Solubility of nitrogen in methane, ethane, and mixtures of methane and ethane at Titan-like conditions: A molecular dynamics study
SSolubility of nitrogen in methane, ethane, and mixtures of methane and ethane atTitan-like conditions: A molecular dynamics study
Pradeep Kumar , ∗ and Vincent F. Chevrier Department of Physics, University of Arkansas, Fayetteville, AR, 72701. Arkansas Center for Space and Planetary Sciences,University of Arkansas, Fayetteville, AR, 72701. (Dated: October 30, 2019)We have studied the temperature dependence of the solubility of nitrogen in methane, ethane, andmixtures of methane and ethane using vapor-liquid equilibrium simulations of binary and ternarymixtures of nitrogen, methane and ethane for a range of temperatures between 90K and 110K at apressure of 1 . PACS numbers:
INTRODUCTION
Besides earth, Saturn’s giant moon Titan is the only other planetary body in our solar system that has stable andaccessible liquid on its surface, an active hydrologic cycle similar to Earth, and a dense atmosphere with a pressureof about 1 . . Magic Islands ” [9, 10]. Hofgartner et. al. have argued thatthese features could not be image artifacts or permanent geophysical structures but are consistent with ephemeralphenomena such as suspended solids, bubbles, waves and tides on Titan[10]. Using a numerical model [11] andexisting solubility data of nitrogen in methane and ethane [12, 13], Cordier et. al. have hypothesized that thesetransient bright features may correspond to dissolution of nitrogen, giving rise to large nitrogen bubbles detected asbright objects by Cassini’s RADAR. Motivated by this, laboratory measurements of solubility of nitrogen in liquidmethane, ethane, and mixtures of methane and ethane, the primary components of seas and lakes on Titan, havebeen carried out [14, 15]. They find that the nitrogen’s solubility in methane is much larger as compared to ethane.Furthermore, the solubility of nitrogen in both methane and ethane decreases upon decreasing temperature. Basedon these results, various scenarios regarding the exsolution of nitrogen from the lakes has been proposed includingexsolution of nitrogen with increasing temperature and exsolution of nitrogen when a methane-rich region meetsan ethane-rich region. While thermodynamic models have been developed and experiments have been performed,a microscopic picture of the dissolution of nitrogen is still lacking. Standard methods for calculating solubility inthe limit of infinite dilution is Widom insertion method [16]. Since the solubility of nitrogen is very high in bothmethane and ethane, we have performed molecular dynamics vapor-liquid equilibrium (VLE) simulations [17–22] ofbinary mixtures of methane and nitrogen, ethane and nitrogen as well as ternary mixtures of methane, ethane, andnitrogen to investigate the solubility of nitrogen in methane, ethane, and mixtures of methane and ethane at Titan-likeconditions. The paper is organized as follows: In the method section, we describe our simulation approaches for all a r X i v : . [ phy s i c s . c h e m - ph ] O c t -20 -10 0 10 20 z [nm] ρ ( z )[ n m - ] EthaneNitrogen
FIG. 1: A snapshot of the nitrogen-ethane binary mixture at T = 90K (Top). Average number density profile, ρ ( z ), for ethaneand nitrogen along the z -direction (Bottom). The blue shaded represents the liquid phase region and the green shaded regionrepresents the vapor phase. the thermodynamic conditions studied here. In the Results section, we present the results for both the binary andternary systems. Finally, we conclude our studies with summary and discussion. METHOD
We have carried out vapor-liquid equilibrium (VLE) simulations of (i) binary mixtures of nitrogen and methane,(ii) nitrogen and ethane for a range of temperatures between 90K and 110K, and (iii) ternary mixtures of nitrogen,methane, and ethane containing different fractions of methane and ethane at temperature T = 90 K. All the simulationswere performed in Gromacs4.6.5 [23–25]. The trappe-UA force field [26–29] was used to model methane, and ethanewas modeled using an improved parameterization of ethane, trappe-UA2 [30]. Trappe-small parameterization wasused to model nitrogen. Lorent-Bertloth [31] rule was used to model the cross interactions of nitrogen with methaneand ethane. The short-range van der Wall interactions were treated with a cut-off of 1 . P = 1 . Lx = Ly = 5 . < L z . In such a box the interface is stable and forms alongthe smallest surface area in the xy -plane, perpendicular to the long-axis. The number of molecules of methane andethane was fixed to 3000 and 2000 for all the binary mixture simulations (nitrogen-methane, nitrogen-ethane) and thenumber for nitrogen molecules varied for different temperatures depending on the solubility and the gas phase density.If the same number of molecules of nitrogen are used for the temperatures where the solubility is small, the box sizewould be enormously large and the computational load of the Ewald summation would be very high. The equationsof motion are integrated with a time step of 2 fs and velocity rescaling is used to attain constant temperature andanisotropic Berendsen barostat for constant pressure, P zz = 1 . z -direction. After the equilibration for60 ns, we ran the simulations for additional 80 ns for each state point and the equilibrium averages are calculatedfrom these trajectories.
90 95 100 105 110
T [K] χ N SimulationsRef. 14Ref. 15Methane
90 95 100 105 110
T [K] χ N Ethane-UA2Experiments Ethane
A. B.
FIG. 2: (A) Mole-fraction, χ N , of the nitrogen in methane as a function of temperature. To compare with experiments, wealso show the data from Refs. [14, 32]. Simulation results are in reasonable agreement with the experimental data with a slightoverestimation of solubility at low temperatures. (B) Mole-fraction, χ N , of the nitrogen in ethane as a function of temperature.For a comparison, we also plot the data from Ref. [14]. Nitrogen in ethane exhibits a decrease of solubility with temperatureand are in quantitative agreement with experimental values in Ref. [14]. The error in the solubility is estimated from the errorsin the mole fractions in the liquid phase. RESULTS
Solubility of nitrogen in methane and ethane
To compute the solubility, we measure the mole-fraction of nitrogen in the liquid phase of methane/ethane in thenitrogen-methane and nitrogen-ethane binary mixtures at equilibrium. To avoid the interface, we define the liquidphase (or the gas phase) as the region where the z -derivative of the density of methane/ethane and nitrogen is zero(ses Fig. 1). We count the number of molecules of nitrogen, N (cid:96)N , in the liquid phase of methane/ethane and similarlycount the number of molecules of methane/ethane, N (cid:96)M or N (cid:96)E , in the liquid-phase for the corresponding binarymixture simulations. The solubility as measured by the mole-fraction, χ N , of nitrogen in the liquid phase is definedas χ N = N (cid:96)N (cid:0) N (cid:96)N + N (cid:96)M (cid:1) For nitrogen-methane binary system (1)= N (cid:96)N (cid:0) N (cid:96)N + N (cid:96)E (cid:1) For nitrogen-ethane binary system (2)In Fig. 2(A), we show the solubility of nitrogen in methane for temperatures T = 90 , , , T = 90K is 0 . ± .
005 [14] as compared to thevalue 0 .
278 + 0 .
002 in our simulations. In Fig. 2(B), we show the Mole-fraction, χ N , of the nitrogen in ethane asa function of temperature. For a comparison, we also plot the data from Ref. [14]. Nitrogen in ethane exhibits adecrease of solubility with temperature and are in quantitative agreement with experimental values. The solubilityvalues of nitrogen for the nitrogen-methane and nitrogen-ethane systems are also listed in Table I. Surface Tension χ M ∆ χ M χ E ∆ χ E χ N ∆ χ N T (K) M E T H A NEE T H A N EM I X T U R E χ M , ethane, χ E , and nitrogen, χ N for nitrogen-methane, nitrogen-ethane, and nitrogen-methane-ethane. The mole fractions χ M and χ E for the ternary mixture are the liquid-phase mole fractions. We also list theestimated errors, ∆ χ X for all the state points studied here.
90 95 100 105 110
T [K] σ [ m N / m ] Nitrogen-Methane
90 95 100 105 110
T [K] σ [ m N / m ] Nitrogen-Ethane
A. B.
FIG. 3: Temperature dependence of the surface tension, σ , for (A) nitrogen-methane and (B) nitrogen-ethane systems. Surfacetension for the nitrogen-methane system is smaller as compared to surface tension of the nitrogen-ethane system. Moreover, thesurface tension for the nitrogen-ethane system decreases monotonically upon decreasing temperature, while the surface tensionfor the nitrogen-methane system exhibits a non-monotonic behavior with temperature. To understand the temperature dependence of the solubility, we next studied the temperature dependence of thesurface tension, which is readily available from the molecular dynamics simulations. The surface tension, σ , is definedas σ = L z P zz − . P xx + P yy )] (3)where L z is the box-length in the z -direction and P xx , P yy , P zz are the diagonal components of the pressure tensorin the x , y , and z -directions, respectively. A factor of 2 accounts for the presence of two interfaces in the simulationbox. Figures 3 (A) and (B) show the temperature dependence of the surface tension, σ , for the nitrogen-methane and χ M χ N Simulations [T=90K]Experiments [T=91±0.5 K] A. B. χ M σ [ m N / m ] T=90K
FIG. 4: (A) Solubility of nitrogen as a function of mole-fraction of methane in the liquid state for the nitrogen-methane-ethanesystem. For comparison, we also plot the data from Ref. [14] at T = 91 ± . K . We find that solubility of nitrogen increaseswith increasing mole-fraction of methane in the liquid-phase, similar to the experimental observations [14]. (B) Surface tensionas a function of mole fraction of methane in the liquid state in a mixture of methane and ethane. Solubility of nitrogen increasesas the mole fraction of methane in the mixture increases while the surface tension decreases upon increasing mole fraction ofmethane. Our values of surface tension are agree well with the experimental values [33–36]. the nitrogen-ethane binary mixtures, respectively. We find that the surface tension of nitrogen-methane interface isabout 8 mN/m smaller than the surface tension of the nitrogen-ethane interface for all the temperatures investigatedhere. Furthermore, we find that σ decreases with decreasing temperature for the nitrogen-ethane system. For thenitrogen-methane system, surface tension shows a non-monotonic dependence on temperature owing to larger partialpressure of methane at T ≥ Solubility of nitrogen in a mixture of methane and ethane
We next studied the solubility of nitrogen in a ternary mixture of methane, ethane, and nitrogen containing differentfractions of methane/ethane at T = 90 K and P = 1 . χ M , in the liquidphase is defined as χ M = N (cid:96)M N (cid:96)M + N (cid:96)E (4)where, N (cid:96)M , N (cid:96)E , and N (cid:96)N are the numbers of molecules of methane, ethane, and nitrogen in the liquid-phase, re-spectively. Similarly the solubility as measured by the mole-fraction, χ N , of nitrogen in the liquid-phase is definedas χ N = N (cid:96)N N (cid:96)M + N (cid:96)E + N (cid:96)N (5)To compare our simulation results with experiments, we also show the solubility data from Ref. [14]. We find thatsolubility of nitrogen increases with increasing mole-fraction of methane in the liquid-phase, similar to the experimentalobservations [14]. Simulation results are in good agreement with the experimental data with slight overestimationof the solubility for all the mole-fractions of methane. In Fig. 4(B), we show the behavior of surface tension as afunction of mole-fraction of methane in the liquid-phase. We find that the surface tension decreases with increasingmole-fraction of methane. The solubility values of nitrogen for the nitrogen-methane-ethane system are summarizedin Table I. -8 -6 -4 -2 0 2 4 6 8 z [nm] ρ ( z )[ n m - ] T=90K -8 -6 -4 -2 0 2 4 6 8 z [nm] ρ ( z )[ n m - ] T=95K -6 -4 -2 0 2 4 6 z [nm] ρ ( z ) [ n m - ] T=90K
Nitrogen-MethaneNitrogen-Ethane -6 -4 -2 0 2 4 6 z [nm] ρ ( z )[ n m - ] T=95K
FIG. 5: Number density, ρ ( z ), of nitrogen (solid red curve) and methane/ethane (solid black curve) for the nitrogen-methaneand nitrogen-ethane system for two different temperatures T = 90 K and T = 95 K. A strong temperature-dependent surfaceadsorption of nitrogen is observed in both systems. Adsorption of nitrogen at the interface
In this section, we investigate the adsorption of nitrogen at the nitrogen-methane and the nitrogen-ethane interface.In Fig. 5, we show the density profile of nitrogen and methane/ethane as a function of temperature for both nitrogen-methane and nitrogen-ethane systems at two different temperatures T = 90K and T = 95K . We find that adsorptionof nitrogen at the interface between vapor and liquid phase is very high and increases upon decreasing temperature.Furthermore, the degree of adsorption of nitrogen at the nitrogen-ethane interface is much higher as compared tothe nitrogen-methane interface whereas the number density of nitrogen reaches approximately the number density ofliquid ethane. Many other studies of liquid-gas interfaces have also found a strong adsorption of gases at the gas-liquidinterface [18, 37].The partition coefficient, K , of two phases is defined as the ratio of the number density of the phases. Here we candefine a z -dependent partition coefficient, K ( z ) K ( z ) = ρ ( z ) ρ v (6)where ρ ( z ) is the density profile along the z -direction and ρ v is the density of the nitrogen in the vapor phase.
90 95 100 105 110
T[K] -3.6-3.4-3.2-3.0-2.8 ∆ G a d s / R T -7 -6 -5 -4 -3 -2 z [nm] -4-3-2-10 ∆ G ( z ) / R T T=90KT=95KT=100KT=105KT=110KLiquidVapor Interface -6 -5 -4 -3 -2 -1 z [nm] -4-3-2-10 ∆ G ( z ) / R T T=90T=95KT=100KT=105KT=110KLiquidInterfaceVapor
90 95 100 105 110
T[K] -3.6-3.4-3.2-3.0 ∆ G a d s / R T A. B.C. D.
FIG. 6: Free energy profile, ∆ G ( z ), at different temperatures for (A) nitrogen-methane and (B) nitrogen-ethane system. Thedata is only shown for the z -values close to the interface so that one can observe the vapor, the interface and the liquid regions.Free energy of adsorption, ∆ G ads /RT , as a function of temperature for (A) nitrogen-methane, and (D) nitrogen-ethane systems. Consequently, one can define excess free energy, ∆ G ( z ), over the free energy of the vapor-phase as∆ G ( z ) = − RT log K ( z ) (7)where R is the universal gas constant and T is the temperature. In Figs. 6(A)(B), we show ∆ G ( z ) for differenttemperature for nitrogen-methane and nitrogen-ethane systems We find that ∆ G ( z ) decreases upon entering theadsorbate region and increases slightly and levels off in the liquid region, suggesting the free change in adsorbateregion is larger compared to that in the liquid-region and hence adsorption at the interface. Moreover, we find thatthe value of ∆ G ( z ) decreases upon decreasing temperature and hence stronger adsorption at lower temperatures. Tofind the free energy associated with the exchange with the adsorbate region, ∆ G ads , one must define the effectivedensity, ¯ ρ I , of the adsorbate region. We define the adsorbate region as the region between the vapor and liquid phasesin which the derivative of nitrogen-density is non-zero. This definition of characterizing interface, gas, and liquidregions are adopted from Ref. [17] . ∆ G ads is defined as∆ G ads = − RT log ¯ ρ I ρ v (8)In Figs. 6(C)(D), we show ∆ G ads as a function of temperature for nitrogen-methane and nitrogen-ethane systems,respectively. We find that ∆ G ads values for both nitrogen-methane and nitrogen-ethane are similar and decreaseupon decreasing temperature, and consequently increasing propensity of adsorption of nitrogen at the interface. Notethat the value of ∆ G ads is smaller than the free energy difference between the gas and the liquid phase. We havesummarized the values of ∆ G ads at different temperatures for both systems in Table II. Orientational ordering of the ethane and nitrogen at the interface
T (K) ∆ G ads /RT ∆ G ads /RT Nitrogen-Methane Nitrogen-Ethane90 -3.59 -3.5695 -3.21 –3.41100 -3.00 -3.26105 -2.90 -3.17110 -2.77 -3.02TABLE II: ∆ G ads for nitrogen-methane and nitrogen-ethane systems, respectively. -6 -5 -4 -3 -2 -1 z [nm] -0.050.000.050.10 S EthaneNitrogenT=90K
FIG. 7: Orientational order parameter, S , for ethane (black circles) and nitrogen (red squares) for nitrogen-ethane mixture at T = 90K. Also shown are the rescaled density profiles of nitrogen and ethane along the z -axis in red and black solid curves,respectively. For clarity we only show the region near the interface. Both ethane and nitrogen lying in the adsorbate regiontend to have a preferential orientation minimum and a maximum in S in the adsorbate region. We next investigate if the nitrogens and ethanes lying in adsorbate region exhibit any preferential orientationalordering. Recent studied of hexane-water system suggest that hexane tend to exhibit a preferred orientation in theadsorbate region [18]. To measure the degree of orientational ordering of of nitrogen and ethane at the interface, weuse an orientational order parameter, S, as [18] S = 12 (cid:10) cos θ − (cid:11) (9)where θ is the angle formed by the nitrogen-nitrogen bond (nitrogen molecule) or carbon-carbon bond (ethanemolecule) with the normal to the interface. This order parameter is closely related to the NMR order parame-ter [38–40]. If the molecules are oriented parallel to the interface then the value of S reaches its maximum value 1,and if they are oriented perpendicular to the interface, the value of S reaches its minimum value − .
5. The value of S is zero if they are oriented randomly or are oriented at the magic angle, 57 . ◦ . In Fig. 7, we show the value of S asa function of z for nitrogen-ethane system at T = 90K. We find that S for nitrogen is zero both in the liquid and thevapor-phase suggesting random orientation of nitrogen in these phases. However, nitrogens are weakly aligned alongthe normal to the interface close to the gas-phase whereas S has weak negative minimum and weakly aligned alonginterface in the region close to the liquid-phase. The value of S for the ethane shows a similar behavior but withlarger preference of alignment normal to the surface in the region close to the gas-phase. The value of S for ethanein the liquid-phase is zero suggesting a random orientation of the molecules. Summary and Discussion
We have studied the temperature dependence of the solubility of nitrogen in methane, ethane, and mixtures ofmethane and ethane by performing extensive vapor-liquid equilibrium simulations of binary and ternary mixturesof nitrogen, methane and ethane for a range of temperatures between 90K and 110K at a pressure of 1 . Author Contributions
PK designed, performed the research, and analyzed the data. PK and VFC wrote and revised the manuscript.
Acknowledgment
Authors would like to thank University of Arkansas High Performance Computing Center for providing com-putational time. V. F. Chevrier acknowledges funding from NASA Cassini Data Analysis Program grant no.NNX15AL48G. ∗ Electronic address: [email protected][1] Jonathan I Lunine and Sushil K Atreya. The methane cycle on titan. nature geoscience , 1(3):159 2008.[2] Ralph Lorenz and Jacqueline
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