Diffusion of CH_4 in amorphous solid water
Belén Maté, Stephanie Cazaux, Miguel Angel Satorre, Germán Molpeceres, Juan Ortigoso, Carlos Millán, Carmina Santonja
AAstronomy & Astrophysics manuscript no. aanda_revised_2 © ESO 2021March 1, 2021
Diffusion of CH in amorphous solid water Belén Maté , Stephanie Cazaux , , Miguel Angel Satorre , Germán Molpeceres , Juan Ortigoso , Carlos Millán , andCarmina Santonja Instituto de Estructura de la Materia, IEM-CSIC, Calle Serrano 121, 28006 Madrid, Spaine-mail: [email protected] Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands Leiden Observatory, Leiden University, P.O. Box 9513, NL 2300 RA Leiden, The Netherlandse-mail: [email protected] Escuela Politécnica Superior de Alcoy, Universitat Politècnica de València, 03801 Alicante, Spaine-mail: [email protected] Institute for Theoretical Chemistry, University of Stuttgart, 70569 Stuttgart, Germanye-mail: [email protected]
March 1, 2021
ABSTRACT
Context.
The di ff usion of volatile species on amorphous solid water ice a ff ects the chemistry on dust grains in the interstellar mediumas well as the trapping of gases enriching planetary atmospheres or present in cometary material. Aims.
The aim of the work is to provide di ff usion coe ffi cients of CH on amorphous solid water (ASW), and to understand how theyare a ff ected by the ASW structure. Methods.
Ice mixtures of H O and CH were grown in di ff erent conditions and the sublimation of CH was monitored via infraredspectroscopy or via the mass loss of a cryogenic quartz crystal microbalance. Di ff usion coe ffi cients were obtained from the experi-mental data assuming the systems obey Fick’s law of di ff usion. Monte Carlo simulations modeled the di ff erent amorphous solid waterice structures investigated and were used to reproduce and interpret the experimental results. Results. Di ff usion coe ffi cients of methane on amorphous solid water have been measured to be between 10 − and 10 − cm s − for temperatures ranging between 42 K and 60 K. We showed that di ff usion can di ff er by one order of magnitude depending on themorphology of amorphous solid water. The porosity within water ice, and the network created by pore coalescence, enhance thedi ff usion of species within the pores.The di ff usion rates derived experimentally cannot be used in our Monte Carlo simulations toreproduce the measurements. Conclusions.
We conclude that Fick’s law can be used to describe di ff usion at the macroscopic scale, while Monte Carlo simulationsdescribe the microscopic scale where trapping of species in the ices (and their movement) is considered. Key words. Di ff usion – solid state:volatiles – Methods:laboratory:molecular – Methods:numerical – Planets and satellites:surfaces– ISM:molecules
1. Introduction
Icy mantles covering dust grains in dense clouds of the interstellar medium are known to be responsible for the large molecularcomplexity of our universe. Within those mantles, atoms and molecules can meet and react with higher probability than in the gasphase. The chemical reactivity of interstellar ice is limited by the di ff usion of reacting atoms or molecules in water ice, its majorcomponent (Tielens & Hagen 1982). In particular the di ff usive Langmuir-Hinshelwood mechanism is believed to be the primaryformation process on ice mantles (Irvine (2011)). Therefore knowing the barriers to di ff usion of the precursors of a reaction isof key importance. However, due to the multiple factors that a ff ect this process, di ff usion coe ffi cients are di ffi cult to obtain, bothexperimentally or theoretically. Due to the lack of accurate information, it is frequently assumed in astrochemical models that thedi ff usion energy is a fraction of surface binding energy of the molecule (Herbst 2001; Garrod et al. 2008). This fraction is verypoorly constrained and values between 0.3 and 0.8 are used by the modeling community (Hasegawa et al. 1992; Cuppen et al. 2017;Garrod 2013).Despite the complexity of the problem, di ff erent experiments have been carried out to obtain information about di ff usion (Heet al. 2018; Ghesquière et al. 2015). In particular, water ice is one the most investigated systems. Laser-induced Thermal Desorptiontechniques (LITD) were employed to investigate isotopic di ff usion (HDO or H
0) or molecular di ff usion (HCl, NH , CH OH)in and on crystalline water ice (see Livingston et al. (2002) and references therein). More recently, other methodologies havebeen developed, based on infrared spectroscopy, to study di ff usion in amorphous solid water (ASW) (Mispelaer et al. (2013),Karssemeijer et al. (2014), Lauck et al. (2015), Ghesquière et al. (2015), Cooke et al. (2018), He et al. (2018)). Surface di ff usioncoe ffi cients for CO, NH , H CO and HNCO in ASW were given by Mispelaer et al. (2013). The di ff usion of CO in ASW at Article number, page 1 of 17 a r X i v : . [ a s t r o - ph . I M ] F e b & A proofs: manuscript no. aanda_revised_2 temperatures between 12 and 50 K was investigated by Karssemeijer et al. (2014) and Lauck et al. (2015), and information aboutCO di ff usion was given by Ghesquière et al. (2015), He et al. (2017), and Cooke et al. (2018). Activated energies of di ff usion ofCO, O , N , CH and Ar on ASW were provided in a recent work by He et al. (2018). In the discussions presented in those papersquestions have raised on how the di ff usion is a ff ected by factors like amorphous water ice reorganization, porosity, layer thickness,or sublimation processes.In this work, the experimental methodology proposed by Mispelaer et al. (2013) based on infrared spectroscopy and a new exper-imental approach based on quartz crystal microbalance (QCMB) measurements, have been combined with Monte Carlo simulationsto investigate the di ff usion of CH in amorphous solid water (ASW).
2. Experimental part
The experiments were carried out in two experimental setups, one at IEM-CSIC-Madrid and the other at UPV-Alcoy. The new exper-imental approach based in QCMB detection was performed at UPV-Alcoy. Similar experiments were conducted in both laboratoriesto test the viability of the new approach, which has the advantage of allowing measuring di ff usion coe ffi cients of non-IR-activespecies. This experimental setup has been described in detail in previous publications (Maté et al. (2018), Gálvez et al. (2009), Herrero et al.(2010)). It consists of a high vacuum chamber evacuated to a base pressure of 1 x 10 − mbar and provided with a closed cycle Hecryostat. A silicon substrate 1 mm thick is placed in a sample holder in thermal contact with the cold head of the cryostat, and itstemperature can be controlled between 15 K and 300 K with 0.5 K accuracy. Infrared spectra are recorded in normal transmissionconfiguration with a FTIR spectrometer (Bruker vertex70) provided with an MCT detector. In this work we have recorded spectraat 4 cm − resolution averaging 100 scans. Controlled flows of CH (99.95 percent, Air Liquide) and H O (distilled water, threetimes freeze-pump-thawed) can be admitted through independent lines to back-fill the chamber. The CH line is provided with amass-flow controller (Alicat) while the H O gas flow is controlled with a leak valve. Ices were grown by vapor deposition on bothsides of the cold Si substrate, at a rate of approximately 1 nm / s.In order to investigate CH di ff usion on ASW a procedure inspired by Mispelaer et al. (2013) has been followed. Initially, icesof methane and water were grown at 30 K at rates that range between 1 and 1.5 nm / s. In some cases a two layer system (firstCH , then H O on top) was generated; in other cases, mixed ices were formed by simultaneous deposition of both gases. Then,the CH :H O system was warmed at a controlled rate, between 5 K / min and 20 K / min, to a temperature, T iso , above methanesublimation temperature. In most of the experiments performed in this work T iso =
50 K. At this temperature CH molecules di ff usethrough the pores of the amorphous solid water structure until they reach the surface of the ice and sublime instantly. Infraredspectra were recorded as function of time elapsed at T iso (time zero is set when T iso is reached). The time decay of the intensityof the ν band of CH informs of the number of molecules that have moved through the amorphous water ice layer and have leftthe sample. Di ff erent experiments varying ice thicknesses, CH / H O ratio, heating rates, and growing configurations (sequential orsimultaneous) were performed. A list of them is given in Table 1.The temperature range chosen to perform the experiments is conditioned by the sublimation temperature of CH and experi-mental limitations. The deposition temperature of 30 K was selected to be the highest one where there is no substantial methanesublimation in our experimental conditions. The isothermal experiment temperature of 50 K was chosen to observe most of themethane di ff usion / sublimation in less than one hour. At lower temperatures, CH di ff usion and sublimation takes longer times, andthe growing of a ASW layer due to background water contamination present in the HV setups, will a ff ect the results in a not negligi-ble way. With a base pressure of ∼ (0.7–1) × − mbar, water vapor from the chamber deposited on our samples at an approximaterate of ∼ (3–6) × molecules cm − hr − .”Ice layer thicknesses were determined from the IR absorbance spectra and infrared band strengths. The OH-stretching band at3200 cm − and the ν mode at 1300 cm − were chosen for H O and CH , respectively. The absorption coe ffi cients of those bandsare A = × − cm molec − (Mastrapa et al. (2009)), and A = . × − cm molec − (Molpeceres et al. (2017)),respectively. Densities of 0.65 g cm − (Dohnálek et al. (2003)) for ASW and 0.46 g cm − for CH (Molpeceres et al. (2017)) wereassumed, for ices grown by background deposition at 30 K. In the case of ice mixtures grown by codeposition, the determinationof ASW film thickness is less accurate, due to the lack of information about band strengths and densities for these mixed systems.However, considering the pure species values for the band strengths is expected to lead to less than 20 percentage error (Kerkhofet al. (1999), Gálvez et al. (2009)). For a given molecular ratio, we have taken the corresponding average density. The estimateuncertainty in the codeposited mixtures ice thickness is 20%. The experimental setup has been described in detail in previous publications (Luna et al. (2012)). It consists of a high vacuumchamber that reaches 5 × − mbar background pressure, where a closed He cryostat allows cooling down a sample holder to 13 K.A quartz crystal microbalance (QCMB) is located in thermal contact with the cold head, acting as part of the sample holder. Aresistor permits to vary the temperature of the cold head between 13 K and room temperature, within 0.2 K by means of a ITC-301temperature controller (Oxford Instruments) and two silicon diodes (Scientific Instruments). One of them is located just below aQCMB and the other just at the end of the second stage of the cold finger of the Leybold Cryostat. The system is provided alsowith an He-Ne double laser interference system ( λ = Article number, page 2 of 17elén Maté et al.: Di ff usion of CH in amorphous solid water Exp. L AS W / L CH ramp T iso % CH % CH % CH nm / nm K / min K 30 K ini T iso fin T iso M1 s 146 /
19 5 50 17.3 10.6 7.3M2 s 228 /
17 5 50 6.5 5.9 3.1M3 s 275 /
36 20 50 13.4 10.5 8.2M4 s 344 /
16 10 50 3.6 3.6 2.4M5 s 410 /
40 10 50 8.9 7.7 5.3M6 s 453 /
47 5 50 9.2 8.3 5.8M7 c 185 10 50 13.8 12.1 7.3M8 c 201 5 50 13.2 11.4 7.0M9 c 212 5 50 12.0 8.8 5.7M10 c 567 5 55 7.6 6.3 4.6M11 s 487 /
37 5 55 6.5 6.0 4.9M12 c 467 5 60 5.3 4.8 3.9
Table 1.
List of experiments performed at IEM-CSIC-Madrid. s or c stands for sequential or codeposited experiments, respectively. L CH and L ASW refer to thickness of CH and H O layers (s experiments), or to the mixture ice thickness (c experiments), at 30 K. Columns six, seven andeight present the number of molecules ratio in percentage (100x N CH / N H O ) of CH at 30 K, at the beginning, and at the end of the isothermalexperiment. QCMB quartz surface. By measuring the laser interference patterns during ice growing, it is possible to determine the real part ofthe refractive index of the ices deposited, at the laser wavelength, and the thicknesses of the ice layers generated. When vapors areintroduced in the vacuum chamber an ice layer grows on top of the QCMB. The QCMB measures any variation in mass depositedon the surface of the quartz crystal. As the mass increases (adsorption, deposition) the vibration frequency of the quartz decreases; ifthe mass decreases (desorption) the quartz vibrational frequency increases. There is a linear dependency between ∆ F , the variationof the quartz crystal frequency, and ∆ m , the mass variation on the quartz crystal surface, the so-called Sauerbrey relationship: ∆ F = − S ∆ m (1)where S is the Sauerbrey’s constant.The procedure employed to investigate CH di ff usion is in essence the same than the one used in the Madrid experiments. Icesof CH and H O were grown by codeposition or sequential deposition (first CH , then H O on top) at 30 K, a temperature wellbelow CH sublimation, and then they were warmed up to T iso , above methane sublimation temperature. At this T iso , the amount ofmethane that di ff uses through the porous structure of ASW and leaves the ice is monitored registering the frequency variation of theQCMB. As explained above, the variation of the quartz crystal frequency is proportional to the variation of the mass deposited on itssurface (see equation 1). In particular, a mass loss along time will be observed as a QCMB frequency increases versus time. In thiscase, instead of recording the time evolution of the IR spectra of the ice, the QCMB is providing this information. It is interesting tonotice that this new experimental approach, opens the possibility to study di ff usion of volatile species that do not have IR spectrum,like H , N or O .Special care was made in using similar deposition rates, CH :H O ratios, and heating ramps in the experiments performed inboth laboratories, in order to facilitate comparison of the results. Nonetheless, larger ASW layer thickness, up to 1 micron, werecovered in the experiments performed in Alcoy. This set of Alcoy experiments has been named A1 to A10, and are listed in Table 2.In an attempt to study the influence of the temperature on the di ff usion coe ffi cient, and extract the energy of the di ff usion barrierand the preexponential factor, avoiding the e ff ect of having distinct water ice structures, a di ff erent set of experiments was designed.In this case, ASW was grown at 50 K (background deposition), was kept several minutes at that temperature, cooled down to 30K to deposit a CH layer on top, and finally warmed to the desired T iso , that ranged between 42.5 K and 52.5 K. At T iso the massloss versus time is monitored with the QCMB. Since ASW ice grown at 50 K do not change its structure for being cooled to 30 Kand then warmed back to 50 K, in these experiments CH di ff usion is measured in ASW ices of identical structure. When CH isdeposited on ASW at 30 K, it di ff uses through ASW pores. During the warm up process to T iso , most of the methane is sublimatedand only the fraction that is within the pores is retained. Since the amount of CH retained is very low, the measurements have largererrors than with the previous methodology. These experiments were labeled AA1-AA5.
3. Experimental results
In this section, a first subsection is dedicated to discuss the morphology of the di ff erent ices investigated via the analysis of theirOH-dangling-bond IR spectra. This analysis gives some qualitative information about the quality of the approximations made forthe determination of the di ff usion coe ffi cient in the following subsections. In a second subsection, the isothermal experiments arepresented, and the last one is devoted to describe the models employed to extract di ff usion coe ffi cients from them. Porosity of amorphous solid water formed by vapour deposition on a cold surface depends on the growing conditions. It is a ff ectedby the surface temperature, the vapour pressure, or the angle at which the molecules hit the surface (Berland et al. (1995), Brown Article number, page 3 of 17 & A proofs: manuscript no. aanda_revised_2
Fig. 1.
IR spectra of the dangling bond vibrations of water for CH :H O mixtures grown at 30 K and warmed to 50 K at 5 K / min, compared withthe spectrum of pure water ice grown at 30 K. Top panel: layered ice:H O on top of ASW. Bottom panel: codeposited ice. All spectra correspondto ASW layers of about 200 nm & George (1995), Dohnálek et al. (2003), Bossa et al. (2014)). The intrinsic density of ASW also changes, it varies between 1.1 gcm − for ices grown below 35 K and 0.94 g cm − for ices grown at T >
70 K. High density amorphous ice evolves to low densityice between 38 K and 68 K, (Narten et al. (1976), Jenniskens et al. (1995)). The measured average density of ASW grown bybackground deposition at 30 K is 0.65 g cm − Dohnálek et al. (2003). Assuming its intrinsic density is that of the low density ice(0.94 g cm − ), its porosity can be estimated via the expression: ρ = − ( ρ a /ρ i ) = in codeposited CH :H O mixtures, will a ff ectASW morphology at 30 K, and upon warming. These e ff ects will be investigated in this work by Monte Carlo simulations (seesection 6).Free OH bonds, know as OH-dangling bonds (DB), appear on the surface of the porous of the amorphous water ice. Theyprovide information about the porosity of the ice, and about how CH molecules will di ff use through the porous structure. Figure1 shows the DB spectral region for sequential and codeposited H O:CH ices grown at 30 K and warmed to 50 K. On one hand,in sequential experiments it is observed that even at 30 K CH molecules di ff use through the ASW structure, as was previouslyreported (Raut et al. 2007; Herrero et al. 2010). In particular, a new peak appears in the water DB region (see top panel, Figure 1).The pure water DB double peak at 3720 cm − and 3696 cm − (blue trace), transforms into a triple peak feature with maxima at 3720cm − , 3692 and 3665 cm − (black trace). An overall increase in DB peaks intensity is also noticed. Since the number of water DBshas not changed, the intensity increase must be due to an increase of the infrared band strength of the dangling OH mode when itis interacting with CH . This e ff ect was already shown in previous publications (Herrero et al. (2010)). The peak at 3665 cm − ischaracteristic of the interaction of CH molecules with dangling bonds of water. Its presence evidences that CH molecules havealready di ff used into ASW pores at 30 K. Subsequently, during the warm up of the ice to 50 K, the pure ASW 3720 cm − peakdisappears, and an intensity decrease of the DB features is noticed (red trace in top panel Figure 1). The 3665 cm − peak stronglydominates the profile at 50 K. The intensity decrease reflects that some CH is lost upon warming, but also it is related to a decreasein the specific surface area (SSA) of the ASW ice when warmed from 30 K to 50 K, as shown by simulations (see section 6).On the other hand, the lower panel in Figure 1 shows the DB features observed when a CH :H O ice is grown by simultaneousdeposition at 30 K (black trace). In particular, the pure ASW DB feature at 3720 cm − and most of the 3696 cm − peak (blue trace)are missing in the 30 K codeposited ice. This indicates that methane molecules are already covering most of the ASW pore surfacein this kind of mixtures. When the mixture is warmed to 50 K, no changes in the DB spectral profile occur, and only a decreasein intensity is observed (red trace). This decrease, similarly to what happened in sequential ices, is due to some CH loss, but alsoreflects a decrease of the ASW specific surface area. In relation with the overall intensity of the DB features in CH :H O codepositedices, it should be pointed out that, as described in previous works (Gálvez et al. (2009), Herrero et al. (2010)), the DB intensity inthe mixtures is always larger than that of the pure ice, and increases with CH concentration.Most of the water ice reorganization takes place during the warm up process from 30 K to T iso , nonetheless ASW morphologychanges may continue during the time elapsed at that T iso . This is a something to be aware with when analyzing the experimentaldata, since it will be assumed (see section on di ff usion modeling) that methane molecules move through the pore surface of an ASWfix structure. The DB evolution at T iso might give an idea of the importance of that reorganization. Figure 2 shows the DB featurefor sequential and codeposited ices at di ff erent times during the isothermal experiment. The main spectral variation shown in bothpanels of Figure 2 is a decrease of the 3665 cm − band, that is associated with CH loss. The rest of the DB feature su ff ers minormodifications indicating ASW reorganization is not significant during the isothermal experiment.Following the intensity decay of the 1300 cm − band, it was possible to quantify the fraction of CH that leaves the ice duringthe heating of the ice mixture via IR spectroscopy. Table 1 displays the CH / H O number of molecules ratio at 30 K; at T iso at thebeginning of isothermal experiment; and at the end of the isothermal experiment. Looking at columns 6 and 7 it is observed that a
Article number, page 4 of 17elén Maté et al.: Di ff usion of CH in amorphous solid water Fig. 2.
IR spectra of the dangling bond vibrations of water for CH :H O mixtures grown at 30 K and warmed to 50 K. Spectra taken at 50 K, atdi ff erent times during the isothermal experiments. Top panel: layered ice:H O on top of ASW. Bottom panel: codeposited ice. fraction CH is already lost at the beginning of T iso , which indicates CH is distributed through the whole ASW top layer at this time.Comparing columns 7 and 8 it can be seen that the fraction of methane that left the ice during the isothermal experiment is small, a20 % on average. Most methane remains trapped in the ASW structure at the end of the isothermal experiment, both in sequentialand in codeposited ices, as the experiments monitor only the methane that di ff use through the open canals (pores) of this structure.As it is well known from thermal programmed desorption measurements of H O:CH mixtures, a release of methane around 40-50K is followed by a second desorption peak around 140 K (associated to the amorphous-to-crystalline water phase change, "volcano"desorption), and a last desorption peak appears together with water sublimation, revealing that even a small fraction of CH stays inthe water ice structure until its sublimation (see for example May et al. (2013)). in ASW. Figure 3 shows the decay of the intensity of the 1300 cm − band of methane versus time elapsed at 50 K for the set of experimentspresented in Table 1. The integral has been normalized to the band intensity at the beginning of the isothermal experiment. Thepanels in the first two rows correspond to sequential deposited ices, with a water ice thickness layer that varies between 145 nmand 480 nm. Panels in the bottom row display the codeposited ice experiments with ASW thickness of ∼
200 nm. Three isothermalexperiments, performed at 50 K, 55 K and 60 K, corresponding to layers ∼
500 nm thick, are presented in Figure 4.In order to extract di ff usion coe ffi cients from these data we have model the decays presented in Figures 3 and 4 using Fick’ssecond law of di ff usion, as will be described below in subsection 3.3. Figure 5 presents the variation of the frequency of the QCMB, showing the signal increase, which is proportional to mass decrease,versus time elapsed at 50 K. In an experiment, the larger the frequency variation, the larger the fraction of methane molecules thathave been sublimated from the ice (Equation 1). Frequency variations have been scaled to plot these experiments in the same graph.Experiments cover the range of thicknesses studied, between 300 nm and 1000 nm. These experiments have been analyzed usingthe first Fick’s law.
Fick’s second law of di ff usion, that relates the unsteady di ff usive flux to concentration gradient, has been applied to obtain the di ff u-sion coe ffi cient, D ( T ), at a given temperature T. In a one-dimensional system where D ( T ) does not depend on z , i.e., ∂ D ( T ) /∂ z = ∂ n ( z , t ) ∂ t = D ( T ) ∂ n ( z , t ) ∂ z (2)where n ( z , t ) is the concentration of the di ff using species at depth z , in the ASW ice at a given time t . Equation 2 is used to describe themethane molecules di ff usion along a fixed ASW layer of known thickness, i.e., in a direction perpendicular to the surface substrate.This approach was shown to give good results to describe similar systems by other authors (Mispelaer et al. 2013; Karssemeijeret al. 2014; Lauck et al. 2015; He et al. 2018). Some initial conditions must be imposed in order to obtain adequate solutions of thisequation. It will be assumed that all CH molecules that reach the surface of the ice layer desorb immediately, therefore n ( L , t ) = molecules can escape from the bottom of the film, ∂ n (0 , t ) /∂ t =
0. As a last
Article number, page 5 of 17 & A proofs: manuscript no. aanda_revised_2
Fig. 3.
Normalized decay of the intensity of the 1300 cm − band of CH versus time elapsed at 50 K. Diamonds: experimental data, line:fit to thesecond’s Fick di ff usion law. condition, the concentration of CH will be assumed homogeneous within the amorphous solid water ice layer at the beginning ofthe isothermal experiment, n ( z , = n . In particular, for sequential experiments, this last condition implies that during the heatingprocess methane has di ff used into the ASW top layer filling the pores homogeneously. Sequential experiments were modelled alsoimposing a di ff erent boundary condition. It was assumed that methane is no homogeneously distributed in the ASW layer, but thatit is in a bottom layer with the ASW layer on top. The fits obtained in this way were worse than those found with the previousapproximation. The solutions of equation 2 for the constraints of our experiment are (Crank (1975)): n ( z , t ) = ∞ (cid:88) n = n ( − i µ i L cos ( µ i z ) exp (cid:16) − µ i D t (cid:17) (3)where µ i = (2 i + π L . (4)From these expressions, the column density of methane molecules in the ice can be obtained by integrating the concentration c ( z , t )over the ice thickness L . The column density is proportional to the area of a particular CH band in the absorbance spectra, A ( t ),and therefore, integrating over z , the solution can be expressed as (Karssemeijer et al. (2014)): A ( t ) = s + ∞ (cid:88) n = A − s ) µ i L exp (cid:16) − µ i D t (cid:17) (5)where A is the initial band area and s is an o ff set related with the amount of methane that remains trapped in the ice.Unweighted least squares fitting of the experimental data presented in Figures 3 and 4 to equation 5 have been performed. Thesolutions are represented with solid lines in the mentioned Figures. The di ff usion coe ffi cients obtained are given in Table 2. Article number, page 6 of 17elén Maté et al.: Di ff usion of CH in amorphous solid water
500 1000 1500 2000 25000,850,900,951,00
M6M12M10 I n t ( c m - ) /I n t time at T iso (s) Fig. 4.
Normalized decay of the intensity of the 1300 cm − band of CH versus time elapsed at 50 K, 55 K and 60 K. Scatter points: experimentaldata, lines: fit to the second’s Fick di ff usion law. D F r eq ( a . u . ) time at 50 K (s) A5 A3A1
Fig. 5. Di ff usion of methane from ASW ice grown at 30 K. QCMB signal variation versus time elapsed at 50 K. Black lines: experimental data,Red lines:linear fit employed to derive di ff usion coe ffi cients form the first Fick’s law Fick’s first law describes the relation between the flux of di ff usion through a surface and the concentration gradient perpendicularto that surface, under steady-state conditions. Fick’s first law is given by: J = − D ( T ) dn ( z ) dz (6)were J is the flux in molec cm − s − , and n ( z ) is the gas concentration in molec cm − , and both magnitudes do not change with time.Isothermal experiments provide the number of methane molecules, n ( t ), that leave the ice at a particular time. Therefore, it ispossible to calculate the methane flow, J ( t ) = dn ( t ) / dt . Looking at the final steps of the di ff usion experiments, the variation ofthe flux with time is very slow, and it can be considered almost constant. Then, in that region Fick’s first law can be applied toextract D ( T ). Another approximation has to be made in order to estimate the concentration gradient. At times where the methaneflow is considered constant, a constant linear distribution of methane through the ASW layer thickness will be assumed. The CH concentration will be expressed as: n ( z ) = a + bz , where n (0) = a at the cold deposition surface, and n ( L ) = N , (molec cm − ) in the ice will be given by: N = z = a (cid:90) z = ( a + bz ) dz (7) Article number, page 7 of 17 & A proofs: manuscript no. aanda_revised_2
From this expression and Fick’s first law, the di ff usion coe ffi cient is given by: D ( T ) = − J L N (8)where L is the ASW layer thickness and J the methane flow. There is always a fraction of methane molecules that remain trappedin the ice at the end of the di ff usion experiments. For the estimation of N , that fraction is not taken into account, and only theCH that intervenes in di ff usion is considered. To obtain the di ff usion coe ffi cients given in Table 2 for the A1-A10 and AA1-AA5experiments, a linear fit of the QCMB signal at the final part of the isothermal experiments has been performed (see Figure 5). Theslope of the fit gives the methane flow that is necessary to derive D ( T ) with equation 8.Experiment T iso L AS W / L CH D(2 nd Fick) D(1 rst
Fick)(K) (nm / nm) (10 − cm s − ) (10 − cm s − )M1 50 146 /
19 1.7 1.0M2 50 228 /
17 2.7 1.3M3 50 275 /
36 3.7 1.6M4 50 344 /
16 2.5 2.1M5 50 410 /
40 8.1 4.0M6 50 453 /
47 10.3 4.0M7 50 185 1.4 0.9M8 50 201 1.8 0.6M9 50 212 1.8 0.8M10 55 567 26.9M11 55 487 /
37 15.4M12 60 469 44.9A1 50 333 9A2 50 333 1.2A3 50 476 6.96A4 50 520 6A5 50 827 30A6 50 913 39A7 50 1026 57A8 50 1033 42A9 50 223 /
169 1.56A10 50 378 /
33 5.38 L CH / L AS W
AA1 42.5 335 /
951 0.226AA2 45 355 /
946 1.03AA3 47.5 353 /
930 0.929AA4 50 358 /
945 3.62AA5 52.5 356 /
953 4.23
Table 2. CH di ff usion coe ffi cients on ASW grown at 30K (M1-M12 and A1-A10) or at 50 K (AA1-AA5), obtained with two di ff erent approxima-tions, Fick’s second law or Fick’s first law, as indicated. When only one thickness appear in column three it correspond to the ASW ice thicknessin codeposited ices, estimated as if no CH were present.
4. Discussion of experimental results
Inspecting the results presented in Figures 3, 4 and 5, and in Table 2, several conclusions can be extracted.1) First and second Fick’s law di ff usion coe ffi cients. Both approximations, even though first Fick’s law can only be applied tothe final part of the isothermal experiments and imposes stronger restrictions, give comparable di ff usion coe ffi cients. When bothapproximations are applied to the same set of experiments (M1-M9), the di ff usion coe ffi cients found with the first law are slightlysmaller than those found with the second law (see Table 2). The new methodology proposed, based on QCMB detection (expA1-A10), gives results consistent with those obtained from IR spectrosocopy (M1-M12).2) CH di ff usion coe ffi cient dependence on ASW ice morphology. The heating ramp will a ff ect ASW morphology at T iso .However, from the inspection of the experiments presented in Table 1, where the ramp was varied between 5 K / min and and 20K / min, it was not possible to extract any conclusion about its e ff ect on methane di ff usion. It looks like the morphology variationscaused by the di ff erent heating ramps are not significant enough to be manifested in the D ( T ) coe ffi cient. Article number, page 8 of 17elén Maté et al.: Di ff usion of CH in amorphous solid water
200 400 600 800 100001020304050
M1-M9, 2nd Fick’s LawA1-A10, 1rst Fick‘s Law D ( x - c m s - ) L(nm)
Fig. 6. Di ff usion coe ffi cient of CH at 50 K, for di ff erent ASW ice layer thicknesses. Red dots: Experiments M1-M9. Black starts: ExperimentsA1-A10. Dotted blue line: an exponential fit performed only to guide the eye. From inspection of Figure 3, codeposited experiments seem to be better described by the second Fick’s law than sequentialexperiments. Although all fits in Figure 3 are satisfactory, in order to get proper fits of sequential experiments, only times up to3000 seconds were considered. However, codeposited experiments could be properly fitted in all the experimental time interval(up to 8000 s). This behavior is related to di ff erences in the morphology of the ice. Monte Carlo simulations have been performedfor both ASW structures, pure and codeposited ASW, shown in Figure 9 (left panels) and Figure 10, respectively, illustrate thosechanges. Di ff usion coe ffi cient obtained for codeposited experiments (M6-M9) vary between 1.4 and 1.8 10 − cm s − , and thosefor sequential experiments with similar layer thickness (M2 and M3) are slightly larger, varying between 2.7 and 3.7 10 − cm s − .This can be attributed to the e ff ect of methane on ASW porosity upon deposition, since depositing methane together with water willslightly decrease the number of pores, slowing the di ff usion. This is discussed in section 6.The D ( T ) values found for methane di ff using on ASW grown at 50 K (experiments AA1-AA5) are about one order of magnitudesmaller than those found for ASW ice grown at 30 K (exp. A5-A8). It is well known that, for background vapor deposition, thehigher the growing temperature, the higher the average density of the ice obtained (Dohnálek et al. (2003)), i.e., the lower theporosity. Monte Carlo simulations performed in this work corroborate this (see Figure 9). Consequently, the experimental D ( T )values found indicate CH di ff uses slower through the pores of the more compact ASW ice.3) ASW ice layer thickness. Although unexpected, a dependence of D ( T ) on ASW layer thickness was observed in the exper-iments. All the di ff usion coe ffi cients obtained in this work for ASW grown at 30 K and T iso =
50 K have been plotted in Figure 6versus ASW layer thickness. It can be seen that below 600 nm, the di ff usion coe ffi cients fluctuations could be considered withinexperimental error. However, for layers above 600 nm, there is a clear increase in the di ff usion coe ffi cient with thickness. This be-haviour has been observed by other authors, like Ghesquière and colaborators (Ghesquière et al. (2015)) in analogous experimentsperformed to study CO di ff usion in water ice. Also May and coworkers (May et al. (2013)), investigating CH di ff usion throughASW layers grown on top, concluded that the CH releasing mechanism was di ff usive only up to a certain ASW layer thickness. Apossible reason for this behaviour could be the failure of the approximations considered in the Fick’s law. In particular, to considerthat D ( T ) is constant along the z direction within the whole ice layer thickness. diffusion barrier Although only a few experiments were performed at temperatures di ff erent from 50 K, they could be used to constrain the di ff usionbarrier, assuming a single-barrier Arrhenius process and a D ( T ) following Arrhenius equation: D ( T ) = D exp ( − E d / T ) , (9)where D is the pre-exponential factor, E d the di ff usion energy barrier and T the temperature. With the di ff usion coe ffi cients obtainedfor experiments M3, M4, M10, M11 and M12 at 50, 55 and 60 K, an Arrhenius-type plot has been performed to extract a di ff usionbarrier and pre-exponential factor. Also, experiments AA1-AA5, that give di ff usion coe ffi cients at temperatures between 42.5 and52.5 K for ASW ice grown at 50 K, have been fitted to the Arrhenius equation. Both fits are presented in Figure 7.Energy barriers of 477 ±
91 K and 639 ±
118 K, and pre-exponential factors of ln( D ) = -18 ± D ) = -8.0 ± D ) = -16 ± D ) = -7.0 ± ff usion coe ffi cient of CH on ASW grown at 10 K andannealed to 70 K for 30 min. From their experiments, performed at temperatures between 17 K and 23 K, they found a E d = ±
10 K and ln(D ) = -14.3 ± ) = -6.23 ± ff erences between the three set of data couldbe due to di ff erences in the morphology of the ASW ice, that were grown at 10 K, 30 K, or 50 K. Article number, page 9 of 17 & A proofs: manuscript no. aanda_revised_2 l n ( D ) -1 )T (K)60 55 50 45 42.5 Fig. 7.
Arrhenius-type plot of the di ff usion coe ffi cients of CH on ASW grown at 30 K (black triangles) and at 50 K (red dots).
15 20 25 30 35 40 45 50 55 60 d =547 K, log(D )= -6.23 This work, E d =477 K, log(D )= -8.0 This work, E d =639 K, log(D )= -7.0 D ( c m s - ) T(K)
Fig. 8. CH di ff usion coe ffi cients obtained in this work compared with those in (He et al. (2018)). Scattered points: experimental data. Thetemperatures indicated in the legend refer to ASW generation temperature. Straight lines: fits extracted form Arrhenius plots.
5. Monte Carlo simulations
We used a step-by-step Monte Carlo simulation to follow the formation of H O ices with CH trapped gas through co-deposition.Our model is described in Cazaux et al. (2015, 2017). H O and CH molecules originating from the gas phase arrive at a randomtime and location on the substrate, and follow a random path within the ice. The arrival time depends on the rate at which gas speciescollide with the surface (section 5.1). The molecules arriving on the surface can be bound to the substrate and to other H O andCH molecules through hydrogen bound and van der Waals interactions. In the present study, we used on-lattice KMC simulations.Other types of simulations, such as o ff -lattice KMC method have also been used to compute the porosity in ices Garrod (2013).While this method is more sophisticated than the present method since it allows to determine the distance of the species explicitly,we here consider the distance between water molecules to be equal, and concentrate on defining the binding energies as functionof neighbours. Because our method does not compute the distance between molecules, the di ff usion could be slightly di ff erent.However, we can provide di ff usion barriers that can be compared with experimental results and used in theoretical models.In the present study we consider two distinct cases on how to calculate the binding energies, depending on whether it concerns awater or a methane molecule. For water, the binding energy of each H O molecule depends linearly on its number of neighbours, asdescribed in Cazaux et al. (2015). For methane, we consider that the binding energy of these molecules is already very high for oneneighbor, and very close to the binding energies measured experimentally (Escribano et al., private communication). We thereforeconsider that the di ff usion energy of methane, which is a fraction of the binding energy, is not depending on the number of neighbors,and use E d ∼ K , as determined in He et al. (2018). This value was chosen because we will compare our experimental resultswith those of He et al. (2018) (see section 6.3), despite di ff erent binding energies appear in the literature (see Luna et al. (2014) Article number, page 10 of 17elén Maté et al.: Di ff usion of CH in amorphous solid water and references therein). Depending on their di ff usion energies, the H O and CH molecules di ff use on the surface / in the ices. Thedi ff usion is described in section 5.2. During warming-up and the waiting time, the molecules can evaporate from the substrate / ices.In order to reproduce experimental measurements, we deposit water and methane with a H O:CH ratio of 10:1. We then increasethe temperature after deposition with a ramp of 1 K every 10 seconds (which is 6 K / min, providing a ramp close to experimentalvalues ranging between 5 to 10 K per minute). When the temperature of 50 K is reached, we let the system evolve with time atconstant temperature and determine the number of molecules evaporating and staying in the ice. This allows a direct comparisonwith the measurements of section 3. In our model, we defined the surface as a grid with a size of 60 ×
60 sites. Low-density amorphous ice consists mainly of fourcoordinated tetrahedrally ordered water molecules. As already discussed in Cazaux et al. (2015), amorphous water ice is modeledusing a grid in which the water molecules are organized as tetrahedrons, which implies that each water molecule has four neighbours.Molecules from the gas-phase arriving on the grid can be bound to the substrate. The accretion rate (in s − ) depends on the densityof the species, their velocity, and the cross section of the surface, and can be written as: R H O = n H O v H O σ S , (10) R CH = n CH v CH σ S , (11)where v H O = (cid:113) k T gas π m H2O ∼ . × (cid:113) T gas cm s − and v CH = (cid:113) k T gas π m CH4 ∼ . × (cid:113) T gas cm s − are the thermal velocity of water andmethane respectively, and σ , the cross section of the surface. S is the sticking coe ffi cient that we consider to be unity in this study.The distance between two sites is 1.58 Å, but each water molecule occupies 1 site over 4 because of its four coordinates tetrahedralorder (see Cazaux et al. (2015)). The surface density of water molecules is what is typically assumed, i.e. ∼ (1 / × . − )Å − ∼ cm − . The cross section scales with the size of the grid considered in our calculations, which is 60 ×
60 sites, as σ ∼ (1 .
58 10 − × cm = − cm . The deposition rate is therefore: R acc(H O) = − n H O s − , and R acc(CH ) = . − n CH s − , for T gas =
100 K.In order to mimic experimental conditions with deposition rates of 1 nm / s ∼ / s ∼
450 molecules / s (we have 60 × / O molecules in the gas in cm − as being n H O = × cm − . The density of CH is scaled to thedensity of water to reproduce the experiments with H O:CH We use a method to simulate di ff usion similar than in Cazaux et al. (2015). The di ff usion barrier is usually considered in models asa fraction of the binding energy. For water, the binding energy increases with the number of neighbors, and therefore the di ff usionbarrier also increases with the number of neighbors. For CH , on the other hand, we consider only one binding energy with water,and that the binding energy does not increase with the number of neighboring water molecules (consistent with the very smalldi ff erences in binding energies seen in Luna et al. (2014)). The di ff usion rate for methane is therefore simply R di ff = ν exp (cid:16) − E d T (cid:17) ,were ν is the pre-exponential factor and E d the di ff usion barrier for methane.For water, we compute the binding energy by adding the number of neighbors nn as E b = nn × E b (H O-H O), with E b (H O-H O) ∼ ∼ s − Fraser et al.(2001). We define the di ff usion rates by calculating the initial energy of a molecule and its final energy in the possible sites whereit can move. For a position (i,j,k) of a molecule in the grid, we calculate the associated binding energy E i and identify the possiblesites where the molecule can di ff use to as i ±
1; j ±
1; k ±
1. The final binding energy E f is calculated as function of the neighbourspresent around this site. The di ff usion rate, from an initial site with an energy E i to a final site with an energy E f , is illustrated inCazaux et al. (2017). The barrier to go from E i to E f is defined as follows if E i ≤ E f : E d = α × min(E i , E f ) , if E i < E f (12)If E i > E f , on the other hand, the barrier becomes: E d = α × min(E i , E f ) + ∆ E , if E i > E f (13)with ∆ E = max( E i , E f ) - min( E i , E f ). By defining the barriers in such a manner, we do take into account microscopic reversibilityin this study (Cuppen et al. 2013), which implies that barriers to move from one site to another should be identical to the reversebarrier. The di ff usion barriers scale with the binding energies with a parameter α . For water, α sets the temperature at which watermolecules can re-arrange and di ff use in the ices to form more dense ices. This parameter is found to be around 30 % for the water-on-water di ff usion derived experimentally (Collings et al. 2003), and we obtained an alpha lower than 40 % in a previous study(Bossa et al. 2015). For methane, He et al. (2018) derived an α ranging between 34 and 50 % and a binding energy ranging from1100 K to 1600 K. In this study, we will use our simulations to constrain the di ff usion barrier.The di ff usion rate for water, in s − , can be written as: R di ff = × (cid:113) E f − E s E i − E s (cid:18) + (cid:113) E f − E s E i − E s (cid:19) × ν exp (cid:18) − E d T (cid:19) , (14) Article number, page 11 of 17 & A proofs: manuscript no. aanda_revised_2 where ν is the pre-exponential factor, T is the temperature of the substrate (water ice or CH ice) and E s is the energy of the saddlepoint, which is E s = (1- α ) × min( E i , E f ). This formula di ff ers from typical thermal hopping because the energy of the initial and finalsites are not identical (Cazaux et al. 2017).The pre-exponential factor is related with the vibrational frequency of a species in its site, and can be derived form Landau &Lifshitz (1966) expresion: ν = (cid:113) N s E i π × m CH , where N s is the number of sites per surface area (10 ), m CH is the mass of methane andE i the binding energy. In this work we have taken ν = s − for water and ν = s − for CH (He et al. (2018)). The molecules present on the surface can return into the gas phase because they sublimate. This desorption rate depends on thebinding energy of the species with the surface / ice. As mentioned previously, the binding energy of a H O molecule depends on itsnumber of neighbours, while we consider a unique binding energy for CH independent of the number of neighbors. The bindingenergy E i of a molecule sets the sublimation rate as: R subl ( X ) = ν exp (cid:18) − E i T (cid:19) , (15)where ν is the pre-exponential factor, which is taken as ν = s − for water, and ν = s − for CH (He et al. (2018)). We useda pre-exponential factor similar than the one used for di ff usion for simplicity. However, we also used the lowest binding energy ofmethane derived from experiments of 1100 K. In this sense, the desorption rate at 30 K is around R subl ( CH ) = − s − , which issimilar to a rate with a pre-exponential factor of 10 and a binding energy of 1300 K. We therefore use a sublimation rate in therange of what has been derived in other studies (He et al. (2018)). Using di ff erent pre-exponential factor and binding energy forsublimation do not change our results after deposition. Methane molecules desorb once they have di ff used through the ice to reachthe surface, and therefore our results depend on the di ff usion rate rather than sublimation rate.
6. Theoretical results
The experiments have been performed for a deposition of water and methane at 30 K with a rate of around 1 nm / s and a ratio ofH O:CH of 10:1. The ice was then heated until 50 K and the diminution of the IR absorbance spectra has given the amount of CH desorbing from the ice. Measuring the concentration of CH with time can directly give a constraint on the di ff usion coe ffi cient, asshown in the previous sections. CH di ff uses through the pores within the water ice, which implies that the porosity of water icecould play a role in the di ff usion. To grasp how water ice changes within the conditions explored in the present experiments, weshow the morphology of water ice with our simulations when deposited at 30 K (Figure 9, top left panel) and heated at 50 K (Figure9, top middle panel), and being deposited at 50 K (Figure 9, top right panel). Each colored square represents a water molecule, andthe color indicates the number of neighbours for each of them, which defined the biding energy. Blue corresponds to 1 neighbor(binding energy of ∼ ∼ ∼ ∼ ff erent panels show the pores network after deposition at 30 K (left panel), after the ice deposited at30 K being heated to 50 K (middle panel) and for ice being deposited at 50 K (right panel). The pores in the case of deposition at30 K, are less and less connected than if the water is subsequently heated to 50 K. The evolution of the pores network show howdi ff usion within the pores will take place when methane is added into the water ice. When water ice is deposited at 50 K, the poresare smaller and less connected, which indicates that di ff usion would be less e ffi cient. In order to understand the trapping and di ff usion of methane in water ices, we performed Monte Carlo simulations for thin icesdeposited at 30 K, heated until 50 K and with a waiting time similar to those observed experimentally. Methane is depositedsimultaneously with the water, with a ratio of H O:CH of 10:1. In this section we use thin ices of 10 nm thick to study the e ff ectof the di ff usion barriers and of the water ice morphology. In Figure 10 we illustrate how methane is mixed within water ice inone of our simulations. In the left panel, the blue boxes represent the water molecules, while the red boxes show the presence ofmethane molecules. The ice mixture is represented just after deposition at 30 K. In this simulations, we use a di ff usion barrier of E d = i for water, and a di ff usion barrier of E d =
660 K for methane (note that for water E i depends on the number of neighbors,while the di ff usion barrier for methane is always the same). The thickness of the ice is slightly larger when methane is included inthe simulations. This is due to the fact that at 30 K methane di ff uses as much as water with one neighbor, which implies that the Article number, page 12 of 17elén Maté et al.: Di ff usion of CH in amorphous solid water Fig. 9.
Water morphology and porosity under our experimental conditions. Top panels show water ice being deposited at 30 K (left panel), andthen being heated to 50 K (middle panel) and water ice being deposited at 50 K (right panel). Each square represent a water molecule and thecolor represent the number of neighbour for each molecule. Blue squares represent water molecules with 1 neighbour, green with 2 neighbours,yellow with 3 and red with 4 neighbors. The bottom panels show the size of the pores in the water ice, which is the negative image of the toppanel (showing the emptiness in ice). The color shows the number of grid cells empty around the cell (corresponding to 1.6 Å). The figures showdeposition at 30 K (left panel), after being heated to 50 K (middle panel) and being deposited at 50 K (right panel). reorganisation is faster than if the ice was composed of 100% water ice. The porosity, presented in Figure 10, right panel, seems tobe less important than in the case of pure water (left bottom panel in Figure 9). This could be due to the fact that the molecules dodi ff use slightly more than for 100 % water, which decrease the coalescence of pores to form networks. It is possible to relate thiswith the di ff usion coe ffi cients found for sequential and codeposited ices, and commented in section 4. In sequential ices, CH willbe di ff using through a pure AWS layer, slightly more porous than the ASW generated by codepositon, and therefore a faster CH di ff usion is expected.The e ff ect of the reorganization of water is illustrated in Figure 11, left panel. In this figure we fix the di ff usion of methane withE d =
990 K. The di ff usion of water is set to two di ff erent values α (H O) = ff usion barrier ofE d = α (H O) E i . This figure shows that the reorganization of water has an e ff ect on the di ff usion of methane. This is due to the factthat methane di ff uses in pores up to the surface of the ice. The reorganization of water a ff ects the presence of pores and the waythey are connected. A slower di ff usion of water will create less pores, and consequently, the di ff usion of methane will be somewhatslower. The di ff usion of methane is presented in Figure 11, right panel, for E d =
660 K (red) and 990 K (green). The di ff usion ofwater is set to α (H O) = d = i . A lower mobility of methane (E d =
990 K) implies that the di ff usion is slowerand that the number of methane desorbing from the ices is smaller. This is seen in the figure when comparing to a faster di ff usiongiven by a smaller barrier of E d =
660 K, as shown in red. Therefore, both the mobility of water and methane are key parameters toconstrain the di ff usion coe ffi cient of methane in water ices. Most of the experimental measurements used to determine the di ff usion have been obtained on ices with thickness larger than 180nm. Our simulations can be made for thin ices, or for thick ices when di ff usion is slow, but cannot be performed for fast di ff usionin thick ices due to large computing times. In the present section, we show an alternative solution to simulate the amount of CH remaining in thick ices. We made calculations for ices of 10 nm thick and used several of these thin ices to scale the results to a thickice. Since we can compute the number of CH evaporating from a 10 nm ice, we can compute the amount of CH evaporating fromseveral blocks of thin ices attached to each other to mimic a thick ice. If the fraction of CH evaporating from the 10 nm ice thick isNEV CH , then the fraction of CH evaporating from two blocks of 10 nm thick ice is NEV CH + NEV CH (the second term shows the Article number, page 13 of 17 & A proofs: manuscript no. aanda_revised_2
Fig. 10.
Water ice (blue) with methane (red) after deposition at 30 K (left panel). The right panel shows the size of the pores in the water ice mixedwith methane. Pore sizes vary from ∼ F r a c t i on CH r e m a i n i ng Time (seconds)30KT increase 50KWaiting time 0.40.9 0 0.2 0.4 0.6 0.8 1 0 200 400 600 800 1000 F r a c t i on CH r e m a i n i ng Time (seconds)30KT increase 50KWaiting time 660 K990 K
Fig. 11.
Left panel: e ff ect of water di ff usion on the amount of methane remaining in the ice. In red E d (H O) = i while in blue E d (H O) = i .Right panel: e ff ect of methane di ff usion on the amount of methane remaining in the ice. In red E d (CH ) =
660 K while in green E d (CH ) =
990 K .fraction reaching the top block from the bottom block, and then crossing the top block to evaporate). If we would have n blocks ofice attached to each other, we could estimate the fraction of CH evaporating as NEV CH + NEV CH + NEV CH + .. + NEV nCH . In orderto validate our method, we compute the amount of CH staying in the ice with one block of 10 nm, one of 25 nm and one of 50 nm.The amount of CH remaining in the ice is computed for 2 ×
25 nm as 1- 1 / × (NEV CH + NEV CH ), with NEV CH is the numberof CH evaporating from a block of 25 nm thick, while for 5 ×
10 nm as 1- 1 / × (NEV CH + NEV CH + NEV CH + NEV CH + NEV CH ),with NEV CH is the number of CH evaporating from a block of 10 nm thick. The simulations for a 50 nm thick ice are shownin figure 12 in blue, while the simulations for 2 ×
25 nm are shown in red, and 5 ×
10 nm, in green. We can see that this methodshows very good agreement to estimate the amount of CH remaining, with a very small di ff erence on short timescales below 1000seconds, and a di ff erence of a few percent for larger timescales. We can therefore use this method to reproduce the amount of CH remaining in thick ices in the experiments. These simulations were made for slow di ff usion where E d =
990 K for methane and E d = i for water.In the simulations presented in this section, we aim to reproduce the experimental results from section 3 in order to constrainthe di ff usion coe ffi cient of methane on ASW deposited at 30 K. We use in our simulations the results obtained in the experimentM7, and use di ff erent di ff usion rates from He et al. (2018) (mentioned as R1) and the ones derived in this work. We converted theD obtained previously with the formula: D = ν a , as shown in He et al. (2018), where ν is the pre-exponential factor, and a =
3Å isthe distance between two sites. We used the rate derived by He et al. (2018) of R1 = exp − T s − , and derived the rates fromthe present study, R2 = exp − T s − and R3 = exp − T s − . We also consider another rate R4, in order to show the e ff ectof the increase of the pre-exponential factor on the fraction of CH trapped. In this case, R4 = exp − T s − . Figure 13, leftpanel, shows the fraction of CH remaining after heating, using the rates mentioned above. In the experiments, 0.88 % of methaneis still present in the ice when reaching 50 K (thick red stripe). Our model cannot account for such a loss with the rates considered. Article number, page 14 of 17elén Maté et al.: Di ff usion of CH in amorphous solid water F r a c t i on CH r e m a i n i ng Time (seconds)5 times 10 nm2 times 25 nm50 nm
Fig. 12.
Comparison of the fraction of CH remaining for a 50 nm (in blue) compared with the fraction in 2 blocks of 25 nm attached (red) and 5block attached (green). F r a c t i on CH r e m a i n i ng Temperature (Kelvins)experimentR1R2R3R4 F r a c t i on CH r e m a i n i ng Time (Seconds)experimentR1R2R3R4
Fig. 13.
Left panel: fraction of CH remaining in the ices during the TPD. The experiments show that 13.8 % of CH was present initially at 30 Kwhile 12.1 % remained at 50 K. Right panel: fraction of CH remaining in the ices during the waiting time after reaching 50 K.The green pointsshow the experimental values. In order to reproduce the measurements, a larger rate (larger ν or lower activation energy) should be used. This implies that in thepresent case, the di ff usion is hindered because many molecules are di ff using back and forth in pores before being able to arriveto the surface and desorb. In other words, the microscopic di ff usion of methane is many times faster than the di ff usion measuredexperimentally. Indeed, methane molecules are constantly di ff using within the ices, which can add up to a very long path while theactual distance travelled is orders of magnitude lower. This is illustrated by Figure 14, which shows the movement of three methanemolecules taken randomly in the ices. The three methane molecules move within the pores without reaching the surface, and thetotal distance they explored in the ice is very di ff erent from the actual path they follow. The symbols show the position of eachmolecule (each molecule represented by a colored cross) during the entire simulation. These are the results from simulations using ν = s − and E d =
660 K, for a 10 nm thick ice. The simulations are following the movement of individual molecules during theheating ramp and waiting time, which lasts in total ∼ remaining in the ice for the four ratesconsidered. Note that for R4, the calculation is computationally expensive, we only show the results of the simulations until 2000seconds. This implies that to reproduce the experimental results, the di ff usion should be faster (larger pre-exponential factors orlower activation energy) than the di ff usion obtained using the Fick’s law. Our results show that the rates extracted from the presentstudy as well as the rate from He et al. (2018) do not reproduce the experimental data with our Monte Carlo simulations. In this section we investigate the e ff ect of porosity on the di ff usion of methane in the ice. We deposited ices at 30 K and 50 K inour simulations, and determined the fraction of methane in the ices as a function of time after 50 K. These results are presented inFigure 15. For the ices deposited at 30 K, we increased the temperature until 50 K, and then determine the amount of CH withwaiting time (red line), while for the ices deposited at 50 K, we determine the amount of CH after the deposition is completed Article number, page 15 of 17 & A proofs: manuscript no. aanda_revised_2
Fig. 14.
Selection of three methane molecules moving in the grid during the entire experiment. The path clearly illustrates a preferred mobility inthe pores. The fact that the green CH moves form a side of the box to the other side is due to side e ff ects. F r a c t i on CH r e m a i n i ng Time (seconds) 30K50K
Fig. 15.
Fraction of methane remaining in the ice after 50 K when deposited at 30 K and at 50 K. (green line). In these simulations, the mobility for water ice is set as E d = i for water and E d =
990 K for methane. Note that onboth simulations the di ff usion rates are the same and only the deposition temperature changes. Our results show that the porositychanges the di ff usion of methane within the ice. A more porous medium, such as the one created when deposition occurs at 30 K,allows for a faster di ff usion and therefore less CH remain in the ice. On the other hand, a less porous medium, such as the onecreated when deposition occurs at 50 K, slows down the di ff usion and consequently more CH molecules remain in the ice. Thisresult is in agreement with experimental findings shown in Figure 8, where the di ff usion coe ffi cient obtained for ices grown at 50 Kis smaller (slower di ff usion) than the di ff usion of ices grown at 30 K. This shows that the porosity plays a role in the di ff usion ofmolecules within the ices, larger and connected pores favoring a higher mobility.
7. Conclusions
We have investigated the surface di ff usion of methane on amorphous solid water, and how it is a ff ected by the porosity of the ASWstructure.Methane di ff usion coe ffi cients have been measured to vary between 10 − and 10 − cm s − , for temperatures between 42 Kand 60 K. It was observed that di ff erent ASW structures modify the CH di ff usion coe ffi cient up to one order of magnitude.Experiments and simulations show periods of time where the rate of sublimation of methane is constant. This implies that themethane concentration gradient remains constant for these time intervals, and the first Fick’s law can be used.Monte Carlo simulations of simultaneous and sequential H O:CH ices at 30 K, show that a more compact ASW structure isformed in codeposited experiments. The di ff usion coe ffi cients measured for CH trapped in codeposited ASW are smaller (slowerdi ff usion) than those found in sequential experiments. Therefore, it is possible to say that when a CH reservoir is di ff using througha pure ASW layer formed on top, the di ff usion will be faster than in homogeneously mixed H O:CH ices. Article number, page 16 of 17elén Maté et al.: Di ff usion of CH in amorphous solid water Measured di ff usion coe ffi cients indicate that CH di ff use faster on ices grown at 30 K than in ices grown at 50 K. Monte Carlosimulations show that ices grown at 50 K present smaller and less interconnected pores than ices grown at 30 K. Therefore, it canbe concluded that larger and more interconnected pores favour methane di ff usion.We used Monte Carlo simulations in order to better understand the experimental results, and to estimate the e ff ect of porosityon the di ff usion rates. We showed the di ff usion rates derived experimentally using Fick’s laws do not reproduce the experimentalresults with our Monte Carlo simulations, considering the range of di ff usion values within the experimental uncertainty. The ex-perimental results can be reproduced only using a di ff usion rate at least 50 times higher, which would imply for the present studya pre-exponential factor ranging between 10 -10 s − . They are consistent with factors derived from Landau & Lifshitz (1966)expression (see section 5.2.). Our work therefore shows that the di ff usion coe ffi cient obtained from the Fick’s law is a di ff usion atthe macroscopic level. This can be used to determine the overall loss of methane in ices on large timescales (for example whenmodelling ices of comets, or moons). However, when modelling reactivity or ice evolution in microscopic level with kinetic mod-els (Monte Carlo simulations, rate equations, etc.) the di ff usion used should be much higher and the pre-exponential factor canbe derived as in Landau & Lifshitz (1966). The discrepancy between macroscopic (Fick’s law) and microscopic view of di ff usioncomes from the fact that microscopic models account for back and forth di ff usion, counting single di ff usion hops in all directionsin the medium. In contrast, macroscopic di ff usion sees the e ff ective distance that methane molecules travelled, describing CH4 as acontinuum medium di ff using due to a concentration gradient.In summary, our work shows that the microscopic di ff usion of methane is many times faster than the macroscopic di ff usionmeasured experimentally. Acknowledgements.
Funds from the Spanish MINECO / FEDER FIS2016-77726- C3-1-P and C3-3-P projects are acknowledged.
References
Berland, B. S., Brown, D. E., Tolbert, M. A., & George, S. M. 1995, Geophysical Research Letters, 22, 3493Bossa, J., Isokoski, K., Paardekooper, D. M., et al. 2014, 136, 1Bossa, J.-B., Maté, B., Fransen, C., et al. 2015, ApJ, 814, 47Brill, R. & Tippe, A. 1967, Acta Crystallographica, 23, 343Brown, D. E. & George, S. M. 1995, 22, 3493Cazaux, S., Bossa, J. B., Linnartz, H., & Tielens, A. G. G. M. 2015, A&A, 573, A16Cazaux, S., Martín-Doménech, R., Chen, Y. J., Muñoz Caro, G. M., & González Díaz, C. 2017, The Astrophysical Journal, 849, 80Collings, M. P., Dever, J. W., Fraser, H. J., McCoustra, M. R. S., & Williams, D. A. 2003, The Astrophysical Journal, 583, 1058Cooke, I. R., Öberg, K. I., Fayolle, E. C., Peeler, Z., & Bergner, J. B. 2018, The Astrophysical Journal, 852, 75Crank, J. 1975, The Mathematics of di ff usion, 2nd edn. (Oxford University Press.)Cuppen, H. M., Karssemeijer, L. J., & Lamberts, T. 2013, Chem. Rev., 113, 8840Cuppen, H. M., Walsh, C., Lamberts, T., et al. 2017, Space Science Reviews, 212, 1Dartois, E., Ding, J. J., de Barros, A. L. F., et al. 2013, Astronomy and Astrophysics, 557, A97Dohnálek, Z., Kimmel, G. A., Ayotte, P., Smith, R. S., & Kay, B. D. 2003, Journal of Chemical Physics, 118, 364Fraser, H. J., Collings, M. P., McCoustra, M. R. S., & Williams, D. A. 2001, Monthly Notices of the Royal Astronomical Society, 327, 1165Gálvez, Ó., Maté, B., Herrero, V. J., & Escribano, R. 2009, Astrophysical Journal, 703, 2101Garrod, R. T. 2013, Astrophysical Journal, 778 [ arXiv:1310.2512 ]Garrod, R. T., Weaver, S. L. W., & Herbst, E. 2008, 283Ghesquière, P., Mineva, T., Talbi, D., et al. 2015, Physical Chemistry Chemical Physics, 17, 11455Hasegawa, T. I., Herbst, E., & Leung, C. M. 1992, The Astrophysical Journal Supplement Series, 82, 167He, J., Clements, A. R., Emtiaz, S., et al. 2019, ApJ, 878, 94He, J., Emtiaz, S., & Vidali, G. 2018, The Astrophysical Journal, 863, 156He, J., Emtiaz, S. M., & Vidali, G. 2017, The Astrophysical Journal, 837, 65Herbst, E. 2001, Chemical Society Reviews, 30, 168Herrero, V. J., Gálvez, O., Maté, B., & Escribano, R. 2010, Physical chemistry chemical physics : PCCP, 12, 3164Irvine, W. M. 2011, Langmuir-Hinshelwood Mechanism, ed. M. Gargaud, R. Amils, J. C. Quintanilla, H. J. J. Cleaves, W. M. Irvine, D. L. Pinti, & M. Viso (Berlin,Heidelberg: Springer Berlin Heidelberg), 905–905Isaacs, E. D., Shukla, A., Platzman, P. M., et al. 1999, Phys. Rev. Lett., 82, 600Jenniskens, P., Blake, D. F., Wilson, M. A., & Pohorille, A. 1995, The Astrophysical Journal, 455, 389Karssemeijer, L. J., Ioppolo, S., van Hemert, M. C., et al. 2014, The Astrophysical Journal, 781, 15ppKerkhof, O., Schutte, W. A., & Ehrenfreund, P. 1999, The infrared band strengths of CH 3 OH, NH 3 and CH 4 in laboratory simulations of astrophysical icemixtures, Tech. rep.Landau, L. D. & Lifshitz, E. M. 1966, Statistische PhysikLauck, T., Karssemeijer, L., Shulenberger, K., et al. 2015, Astrophysical Journal, 801, 118Livingston, F. E., Smith, J. A., & George, S. M. 2002, Journal of Physical Chemistry A, 106, 6309Luna, R., Satorre, M. Á., Domingo, M., Millán, C., & Santonja, C. 2012, Icarus, 221, 186Luna, R., Satorre, M. Á., Santonja, C., & Domingo, M. 2014, Astronomy and Astrophysics, 566, A27Mastrapa, R. M., Sandford, S. A., Roush, T. L., Cruikshank, D. P., & Ore, C. M. D. 2009, 1347Maté, B., Molpeceres, G., Tanarro, I., et al. 2018, The Astrophysical Journal, 861, 61May, R. A., Smith, R. S., & Kay, B. D. 2013, Journal of Chemical Physics, 138Mispelaer, F., Theulé, P., Aouididi, H., et al. 2013, Astron. Astrophys., 555, A13Molpeceres, G., Satorre, M. A., Ortigoso, J., et al. 2017, Monthly Notices of the Royal Astronomical Society, 466, 1894Narten, A. H., Venkatesh, C. G., & Rice, S. A. 1976, The Journal of Chemical Physics, 64, 1106Raut, U., Teolis, B. D., Loe ffl er, M. J., et al. 2007, Journal of Chemical Physics, 126Tielens, A. G. G. M. & Hagen, W. 1982, A&A, 114, 245er, M. J., et al. 2007, Journal of Chemical Physics, 126Tielens, A. G. G. M. & Hagen, W. 1982, A&A, 114, 245