Solid-phase equilibria on Pluto's surface

Abstract

1 INTRODUCTION

Pluto is the outermost body in the Solar system known to have an atmosphere. The tenuous atmosphere is even believed not to collapse during its orbit around the Sun (Olkin et al. 2015), though others expect it to collapse completely at aphelion in year 2113 (Lellouch et al. 2011). It consists mostly of nitrogen with small fractions of methane, carbon monoxide, and other heavier molecules. At extremely low temperatures on the surface, Pluto's atmosphere may be considered in thermodynamic equilibrium with its solid phases termed as ices.

At this extreme condition, the ices must be multicomponent solid solutions (see overview by Cruikshank et al. 2015). It has been realized that an assumption of pure nitrogen frost in a Pluto thermal model, thus neglecting the presence of other species (mainly CH₄ and CO) in the solid solutions, must be viewed as a serious limitation of the model (Hansen & Paige 1996). However, due to the difficulties in describing multicomponent solid phases, models still treat the chemical species individually for the solid phase (e.g. in Global Climate Model by Forget et al. 2017). Attempts to include methane in nitrogen-rich solid of binary N₂/CH₄ have been made (Trafton 2015) but without satisfactory explanation of the methane-rich solid that was proposed (Protopapa et al. 2015) and later directly observed on mountain ridges at low latitudes and high northern latitudes (Grundy et al. 2016; Protopapa et al. 2017; Schmitt et al. 2017).

Trafton (2015) tried to explain the presence of both methane-rich and nitrogen-rich solids on Pluto by applying an ideal solution approach for the binary N₂/CH₄. However, the resulting discontinuity in the partial pressures is likely due to the non-ideality of the chemical system at the extremely low temperatures.

All difficulties above can readily be solved by an equation of state (EOS) that can carry out phase-equilibria calculations for solid phases at very low temperatures and low pressures. Such an EOS has been recently developed (Tan et al. 2013), referred to as CRYOCHEM. It does not need an assumption of ideal solution and can work with multicomponent solid solutions.

In this paper, an improvement and extension of the EOS is presented to include N₂ and CO in the data base. The purpose of this paper is to establish the thermodynamically consistent solid solution mixing ratios needed to account for the observed atmospheric composition. Therefore, the study is a snapshot in time at New Horizons’ arrival date, where the data for Pluto's condition may be taken from most recent works that derived them from direct observations (Gladstone et al. 2016; Grundy et al. 2016).

Though the snapshot does not consider the dynamics of Pluto's surface pressure, which is effectively the vapour pressure of the volatile ices, the quasi-equilibrium condition does allow thermodynamic calculations that provide us with the limiting phase behaviour of Pluto's system. No attempts were made to conduct complete modelling, such as that in global simulations, because many other factors need to be included for such attempts, which are beyond the scope of our thermodynamic approach.

The phase behaviour analysed here include the equilibrium compositions, phase densities, phase mole fractions, sublimation and condensation curves, as well as comparison between treatments using binary (N_2/CH₄) and ternary mixtures (N₂/CH₄/CO).

2 PHASE EQUILIBRIA

The atmosphere may be modelled as a ternary mixture of the three major components: N₂, CH₄, and CO. This also means that, according to the Gibbs’ phase rule, there are up to four possible solid phases that may exist simultaneously in equilibrium with the vapour phase. At Pluto's conditions, no liquid phase can exist so that all condensates are solid. Therefore, the phase equilibria that may occur are the two-phase solid–vapour (SV), two-phase solid–solid (SS), three-phase solid–solid–vapour (SSV), and three-phase solid–solid–solid (SSS). In this paper, the phase labelling may be in any order, for example, SV or VS means the vapour phase is in equilibrium with a solid phase. If using subscripts, it means to be rich with (1: N₂, 2: CH₄, 3: CO), for example, S₁ is nitrogen-rich solid phase and S₃₁ is a solid phase rich with both CO and N₂or either of them.

After solving equation (1b), the phase mole fractions, ζ and ξ, can be solved from equation (3). In this case, the resulting phase fractions must be between 0 and 1; otherwise z must be outside the three-phase region, which means the mixture is not in a three-phase equilibrium at the specified condition, but either in a homogeneous one-phase or heterogeneous two-phase regions.

The fugacity of individual components in the phases in equations (1a) and (1b) must be obtained from an EOS. As stated before, we will use CRYOCHEM to calculate all fugacities in the equations.

3 CRYOCHEM

The EOS consists of two parts, i.e. the fluid part, which is the Perturbed-Chain Statistical Associating Fluid Theory EOS (Gross & Sadowski 2001) that has been widely used in petrochemical applications (Tan, Adidharma & Radosz 2008), and the solid part, which applies the Lennard–Jones Weeks–Chandler–Andersen (LJ-WCA) approach as used in the earlier version of CRYOCHEM (Tan et al. 2013). The readers are referred to the respective papers for details. Here, for the new improved and extended version, we will only provide a brief description of the solid part of the EOS that has been further developed since the previous version.

The subscripts refer to solid phase, while the superscripts hs, LJ, and chain refer to the WCA hard sphere, LJ, and chain-forming covalent bonds terms, respectively.

The difference between the new version CRYOCHEM 2.0 and the previous one is mainly in the equations to obtain the contributing energy terms. First, the LJ term can now account for the face-centred cubic (FCC) crystalline structure and the hexagonal close-packed (HCP), based on the recent Monte Carlo (MC) simulation results for the LJ solids (Adidharma & Tan 2016). The HCP term allows the EOS to include N₂ and CO into the data base; CH₄ has FCC structure. Secondly, the operational temperature range of the EOS is extended to a reduced temperature T* = 0.1 (see Appendix  A for the definition of T*), which well includes the low temperatures on Pluto. The new equations for the LJ term are correlated against the simulation results within the WCA perturbation framework used in CRYOCHEM. The new equations for all terms in equation (4) are included in Appendix A.

All together CRYOCHEM has six major parameters for each chemical species, three for the fluid part and three for the solid part (see Appendix  A). In addition to the pure-component parameters, there are also binary-interaction parameters for each of the binaries so that each binary has two interaction parameters, one in fluid phase and the other in solid phase. All parameters were correlated from phase-equilibrium data: vapour–liquid (VL) for the former and solid–liquid (SL) for the latter (or SV if SL is not available). The parameters used in this paper are included in Appendix  B, as well as the excellent performance of the EOS using the parameters.

Special note on the EOS performance is given to a relatively recent experimental data of SL equilibria of binary N₂/CH₄ (Roe & Grundy 2012). It was found that this binary undergoes density inversion, in which the solid density is less than that of its liquid, thus the solid floats in its liquid. The density inversion was bracketed between 83.8 and 84.1 K at 1.5 bar, while the prediction of CRYOCHEM 2.0 is 84.0 K as shown in Fig. 1. Based on the excellent performance, CRYOCHEM 2.0 is reliable to analyse the solid-phase equilibria on Pluto's surface in the next section.

Figure 1.

The densities of equilibrium liquid and solid of binary CH₄/N₂ at 1.5 bar as calculated using CRYOCHEM 2.0; the density inversion is experimentally between 83.8 and 84.1 K where the solid phase starts floating in the liquid as the temperature drops.

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4 RESULTS AND DISCUSSION

Most calculations in this work used a fixed overall composition and a fixed atmospheric pressure of 11 μbar (Gladstone et al. 2016) as if we were at the condition when New Horizons arrived in 2015. The main analyses were then carried out by varying the temperature. Only when the sublimation and freezing points are discussed, it will be shown how the change of pressure or composition may affect those phase-transition points.

The overall composition used for calculation in this work is chosen close to that of the atmosphere, thus close to the saturation: 0.60 per cent methane (Gladstone et al. 2016) and 0.05 per cent carbon monoxide (Lellouch et al. 2011) with nitrogen as the balance. Though this choice may have to be adjusted to reflect the quantitative situation more exactly, the composition serves the purpose of this study very well as shown later in the paper.

The phase equilibria of the atmospheric ternary mixture need to be presented on ternary composition phase diagrams for analyses purposes. For this, one should start with the pure gases. Due to the low temperature and pressure ranges on Pluto, the phase transitions of the gases must be on the sublimation line, i.e. equilibria between vapour and solid. At the atmospheric pressure of 11 μbar (Gladstone et al. 2016), the calculated sublimation temperatures of the pure components (36.9 K for nitrogen, 53.0 K for methane, and 40.8 K for CO) are shown in Fig. 2 at mole fractions 0 and 1. Fig. 2 shows the temperature–composition phase diagrams of the binary mixtures that constitute the ternary at 11 μbar. The phase behaviour of these binary mixtures at five temperatures is listed in Table 1 and useful to construct the ternary phase diagram. The temperatures for later analyses are shown as horizontal dotted lines in Fig. 2, from top to bottom: 41.5, 40.5, 37.5, and 36.5 K.

Figure 2.

Binary mixtures of Pluto's system at 11 μbar; horizontal dotted lines are the temperatures for analyses, from top to bottom: 41.5, 40.5, 37.5, and 36.5 K. The small two-phase VS₁ and VS₃ regions of CH₄/N₂ and CH₄/CO, respectively, are below the three-phase lines.

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Note that the phase diagram of binary CH₄/N₂ in Fig. 2(b) is to be used for analyses here for Pluto, not the one at a much higher pressure commonly used to describe Pluto's condition (e.g. Trafton 2015; Protopapa et al. 2015), which still has liquid phase at high-temperature region. At 11 μbar, there is no liquid phase at any temperatures. Concerning phase diagrams that involve solid phases in comparison with the experimental data, the readers are referred to our previous work (Tan et al. 2013); for this binary CH₄/N₂, see Appendix  C for details. The phase transition of solid N₂ from HCP to FCC crystalline structure at about 35.6 K gives a three-phase SSS (all solids) line on the diagram, but this needs data of FCC N₂ to derive the solid parameters; such data are not available. However, as described later, we do not need it, as we will never go to that low temperature in our work here. Furthermore, Pluto's absorption spectra were also found consistent with the HCP N₂ (Cruikshank et al. 2015).

For T = 40.5 and 37.5 K in Fig. 3, as seen in Table 1, all binary mixtures have two-phase equilibria in some ranges of their compositions: VS for nitrogen-containing mixtures and SS for CH₄/CO. The equilibria also appear on the sides of the ternary composition diagrams in Fig. 3. It turns out that the ternary exhibits three-phase equilibrium SSV at these conditions, shown as the shaded triangle on the diagrams. The region disappears at higher or lower temperatures as shown in Fig. 4 at 41.5 and 36.5 K, where only two-phase VS and SS regions can exist, respectively. Any mixtures with overall composition lying inside the three-phase region will immediately split into three equilibrium phases: vapour phase (V), methane-rich solid (S₂), and CO-rich or CO–N₂-rich solid (S₃ or S₃₁, respectively) with compositions at the respective vertices of the three-phase triangle.

Figure 3.

Composition phase diagrams of N₂/CH₄/CO at 40.5 and 37.5 K; the star close to the nitrogen corner is the overall composition. Vapour region (V) almost coincides with the CO–N₂ side. Dotted lines are tie lines connecting equilibrium compositions.

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Figure 4.

Composition phase diagrams of N₂/CH₄/CO at 41.5 and 36.5 K; the three-phase region is gone as binary N₂/CO lies well in the vapour region and in the solid region, respectively (see Fig. 2a).

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The overall composition of Pluto's atmosphere (star near the N₂ corner) lies in the S₂V region at those temperatures except at 36.5 K where it is in the homogeneous solid-phase region. Table 2 provides the phase state at the temperatures in Table 1 along with the results of phase-equilibrium calculation, which include the equilibrium composition, densities, and the amount of the equilibrium phases.

If the temperature increases to 43.3 K, all solid phase that is in equilibrium with the atmosphere will be gone; the atmospheric fluid is entirely in gaseous phase. In other words, the condensation point (snow or frost point) at 11 μbar for this atmospheric composition is 43.3 K, i.e. when the first miniscule amount of solid phase appears upon decreasing temperature. This is consistent with the fact that Cthulhu Regio, the temperature of which never falls below 42.5 K (Earle et al. 2017), does not show volatile abundances as solid phase on the surface (Grundy et al. 2016).

On the other hand, decreasing temperature to below 36.9 K, such as 36.5 K in Table 2 and Fig. 4(b), the overall composition falls in a homogeneous solid region. In other words, the sublimation point at 11 μbar is 36.9 K, where the last bubble disappears in solid phase upon decreasing temperature. The solid is nitrogen-rich; S₃₁ in Fig. 4(b) is effectively S₁ for the evaluated overall composition. Again, this is consistent with the fact that Tombaugh Regio, temperature of which never rises above 37.0 K (Earle et al. 2017), has solid that is rich with nitrogen on the surface (Grundy et al. 2016). The methane-rich solid must be formed at higher temperatures until it completely sublimes at the condensation point, as can be seen in Table 2.

Of course, these isobaric processes do not happen on Pluto, because any temperature changes must be accompanied by sublimation or condensation that eventually changes the pressure. The latter process is governed by other external factors, e.g. the gravitational field that leads to hydrostatic effects, thus not accounted for in our thermodynamic approach. Even though the isobaric analyses represent only a snapshot in time on Pluto, the agreement of the observations with the calculation results, the conditions of which are derived from New Horizons data, validates the EOS as a robust thermodynamic tool for future studies, which could potentially include a dynamic climate and meteorology.

To complete our discussion, the sublimation and condensation curves for the overall composition are presented in Fig. 5. At the pressure measured by New Horizons, 11 μbar, the temperature range where solid and vapour may coexist in equilibrium is from 36.9 to 43.3 K, thus within 6.4 K wide range. It is clear on the figure that the temperatures increase if the pressure increases. At a pressure of 17 μbar, previously predicted by Lellouch et al. (2015), each of the temperatures would increase by 0.7 K. This means that the width of the coexistence temperature range is not sensitive to the change of pressure. On the other hand, if the overall CH₄ fraction increases, the temperature range gets wider. For example, if 0.84 per cent CH₄ (the upper bound by Gladstone et al. 2016) was used instead of 0.60 per cent, then the condensation temperature would increase by 0.5 K while the sublimation temperature would be nearly unchanged.

Figure 5.

The sublimation and condensation curves of binary N₂/CH₄ coincide with those of ternary N₂/CH₄/CO at their respective overall compositions; the three-phase curve belongs to the binary.

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As seen from the compositions of the equilibrium vapour phase in Table 2, the atmospheric fraction of carbon monoxide hardly changes with the temperature change. Furthermore, solid phases on Pluto's surface will never be CO rich; they are nitrogen-rich below the sublimation point and methane-rich in between the sublimation and condensation points. These two properties justify the use of binary N_2/CH₄ in modelling Pluto's atmosphere at small risks, unless high accuracy is needed. The justification is further verified in Fig. 4, where the condensation and sublimation curves of the ternary mixture at the overall composition is almost identical to those of the binary N₂/CH₄ at an overall composition (99.40 per cent N₂, 0.60 per cent CH₄).

The three-phase curve of the binary N₂/CH₄, where the solid phase switches from nitrogen rich to methane rich is also shown in the figure. At 11 μbar, the three-phase temperature is shown in Fig. 2(b) as the horizontal line. The three-phase region of the ternary cannot be calculated using the current technique, but must be very narrow and close to the three-phase curve of the binary. Only after crossing this three-phase region to lower temperature, the solid phase changes to nitrogen-rich S₁. If the binary N₂/CH₄ is used to represent the atmosphere, it can be seen from Fig. 2(b) that the sublimation point at 0.60 per cent CH₄ is on the solid-phase boundary of VS₁ region. At 36.9 K the overall composition is on the sublimation point before completely enters the homogeneous S₁ at lower temperatures. This is also the case with the ternary, but labelled as S₃₁V in Fig. 3(b). As the temperature further decreases from 37.5 K, the three-phase region gets narrower and shifts to the right, while the S₃₁V region shrinks to effectively become S₁V (closer to pure nitrogen).

5 CONCLUSIONS AND REMARK

The phase equilibria that can occur on Pluto's surface were analysed and discussed. The results provide some important conclusions on the phase behaviour of Pluto's surface ices that are in equilibrium with the atmosphere.

Due to the small fraction of CO, as expected, it is confirmed that the phase behaviour of the ternary N₂/CH₄/CO is very similar qualitatively and quantitatively to that of binary N₂/CH₄. Therefore, in many cases, the binary mixture may well represent the atmosphere.

The sublimation point, i.e. the lowest temperature at which the sublimation to occur, is close to the three-phase line (VSS) of the binary N₂/CH₄. Therefore, while the solid phase below the sublimation point is nitrogen rich, the solid formed above the point is methane rich. The condensation point, i.e. the temperature at which solid phases in equilibrium with Pluto's atmosphere disappear upon increasing temperature, is close to the condensation curve of the binary N₂/CH₄.

In summary, methane-rich ice is formed between the sublimation and condensation points, while nitrogen-rich ice must be formed below the sublimation point. CO is never dominant in the solid phase.

For an overall composition of 0.60 per cent methane at 11 μbar, the sublimation point is about 36.9 K while the condensation point is about 43.3 K. Therefore, the width of the temperature range where solid and vapour may coexist in equilibrium is about 6.4 K. This width is not sensitive to changes in pressure but in composition, i.e. becomes larger with larger fraction of CH₄. Larger fraction of CH₄ increases the condensation point while the sublimation point is unchanged.

As mentioned before, even though the isobaric processes do not reflect the dynamic situation on Pluto, they prove useful for the EOS validation, and at the same time, provide the limiting phase behaviour of Pluto's system as presented in the conclusions above.

Consistencies shown in this work open the way to the applications of the EOS in climate studies, such as distribution of volatile ices and atmospheric behaviour as Pluto moves around the Sun, within a larger scheme of global simulations. The applications can be further extended to other cold bodies that have atmospheric chemistry similar to Pluto, such as Triton (Hansen & Paige 1996), or even Titan, which is somewhat different. The work on Titan will be presented in a separate publication (Tan & Kargel 2018). Further development of CRYOCHEM could include additional crystal symmetries and additional trace constituents, which may be relevant to photolytic and radiolytic alteration and various geochemical and atmospheric evolution studies.

ACKNOWLEDGEMENTS

This work was supported by NASA Outer Planets Research Program: # NNX14AC64G. We also thank Dr. Candice Hansen for her review and suggestions that improved the presentation of this paper.

REFERENCES

APPENDIX A: THE SOLID-PHASE FREE ENERGY TERMS (Tan & Kargel 2018)

A.1. WCA hard-sphere term

The coefficients a_ij are listed in the table below:

All EOS parameters appear in equations (A2) and (A3), i.e. m (the number of hard-sphere segments in a molecule), σ (the hard-sphere diameter), ε (the LJ potential depth), and k_ij (the binary-interaction parameter between molecules of species i and j). The averaging of σ and ε in the equations are according to the Van der Waals’ one-fluid theory for mixture solutions. Even though the hard-sphere term is for FCC structure, the combination between it and the LJ perturbation term determines the structure of the LJ solids.

A.2. LJ perturbation term

This new correlation of the residual energy term is good for the reduced-temperature and reduced-density ranges of 0.1 ≤ T* ≤ 1.2 and 1.0 ≤ ρ* ≤ 1.40 to fit the MC simulations (Adidharma & Tan 2016). The coefficients q_ij are listed in the tables below for FCC and HCP structures, respectively:

The summations are carried out over the components that have FCC structures and HCP structures, respectively.

A.3. Chain term

The coefficients B_ij are correlated over the MC simulation results (Adidharma & Tan 2016):

APPENDIX B: CRYOCHEM PARAMETERS

Fluid parameters

Solid parameters

Binary-interaction parameters (k_a, k_b):

Binary parameters for fluid phases are fitted to VL equilibria data that are available in the literature. Fig. A1 shows the performance of the EOS using the parameters. For binary CH₄/N₂, see the supplementary materials of our previous paper (Tan et al. 2015).

Figure A1.

The EOS performance in representing the VL equilibria of binary mixtures: (a) N₂/CO and (b) CO/CH₄.

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The performance of the EOS using the solid parameters is presented next. The SV (sublimation) curves for pure gases are all predicted. They show excellent fits to the experimental data even though the parameters were derived from SL behaviour. The calculated triple points are in excellent agreement with the literature values in the parentheses (Fray & Schmitt 2009): 63.14 K/0.126 bar (63.14 K/0.126 bar) for N₂, 90.69 K/0.118 bar (90.69 K/0.117 bar) for CH₄, and 68.10 K/0.155 bar (68.10 K/0.154 bar) for CO. It is worth noting here that using SV data for deriving pure-component and binary parameters is not advised for accuracy reason unless there are no SL data available (see also Tan et al. 2013).

For binaries, as described in the literature, solidus data are not reliable due to the difficulties in observing the phase transition (Moran 1959; Omar et al. 1962). Therefore, the binary parameters are derived from the solubility data only in this work, liquidus for CH₄/N₂ (SL) and saturation vapour for N₂/CO (SV).

Fig. B1 shows the performance of CRYOCHEM in representing pure N₂ and pure CO, and their mixtures. The representation of pure CH₄ is the same as that in the previous work (Tan et al. 2013). The parameters of CH₄ in this work are different from before merely due to changes by the improvement of the solid part of the EOS.

Figure B1.

The performance of CRYOCHEM 2.0 in representing: (a) pure N₂ and pure CO and (b) the sublimation curves of binary CO/N₂. All symbols are data and curves are calculations.

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The two-phase SL data of N₂/CO are available in the literature but of poor quality (Ruhemann et al. 1935), so that the SV data in Fig. B1 were used instead. The performance of CH₄/CO is not shown since no data are available in the literature (see the footnote of the table of k_ij).

APPENDIX C: THE BINARY CH₄/N₂

Fig. C1(a) shows the EOS performance for the binary CH₄/N₂. The experimental data were measured under the mixture's own vapour pressure (Omar et al. 1962), i.e. at the three-phase solid–liquid–vapour (SLV) equilibrium, which is always less than 1 bar (Fig. C1b). However, since the SL equilibrium is not sensitive to the pressure, the isobaric calculations at 1 bar in Fig. C1(a) are representative for the data.

Figure C1.

The performance of CRYOCHEM 2.0 in representing the binary CH₄/N₂: (a) phase equilibria at 1 bar and (b) three-phase SLV equilibrium. Symbols are experimental data; in figure (a), open symbols are the liquidus and filled symbols are the solidus; L: liquid phase.

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Nevertheless, the pressure effect is very important for SV equilibria, as illustrated for CH₄/N₂ in Fig. C2 where the pressure is varied down to 0.1 bar. At 1 bar in Fig. C1(a), the two-phase VL region is well above the SL region, so that the vapour phase has no direct contact with solid phase at any temperatures. When the pressure decreases to 0.5 bar, the VL overlaps with SL creating the new VS region in between. The VS region expands and shifts to lower temperatures as the pressure further decreases, swallowing almost all liquid regions at 0.1 bar and removing completely the liquid phase as in Fig. 2(b) at 11 μbar. Note that the SL boundaries are not sensitive to pressure change as mentioned before.

Figure C2.

The isobaric phase diagrams of CH₄/N₂ at 0.5, 0.2, and 0.1 bar calculated using CRYOCHEM 2.0.

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The three-phase SLV equilibrium appears at low pressures starting at about 0.7 bar (Fig. C1b) when the VL touches the SL boundary. There are even two SLV's at different temperatures bracketing the two-phase VS region as the pressure further decreases, which are also shown in Fig. C2(a, b) at 0.5 and 0.2 bar. The upper SLV disappears at lower pressures, e.g. at 0.1 bar in Fig. 4(c) and 11 μbar in Fig. 2(b). Further pressure drop removes all the liquid phase leaving two-phase VS equilibria at high-temperature range while the three-phase SLV is replaced by VSS as can be inferred from the progression in Fig. C2.

APPENDIX D: REFERENCES THAT APPEAR ONLY IN THE APPENDIX AND THE FIGURES

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Din F., 1956, Thermodynamic Functions of Gases, v. 1. Butterworth, Washington, DC

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