atoms (i.e., the radii of the noble gas atoms are in essence empiri- Re s u lt s cal inputs to the simulation, not results of the simulation). Our simulations avoid this issue by utilizing non-empirical potentials Radial distribution functions based on density functional theory. The methods have been The local structure in liquid systems can be characterized well established to provide accurate results for a wide range of by the radial distribution function, g(r), which represents the materials, but have not been applied previously to noble gases relative probability of finding an atom at the distance r from a dissolved in condensed phases. Based on the results, we derive given atom. The values of g(r) go to zero at small r due to the atomic radii for He, Ne, Ar, Kr, and Xe and evaluate their changes steric repulsion between atoms, and the values of g(r) go to 1 with pressure and local structural environment. The noble gases for large r (in disordered systems) indicating the absence of cor- dissolved in liquid silica are found to be far more compressible relation in the positions of atoms at large r. Peaks and valleys than silicon and oxygen, and cannot be considered to have fixed in g(r) indicate increased and decreased probabilities of finding radii under mantle conditions. The effective size of a noble gas atoms at that distance r. atom depends significantly on pressure, and also on the type of The local environment surrounding a noble gas atom in liquid site it occupies. silica can be understood by comparing the radial distribution 602 Zhang et al.: Effective radii of noble gas atoms functions for the interactions X-Si and X-O, where X represents fit well the sharp increase of g(r) at small r, the fitting is carried a noble gas atom. Figure 1 shows these radial distribution func- out only for values of g(r) > 0.8. The uncertainty in determining tions for X = Ar. Note that the initial increase of g(r) occurs at these positions is approximately 0.05 Å at larger volumes, and smaller r for O than for Si, which indicates that the Ar atom has 0.01 Å at smaller volumes [this uncertainty is estimated by the O atoms as its nearest neighbors. This result is generally valid variability in the results obtained by fitting to different ranges for the other rare gas atoms as well, and is in agreement with of g(r)]. the results of previous molecular dynamics simulations based The coordination numbers for the silicon and noble gas atoms on empirical interatomic forces (Angell et al. 1988; Guillot and are shown in Figure 4. For silicon, the results are similar to those Guissani 1996). of previous studies on the pure liquid silica system (Karki et al. The dependence of the X-O radial distribution functions on 2007), indicating that the presence of the noble gas atom has little the type of noble gas atom is shown in Figure 2. As the atomic effect on silicon coordination. For decreases in volume <20%, number of the noble gas atom increases, the initial increase of g(r) shifts to larger r; i.e., the steric repulsion of the atom 2 2.5 increases as its atomic number increases because of the larger 1.5 He V/V =1 2 V/V =0.5 0 1.5 0 1 number of electrons. 1 The radial distribution functions change significantly with 0.5 0.5 0 0 changing volume. As an example, the effects of volume on 2 2.5 2 the g(r) results for Ar-O are shown in Figure 3. As the volume 1.5 Ne V/V0=1 V/V0=0.5 1.5 1 decreases, the radial distribution function develops a much 1 more pronounced first peak and valley, both of which move to 0.5 0.5 0 0 smaller values of r. This result indicates that with decreasing 2 2.5 volume the O atoms surround the argon atom more closely and 1.5 Ar V/V0=1 2 V/V0=0.5 1.5 in a more ordered way. 1 g (r) 1 0.5 0.5 Coordination number 0 0 2 2.5 The coordination number is the number of nearest neighbor 1.5 Kr V/V =1 2 V/V =0.5 0 1.5 0 1 atoms that surround a central atom. The nearest neighbors are 1 represented by the first peak of the radial distribution function, 0.5 0.5 0 0 the upper limit of which is usually defined to be the position of 2 2.5 the first valley. The coordination number can be obtained by 1.5 Xe V/V =1 2 V/V =0.5 0 1.5 0 1 integrating the area under the first peak of the g(r); i.e., integrat- 1 ing g(r) from 0 to the position of the first valley. We determined 0.5 0.5 0 0 the coordination numbers for the silicon and noble gas atoms 01234 5 01234 in this way, using the radial distribution functions for Si-O and r (Å) X-O, respectively. Fi g u r e 2. g(r) for a noble gas atom and oxygen in liquid silica: He- Numerical noise obscures the precise positions of the first O, Ne-O, Ar-O, Kr-O, and Xe-O at T = 4000 K and V = 27.57 cm3/mol valley in the g(r). To unambiguously estimate these positions, (left panels) and 13.79 cm3/mol (right panels). the g(r) results are fitted to a sixth-order polynomial, from which the positions can be determined. Since the polynomial would not 2.5

2 2

1.5 1.5 g (r)

1 1 g (r)

0.5 0.5

0 2 2.5 3 3.5 4 r (Å) 0 01234 5 Fi g u r e 3. Volume dependence of g(r) for Ar and O in liquid silica r (Å) at T = 4000 K: V = 27.57 cm3/mol (solid line), V = 26.19 cm3/mol (long Fi g u r e 1. Comparison of g(r) for Ar-O (solid line) and Ar-Si dashed line), V = 22.06 cm3/mol (dot-dashed line), V = 19.30 cm3/mol (dashed line) in liquid silica at T = 4000 K and V = 27.57 cm3/mol (dash-dot-dashed line), V = 16.54 cm3/mol (short dashed line), and V = (ambient pressure). 13.79 cm3/mol (dotted line). Zhang et al.: Effective radii of noble gas atoms 603 the silicon coordination number remains essentially constant at “atomic size” are continuous. In this paper, we define two types 4, but for greater decreases in volume the coordination number of effective atomic radii in terms of the potential of mean force. increases toward a value of 6. The potential of mean force w(r), which is related to the radial The coordination of the rare gas atoms exhibits very different distribution function by w(r) = –kTlng(r) (Chandler 1987), rep- behavior. First, these coordination numbers range from 5 to 17, resents an effective interatomic potential that accounts for the and are thus usually much larger than those for silicon; a snap- effects of all other atoms in the system. The two types of effective shot of the local environment of a noble gas atom (Xe), which radii we define are an “equilibrium radius” Re that corresponds shows this large coordination, is shown in Figure 5. Second, for to the optimal atomic separations, and a “repulsive wall” radius decreases in volume <20%, the coordination number increases σ that corresponds to the distance at which the potential of mean with decreasing volume, but for greater decreases in volume the force becomes repulsive. The relationships of these two types of coordination number remains essentially constant; this volume radii to the potential of mean force are shown in Figure 6. These dependence is opposite to that found for silicon. two types of radii convey different information: Re is related to Thus there are two distinct regimes of compression. In the the interatomic distance in situations where the noble gas atom first regime (volume decreases <20%), the coordination number environment is fully relaxed (e.g., in a melt), and σ is related to of the silicon remains constant but the coordination number of the interatomic distance in situations where the noble gas atom the noble gas atom increases. In the second regime (volume environment is highly constrained (e.g., in an interstitial site of decreases >20%), the coordination number of silicon increases a crystal). Again, we stress that these effective radii are param- but the coordination number of the noble gas atom remains eters that partially characterize the spatially continuous atomic constant. interactions; they serve as approximate, rather than rigorous, predictors of interatomic distances (rigorous predictions require Effective radii of noble gas atoms detailed simulations such as carried out here).

Any definition of the radius of an atom is essentially arbitrary, The value of Re is based on the position of the first peak of the since the electron distributions that are the physical basis of any radial distribution function for atoms that are nearest neighbors; the position of this peak represents the most probable distance of separation, and so this definition is such that the sum of the R 20 e a of two atoms is their most probable nearest neighbor separation.

X-O The Re of an O atom is obtained as half the distance of the first peak of the O-O radial distribution function (note that O atoms in 15 silica are much larger than Si atoms, and so O atoms are nearest

neighbors in silica). The Re of the noble gas atom is obtained as the distance of the first peak of the X-O radial distribution 10

r function minus the value of Re for the O atom. The value of σ is based on the position in which the radial distribution function passes through the value of 1; since the 5 potential of mean force w(r) is related to the radial distribution function by w(r) = –kTlng(r) (Chandler 1987), this definition 14 16 18 20 22 24 26 28 is such that the sum of the σ of two atoms is the separation at 6 5.5 Si-O coordination numbe 5 4.5 4 3.5 b 3 14 16 18 20 22 24 26 28 3 molar volume (cm /mol)

Fi g u r e 4. Coordination numbers in liquid silica at T = 4000 K, for (a) noble gas atoms; (b) Si atoms. Results are for liquid silica systems with one atom of: He (circles), Ne (squares), Ar (diamonds), Kr (up-pointing Fi g u r e 5. Snapshot of the local environment of a Xe atom in liquid triangles), or Xe (left-pointing triangles). Uncertainties were estimated silica at 4000 K and V = 13.79 cm3/mol, obtained using the VMD program by calculating the average value of CN for separate time segments, each (Humphrey et al. 1996, 2008). The Xe atom is at the center, and the ~4 ps in duration. The error bars show ±2 standard deviations of these surrounding spheres are O atoms. The 16 O atoms are within the distance average values. of the first valley of the radial distribution function. 604 ZHANG ET Al.: EFFECTIvE RADII OF NOBlE GAS ATOMS

a 2.2

2

1.8

1.6 (Å ) e

R 1.4

1.2

1

0.8

b 2.2

Fi g u r e 6. Schematic showing the relationship of the effective radii 2 σ and Re to the potential of mean force. 1.8

1.6 which the potential of mean force becomes repulsive. These 1.4 values are obtained from the O-O and X-O radial distribution (Å ) σ functions in a way analogous to the method for obtaining Re, as 1.2 described above. The results for the effective radii of the noble gas atoms in 1 silica are shown in Figure 7. Note that values of Re for He and 0.8 σ for He and Ne at V = 27.57 cm3/mol are not reported because 0.6 numerical noise coupled with relatively flat g(r) curves render 14 16 18 20 22 24 26 28 their values unreliable. The effective radii decrease with de- 3 creasing volume (as also evident in Fig. 3). The oxygen radii molar volume (cm /mol) also decrease, but over a much smaller range (e.g., the radii for oxygen range are R = 1.31 Å and σ = 1.14 Å at V = 27.57 cm3/ Fi g u r e 7. Equilibrium radii Re (a) and repulsive radii σ (b) of noble e gas atoms in liquid silica at 4000 K, as a function of the system volume; mol, and R = 1.19 Å and σ = 1.04 Å at V = 13.79 cm3/mol). e He (circles), Ne (squares), Ar (diamonds), Kr (up-pointing triangles), The results for the noble gas radii as a function of volume are Xe (left-pointing triangles). correlated with the following equations:

Re(V) = Re,0 + R′e (V/V0 – 1) (1) vite, i.e., space group P42/mnm, (Baur and Kahn 1971), and the

σ(V) = σ0 + σ′ (V/V0 – 1) (2) simulation cell dimensions appropriate for stishovite at ambient conditions (V = 14.05 cm3/mol); these systems correspond to 2 where Re,0 and σ0 are the effective noble gas radii at V = V0 = × 2 × 4 unit cells of stishovite. 3 27.57 cm /mol [V0 is the volume at ambient pressure (Karki et The single noble gas atom is inserted into the largest inter- al. 2007)], while R′e and σ′ are parameters describing how the stitial site (Wyckoff special position c) in the structure. This effective noble gas radii change with volume. The simulation interstitial site has two neighboring O atoms at a distance of results are fitted to Equations 1 and 2, and the fitted values for 1.51 Å, and two neighboring Si atoms at a distance of 2.09 Å;

Re,0, σ0, R′e, and σ′ are given in Table 1. all other atoms are at distances >2.8 Å. Thus, the insertion of the noble gas atom will cause local distortion of the stishovite Results for a noble gas atom in stishovite structure, which is of course dependent on the type of noble The relevance of the melt results to crystalline systems is gas atom. Energy minimizations are carried out to determine examined by carrying out simulations of noble gas atoms in the structures that minimize the energy of the system (note that interstitial sites of stishovite, which is the stable crystal phase these are not molecular dynamics simulations, as were carried of silica at high pressure. Stishovite was chosen for two reasons. First, since stishovite is a very dense phase, it provides a stringent TABLE 1. Noble gas equilibrium and repulsive radii parameters (see test for ideas concerning the incorporation of interstitial atoms. Eqs. 1 and 2)

Second, the high symmetry of the stishovite structure allows a Re,0 (Å) R′e (Å) σ0 (Å) σ′ (Å) straightforward analysis of the deformation of the crystal result- He 1.32 1.12 1.14 1.02 ing from the incorporation of the noble gas atom. Ne 1.50 1.00 1.27 0.82 Ar 1.95 1.42 1.73 1.33 Simulations are carried out with the silicon and O atoms Kr 2.00 1.28 1.82 1.23 initially in the appropriate crystallographic positions for stisho- Xe 2.22 1.49 2.04 1.52 ZHANG ET Al.: EFFECTIvE RADII OF NOBlE GAS ATOMS 605 out for the melt systems). is more significant for the larger noble gas atoms because these The results for the smallest X-O distances are shown in Figure atoms require larger lattice distortions. 8. For all noble gas atoms, there is some local distortion of the crystal lattice in the neighborhood of the occupied interstitial di s c u s s i O n site—i.e., the X-O distance is greater than the value 1.51 Å that would occur in the absence of the noble gas atom. As expected, Comparison with experiment the magnitude of the lattice distortion increases as the atomic We are aware of one experimental investigation of the number of the noble gas atom increases. In all cases, the integrity structural environment of noble gas atoms in liquid silicate of the crystal structure is maintained, and the lattice distortions systems (Wulf et al. 1999). In this study, Kr was dissolved in a are relatively minor, localized, and preserve the symmetry. supercooled silica melt at 1473 K and 0.7 GPa; after allowing Figure 8 shows that these smallest X-O distances in stishovite the system to equilibrate, the supercooled liquid was quenched are almost exactly equal to the sum of the repulsive wall radii to form a glass, and X-ray absorption experiments were used to for the noble gas atom and oxygen obtained from the simula- address the structural environment of Kr in the glass. From these tions on liquid silica. Recall that the “repulsive wall” radius σ experiments, the average Kr-O nearest neighbor separation was corresponds to the distance at which the interatomic potential determined to be 3.45 ± 0.1 Å. becomes repulsive; therefore, there is a large energy penalty for To compare with this experimental result, the average Kr-O

X-O distances <(σX + σO). The results in Figure 8 demonstrate nearest neighbor separation is determined from the present that σ indeed determines the interatomic distance in situations simulation results [note that the average separation differs in where the noble gas atom environment is highly constrained general from the position of the peak of g(r), and is obtained (e.g., in an interstitial site of a crystal). We note that the near- by integration of g(r) from r = 0 up to the distance of the first exact agreement between the smallest X-O distances and σX valley of g(r)]. The present result for the average Kr-O nearest

+ σO is fortuitous—as pointed out above, the values of σ are neighbor separation is 3.50 Å, which is in good agreement with measures of the distance where repulsive interactions begin to the experimental result. dominate, but since the interatomic potentials are continuous the onset of interatomic repulsion occurs gradually over a range Coordination numbers of the noble gas atoms of distances. The coordination number results for the noble gas atoms can As evident in Figure 8, for the smaller atoms the smallest X-O be understood in terms of the packing of “hard” (impenetrable) distances are greater than σX + σO, while for the larger atoms the spheres. More specifically, we address the following question: opposite is true. This result is understandable because the local if the noble gas atom is modeled as a hard sphere of radius R1, distortion of the lattice surrounding the rare gas atom, which and the O atoms are modeled as hard spheres of radius R2, what is required in order for the smallest X-O distances to become is the largest number of O atoms that can be in contact with the comparable to σX + σO, introduces a lattice energy penalty that noble gas atom? The problem of the maximum number of hard increases strongly with the magnitude of the lattice distortion. spheres of radius R2 that can pack around a hard sphere of radius

This energy penalty, which acts to reduce the lattice distortions, R1 has been solved (Clare and Kepert 1986; Kottwitz 1991). Note that this problem is equivalent to the problem of how many

two-dimensional circles of radius R2 can pack on the surface

of a sphere of radius R1 + R2. The solutions to this problem are shown as the solid line in Figure 9 as a function of the ratio of the diameter of the circles relative to the radius of the sphere. This theoretical picture is used to address the packing of O atoms around the noble gas atom. The diameter of the two-

dimensional circle (Dcircle) corresponds to the diameter of the O atom, obtained as the distance of the first peak of the O-O radial

distribution function. The radius of the central sphere (Rsphere) cor- responds to the sum of the radii of the noble gas atom and the O atom, obtained as the distance of the first peak of the X-O radial distribution function. Figure 9 shows the coordination number results obtained in the simulations (as described in the Results

section) as a function of the ratio Dcircle/Rsphere, where Dcircle and

Rsphere are obtained as described in this paragraph. Figure 9 shows that, at smaller volumes (volume decreases greater than 20%), the simulation results are very similar to the

hard-sphere closest-packing results; i.e., the packing of O atoms around the noble gas atom is well described as closest packing. Fi g u r e 8. Relationship of closest X-O distance in stishovite to the sum of the repulsive wall radii for X and O determined from the In contrast, at larger volumes (volume decreases <20%) this is liquid silica simulations. These repulsive wall radii were obtained not the case; there are fewer O atoms surrounding the noble gas from Equation 2 using the molar volume appropriate for stishovite (V atom than expected in the case of close packing. = 14.05 cm3/mol). The approach toward closest packing can explain the origin 606 Zhang et al.: Effective radii of noble gas atoms of the two regimes of compression evident in Figure 4. Compres- make the present results more reliable. sion of the system at larger volumes causes more O atoms to be The ambient pressure values of Re determined here are very “pushed” near the noble gas atom; in this regime the coordination close to the spherical noble gas radii determined from the lat- number increases with compression. After sufficient compres- tice parameters for fcc Ne, Ar, Kr, and Xe crystals at 4–58 K sion there is no more available space around the noble gas atom (Wyckoff 1963). In each case the agreement is better than 5%, because O atoms have become close packed around the noble gas and the noble gas crystal radii are not systematically larger or atom; in this regime the coordination number remains constant smaller than the equilibrium radii determined here. However, as the system is compressed. at higher pressures (Jephcoat 1998; Dewaele et al. 2008) the radii of noble gases in the pure fcc crystals are systematically Comparison with previous estimates of rare gas atom size larger than the equilibrium radii determined here. The noble Our results for the effective radii of the noble gas atoms in gas crystal radii are ~10% larger for Ar, Kr, and Xe at 50 GPa liquid silica at 4000 K, and a volume corresponding to ambient (corresponding to V/V0 ~ 0.5 in the present study). At first glance pressure, are shown in Table 1. Zhang and Xu estimated radii of this seems surprising, because the noble gas radii scale so well the noble gas atoms dissolved in other materials from interpola- with volume, and the noble crystals are more compressible than tion or extrapolation of the radius-charge relations for isoelec- liquid silica (Jephcoat 1998; Karki et al. 2007). However, in tronic ions in crystals (Zhang and Xu 1995). Their estimated contrast to the noble gas crystals, compression of liquid silica radii are significantly smaller than the values of Re found here, is accomplished by a collapse of the melt structure, with very being on average 17% smaller (note that their radii are based on little compression of the Si and O atoms themselves. Because the equilibrium bond distances of crystals, and are thus analogous the noble gases occupy spaces in the melt framework rather to our “equilibrium radii,” Re, rather than our “repulsive wall” than Si or O sites, it is the compressibility of the “free space” radii, σ). A factor that leads to this discrepancy is that the crystal within the melt structure, or “ionic porosity” (Carroll and Stolper structures used to derive these effective radii are composed of 1993), that is expected to be relevant. This “free space” in the ions with coordination numbers ranging from 4 to 8; however, liquid silica (i.e., the volume that is not occupied by Si and O the coordination numbers of the noble gas atoms in liquids are atoms) is more compressible than any of the noble gas crystals, (in almost all cases) significantly larger. and closely matches the volume compression of the noble gas Guillot and Guissani (1996) determined X-O radial distri- atoms dissolved in liquid silica (Fig. 10). butions in silica using molecular dynamics simulations with empirical interatomic potentials. The distances they obtain for Comments on the variation in noble gas solubility the first peaks of the X-O radial distribution function, at 4000 K with pressure and a density 2.2 g/cm3, are significantly larger than found here: There have been several reports of a sharp decrease in noble 10% larger for He, and 25% larger for Ar. The first principles gas solubility with pressure in silicate melts (Chamorro-Pérez et methodology used here for the interatomic forces is expected to al. 1996, 1998; Bouhifd and Jephcoat 2006). In liquid silica, the data for Ar show a continuous increase in solubility up to 5 GPa, then a sharp decrease in the solubility, one order of magnitude

25 over the pressure interval from 5 to 6 GPa (Chamorro-Pérez et al. 1996). There is some controversy over whether this drop in solubility is real. The results of Schmidt and Keppler (2002) for a 20 fully polymerized haplogranitic melt are consistent with those of

r Chamorro-Pérez et al. (1996) up to 5 GPa, but beyond this pres-

15 sure the solubility remains approximately constant up to at least 8 GPa. Bouhifd and Jephcoat (2006) suggested that the Al content of the melt controls the pressure at which the drop in solubility 10 occurs, and that this may explain the apparent discrepancy in the

coordination numbe results for polymerized melts. Sarda and Guillot (2005) modeled

5 the solubility of noble gases, using a simple hard sphere model for the noble gas and for the silicate melt. The results were found to provide a good description for the solubility as a function of 0 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 pressure for a range of silicate melts at relatively low pressures, D / R circle sphere including a plateau in the solubility in the ~5–10 GPa range, but could not explain a sharp drop in solubility with pressure. Fi g u r e 9. Dependence of the coordination number for noble gas Sarda and Guillot (2005) suggested that such a change might be atoms in liquid silica on the parameters relevant to the packing of circles accounted for by an abrupt change in the melt structure that was on a sphere. The line represents the theoretical closest-packing results for not accounted for in the hard sphere model. hard spheres (Clare and Kepert 1986; Kottwitz 1991), and the symbols are simulation results at T = 4000 K and the following volumes: V = The solubility of noble gases in silicate melts is strongly cor- 27.57 cm3/mol (circles), V = 26.19 cm3/mol (squares), V = 22.06 cm3/ related with the size of the noble gas atom and the openness of mol (diamonds), V = 19.30 cm3/mol (up-pointing triangles), V = 16.54 the melt structure, in particular the ionic porosity (Carroll and cm3/mol (left-pointing triangles), and V = 13.79 cm3/mol (down-pointing Stolper 1993). The radii of the noble gases vary linearly with the triangles). molar volume of the melt (Fig. 7), and with the ionic porosity, Zhang et al.: Effective radii of noble gas atoms 607

pressure range. The much higher compressibility of the noble gas 1 atoms in comparison to the other components in silicates means 0.9 that more care needs to be taken in choosing the radii that are

0.8 relevant to the particular problem being considered. Different definitions of the effective radii are necessary depending on the 0.7 3 ) extent to which the surrounding atoms can relax in response to e, 0

/R 0.6 e the presence of a noble gas atom. In a melt, the atoms can move (R 0.5 to accommodate the noble gas atom to the same extent that they accommodate all other atoms in the melt, and so the radius of 0.4 the noble gas atom is determined by the distance for the most 0.3 favorable interactions with its neighbors, and the radius of inter-

0.2 est is the equilibrium radius given in Table 1. In contrast, the

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 long-range lattice structure of a crystal tethers the atoms to lattice V /V free free,0 sites, with energy penalties for displacements from these sites. As a result, the radius of a noble gas atom in a highly constrained Fi g u r e 10. Compression of noble gas atoms vs. compression of the lattice site is determined by the distance where the interaction free space in liquid silica. The cube of the equilibrium radii, relative to with its neighbors becomes repulsive. The repulsive radii given the radii at ambient pressure, are plotted against the “free” volume of the melt (i.e., the volume unoccupied by silicon and O atoms) relative in Table 1, or the radii estimated by Zhang and Xu (1995) for to the free volume at ambient pressure. Symbols are plotted as: He neutral noble gas atoms in particular coordination environments, (circles), Ne (squares), Ar (diamonds), Kr (up-pointing triangles), and are more relevant to the solubility of noble gas atoms in minerals, Xe (left-pointing triangles). and to the prediction of mineral/melt partition coefficients.

Ac k n o w l e d g m e n t s This material is based upon work supported by the National Science Founda- calculated as (1 – V /V ), where V is the volume occupied by tion under grant number EAR-0635820, and a grant of computing time from the ca m ca Ohio Supercomputer Center. The constructive comments of Frank Spera and an silicon and O atoms, and Vm is the melt volume (Fig. 10). These anonymous reviewer helped improve the manuscript. parameters vary continuously, rather than abruptly, with melt volume (and thus with pressure). As discussed above, the local Re f e r e n c e s s i t e d environment surrounding the noble gas atom evolves to a more Allègre, C.J., Staudacher, T., and Sarda, P. (1987) Rare gas systematics: Formation of the atmosphere, evolution and structure of the Earths mantle. Earth and densely packed structure as the melt volume decreases, with Planetary Science Letters, 81, 127–150. closest packing achieved after ~20% compression. 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