First-Principles Diffusivity Ratios for Atmospheric Isotope Fractionation on Mars and Titan

Abstract

Recent work used the kinetic theory of molecular gases, along with state-of-the-art intermolecular potentials, to calculate from first principles the diffusivity ratios necessary for modeling kinetic fractionation of water isotopes in air. Here, we extend that work to the Martian atmosphere, employing potential-energy surfaces for the interaction of water with carbon dioxide and with nitrogen. We also derive diffusivity ratios for methane isotopes in the atmosphere of Titan by using a high-quality potential for the methane-nitrogen pair. The Mars calculations cover 100 K to 400 K, while the Titan calculations cover 50 K to 200 K. Surprisingly, the simple hard-sphere theory that is inaccurate for Earth’s atmosphere is in good agreement with the rigorous results for the diffusion of water isotopes in the Martian atmosphere. A modest disagreement with the hard-sphere results is observed for the diffusivity ratio of CH₃D in the atmosphere of Titan. We present temperature-dependent correlations, as well as estimates of uncertainty, for the diffusivity ratios involving HDO, H₂¹⁷O, and H₂¹⁸O in the Martian atmosphere, and for CH₃D and ¹³CH₄ in the atmosphere of Titan, providing for the first time the necessary data to be able to model kinetic isotope fractionation in these environments.

Plain Language Summary

Different isotopes distribute unevenly between the vapor phase and liquid or solid phases during precipitation and evaporation, and the resulting changes in isotope ratios are used in the study of climate and other geophysical processes. While equilibrium aspects of this fractionation are fairly well understood, in some circumstances there is also a kinetic component that depends on the relative diffusivities of different isotopic species in the atmosphere. We used rigorous molecular collision calculations to model this effect for water isotopes in the CO₂-rich Martian atmosphere and for methane isotopes in the nitrogen atmosphere of Titan. For the Martian atmosphere, the results are not significantly different from those obtained by a simple theory that assumes the molecules to be hard spheres; this is surprising since previous work showed that the hard-sphere approach is significantly in error for water in Earth’s atmosphere. For methane in the atmosphere of Titan, a small improvement is obtained for the diffusivity ratio of CH₃D. We provide simple correlations that allow these diffusivity ratios to be used in planetary modeling.

1. Introduction

On Earth, the fractionation of stable water isotopes such as HDO and H₂¹⁸O is important for understanding the hydrologic cycle (Gat, 1996). In addition to the equilibrium fractionation that occurs between water vapor and ice or liquid water, in some cases a significant role is played by kinetic fractionation that depends on the ratio of the atmospheric diffusivity of the isotopologue of interest to that of H₂O. Recently, we reported temperature-dependent diffusivity ratios in air calculated from the kinetic theory of molecular gases applied to state-of-the-art intermolecular potentials based on ab initio calculations for the interaction between water and molecular nitrogen and oxygen (Hellmann & Harvey, 2020). We found that, especially for HDO, the often-assumed constant diffusivity ratios from simple hard-sphere kinetic theory were in error, and that the temperature dependence of the diffusivity ratios was not negligible. We supplied recommended diffusivity ratios and their uncertainties from 190 K to 500 K, greatly exceeding the range in which the (scattered) experimental data exist.

Earth is not the only place in our solar system with a hydrologic cycle. The distribution and seasonal variation of isotopic water species (especially HDO) has been used to study the climate of Mars (Montmessin et al., 2005; Villanueva et al., 2015; Krasnopolsky, 2015; Encrenaz et al., 2016; Vos et al., 2019). Current modeling of isotopic cycles on Mars neglects kinetic fractionation (Montmessin et al., 2005). To our knowledge, no experimental measurements of the relevant diffusivity ratios exist. Based on our results for Earth’s atmosphere (Hellmann & Harvey, 2020), a hard-sphere estimate for the diffusivity ratios would be expected to be in error. Because the Martian atmosphere consists primarily of carbon dioxide, diffusivity ratios in terrestrial air are not applicable, requiring new calculations.

Another “hydrologic” cycle of interest is found on Titan, where methane precipitates in a nitrogen atmosphere (Roe, 2012; Mitchell & Lora, 2016; Hayes et al., 2018). There has been some study of methane isotopes (especially CH₃D) in the atmosphere of Titan (Nixon et al., 2012; Ádámkovics & Mitchell, 2016; Hörst, 2017; Thelen et al., 2019). Information exists on equilibrium fractionation of CH₃D (Armstrong et al., 1955; Calado et al., 1997), but the relative diffusivities have not been studied. While we are unaware of any current plans to include kinetic fractionation in models of Titan’s atmosphere, we calculate the diffusivity ratios so that they will be available if needed.

We follow the notation of Hellmann and Harvey (2020) in defining the relative diffusivities for water isotopologues as D_r,HDO ≡ D_HDO/D_H2O, D_r,17 ≡ D_H2¹⁷O/D_H2O, and D_r,18 ≡ D_H2¹⁸O/D_H2O, where D_i is the diffusivity of species i in the atmosphere of interest. Similarly, for the diffusion ratios for methane, D_r,_CH3D ≡ D_CH3D/D_CH4 and D_r,13 ≡ D_¹³CH4/D_CH4.

For hard spheres at low density, simple kinetic theory yields

$$D_{\text{r},i}={\left[\frac{M_0\left(M_i+M_{\text{G}}\right)}{M_i\left(M_0+M_{\text{G}}\right)}\right]}^{1/2}$$

for the ratio of the diffusivity of an isotopic species i to that of the reference species (subscript 0) in a gas G, where M is the molar mass. In this relation, the diameters of the isotopic species and of the reference species are assumed to be equal. If minor components (those other than CO₂, N₂, and Ar) are ignored, the composition of the Martian atmosphere (Trainer et al., 2019) yields a molar mass of 43.5 g mol⁻¹, and for Titan we use the molar mass of N₂, which is 28.0134 g mol⁻¹. In this simplification, Equation 1 yields for the Martian atmosphere 0.9811 for D_r,HDO, 0.9812 for D_r,17, and 0.9640 for D_r,18. For a nitrogen atmosphere (Titan), the result is 0.9811 for D_r,CH3D and 0.9813 for D_r,13. [After our calculations were completed, it was brought to our attention that molar masses slightly different from these terrestrial values would have been appropriate, due to different isotopic compositions on Mars and Titan. For example, N₂ on Titan has more ¹⁵N than on Earth (Niemann et al., 2010), leading to a molar mass of 28.018 g mol⁻¹. These differences are negligible in the context of the present calculations.]

As shown by Hellmann and Harvey (2020), the simple hard-sphere kinetic theory is inaccurate for water, because it ignores the rotational dynamics that are affected by isotopic substitution (especially H/D substitution). Modern kinetic theory can significantly improve on the hard-sphere results. The relevant collision integrals (sometimes called generalized cross sections) can be calculated essentially exactly from the full intermolecular potential-energy surface. In this work, we use state-of-the-art pair potentials for the H₂O–CO₂ and H₂O–N₂ interactions to calculate the diffusivity ratios D_r,HDO, D_r,17, and D_r,18 in the Martian atmosphere as a function of temperature. Similarly, we use a high-accuracy potential for the CH₄–N₂ interaction to calculate D_r,CH3D and D_r,13 at conditions relevant to Titan.

2. Methods and Results

2.1. Intermolecular Potentials

The pair potentials used in this work for modeling H₂O–CO₂ (Hellmann, 2019a), H₂O–N₂ (Hellmann, 2019b), and CH₄–N₂ (Hellmann et al., 2014) interactions were developed using state-of-the-art quantum-chemical ab initio approaches, see these papers for full details. We used the potentials without modifications for the calculations with HDO, H₂¹⁷O, H₂¹⁸O, CH₃D, and ¹³CH₄. This is a valid approximation only if the isotopic substitutions of water and methane affect the collisional dynamics predominantly through the changes to the molecular masses and to the moment of inertia tensors and not through changes to the pair potential-energy surfaces. Here, we provide justifications for this assumption.

If the geometries of HDO, H₂¹⁷O, and H₂¹⁸O were identical to that of H₂O and the geometries of CH₃D and ¹³CH₄ were identical to that of CH₄, the respective interaction potentials would also be identical apart from very small effects beyond the Born–Oppenheimer approximation. While the equilibrium geometries are indeed identical (at least within the Born–Oppenheimer approximation), the zero-point vibrationally averaged geometries, which better represent the molecules in a rigid-rotor treatment of thermophysical properties, are not. For the development of the H₂O–CO₂, H₂O–N₂, and CH₄–N₂ pair potentials, vibrationally averaged geometries of H₂O and CH₄ were used.

The changes in the vibrationally averaged molecular geometries for the three different isotopic substitutions of H₂O were obtained in our previous work (Hellmann & Harvey, 2020) from cubic force field calculations at the frozen-core CCSD(T)/cc-pVQZ level of theory using the CFOUR quantum chemistry code (Stanton et al., 2019). For convenience, the results are provided here again, see Table 1. Only the geometry of HDO changes appreciably, with the OH bond length differing by −0.14% and the OD bond length differing by −0.32% from the OH bond length in H₂O. In the present work, we performed a similar analysis of the geometries of the CH₄ isotopologues at the frozen-core CCSD(T)/cc-pVTZ level using CFOUR. The results are shown in Table 2. While the geometry of ¹³CH₄ is virtually identical to that of CH₄, the lengths of the three CH bonds and the single CD bond in CH₃D differ by −0.08% and −0.17%, respectively, from the length of the CH bonds in CH₄.

Table 1.

Zero-Point Vibrationally Averaged Geometries of H₂O, HDO, D₂O, H₂ ¹⁷O, and H₂ ¹⁸O in Terms of the Bond Lengths r_OX (with X=H or X=D) and the Bond Angle θ from Quantum-Chemical Cubic Force Field Calculations at the Frozen-Core CCSD(T)/cc-pVQZ Level of Theory, as Reported by Hellmann and Harvey (2020)

Isotopologue	Geometric Parameter	Value	Deviation from H2O
H2O	r OH	97.262 pm
	θ	104.00°
HDO	r OH	97.126 pm	−0.135 pm
	r OD	96.947 pm	−0.315 pm
	θ	104.01°	0.01°
D2O	r OD	96.861 pm	−0.401 pm
	θ	104.00°	0.00°
H2 17O	r OH	97.259 pm	−0.003 pm
	θ	104.00°	0.00°
H2 18O	r OH	97.257 pm	−0.005 pm
	θ	104.00°	0.00°

Table 2.

Zero-Point Vibrationally Averaged Geometries of CH₄, CH₃D, CD₄, and ¹³CH₄ in Terms of the Bond Lengths r_CX (with X=H or X=D) and the Bond Angles θ_XCY (with X=H or X=D and Y=H or Y=D), as Obtained from Quantum-Chemical Cubic Force Field Calculations at the Frozen-Core CCSD(T)/cc-pVTZ Level of Theory

Isotopologue	Geometric Parameter	Value	Deviation from CH4
CH4	r CH	110.209 pm
	θHCH	109.47°
CH3D	rCH	110.122 pm	−0.087 pm
	r CD	110.018 pm	−0.192 pm
	θ HCH	109.48°	0.01°
	θ HCD	109.46°	−0.01°
CD4	r CD	109.853 pm	−0.356 pm
	θ DCD	109.47°	0.00°
13CH4	r CH	110.206 pm	−0.003 pm
	θ HCH	109.47°	0.00°

To take the differing bond lengths between H₂O and HDO properly into account, one would have to compute the differences in the interaction energies with CO₂ and N₂ between the two geometries in order to construct dedicated HDO–CO₂ and HDO–N₂ potentials, but this is quite complicated due to the lower symmetry of HDO. However, this complication does not occur with D₂O, whose bonds are 0.41% shorter than those of H₂O (see Table 1). Calculations of D_D2O/CO2 with both the H₂O–CO₂ potential and a dedicated D₂O–CO₂ potential derived from it would thus be a suitable proxy for estimating the influence of the water geometry, an approach that we already applied in our previous work (Hellmann & Harvey, 2020) for the interactions with N₂. Therefore, we constructed a D₂O–CO₂ potential by first computing the differences in the interaction energies due to the differences between the D₂O and H₂O geometries at the reasonably accurate RI-MP2/aug-cc-pVQZ (plus bond functions) level of theory using the ORCA program (Neese, 2012); see Hellmann (2019a) for more details on these types of calculations. In the next step, we added these differences to the H₂O–CO₂ interaction energies of Hellmann (2019a) and refitted the potential. Kinetic-theory calculations of D_D2O/_CO2 with the two potentials yielded values that differ by only 0.13% at 100 K, 0.11% at 200 K, and again 0.13% at 400 K, with the values for the D₂O–CO₂ potential being consistently larger. For D_HDO/CO2 , the effect is expected to be only about half as large.

For CH₃D in N₂, the complications are similar due to the lower symmetry of CH₃D compared with CH₄. Using CD₄ to estimate the influence of the different geometries circumvents these problems. The bonds in CD₄ are 0.32% shorter than those in CH₄ (see Table 2). We constructed a CD₄–N₂ potential by calculating the differences in the interaction energies resulting from the differences between the CD₄ and CH₄ geometries at the frozen-core CCSD(T)/aug-cc-pVTZ [plus bond functions, see Hellmann et al. (2014) for details] level of theory using the CFOUR program. The further steps are analogous to the D₂O–CO₂ case. The differences between the D_CD4/N₂ values obtained with the CD₄–N₂ and CH₄–N₂ potentials are 0.24% at 50 K, 0.23% at 100 K, and 0.18% at 200 K, with the values for the CD₄–N₂ potential being always larger. With only one CD bond in CH₃D, the respective differences for D_CH3D/N2 should be roughly four times smaller.

Thus, we can conclude that the errors introduced by using the pair potentials without further adjustments likely do not exceed 0.1% for both D_r,HDO and D_r,CH3D and are completely negligible for D_r,17, D_r,18, and D_r,13. A similar analysis with similar conclusions was presented in our work on water diffusivity ratios in Earth’s atmosphere (Hellmann & Harvey, 2020). An important advantage of using the same pair potentials for different isotopologues is that any inaccuracies in the potentials that would cause D_H2O and D_CH4 to be in error would affect D for the substituted isotopologues in a similar manner, making the diffusivity ratios insensitive to such errors.

2.2. Kinetic-Theory Calculations

The kinetic-theory calculations performed in this work are only briefly summarized here. The interested reader is referred to Hellmann (2019a, 2019b) and Hellmann et al. (2014) for more detailed descriptions of the methodology.

The required collision integrals (generalized cross sections) for computing the diffusivity ratios were obtained from classical trajectories for binary collisions of H₂O, HDO, H₂¹⁷O, and H₂¹⁸O with CO₂ and N₂ and of CH₄, CH₃D, and ¹³CH₄ with N₂. Rigid molecules were assumed, and the trajectories were calculated from Hamilton’s equations solved numerically from pre-collisional to post-collisional asymptotic conditions. For a given constant collision energy, the generalized cross sections are expressed as 11-dimensional integrals over the initial states of the trajectories, which are characterized by the spatial orientations of the two molecules, their angular momentum vectors, and the impact parameter (the intermolecular separation at closest approach if there were no interactions between the molecules). This high dimensionality necessitated using a Monte Carlo integration approach, involving, for each collision energy, the computation of typically several million trajectories. An appropriate thermal averaging procedure was used to convert the generalized cross sections at fixed collision energies to values as a function of temperature, from which the diffusivities can be obtained in a straightforward manner. The range of investigated collision energies was chosen such that the diffusivities of the H₂O and CH₄ isotopes could be obtained at temperatures as low as 100 K and 50 K, respectively. The computations of the generalized cross sections at different energies (and temperatures) were performed with an in-house version of the TRAJECT code (Heck & Dickinson, 1996). Unlike the original program, our version is not limited to linear molecules.

The diffusivities were calculated for mole fractions of water and methane approaching zero. For the atmospheres of Mars and Titan, this is a sensible choice, which has the additional advantage that in this limit the diffusivities depend only on the interaction potentials between unlike species. Thus, no models were needed for H₂O–H₂O, CH₄–CH₄, CO₂–CO₂, and N₂–N₂ interactions. We note that the mole-fraction dependencies of the diffusivities do not exceed a few tenths of a percent at any temperature; this small effect would almost completely vanish in the diffusivity ratios.

To obtain the diffusivities of water isotopes in Mars’ atmosphere, the respective diffusivities in CO₂ and N₂ were weighted with the use of a result from first-order kinetic theory (Marrero & Mason, 1972),

$$D_i={\left(\frac{x_{{\text{CO}}_2}}{D_{i/{\text{CO}}_2}}+\frac{x_{{\text{N}}_2}}{D_{i/{\text{N}}_2}}\right)}^{-1},$$

where the mole fraction of CO₂ is x_CO2 = 0.9545 (Trainer et al., 2019) and that of N₂ accounts also for argon and all other minor components, so that x_N2 = 1−x_CO2. This simplification is justified because the diffusivity of water vapor is very similar in argon and in nitrogen (O’Connell et al., 1969).

The calculated diffusivity ratios D_r,HDO, D_r,17, and D_r,18 in the Martian atmosphere are given for selected temperatures up to 400 K in Table 3 and are shown graphically in Figure 1, while the calculated ratios D_r,CH3D and D_r,13 in N₂ are provided at selected temperatures up to 200 K in Table 4 and are shown in Figure 2. The ratios have expanded statistical uncertainties (k = 2, roughly equivalent to a 95% confidence interval) of less than 0.1% and 0.05% for the atmospheres of Mars and Titan, respectively. This uncertainty component arises from the Monte Carlo integration over the randomly chosen initial conditions of the trajectories for the binary collisions. The expanded uncertainties listed in the tables and displayed in the figures also take into account that we neglected the influence of isotopic substitutions on the pair potentials and that quantum effects on the generalized cross sections are not accounted for due to calculating the collision trajectories classically.

Table 3.

Calculated Diffusivity Ratios for Water Isotopologues in Mars’ Atmosphere at Selected Temperatures and Estimates of Their Expanded Uncertainties at the 95% Confidence Level

T/K	D r,HDO	D r,17	D r,18
100	0.9772 ± 0.0100	0.9797 ± 0.0020	0.9615 ± 0.0040
110	0.9779 ± 0.0093	0.9796 ± 0.0020	0.9614 ± 0.0038
120	0.9785 ± 0.0086	0.9796 ± 0.0019	0.9613 ± 0.0036
130	0.9791 ± 0.0079	0.9795 ± 0.0019	0.9612 ± 0.0034
140	0.9797 ± 0.0072	0.9795 ± 0.0018	0.9612 ± 0.0032
150	0.9802 ± 0.0065	0.9795 ± 0.0018	0.9613 ± 0.0030
160	0.9806 ± 0.0058	0.9796 ± 0.0017	0.9614 ± 0.0028
170	0.9810 ± 0.0051	0.9796 ± 0.0017	0.9615 ± 0.0026
180	0.9813 ± 0.0044	0.9797 ± 0.0016	0.9617 ± 0.0024
190	0.9815 ± 0.0037	0.9798 ± 0.0016	0.9619 ± 0.0022
200	0.9816 ± 0.0030	0.9799 ± 0.0015	0.9621 ± 0.0020
220	0.9818 ± 0.0027	0.9801 ± 0.0014	0.9626 ± 0.0018
240	0.9818 ± 0.0024	0.9803 ± 0.0013	0.9630 ± 0.0016
260	0.9818 ± 0.0021	0.9806 ± 0.0012	0.9635 ± 0.0014
280	0.9816 ± 0.0018	0.9808 ± 0.0011	0.9639 ± 0.0012
300	0.9815 ± 0.0015	0.9811 ± 0.0010	0.9643 ± 0.0010
350	0.9811 ± 0.0015	0.9816 ± 0.0010	0.9652 ± 0.0010
400	0.9806 ± 0.0015	0.9820 ± 0.0010	0.9659 ± 0.0010

Figure 1.

Calculated diffusivity ratios D_r,HDO, D_r,17, and D_r,18 in the atmosphere of Mars as a function of temperature. The shaded areas indicate the estimated expanded uncertainty of the calculations at the 95% confidence level. The dotted horizontal lines correspond to the diffusivity ratios resulting from the simple hard-sphere kinetic theory, Equation 1.

Table 4.

Calculated Diffusivity Ratios for Methane Isotopologues in Nitrogen at Selected Temperatures and Estimates of Their Expanded Uncertainties at the 95% Confidence Level

T/K	D r,CH3D	D r,13
50	0.9775 ± 0.0040	0.9818 ± 0.0020
55	0.9776 ± 0.0038	0.9816 ± 0.0019
60	0.9777 ± 0.0036	0.9815 ± 0.0018
65	0.9779 ± 0.0034	0.9814 ± 0.0017
70	0.9780 ± 0.0032	0.9813 ± 0.0016
75	0.9783 ± 0.0030	0.9813 ± 0.0015
80	0.9785 ± 0.0028	0.9812 ± 0.0014
85	0.9787 ± 0.0026	0.9812 ± 0.0013
90	0.9790 ± 0.0024	0.9811 ± 0.0012
95	0.9792 ± 0.0022	0.9811 ± 0.0011
100	0.9794 ± 0.0020	0.9811 ± 0.0010
110	0.9799 ± 0.0018	0.9811 ± 0.0009
120	0.9802 ± 0.0016	0.9811 ± 0.0008
130	0.9806 ± 0.0014	0.9810 ± 0.0007
140	0.9809 ± 0.0012	0.9810 ± 0.0006
160	0.9813 ± 0.0010	0.9810 ± 0.0005
180	0.9817 ± 0.0010	0.9810 ± 0.0005
200	0.9819 ± 0.0010	0.9810 ± 0.0005

Figure 2.

Calculated diffusivity ratios D_r,CH3D and D_r,13 in N₂ as a function of temperature. The shaded areas indicate the estimated expanded uncertainty of the calculations at the 95% confidence level. The dotted horizontal lines correspond to the diffusivity ratios resulting from the simple hard-sphere kinetic theory, Equation 1.

The quantum effects on the collision integrals depend not only on the pair potential and temperature, but also on the masses and moments of inertia of the two colliding molecules, so they will not fully cancel out in the diffusivity ratios. Our estimates for the magnitude of quantum effects are educated guesses based on our experience. Support for our estimates comes, for example, from the fact that the viscosity of dilute water vapor obtained from classical calculations such as those performed here for H₂O–H₂O collisions differs from reliable experimental data near room temperature by less than 1% (Hellmann & Vogel, 2015). Although viscosity and diffusivity in dilute gases are closely related quantities for which quantum effects should be roughly similar on a relative scale, the H₂O–H₂O system is quite different from those considered here, and our lowest temperatures are significantly below those of the water data. However, we can draw some insight from the second virial coefficient, which is a thermophysical property that, like the dilute-gas viscosity and diffusivity, is determined solely by pair interactions. Quantum effects are much more important for second virial coefficients than for diffusivities, so we cannot directly compare their influence on the two properties. But a comparison of the relative magnitudes of the quantum effects on the second virial coefficient of water at ambient temperature with those on the second virial coefficients of the H₂O–CO₂ and CH₄–N₂ systems at our lowest temperatures should provide a qualitative measure of the quantum character of these systems. For water, quantum effects change the second virial coefficient at 300 K by about 20% (Garberoglio et al., 2018), while for H₂O–CO₂ at 100 K and CH₄–N₂ at 50 K the changes [calculated semiclassically as done by Hellmann (2019a) and Hellmann et al. (2014)] are about 44% and 12%, respectively. Thus, by this rough measure, the H₂O–CO₂ system at 100 K can be said to have a stronger quantum character than the H₂O–H₂O system at 300 K, but this is taken into account in our uncertainty estimates, which we believe to be conservative.

Table S1 of the supporting information provides the calculated absolute diffusivities of H₂O, HDO, H₂¹⁷O, and H₂¹⁸O in CO₂, N₂, and Mars’ atmosphere at a large number of temperatures from 100 K to 1000 K, normalized to a standard pressure of 101.325 kPa (1 atm). For CH₄, CH₃D, and ¹³CH₄ in N₂, the respective diffusivities are provided from 50 K to 500 K in Table S2. Note that the diffusivities of H₂O in CO₂ and N₂ at several temperatures from 250 K to 1000 K and the diffusivities of CH₄ in N₂ from 70 K to 500 K were previously provided by Hellmann (2019a, 2019b) and Hellmann et al. (2014) and are only listed here for convenience.

3. Discussion

The results shown in Figure 1 for water isotopes on Mars are surprising, in that they show good agreement between our rigorous kinetic-theory calculations and the results from a simple hard-sphere calculation. That was clearly not the case for water isotopes in Earth’s atmosphere, as can be seen in Figure 3 which we reproduce from the paper of Hellmann and Harvey (2020). In air, D_r,HDO differs from unity by roughly 50% more than does the hard-sphere result, and displays a clear temperature dependence; D_r,18 differs less dramatically from the hard-sphere result but the difference is still significant compared to the uncertainty of the calculations. In that previous work (Hellmann & Harvey, 2020), we attributed the failure of the hard-sphere approach to the significant anisotropy of the H₂O–N₂ and H₂O–O₂ interactions and to the degree to which isotopic substitution changes the rotational dynamics. The results for Mars are dominated by the H₂O–CO₂ interaction, where these factors in the collision dynamics should be at least as large as for H₂O with the components of air. It therefore seems likely that there are compensating effects that cause the net diffusivity ratios for H₂O–CO₂ to be similar to the hard-sphere results. There is some deviation from the hard-sphere result for D_r,HDO at the lowest temperatures examined, but the difference is within the uncertainty of our calculations. Our results for D_r,18 show a temperature dependence that seems to be (barely) significant compared to the uncertainties of the calculations.

Figure 3.

Calculated diffusivity ratios D_r,HDO (a) and D_r,18 (b) in air and available experimental data, as a function of temperature. The shaded areas indicate the estimated expanded uncertainty of the calculations at the 95% confidence level. The figure was taken from the paper of Hellmann and Harvey (2020).

The results for methane diffusion ratios in the N₂ atmosphere of Titan, shown in Figure 2, are less surprising. Interactions involving methane are much less anisotropic than those involving water, so one would expect less deviation from hard-sphere results for the CH₄–N₂ interaction than for the H₂O–N₂ interaction that is the main component of the water-air system shown in Figure 3. Nevertheless, the mass asymmetry induced by the deuterium substitution in CH₃D is enough to produce a clear temperature dependence and a noticeable difference in D_r,CH3D from the hard-sphere results at low temperatures. As expected, isotopic substitution on the central carbon atom of CH₄ has negligible influence on the rotational collision dynamics, producing a diffusivity ratio D_r,13 nearly indistinguishable from the simple mass-based result of Equation 1.

For convenience in applications, we fitted simple correlating equations to our calculated diffusivity ratios using the symbolic regression software Eureqa (Schmidt & Lipson, 2009). They are valid for temperatures from 100 K to 400 K for D_r,HDO, D_r,17, and D_r,18 and from 50 K to 200 K for D_r,CH3D and D_r,13. The resulting expressions are

$$D_{\text{r},\text{HDO}}\equiv D_{\text{HDO}}/D_{{\text{H}}_2\text{O}}=0.97041+\frac{0.03842}{{\left(T^{\ast }\right)}^{3/4}}-\frac{0.06168}{{\left(T^{\ast }\right)}^2}+\frac{0.03010}{{\left(T^{\ast }\right)}^3},$$

$$D_{\text{r},17}\equiv D_{{\text{H}}_2^{17}\text{O}}/D_{{\text{H}}_2\text{O}}=0.98644-\frac{0.02300}{T^{\ast }}+\frac{0.02098}{{\left(T^{\ast }\right)}^2}-\frac{0.00470}{{\left(T^{\ast }\right)}^4},$$

$$D_{\text{r},18}\equiv D_{{\text{H}}_2^{18}\text{O}}/D_{{\text{H}}_2\text{O}}=0.97105-\frac{0.08330}{{\left(T^{\ast }\right)}^{3/2}}+\frac{0.08499}{{\left(T^{\ast }\right)}^2}-\frac{0.01127}{{\left(T^{\ast }\right)}^4},$$

$$D_{\text{r},{\text{CH}}_3\text{D}}\equiv D_{{\text{CH}}_3\text{D}}/D_{{\text{CH}}_4}=0.98379-\frac{0.00533}{{\left(T^{\ast }\right)}^{3/2}}+\frac{0.00143}{{\left(T^{\ast }\right)}^4}-\frac{0.00044}{{\left(T^{\ast }\right)}^5},$$

$$D_{r,13}\equiv D_{{13}_{{\text{CH}}_4}}/D_{{\text{CH}}_4}=0.98106-\frac{0.00043}{{\left(T^{\ast }\right)}^2}+\frac{0.00059}{{\left(T^{\ast }\right)}^3}-\frac{0.00014}{{\left(T^{\ast }\right)}^4},$$

where T* = T/(100 K). The correlations reproduce the calculated ratios within ±4 × 10⁻⁵ and thus well within their uncertainties.

A similar approach could in principle be applied to other planetary atmospheres, but the gas giants would present additional complication. For example, Jupiter has ammonia and water clouds in an atmosphere consisting primarily of hydrogen (Atreya et al., 1999; Young et al., 2019). For water, accurate H₂O–H₂ potential-energy surfaces exist (Hodges et al., 2004; Valiron et al., 2008; Homayoon et al., 2015). However, the highly quantum nature of H₂, and the low temperatures involved, mean that classical trajectory calculations such as those in this paper would likely be significantly in error. Quantitative accuracy would require fully quantum scattering calculations; the framework for such calculations is known (McCourt et al., 1990), but for molecules of this complexity the effort might be prohibitive.

4. Conclusion

We have employed state-of-the-art intermolecular potentials for H₂O–CO₂ and H₂O–N₂ to perform rigorous kinetic-theory calculations for the diffusivity of water isotopologues in the atmosphere of Mars. The resulting temperature-dependent diffusivity ratios provide the first data for these quantities, enabling the inclusion of kinetic isotope fractionation in future modeling of the hydrologic cycle on Mars. In contrast to similar calculations for diffusivity ratios in Earth’s atmosphere, the difference from a simple hard-sphere calculation is at most of only marginal significance.

We performed similar calculations for the diffusion of methane isotopologues in nitrogen, representing the atmosphere of Titan. In that case, a small but not insignificant deviation from the hard-sphere result is obtained for the diffusivity ratio of CH₃D.

For convenience in modeling, we have provided temperature-dependent correlations for each diffusivity ratio studied. These are valid from 100 K to 400 K for water in the atmosphere of Mars (Equations 3–5), and from 50 K to 200 K for methane in the atmosphere of Titan (Equations 6 and 7).

In some ways, this paper reports a negative result. The simple hard-sphere theory predicts values almost identical to our more rigorous results for diffusivity ratios of water isotopologues in the Martian atmosphere. However, this negative result is itself unexpected (suggesting some cancellation of errors in the hard-sphere model), because our previous work (Hellmann & Harvey, 2020) demonstrated that the hard-sphere theory is significantly in error for water species in Earth’s atmosphere. The difference is marginally significant for CH₃D in Titan’s atmosphere, but realistically the effect is probably smaller than other uncertainties would be for any isotopic modeling of Titan in the foreseeable future. However, because these ratios have now been computed, and because Equations 3–7 are simple, the more rigorous diffusivity ratios developed here can be used in modeling with an insignificant increase in computational effort.

Key Points:

• Kinetic isotope fractionation depends on diffusivity ratios that have not been measured for conditions corresponding to Mars or Titan.

• The relative diffusivity of isotopologues can be calculated as a function of temperature from high-accuracy intermolecular potentials.

• Rigorous results agree well with simpler kinetic theory for water on Mars, but less well for methane on Titan.

Data Availability Statement

The intermolecular H₂O–CO₂, H₂O–N₂, and CH₄–N₂ potentials used in this study are fully documented in the supporting information of (Hellmann, 2019a, 2019b) and Hellmann et al. (2014). Tables S1 and S2 of the present supporting information provide the calculated diffusivities of H₂O, HDO, H₂¹⁷O, and H₂¹⁸O in CO₂, N₂, and Mars’ atmosphere and the calculated diffusivities of CH₄, CH₃D, and ¹³CH₄ in N₂ at a large number of temperatures. The data in Tables S1 and S2 are also provided in the NIST Public Data Repository (Harvey & Hellmann, 2021).