SUPPLEMENTAL INFORMATION
1. 36S measurements
We report here the
36S/32S
ratio (i.e., 36S values) measured in our
experimental run products (Table SI1). The external error for an individual analysis
of 36S averages 0.042 ‰ (1 standard error). Analyses of interlaboratory standards
and common natural sulfur-bearing minerals in the IPGP laboratory yield a
‘terrestrial fractionation line’ in a plot of 36S* vs. 34S* having a slop of 1.9004. This
compares well to a canonical value of 1.90 (e.g., Otake et al., 2008).
The procedural blanks (Experiments 1 and 2) yielded values of 36S that are
comparable to the DD-1 starting material gas, though at the upper and lower limit of
the observed range. We conclude on this basis that fractionations of 36S associated
with transferring and storing SF6 in our experimental apparatus are detectable but a
small multiple of analytical precision.
36S
is by far the lowest abundance stable S
isotope (~ 0.02 %, relative) and is relatively susceptible to isobaric interferences
from trace contaminants (including species, such as C3F5, that are generated in the
ion source); for this reason, we find it unsurprising that our experimental
procedures generate variations in measured 36S on the order of a few tenths of per
mil.
Our shortest duration ice synthesis experiment (Experiment 4) yielded a
significant fractionation of 36S, with vapor 1.20 ‰ higher than ice, despite
negligible fractionations of 33S and 34S. It is imaginable that this represents an
actual mass-independent fractionation, but this seems unlikely in light of the
subsequent gradual growth of the fractionation in pace with equilibration of 33S
and 34S (Tables 1, SI1). It would be reasonable to suspect that the anomalous
result for 36S reflects the presence of a trace contaminant that forms an isobaric
interference with the
36SF5+
signal detected in the mass spectrometer (131 AMU);
this could occur if the contaminant were a high-vapor-pressure species that is
quantitatively retained in the gas phase at the temperature of this experiment (150
K).
131Xe
is a reasonable candidate. The vapor pressure of Xe at ~ 150 K is
hundreds of millibars, and a contaminant of only 1 ppm in the SF6 vapor fraction
would explain this result, so this is a physically plausible scenario (though it is
surprising that this species could be retained through freeze/thaw cycles involved
in transferring analyte to the mass spectrometer bellows). If so, we should expect
that a ~ 1 ‰ apparent enrichment in 36S should be accompanied by enrichments
of ~ 0.002 and 0.006 ‰ in 33S and 34S, respectively, based on natural abundances
of
128Xe
and
129Xe
and relative abundances of the S isotopes. These are negligible
values relative to our analytical and experimental errors and consistent with the
lack of observed fractionation of these isotope ratios during Experiment 4.
Alternatively, it is imaginable that our
36S
results are compromised by an organic
contaminant (or species made in the ion source) that makes an interference at 131
AMU.
The weighted average of run products (vapor and condensate) for ice
synthesis experiments yielded an average 36SCDT value of -0.288±0.092. This is
subtly (~ 0.2 ‰) enriched relative to the starting composition. We think that this
difference likely reflects subtle, consistent increases due to trace contaminants in
most or all experiments. However, it is also possible that this difference reflects
removal of a contaminant from the starting gas due to exposure to cold traps and
sorption substrates; i.e., it is possible that the run products are the more accurate
results. Nevertheless, we have less confidence in our 36S results than in our results
for 33S and 34S.
We suggest that a confident study of the
36S
systematics associated with
subtle fractionations, such as the vapor pressure isotope effect experiments
presented here, may have to await the development of a class of gas source isotope
ratio mass spectrometers capable of mass resolving
36S-bearing
analytes from
contaminants while retaining the ~0.1 ‰ precision needed for useful
measurements.
2.1. Density-functional and Lattice-dynamics models of the vapor pressure isotope
effect of SF6
In this section we present our efforts model the vapor pressure isotope effect
of SF6 ice using density function theory and lattice dynamics models. These models
failed to quantitatively reproduce our experimental data; we suspect this failure
largely stems from the inadequacy of our lattice-dynamics model of the solid
phonon spectrum. Nevertheless, this work may provide a helpful starting point for
future research on this problem and so we present it in some detail.
It is relatively straightforward to estimate the reduced partition function
ratio of SF6 vapor using estimates of its fundamental modes of vibration, taken
either from spectroscopic measurement or ab initio models, combined with UreyBiegeleisen theory of isotope effects on harmonic molecular vibrations:
eqn. SI1
where m and m’ are the masses of two isotopes of interest (e.g.,
32S
and
33S,
respectively),  and ’ are the symmetry numbers of the isotopologues of interest
(e.g.,
32SF6
and
33SF6,
respectively), i denotes the vibration frequency i (where i
corresponds to 1 of the 15—including degenerate modes—intramolecular
vibrational frequencies for SF6, Ui = hi/kbT, h is the Plank constant, kb is the
Boltzmann constant and T is temperature; Bigeleisen and Mayer, 1947).
Frequencies of vibration for all fundamental modes of all isotopic versions of
vapor were calculated by density functional theory (DFT), using a coupled cluster
singles and doubles (ccsd) technique with a correlation-consistent polarized double
zeta (cc-pVDZ) basis set. The structure and vibrational frequencies were optimized
in the gas phase as a single molecule; a separate optimization was performed for
each individual isotopologue. Our calculated fundamental mode frequencies for
32SF6
(Table SI2) closely resemble those of previous, generally similar theoretical
studies (Computational Chemistry Comparison and Benchmark Database, 2011).
The model satisfies the Redlich-Teller spectroscopic theorem; i.e., it is selfconsistent with respect to isotope effects on vibration frequencies.
The fundamental modes of vibration predicted by such models (including
ours) disagree with JANAF table compilations of spectroscopic data (Chase et al.,
1985) by approximately ±5 %, relative (compare Table 2 of the main text with Table
SI2). We have adopted calculated vibrational frequencies rather than using
spectroscopic constraints on fundamental mode frequencies because of the
importance to our study of self-consistent treatments of the frequency shifts of each
mode on isotopic substitution. Spectroscopic measurements of fundamental modes
for materials with approximately natural isotope abundances reflect frequencies of
the abundant species (i.e., 32SF6). Scaling these frequencies to other isotopic species
(33SF6 and 34SF6) requires knowledge of the reduced masses of these vibrations and
the degree of coupling between fundamental modes. If we treat the reduced mass
for each mode of vibration using simple analytical expressions (i.e., of the form, µ =
mimj/(mi+mj)) and do not consider coupling between modes (see the note at the end
of this section for details), we find frequency shifts on heavy isotope substitution
that disagree by a factor of several with spectroscopic observations and DFT-type
models, particularly for the 4 mode (McDowell and Krohn, 1986; Boudon et al.,
2006; see Table SI3). The DFT model we present better matches those spectroscopic
constraints on frequency shifts. We conclude that the modest errors in fundamental
vibration frequency resulting from DFT modeling are compensated by their better
treatment of the effect of isotopic substitution on the 4 vibration mode. The
partition function ratio of vapor was calculated using Equation SI1, above, summing
over all 15 fundamental modes modeled for each isotopologue of vapor (Table SI2).
At an assumed temperature of 150 K, we found values of
33Q/32Q
and 34Q/34Q (i.e.,
partition function ratios) of 1.187452 and 1.398420, respectively.
It is less straightforward to estimate the reduced partition coefficient ratio of
ice because its phonon spectrum is complex and incompletely known. We
approached this problem by constructing lattice dynamics models of the discrete
fundamental vibration modes and their dependence on isotopic substitution, using
assumed force constants for the S-F and F-F interactions. Our model for SF6 ice used
the GULP model for solids (Gale, 1997). Long-range F-F interactions were treated
using a Lennard Jones potential, following the form and constants presented in Dove
and Pawley (1983); the energy for the F-F potential, ELJ:
where  = 0.006302(kJ/mol) and  = 2.7692(Å). (The Lenard-Jones potential
constants are generally similar among models for low temperature solids and
liquids). The energy associated with the bending of the F-S-F bonds (‘Ethree’) is
treated using a three-body equation:
where  = 6.5598 (eV rad-2) and 0 = /2 radians (McDowell et al., 1976; Olivet and
Vega, 2007). Finally, the energy of harmonic stretching of the fluorine sulfur
interaction (‘Eharmonic’) was modeled with a harmonic potential of the form:
2
where r0 = 1.565 Å and the force constant, k2 = 34.1098 eV/Å2 (McDowell et al.,
1976; Dove and Pawley, 1983; Olivet and Vega, 2007). Atomic positions in the lattice
were taken from Olivet and Vega, 2007, and the model was solved for a structure
having the formula: S3F18. The resulting calculated set of fundamental frequencies is
presented in Table SI4, which can be compared with the experimentally measured
phonon spectrum in the Table 3 of the main text. Note that our lattice-dynamics
model is approximately accurate at the low and high end of the frequency range, but
does a poor job in the middle (i.e., the observed fundamental frequencies in the
range ~450-650 cm-1 and their place in the phonon spectrum appears to be replaced
with a group of frequencies in the range ~250-450 cm-1). The model satisfies the
Redlich-Teller spectroscopic theorem, so this model, while poor, appears to at least
be self-consistent with respect to isotope effects on vibration frequencies. We
explored models in which the force constants for the harmonic and bending mode
were varied by ±20 %, relative, or the Lenard-Jones potential was increased over a
factor of 2. None of these corrected this striking disagreement. It is unclear whether
this reflects an over-simplification of the lattice dynamics model we have used, an
error in the force constants we’ve assumed, or some other computational mistake
on our part. Regardless, this is the most fundamental failing of our model and we
suggest it is the best place for any future effort to begin. We suggest the most useful
future steps might be higher-level theoretical treatments of the forms and constants
for inter-atomic potentials, perhaps supplemented by new spectroscopic studies of
the absorption features of isotopically labeled ice. The partition function ratio of ice
was calculated using Equation SI1, above, summing over all 63 fundamental modes
calculated for each isotopologue of ice. At an assumed temperature of 150 K, we
found values of
33Q/32Q
and
34Q/34Q
for S3F18 ice of 1.455991 and 2.078778,
respectively.
Finally, we calculated the fractionation factors (33 and 34) between ice and
vapor at 150 K using the formula: 
ice-vapor
= (Q’/Q)1/3ice/(Q’/Q)vapor (note that
because the formula of the cell modeled by GULP is S3F18, we have taken the cube
root of its partition function; i.e., the correct formulation of the reaction we have
modeled is: 1/3S3F18(solid) = SF6(vapor)). The resulting values of 33 and 34 are 0.9545
and 0.9126, respectively. This model results in a reversed vapor pressure isotope
effect — i.e., the same direction as that observed in our experiments — but
calculated fractionations are about a factor of ~40 greater than observed. It seems
likely to us that this disagreement can be traced (at least in part) to our model’s
poor description of the phonon spectrum of ice (but might also be due to the over
simplifications of harmonic models of vibrational isotope effects; Liu et al., 2010).
The alternate models we explored (i.e., varying temperature and the various
potentials assumed in our lattice dynamics model) yielded qualitatively similar
results.
The model discussed above does not meaningfully test the hypothesis
discussed in the main text and illustrated in Figure 5, principally because the
calculated fractionation is so large that it cannot exhibit the peculiar mass law
behavior we expect to occur as one approaches the ‘crossover’ from reversed to
normal vapor pressure isotope effects. The most obvious next step to improve this
model is a more accurate description of the vibrational energetics of SF6 ice, with
the goal of matching its known phonon spectrum to within reasonable errors.
2.2. Notes regarding the simple analytical models of isotope effects on 3 and 4 in
vapor
The effect of isotopic substitution on the frequency of vibration () of an
harmonic oscillator can be calculated as a function of the reduced mass (µ), through
the expression: ’/ = (µ/µ’)0.5, where the prime indicates an isotopically substituted
compound (i.e., 34SF6). Analytical expressions for the reduced masses of the 3 and
4 modes of vibration in SF6 vapor are: µ3 = (6x31.972x18.998)/(6x18.998 +
31.972) — i.e., motion of 6 S-F bonds; and µ4 = (3x31.972x18.998)/(3x18.998 +
31.972) — motion of 3 S-F bonds. The resulting values of µ3 and µ4 — 24.969 and
20.482, respectively — are similar to the values of 25.61 and 20.18 calculated by the
DFT model we present above. However, the DFT model calculates changes in 3 and,
particularly, 4 with isotopic substitution that differ markedly from those one
would estimate by treating each mode as an independent harmonic oscillator (Table
SI2). We suspect this reflects the effects of anharmonicity and/or coupling between
modes. The frequency shifts on
34S
substitution predicted by our DFT model are
similar to those observed by spectroscopy, leading us to use this model in our
evaluation of the partition function ratio of SF6 vapor.
Supplemental Information References
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