domain: shs sdv

text:
Olivier Morin

Pierre Dutreuil

A model to analyze attempt of Science Education to coincide with Participatory Education by schooling SocioScientific Issue : Case study of French lesson dealing with Vaccine Hesitancy

Keywords: Biology Education, Citizenship Education, Socially Acute Questions, Vaccine Hesitancy

. We offer a model to characterize the strategies adopted a priori by the teachers and to compare them a posteriori with the pupils'reactions. A study of two biology sessions during which 13 to 15-year-old French pupils discussed with their teacher the delicate issue of vaccine hesitancy shows the potential interest of this model to characterize the orchestration of surveys (Simonneaux, 2019) about Socially Acute Questions.

Rationale

Questionning contemporary stakes is questionning the future of a world made up of hybrid objects belonging both to the natural and the cultural, to the scientific, technical, political, economic and social fields. It calls for forging links between knowledge (scientific, technical, professional, local, traditional) distributed among stakeholders and particular to specific situations. It also leads to adopt and defend delicate positions regarding complex problems. Insisting on the scientific dimension of such issues, [START_REF] Irwin | Constructing the scientific citizen : science and democracy in the biosciences[END_REF] names scientific citizenship the possibility given to non-specialist citizens to take part democratically in the choices of society in terms of technoscientific innovations. It requires an education whose modalities, such as the schooling of socioscientific controversies, deserve to be questioned. Considering the orchestrations of investigative approaches [START_REF] Simonneaux | La démarche d'enquête. Une contribution à la didactique des Questions Socialement Vives[END_REF] about Socially Acute Questions (SAQ), we are focusing this study on the articulation of science education and political education approaches.

Framework

We draw on the typology of [START_REF] Levinson | Science education and democratic participation: an uneasy congruence?[END_REF]. He distinguishes four types: the deficit framework where the knowledge taught are that produced by science and delivered to teachers, who in turn deliver them to students, the deliberative framework where inquiries to solve

Transforming Science & Technology Education to Cultivate Participatory Citizens

Daegu, Korea, 23-28 August, 2020 scientific problems are favored, but without debating the societal issues, the science education as praxis model, where the knowledge taught are not only that which generate consensus and the knowledge discussed are not only scientific knowledge, and the dissent & conflict model which questions the socio-political context of the emergence of socioscientific controversies. In order to constitute an interpreting grid of teaching-learning situations, we have translated each of these types into indicators to characterize on the one hand the specificity of strategies adopted a priori by teachers, on the other hand a posteriori the pupils' reaction in these situations. Thus, our grid typically considers four items: the educational objectives targeted by the teachers and the nature of the knowledge to which the investigations they submit to their students relate, the nature of the knowledge the students actually mobilize and their mode of effective apprehension of these knowledge. For this communication, we propose to test this model by analyzing a case study.

Case study

Context

The case study deals with two biology sessions conducted with French pupils aged 13 to 15 (4 ° middle school class), built within the framework of the CEHVAC project (Controversy, Teaching and History of Vaccination) according to a design-experiment approach [START_REF] Cobb | Design Experiments In Educational Research[END_REF][START_REF] Sanchez | Recherche collaborative orientée par la conception. Un paradigme méthodologique pour prendre en compte la complexité des situations d'enseignement-apprentissage[END_REF] aiming to be both collaborative and iterative. For this first iteration, the pupils worked with their teacher on the delicate question of vaccine hesitancy that we acknowledge as a SAQ. Indeed, this question is exceptionnally delicate in France [START_REF] Larson | The State of Vaccine Confidence 2016: Global Insights Through a 67-Country Survey[END_REF][START_REF] Raude | Les attitudes des Français face à la vaccination : une évolution préoccupante[END_REF], as this country is worldwide a specifical case of such a Vaccine Hesitancy: according to Welcome Global Monitor 2018 (p 112) a very large part of French people (33%) disagreed in 2019 that vaccines were safe, and (19%) that they were effective. The class focused for two periods of one hour and a half on the Human Papilloma Virus vaccine recommended for adolescent girls (and soon for adolescents boys), as its effectiveness and safety are questioned in the media.

Method of Analysis

After a cutting-out into episodes based on the activity instructions given to the pupils, we've analyzed each of the two periods. The grid is applied to each of the episodes: i) the wording of the instructions allows us to fill in the first item (the targeted educational objective), ii) the documents provided to the pupils allow us to fill in the second item (the knowledge on which the teachers want the students' investigations to focus), iii) the pupils' questions and remarks are retained for the third item (the nature of the knowledge the students actually mobilize), and iiii) the pupils' written and oral productions are used for the last item (the effective way of understanding this knowledge by the students).

Findings

We will present and discuss the main results of the crossing of these four analyzes in terms of the model's potential to account for an orchestration of investigative approaches. First, the case study shows the differences in the nature of the knowledge-investigative objects perceived by the teacher and the pupils and the possible adjustments that it allows. Second, it also shows that the complementarity and the articulation of the two strategies adopted by the

Transforming Science & Technology Education to Cultivate Participatory Citizens Daegu, Korea, 23-28 August, 2020 teacher helped the pupils to identify stabilized knowledge but also to problematize a controversial knowledge as well as to consider their participation in the debate with dynamic positions subjected to re-consideration.


domain: sdu phys

text:
Julien Joliat

Antoine Patt

Jean-Marc Marc Simon

Sylvain Picaud
email: sylvain.picaud@univ-fcomte.fr

Adsorption of organic compounds at the surface of Enceladus' ice grains. A grand canonical Monte Carlo simulation study

Keywords: Enceladus, ice, adsorption, trace gases, simulation

come

Introduction

Enceladus is an active moon of Saturn which hides a global ocean of liquid salty water beneath its icy crust [START_REF] Porco | Cassini Observes the Active South Pole of Enceladus[END_REF]. This moon has been widely studied by the Cassini-Huygens mission a few years ago, that evidenced, among other discoveries, jets of icy particles from the internal ocean, mixed with a variety of simple organic compounds [START_REF] Hunter | Cassini ion and neutral mass spectrometer: Enceladus plume composition and structure[END_REF][START_REF] Hansen | Enceladus' water vapor plume[END_REF]. The plumes thus formed, extend hundreds of miles into space. Although approximately 5-10% of the emitted icy grains escapes and enters the Saturn's E ring, most of the matter they contain finally falls back onto the surface [START_REF] Kempf | How the Enceladus dust plume feeds Saturn's E ring[END_REF].

Since the ocean in Enceladus supplies the jets, knowing the exact plume composition would allow a better quantification of the abundance of organics at Enceladus, which is a prerequisite to understand its ocean's chemistry and, thus, its possible habitability [START_REF] Bouquet | How Adsorption Affects the Gas-Ice Partitioning of Organics Erupted from Enceladus[END_REF]. However, the organic molecules present in the vapour phase of the plumes are in contact with the icy grains that are concomitantly ejected. They also interact, before ejection, with the ice walls of the cracks through which the jets have erupted. As a consequence, these organics/ice interactions have an important impact on the measured composition of the plumes [START_REF] Khawaja | Low-mass nitrogen-, oxygenbearing, and aromatic compounds in Enceladean ice grains[END_REF], due to the differences in depletion vs release mechanisms that are driven by the characteristics of the involved species [START_REF] Bartels-Rausch | A review of air-ice chemical and physical interactions (AICI): liquids, quasi-liquids, and solids in snow[END_REF]. Indeed, adsorption of organic molecules on ice surfaces has been widely studied in the Earth's context, where it has been recognised as an important phenomenon able to modify the fate of various species in the troposphere as well as on the ice mantles at the surface [START_REF] Abbatt | Interactions of atmospheric trace gases with ice surfaces: Adsorption and reaction[END_REF][START_REF] Jpd Abbatt | Halogen activation via interactions with environmental ice and snow in the polar lower troposphere and other regions[END_REF].

In a recent study, Bouquet et al. have developed an analytical adsorption/desorption model to quantify the influence of the ice grains and ice walls on the plume composition [START_REF] Bouquet | How Adsorption Affects the Gas-Ice Partitioning of Organics Erupted from Enceladus[END_REF].

This model requires the knowledge of the binding energy of the adsorbed compounds to water ice, most of the corresponding values being based on available experimental values, obtained on Earth's conditions [START_REF] Bouquet | How Adsorption Affects the Gas-Ice Partitioning of Organics Erupted from Enceladus[END_REF]. The general conclusion of this study is that, in conditions relevant to Enceladus, adsorption on ice strongly segregates between compounds that are trapped or released by the ice surface, as a function of their binding energy. According to Bouquet et al., compounds with binding energy greater than -0.5 eV are thus negligibly present at the ice surface, whereas those with binding energy less than -0.7 eV dominates at the surface. Compounds with intermediate binding energies may also stay at the surface in some situations, if the competitive adsorption with those of lower energy allows [START_REF] Bouquet | How Adsorption Affects the Gas-Ice Partitioning of Organics Erupted from Enceladus[END_REF]. As a consequence, the preponderance of a compound at the ice surface is not directly correlated to its abundance in gas phase, and this has to be taken into account in the conclusions that can be drawn from the corresponding measurements.

A thorough characterisation of the uptake processes of organics on ice is thus of fundamental interest to adequately assess the fate of these species on Enceladus. However, although the adsorption of trace gases on ice surfaces has been widely characterised experimentally in the past decades in various contexts, including interstellar ices [START_REF] Bartels-Rausch | A review of air-ice chemical and physical interactions (AICI): liquids, quasi-liquids, and solids in snow[END_REF][START_REF] Abbatt | Interactions of atmospheric trace gases with ice surfaces: Adsorption and reaction[END_REF][START_REF] Jpd Abbatt | Halogen activation via interactions with environmental ice and snow in the polar lower troposphere and other regions[END_REF][START_REF] Huthwelker | The Uptake of Acidic Gases on Ice[END_REF][START_REF] Dominé | Air-snow interactions and atmospheric chemistry[END_REF][START_REF] John | Evaluated kinetic and photochemical data for atmospheric chemistry: Volume V-heterogeneous reactions on solid substrates[END_REF][START_REF] Mcneill | Organics in environmental ices: sources, chemistry, and impacts[END_REF][START_REF] Wa Schutte | Laboratory simulation of physical and chemical processes in interstellar ices[END_REF], some compounds of interest for Enceladus have not been studied yet, or their interaction with ice has been characterised in conditions that are not transferable to Enceladus. This implies that their binding energy to ice is not accurately known [START_REF] Khawaja | Low-mass nitrogen-, oxygenbearing, and aromatic compounds in Enceladean ice grains[END_REF][START_REF] Ba Magee | Neutral gas composition of Enceladus' plume-model parameter insights from Cassini-INMS[END_REF]. Moreover, fundamental questions regarding the location of the adsorbed molecules, their orientation, the number of hydrogen bonds they form at the ice surface as well as their ability to form single or multi-layers on ice often remain unanswered. All these points can be theoretically tackled by means of atomistic simulations, given that the model chosen to represent the organic/water interactions, accurately reproduces known experimental properties [START_REF] Picaud | Molecular dynamics and Monte Carlo simulations of organic compounds adsorbed on ice surfaces[END_REF][START_REF] Hantal | Molecular-scale modeling of the adsorption of partially oxidized organic compounds on ice[END_REF][START_REF] Picaud | Molecularscale simulations of organic compounds on ice: application to atmospheric and interstellar sciences[END_REF]. As far as the structure and the energy of the adsorption process are concerned, the Grand Canonical Monte Carlo (GCMC) method has been shown to be particularly suitable to provide detailed direct insights of the studied system, at the molecular level [START_REF] Picaud | Molecularscale simulations of organic compounds on ice: application to atmospheric and interstellar sciences[END_REF]. Indeed, GCMC simulations allow the theoretical determination of the adsorption isotherm, i.e. the surface coverage as a function of the partial pressure at a given temperature. In addition, the results of the simulations can be statistically analysed for fixed values of the partial pressure, and give access to the average binding energy and the distribution of the molecular orientations at the surface.

Thus, for more than 15 years, we have used the GCMC method to model, at the molecular scale, the interaction between ice and various organic molecules (see Ref. [START_REF] Picaud | Molecularscale simulations of organic compounds on ice: application to atmospheric and interstellar sciences[END_REF] and reference therein). More recently, we also used the same approach to simulate the trapping of various atomic and molecular species at the surface or inside clathrate hydrates, another form of solid water, in the astrophysical context [START_REF] Lectez | A 32-70 K Formation Temperature Range for the Ice Grains Agglomerated by Comet 67 P/Churyumov-Gerasimenko[END_REF][START_REF] Fábián | Ammonia clathrate hydrate as seen from grand canonical monte carlo simulations[END_REF][START_REF] Patt | A grand canonical monte carlo study of the N2, CO, and mixed N2-CO clathrate hydrates[END_REF][START_REF] Petuya | Molecular Selectivity of CO-N2 Mixed Hydrates: Raman Spectroscopy and GCMC Studies[END_REF][START_REF] Patt | Adsorption of CO and N2 molecules at the surface of solid water. A grand canonical Monte Carlo study[END_REF]. In these simulations, large numbers of molecules have been considered and the temperature effects have been included. This is an advantage over other methods like, e.g. optimisation methods or quantum calculations that are usually restricted to a limited number of molecules, the adsorption of which being most often characterised at 0 K.

Here, we have performed GCMC simulations to investigate the behaviour at the surface of crystalline I h ice of three molecules, namely ethylene (C 2 H 4 ), propanol (C 3 H 8 O) and hexanal (C 6 H 12 O) at 236 K, i.e. a temperature relevant for the ice grains and the ice walls that are in contact with organic compounds in the cracks near the surface of Enceladus [START_REF] Bouquet | How Adsorption Affects the Gas-Ice Partitioning of Organics Erupted from Enceladus[END_REF]. These molecules have been selected among others because, to the best of our knowledge, their adsorption on ice has never been simulated before, and available experimental data indicate that their binding energy to ice (E b ) ranges in the three different domains mentioned by Bouquet et al. [START_REF] Bouquet | How Adsorption Affects the Gas-Ice Partitioning of Organics Erupted from Enceladus[END_REF], namely E b (C 2 H 4 ) = -0.16 eV [START_REF] Silva | Interaction of acetylene, ethylene, and benzene with ice surfaces[END_REF], E b (C 3 H 8 O) = -0.71 eV [START_REF] Dj Donaldson | Adsorption of atmospheric gases at the air-water interface. 2. C1-C4 alcohols, acids, and acetone[END_REF] and E b (C 6 H 12 O) = -0.67 eV [START_REF] Sokolov | Adsorption to Ice of n-Alcohols (Ethanol to 1-Hexanol), Acetic Acid, and Hexanal[END_REF]. Notice that ethylene and propanol have been actually detected by the Cassini mission [START_REF] Ba Magee | Neutral gas composition of Enceladus' plume-model parameter insights from Cassini-INMS[END_REF], contrarily to hexanal, the mass of which being out of the instrumental range of measurements. Nevertheless, this molecule has been taken into account in the model developed by Bouquet et al. [START_REF] Bouquet | How Adsorption Affects the Gas-Ice Partitioning of Organics Erupted from Enceladus[END_REF] and thus deserves consideration in the present GCMC simulations.

This paper is organised as follows. In Section 2 details of the GCMC simulations are given, and the corresponding results for the three organic compounds considered here are detailed in Section 3 and discussed in Section 4. Finally, in Section 5 the main conclusions of this study are summarised.

Details of the computer simulations at the molecular scale

The adsorption of C 2 H 4 , C 3 H 8 O and C 6 H 12 O has been studied at the surface of crystalline hexagonal ice, at 236 K. The algorithm of Buch et al. [START_REF] Buch | Simulations of H2O solid, liquid, and clusters, with an emphasis on ferroelectric ordering transition in hexagonal ice[END_REF] has been used to prepare the I h phase of ice by considering water molecules arranged in 18 layers (containing 160 molecules each), along the interface normal axis (z) in such a way that adsorption on the (0001) surface of crystalline ice can be investigated. The ice solid has a proton disordered hexagonal structure that fulfils the Bernal-Fowler rules [START_REF] John | A theory of water and ionic solution, with particular reference to hydrogen and hydroxyl ions[END_REF], with no net charge nor total dipole moment. In the GCMC simulations, this ice slab has been placed in the middle of a rectangular simulation box, the edges of which being equal to 35.926 and 38.891 Å along the x and y axes, respectively. Periodic boundary conditions have been applied to the simulation box in these (x,y) directions, whereas a large vacuum above the surface along the z direction has been created, with three-dimensional periodicity (see Figure 1). The simulated system is thus made of a stack of non-interacting ice slabs of infinite surface area, in such a way that adsorption can be considered at the two gas/ice interfaces present in the simulation box to ensure better statistics especially when the number of adsorbed molecules is very weak. In the present simulations, the edge of the rectangular basic simulation box has been 100 Å along the z direction, a value that represents a good compromise between a sufficiently large space separating two (non-interacting) gas/ice interfaces and a volume to be filled not too large, when the condensation pressure has been reached in the simulations. Notice that, some tests have been performed with a value of 150 Å, which did not show any noticeable differences in the corresponding results.

Z Ethylene

Propan-1-ol The water molecules have been described by the four-site TIP4P/ice model [START_REF]A potential model for the study of ices and amorphous water: TIP4P/Ice[END_REF] which reproduces well the phase diagram of water. With this model, the melting temperature of I h ice is around 272 K [START_REF]A potential model for the study of ices and amorphous water: TIP4P/Ice[END_REF], i.e. well above the temperature considered here, thus avoiding any artificial ice melting in the present GCMC simulations. Simple intermolecular potentials, based on site-site electrostatic and Lennard-Jones dispersionrepulsion contributions, have been used to represent the organic compounds. The parameters of the corresponding models have been taken from the literature and they are gathered in Table 1. Thus, ethylene has been described by an anisotropic united-atoms force field specifically parametrised to predict the equilibrium properties of various olefins [START_REF] Bourasseau | New optimization method for intermolecular potentials: Optimization of a new anisotropic united atoms potential for olefins: Prediction of equilibrium properties[END_REF]. Similarly, the anisotropic united-atoms AUA4 potential has been chosen to model both the propanol [START_REF] Ferrando | Transferable Force Field for Alcohols and Polyalcohols[END_REF] and hexanal [START_REF] Ferrando | Monte Carlo Simulations of Mixtures Involving Ketones and Aldehydes by a Direct Bubble Pressure Calculation[END_REF] molecules. Indeed, these interaction models have been shown to accurately represent experimental data for the thermodynamics properties of these organic compounds. Notice that, while the small ethylene molecule has been kept rigid, flexible propanol and hexanal molecules have been considered in our simulations. The usual Lorentz-Berthelot combining rules have been used to calculate cross interactions between water and the organic molecules [START_REF] Allen | Computer Simulation of Liquids. 1st[END_REF]. In our GCMC simulations, the short-range Lennard-Jones interactions have been truncated using a simple cut-off equal to half the shortest box-length whereas the Ewald summation method has been used to calculate the electrostatic interactions, in accordance with the original parameterisation of the corresponding models.

Hexanal C 1 C 2 C 1 C 2 C 3 O H C 1 C 2 C 3 C 4 C 5 C 6 O
The details of the interaction potential models can be found in the original publications [START_REF] Bourasseau | New optimization method for intermolecular potentials: Optimization of a new anisotropic united atoms potential for olefins: Prediction of equilibrium properties[END_REF][START_REF] Ferrando | Transferable Force Field for Alcohols and Polyalcohols[END_REF][START_REF] Ferrando | Monte Carlo Simulations of Mixtures Involving Ketones and Aldehydes by a Direct Bubble Pressure Calculation[END_REF]. Monte Carlo simulations in the grand canonical ensemble have been performed using the GIBBS code [START_REF] Ungerer | Applications of molecular simulation in the oil and gas industry: Monte Carlo methods[END_REF]. The average number of particles N in the simulation box has been calculated at constant temperature (T = 236 K), volume (V) and fugacity (f ) [START_REF] Patt | A grand canonical monte carlo study of the N2, CO, and mixed N2-CO clathrate hydrates[END_REF]. Different fugacity values have been used to simulate adsorption processes from vanishingly small surface coverage to the condensation situation. For each fugacity value, 10 9 iteration steps have been performed to ensure the equilibration, followed by a production stage of 2.10 8 steps. Data have been collected every 4.10 4 steps for statistical analysis (meaning that 5000 configurations have been included in the analysis, each time). In these MC simulations, particle displacement and insertion/deletion moves have been performed with equal probabilities. For the rigid ethylene molecule, these probabilities have been 25%, 25% and 50%, for translation, rotation, and insertion-deletion moves, respectively. For the flexible propanol and hexanal molecules, changes of the internal molecular geometry have also been considered, and the probabilities for moves have thus been set to 20% for translation, 20% for rotation, 20% for internal changes, and the remaining 40% for insertion-deletion. Notice that uniform probability of insertion has been used for the gas and solid regions, such as no bias has been introduced in the choice of the insertion sites for the organic molecules. In addition, because adsorption process was investigated, only the organic molecules have been inserted or deleted in the simulation box and, accordingly, only translations and rotations have been allowed for the water molecules.

Simulation results

Adsorption of ethylene on crystalline ice

Let us first detail the main properties of ethylene adsorbed on proton disordered crystalline I h ice. The adsorption isotherm, N (f ), for this species is shown in Figure 2, as obtained from the GCMC simulations at 236 K. The obtained curve departs from zero at a fugacity value around 10 5 Pa and then increases with increasing steepness in a quite narrow range of fugacity, up to the point where the condensation of ethylene occurs, at a fugacity value of about 10 6 Pa. Note that above the condensation pressure, the increase of the isotherm is simply related to the high compressibility of the ethylene fluid at high pressure, corresponding to a density increase from 455 kg/m 3 to 577 kg/m 3 , between 1.6 and 101 MPa, at 236 K [START_REF] Pj Linstrom | NIST Chemistry WebBook[END_REF]. This behaviour is similar to what has been simulated for the adsorption of formaldehyde on ice [START_REF] Hantal | Calculation of the adsorption isotherm of formaldehyde on ice by grand canonical Monte Carlo simulation[END_REF] and might indicate that no saturated monolayer of ethylene molecules is particularly stable at the ice surface. To confirm this assumption, we have analysed the results of the GCMC simulations at three fugacity values below the condensation, corresponding to very low (System 1), medium (System 2) and high (System 3) coverages of the ice surface, and just above the condensation (System 4) (as marked by the arrows in Figure 2). As an example of analysis aiming at characterising the translational arrangement of the ethylene molecules, their number density profile ρ(z) is shown in Figure 3(a). This number density profile gives the average number of ethylene molecules (represented by the position of their centre of mass) that are located at a given position along the ice surface normal axis z. Notice that the density profile for the interfacial water molecules is also indicated in this figure, for reference.

In the case of System 1, all the ethylene molecules are adsorbed at the surface of ice, as the density profile nearly drops to zero beyond the z value of about 87 Å , i.e. at the distance that is about one molecular layer away from the ice surface. In systems 2 and 3, most of the ethylene molecules are again in contact with the ice phase, although a weak shoulder at larger z positions indicates that a second peak of the density profile starts to appear at about z = 88.5 Å. Finally, in System 4, the ethylene molecules occupy the whole empty space above the ice phase, because 3D condensation of the adsorbate has already occurred at the corresponding fugacity value. Interestingly, the intensity of the peak located at around 85 Å in Systems 3 and 4, which corresponds to the ethylene molecules that are directly in contact with the ice surface, is much higher than the peaks located at the same position for Systems 1 and 2. This, together with the long tail of the peak observed for System 3, clearly indicates that additional layers of ethylene molecules started to build up before the first adsorption layer is completed. Note that, in this first adsorption layer, the ethylene molecules lie parallel to the ice surface, as shown by the statistical analysis of the molecular orientations (not displayed).

To characterise the energetic background of the adsorption, we have thus analysed the distributions of the interaction energy of an adsorbed ethylene molecule with all the water molecules of the ice phase, U ads-w , and that of a given ethylene molecule with all the other ethylene molecules present in the simulation box, U ads-ads (Figure 3(b,c)), as obtained in Systems 1, 2, and 3.

At low surface coverage (System 1), the P(U ads-w ) and P(U ads-ads ) distributions have their peak at around -9 and 0 kJ/mol, respectively, indicating that the interaction of ethylene with ice is very weak and that the lateral interactions within the adsorbate are vanishingly small. Consequently, for System 1, the mean value of the P(U ads-w ) distribution can be regarded as an estimate of the heat of adsorption at very low surface coverages. At higher coverage (System 2), weak lateral interactions are evidenced by the shift of the peak of P(U ads-ads ) to smaller (i.e. larger in magnitude) energies, which now appears at around -4 kJ/mol. This tendency is reinforced for System 3, where the lateral interactions increase and shift the peak of P(U ads-ads ) down to about -9 kJ/mol. However, this increase of the interaction between coadsorbed ethylene molecules does not significantly change their interaction with the ice phase, the peak of the P(U ads-ads ) distribution being still observed at around -9 kJ/mol. Note that the narrow peak at around 0 kJ/mol in the P(U ads-w ) distributions for Systems 1, 2 and 3 also reveals that some ethylene molecules are located far enough from the ice surface to not significantly interact with the water molecules, in accordance with the long tails of the peaks evidenced in the density profiles of the ethylene centres of mass (Figure 3(a)).

To summarise, the results of the statistical analysis presented above confirm that, as for formaldehyde [START_REF] Hantal | Calculation of the adsorption isotherm of formaldehyde on ice by grand canonical Monte Carlo simulation[END_REF], the shape of the isotherm adsorption of ethylene on ice does not evidence any formation of a stable monolayer, a feature that can be related to the weakness of the interaction between the ethylene molecules and the ice phase.

Adsorption isotherms of propanol and hexanal on crystalline ice

The adsorption isotherms N (f ) of propanol and hexanal, as obtained from the simulations, are plotted in Figure 4(a). In the case of propanol (red curve), this isotherm typically shows three different regions. At low fugacity values, the isotherm exhibits an increase up to about 1.0 Pa that corresponds to the building up process of the adsorption layer during which the possible adsorption sites at the ice surface are gradually occupied. Above this f value, a pseudo-plateau is observed in the isotherm which indicates that the adsorption layer is close to be saturated. Indeed, large changes in the fugacity result in only a small increase (of no more than about 10%) of the number of adsorbed molecules in this region. Finally, at f value of about 35 Pa, the isotherm exhibits a sudden jump, corresponding to the 3D condensation of propanol and above this fugacity, the box is thus entirely filled with propanol molecules. In this respect, the adsorption isotherm for propanol is rather similar to what was previously obtained for methanol [START_REF] Jedlovszky | Determination of the Adsorption Isotherm of Methanol on the Surface of Ice. An Experimental and Grand Canonical Monte Carlo Simulation Study[END_REF].

The adsorption isotherm N (f ) calculated for hexanal (black curve) also exhibits three typical regions, as for propanol, although its behaviour is not exactly similar. Indeed, the adsorption isotherm for hexanal is firstly characterised by a sharp exponential increase at low fugacity value, indicating that the adsorption of an individual hexanal molecule is independent from the presence or absence of other molecules at the surface of ice. The isotherm then exhibits an increasing linear shape rather than a plateau, for fugacity values ranging typically between 0.05 and 9 Pa. This behaviour rather seems correspond to a continuous, progressive saturation of all the adsorption sites at the ice surface, than to the stabilisation of a monolayer in a large range of fugacity values. Finally, above 9.5 Pa, condensation of hexanal occurs, as indicated by the sudden jump observed in the isotherm at this fugacity value.

Notice that we have also checked the influence of the box size on these results by recalculating some points of the isotherms, considering now a box of edge size equal to 150 Å along the z axis, instead of 100 Å. No noticeable difference has been evidenced in the results obtained with these two different box sizes. More specially, the 3D condensation has been shown to occur at the same fugacity values in both cases and for the two species considered, thus precluding any artefact due to the size of the empty space between the ice slabs.

To further investigate the differences between propanol and hexanal behaviours at the surface of ice, their adsorption isotherms have been converted to the Γ(f rel ) form, where Γ is the surface density of the adsorbed molecules expressed in µmol.m -2 , and f rel is the fugacity of the vapour phase f normalised by the fugacity f 0 at which the 3D condensation is observed for the species considered. The value of Γ has simply been calculated, up to the point where condensation occurs, as

Γ = N 2XY ( 1
)
where N is the number of propanol or hexanal molecules that are adsorbed at the ice surface and X and Y are the lengths of the box edges along the x and y directions, respectively. Note that the factor 2 at the denominator accounts for the fact that, due to the periodicity of the system, the ice phase has two surfaces in the basic box. The corresponding calculated isotherms are shown in the Γ(f rel ) form in Figure 4(b,c) for propanol and hexanal, respectively. Following our previous studies on alcohol [START_REF] Jedlovszky | Determination of the Adsorption Isotherm of Methanol on the Surface of Ice. An Experimental and Grand Canonical Monte Carlo Simulation Study[END_REF] and aldehyde [START_REF] Hantal | Calculation of the adsorption isotherm of formaldehyde on ice by grand canonical Monte Carlo simulation[END_REF][START_REF] Petitjean | Adsorption of Benzaldehyde at the Surface of Ice, Studied by Experimental Method and Computer Simulation[END_REF][START_REF] Darvas | Adsorption of Acetaldehyde on Ice As Seen from Computer Simulation and Infrared Spectroscopy Measurements[END_REF] molecules adsorbed on ice, these two isotherms have been tentatively described in terms of the Langmuir isotherm, i.e. by the function:

Γ = Γ max f rel K f rel K + 1 ( 2
)
where the parameters Γ max and K are the saturated surface density and Langmuir partitioning coefficient, respectively.

As is seen from Figure 4(b), the calculated Γ(f rel ) points for propanol can be well fitted by the Langmuir isotherm up to about f rel = 0.1, i.e. a fugacity range for which the lateral interactions between the adsorbed molecules seem to not play any significant role. However, at higher relative fugacity values, the simulated isotherm clearly departs from the Langmuir behaviour, as previously observed for methanol [START_REF] Jedlovszky | Determination of the Adsorption Isotherm of Methanol on the Surface of Ice. An Experimental and Grand Canonical Monte Carlo Simulation Study[END_REF]. For hexanal, no satisfactory Langmuir fit can be achieved on the simulated data, except perhaps when considering the lowest relative fugacity values, only (Figure 4(c)). This could originate from larger lateral interactions within the adsorbate, even at low coverage of the ice surface. The behaviour of hexanal on ice is thus closer to that of benzaldehyde [START_REF] Petitjean | Adsorption of Benzaldehyde at the Surface of Ice, Studied by Experimental Method and Computer Simulation[END_REF] than of acetaldehyde [START_REF] Darvas | Adsorption of Acetaldehyde on Ice As Seen from Computer Simulation and Infrared Spectroscopy Measurements[END_REF] and could be related to molecular size effects (here, the length of the aliphatic carbon chain).

Analysis of the adsorption layers of propanol and hexanal on crystalline ice

To investigate in more details the adsorption characteristics of propanol and hexanal on ice, four different values of the fugacity have been selected for which statistical analyses of the adsorbed phase have been performed. Thus, System 1, corresponding to very low fugacity value, is located at the exponentially rising part of the isotherms, where even the first molecular layer of the adsorbate is far from being saturated. The fugacity value selected for System 2 is located approximately at the position where the quasi-plateau of the propanol isotherm starts or in the middle of the fugacity value range where the linear part of the isotherm of hexanal is obtained. The fugacity value for System 3 is chosen to be just before the condensation occurs. Finally, System 4 corresponds to the condensed phase.

Density profiles

The density profiles of the propanol and hexanal molecules along the interface normal axis z, as obtained in systems 1-4, are presented in Figures 5 and6, respectively. In these density profiles, separated analyses have been done for the positions of the molecular centres of mass and of various atoms of the molecules.

All the profiles shown are averaged over the two interfaces present in the basic box. As is seen, all these profiles present typically one single peak before the condensation occurs (i.e. for Systems 1-3), the intensity of this peak increasing with the coverage of the ice surface. This indicates that the saturated adsorption layer is still monomolecular up to the point where the presence of a rather disordered condensed phase is evidenced at the highest fugacity values considered here (Systems 4). Additional information can be obtained by carefully looking at the peak positions. Indeed, for propanol the density peak of the O atoms is always located closer to the ice surface than that of both the centre of mass and the terminal C atoms, indicating that the adsorbed molecules prefer to point, on average, towards the ice surface by their OH group. Interestingly, the density profile for H atoms exhibits only one peak in the case of System 1, while two peaks are obtained for the other Systems, suggesting that both donor and acceptor hydrogen bonds with the ice surface are thus formed when the propanol coverage increases (Figure 4(d)). Moreover, increasing the propanol coverage from System 1 to System 3 leads to a small shift of both the centre of mass and O atom density peaks to higher distances from the ice surface (Figure 5(a,c)), while the peak position of the terminal C atoms density does not shift (Figure 5(b)). We can thus infer some configurational changes of the propanol molecular orientation upon coverage increase, the molecular axis straightening out on the surface when the number of propanol molecules increases at the surface. For hexanal, the density peaks of the terminal C (C 6 ) and O atoms (Figure 6(b,c)) are always located closer to the ice surface than that of the centres of mass (Figure 6(a)), indicating that the adsorbed molecules prefer to point, on average, towards the ice surface by their aldehyde functional group. Moreover, increasing the coverage of the ice surface does not lead to any significant shift to higher distances of the density peaks of the O (Figure 6(c)) and terminal C atoms (Figure 6(b)), contrarily to the 2 Å shift observed for the peaks of the density of the centres of mass (Figure 6(a)). This suggests that a reorientation of the adsorbed molecules occurs upon saturation of the adsorption layer, the molecular axis becoming less tilted with respect to the surface normal when increasing the number of adsorbed molecules, but without changing the configuration of the carbonyl group with respect to the ice surface. In other word, only the long aliphatic chain seems to be affected by the coverage increase. This is clearly confirmed by the density profile of the methyl carbon atom (labelled C 1 , see Figure 1) given in Figure 6(d), which exhibits a main peak at around 85.5 Å in System 1, that shifts to higher values around 90 Å, when the coverage is increased (Systems 2-4).

These points are further discussed in a following sub-section devoted to the analysis of the molecular orientations of the adsorbed molecules at the ice surface.

Energetic background of the adsorption

In order to get insight into the energetic background of the adsorption, we have calculated the distribution of the U ads-ads and U ads-w contributions to the total binding energy, coming from the interaction between the adsorbed (propanol or hexanal) molecules and the water molecules of the ice phase and from the lateral interaction with the other adsorbed molecules, respectively. The corresponding P(U ads-w ) and P(U ads-ads ) distributions obtained in Systems 1-3 (i.e. before condensation has occurred) are shown on the top and bottom panels of Figure 7, respectively.

As is seen, at low surface coverage (i.e. in System 1 for both propanol and hexanal adsorbates) the P(U ads-w ) distribution exhibits a single, rather large, peak around -67 kJ/mol and -58 kJ/mol, for propanol and hexanal adsorbates, respectively, the corresponding mean values of the distribution being about -70 kJ/mol and -59 kJ/mol. Notice that, as stated above for ethylene, these values can serve as an estimate for the heat of adsorption at infinitely low coverage [START_REF] Darvas | Adsorption of Acetaldehyde on Ice As Seen from Computer Simulation and Infrared Spectroscopy Measurements[END_REF].

Upon increasing the coverage of the ice surface, the peak of the P(U ads-w ) distribution gradually shifts to higher energies (i.e. smaller absolute values), its maximum value being at about -41 and -39 kJ/mol in System 3, for propanol and hexanal molecules, respectively, as a consequence of the increasing competition of the lateral interactions between adsorbed molecules (see below). In addition, for this system that corresponds to a fugacity value just below f 0 (i.e. the value at which condensation occurs), a very small number of molecules starts to be adsorbed farther from the ice surface in the GCMC runs, as indicated by the small peak of the P(U ads-w ) distributions, in the [-10, 0] kJ/mol range. The P(U ads-ads ) distribution (Figure 7, bottom) exhibits a large peak at very small energy values in System 1 for both propanol and hexanal adsorbates, reflecting the fact that, at this low coverage, the adsorbed molecules stay rather far from each other at the ice surface. Notice however that the shoulder of this peak towards lower energy is wider for hexanal than for propanol molecules, indicating larger lateral interactions (albeit still weak) within the hexanal adsorbate, even at relatively low coverage. This could explain the different shape of the corresponding isotherms in the low fugacity range (Figure 4). By contrast, at higher coverages of the surface (Systems 2 and 3), the P(U ads-ads ) distribution shifts toward much lower energy values, indicating the strong increase of the lateral interactions within the adsorbate. Just before condensation (i.e. for System 3), this distribution is characterised by a large single peak around -55 kJ/mol in the case of hexanal, whereas it exhibits one main, large peak at around -50 kJ/mol and another, smaller one, around -25 kJ/mol, in the case of propanol. Considering the fact that the energy of a hydrogen bond fall roughly in the [-25 --20] kJ/mol range, the above findings indicates that the adsorbed molecules might form either one or two hydrogen bonds, or only two hydrogen bonds with their neighbours, at monolayer coverage, in the case of propanol and hexanal, respectively. However, the real situation could be more complicated because part of the lateral interactions may come not only from hydrogen bonding between adjacent polar heads, but also from additional interactions between the long aliphatic chains, depending on their respective orientations.

Then, we have calculated the average number of hydrogen bonds formed between propanol and water molecules, and between neighbouring propanol molecules within the adsorbed layer, by using the standard geometrical criterion for the definition of these H-bonds [START_REF] Arunan | Definition of the hydrogen bond (IUPAC Recommendations 2011)[END_REF]. For System 1, about 3 H-bonds are thus formed between the propanol and the water molecules, in accordance with the calculated value of the heat of adsorption at infinitely low coverage (-70 kJ/mol). Upon increasing the propanol coverage with fugacity, the average number of H-bond between propanol and ice decreases from about 3.11 (System 1) to about 1.6 and 1.3 for Systems 2 and 3, respectively, in connection with the increase to about -40 kJ/mol of the corresponding U ads-w values. Concomitantly, the average number of H-bonds between neighbouring propanol molecules within the adlayer increases from about 0.1 (System 1) to 0.90 (System 2) and 1.05 (System 3), again in accordance with the variations of the U ads-ads values. However, because one H-bond accounts for about -25 kJ/mol, only, the peak at around -50 kJ/mol in the P(U ads-ads ) distribution indicates that the aliphatic chains may also significantly contribute to the intra-adsorbate interaction energy. It is worth mentioning that, for hexanal, the interaction model we use do not consider explicitly the H atom of the aldehyde group, preventing thus any rigorous analysis of the hydrogen bonding for this molecule.

Orientation of the adsorbed molecules

The description of the adsorption process can be completed by the characterisation of the molecular orientations at the surface of ice. Here, we focus on the orientation of the molecular axis, which can be defined by the angle θ 1 formed between the vector normal to the ice surface (z) and the C-C vector joining the C atom (C 1 , see Figure 1) of the terminal aliphatic group (CH 3 ) to the C atom bearing the alcohol (C 3 ) or the aldehyde (C 6 ) function. In this case, a value θ 1 = 0 corresponds to a situation where this C-C vector is oriented perpendicular to the ice surface, the aliphatic group pointing to the ice surface, while the functional group points to the gas phase. In addition, to get more information on the orientation of the functional groups of these molecules with respect to the ice surface, we also characterise the angle θ 2 formed between z and the CO vector and, for propanol, the angle θ 3 formed between z and the OH vector. Note that because the adsorbed molecules are flexible in the simulations, their orientation cannot be unambiguously described by the two independent polar coordinates θ and φ that are usually used when considering the orientation of a rigid body relative to an external direction [START_REF] Jedlovszky | New insight into the orientational order of water molecules at the water/1,2-dichloroethane interface: A Monte Carlo simulation study[END_REF].

The following orientational analysis is limited to the propanol and hexanal molecules that are directly attached to the ice surface, i.e. those corresponding to the first peak in the centre of mass density profiles of the adsorbate shown in Figures 5 and6. The orientation distribution of the angle θ 1 for propanol given in Figure 8 (top left panel) clearly indicates that a reorientation of the molecular axis occurs upon the adsorption of an increasing number of molecules. Indeed, at very low coverage, this distribution is characterised by a single peak with a maximum at the value θ 1 ≈ 103 • which shifts to larger values upon coverage increase, up to a maxi-mum corresponding to θ 1 ≈ 155 • . The propanol molecules thus move from a configuration nearly parallel to the ice surface at low coverage, to a configuration in which their molecular axis is tilted from the surface normal by about 25 • , the polar head pointing to the ice surface, at higher coverages. Meanwhile, the orientation of the polar head significantly changes upon adsorption of an increasing number of propanol molecules, as indicated by the evolution of the distributions of θ 2 and θ 3 angles. Interestingly, at high coverage, P(θ 3 ) is characterised by a double peak (bottom left panel of Figure 8) which corresponds to configurations where the OH bond points either down to the surface or to the gas phase, supporting the assumption on the formation of proton donor and proton acceptor hydrogens bonds with the underlying water molecules, as suggested by the analysis of the density profile for H atoms (see above).

A similar behaviour is observed for the hexanal molecules that move from configurations nearly parallel to the ice surface at low coverage (right top panel of Figure 8) to configurations in which their molecular axis is tilted from the surface normal by about 30 • , so that the aldehyde function points to the ice surface at higher coverages, the long aliphatic chain being oriented in such a way that the terminal CH 3 points towards the gas phase (see also the snapshots given in Figure 6 for illustration). Note however that the orientation of the polar head remains unchanged irrespective of the coverage, the CO bond being always preferentially nearly parallel to the ice surface, as indicated by the corresponding distribution P(θ 2 ) (bottom right panel of Figure 8).

The results of this orientational analysis are thus in clear accordance with the information issued from the evolutions of the density profiles upon coverage increase and of the corresponding distributions of the U ads-ads and U ads-w contributions to the total binding energy, as analysed above.

Discussion

The results of the present GCMC simulations have evidenced that ethylene is quite weakly bonded to ice, with a mean interaction energy of about -9 kJ/mol, i.e. -0.093 eV, with the ice surface, which precludes the formation of any stable adlayer on ice before 3D condensation occurs. Although there is no experimental information at 236 K, the results of the simulation agree quite well with previous experimental conclusions [START_REF] Silva | Interaction of Acetylene, Ethylene, and Benzene with Ice Surfaces[END_REF]. Indeed, on the basis of infrared spectroscopy measurements, it has been concluded that the interaction energy of ethylene on amorphous ice is equal to about 0.16 eV (15 kJ/mol), which corresponds to a measured shift of 75 cm -1 of the dangling-H signal. Because this shift has been measured to be 1/3 less on crystalline ice (50 cm -1 ), and assuming a linear relation between shift and bond energy [START_REF] Silva | Interaction of Acetylene, Ethylene, and Benzene with Ice Surfaces[END_REF], we can infer that the corresponding interaction between ethylene and crystalline ice would be proportionally smaller than on amorphous ice, with a value thus estimated to be about 0.11 eV, i.e. 10.6 kJ/mol, in fair agreement with the value issued from our simulations. Moreover, the experimental spectra do not give any evidence of ethylene acting as proton donor on ice [START_REF] Silva | Interaction of Acetylene, Ethylene, and Benzene with Ice Surfaces[END_REF], which is consistent with the flat orientation of the molecular axis found in the present GCMC simulations.

By contrast, GCMC simulations have revealed that both propanol and hexanal molecules can be strongly adsorbed at the surface of crystalline ice at 236 K, a temperature typical of Enceladus' environment. The corresponding adsorption process is characterised by the stabilisation of a single layer of propanol molecules over a large range of fugacity, and by the progressive saturation of all possible adsorption sites for hexanal, up to the 3D condensation of the adsorbate above the ice phase. These features are in qualitative agreement with experimental conclusions coming from flow tube measurements by Sokolov and Abbatt at 228 K [START_REF] Sokolov | Adsorption to Ice of n-Alcohols (Ethanol to 1-Hexanol), Acetic Acid, and Hexanal[END_REF]. Indeed, in these experiments, it has been concluded that the adsorption process of 1-propanol on ice follows Langmuir behaviour and that, in the pressure range investigated, propanol may form a saturated layer at the surface of ice. Moreover, there are strong indications that the adsorbed molecules are oriented in such a way that their hydroxyl group binds to the ice surface [START_REF] Sokolov | Adsorption to Ice of n-Alcohols (Ethanol to 1-Hexanol), Acetic Acid, and Hexanal[END_REF], in accordance with the results of the present orientational analysis. Notice however, that the simulated isotherm clearly deviates from the Langmuir behaviour, even at relatively moderate pressure, and this deviation could be related to the increasing contribution of the lateral interactions within the adsorbed layer [START_REF] Jedlovszky | Determination of the Adsorption Isotherm of Methanol on the Surface of Ice. An Experimental and Grand Canonical Monte Carlo Simulation Study[END_REF]. This indicates that, for molecules that may strongly interact together even at moderate coverage of the ice surface, experiments should be also conducted (if possible) under relatively high partial pressure conditions. Nevertheless, the adsorption enthalpy of propanol on ice at low coverage has been estimated to be around -68 kJ/mol (-0.7 eV) in the experiments [START_REF] Sokolov | Adsorption to Ice of n-Alcohols (Ethanol to 1-Hexanol), Acetic Acid, and Hexanal[END_REF][START_REF] Buch | Simulations of H2O solid, liquid, and clusters, with an emphasis on ferroelectric ordering transition in hexagonal ice[END_REF], a value which is very close to the one determined in the present GCMC simulations for System 1 (see above).

For hexanal, the uptake at the surface of ice has been shown to exhibit a linear behaviour between 218 and 226 K, in the pressure range experimentally investigated (P < 0.013 Pa), so that no Langmuir equation can be adequately fit on the data of the corresponding measured isotherms [START_REF] Sokolov | Adsorption to Ice of n-Alcohols (Ethanol to 1-Hexanol), Acetic Acid, and Hexanal[END_REF], in qualitative agreement with the non-Langmuir behaviour evidenced by the results of the present GCMC calculations at 236 K. Meanwhile, the adsorption energy of hexanal has been experimentally determined to be around -64.8±5.8 kJ/mol (-0.67 ± 0.06eV) [START_REF] Sokolov | Adsorption to Ice of n-Alcohols (Ethanol to 1-Hexanol), Acetic Acid, and Hexanal[END_REF], i.e. only slightly lower than the value calculated here using Monte Carlo simulations (-0.61 eV).

All these features give thus confidence in the ability of the interaction potential models used in the present GCMC simulations, to accurately represent the adsorption process of ethylene, propanol and hexanal molecules at the surface of hexagonal ice, in thermodynamics conditions relevant for Enceladus' environments. As a consequence, the results of such studies can be used to give reliable input data for 1) of the terminal aliphatic group (CH 3 ) to the C atom bearing the alcohol (C 3 , see Figure 1) or the aldehyde function (C 6 , see Figure 1). Middle row: θ 2 is the angle formed between z and the CO vector; bottom row (for propanol, only), θ 3 is the angle formed between z and the OH vector.

adsorption models such as the one recently developed by Bouquet et al. [START_REF] Bouquet | How Adsorption Affects the Gas-Ice Partitioning of Organics Erupted from Enceladus[END_REF]. This seems particularly true when considering the binding energy at low coverage, which has been shown to be the key parameter in this model, for segregating between compounds that are trapped or released by the ice surface. Thus, according to the conclusions of Bouquet et al., ethylene which has a binding energy greater than -0.5 eV should be negligibly present at the ice surface, in contrast with propanol which has a binding energy of about -0.7 eV. The adsorption energy of hexanal falling just in between these two values, this species may compete with compounds of lower energy and may thus likely stay at the ice surface in some situations [START_REF] Bouquet | How Adsorption Affects the Gas-Ice Partitioning of Organics Erupted from Enceladus[END_REF].

Moreover, the present simulations indicate that the shape of the adsorption isotherm of propanol and hexanal significantly deviates from the Langmuir equation unless only the very low pressure range is taken into account when fitting the corresponding parameters (for propanol). This non-Langmuir behaviour influences the value of the saturation coverage, which thus could be underestimated in the interpretation of the experimental data, with respect to the value calculated in the GCMC simulations when the lateral interactions are taken into account. For instance, this saturation coverage has been experimentally estimated to be around 3.1 × 10 14 molecules.cm -2 for propanol at 226 K, whereas the present simulation results indicate that it is certainly greater than 4.3 × 10 14 molecules.cm -2 at 236 K, i.e. the value issued from our Langmuir fit (see Figure 4, and the deviation from the Langmuir behaviour at high relative pressure). It is thus reasonable to conclude, as previously [START_REF] Jedlovszky | Determination of the Adsorption Isotherm of Methanol on the Surface of Ice. An Experimental and Grand Canonical Monte Carlo Simulation Study[END_REF], that this saturation coverage is likely a meaningless parameter in experimental studies of trace gas-ice interactions conducted only at low pressures, when considering adsorbates for which lateral interactions are non-negligible. This emphasises the need for numerical data issued from, for instance, GCMC simulations based on accurate interaction potential models, to achieve a correct determination of the saturation coverage of organic compounds on ice, when the corresponding experimental data are poorly defined.

Worthwhile to note is that, in addition to the binding energy and saturation coverage, GCMC simulations give also useful information on the orientations of the adsorbed molecules, which is an important parameter for studying chemical reactions that can occur at the surface of ice. This appears crucial especially when addressing the ability of ice surfaces at promoting prebiotic chemistry in Space [START_REF] Silva | Interaction of Acetylene, Ethylene, and Benzene with Ice Surfaces[END_REF], or at influencing on oxidation processes of organic compounds in the troposphere [START_REF] Korcsok | Multiscale Modeling of Interfacial Oxidation Mechanism at Air/Organic Interface: Reactions of CH2=CH-Terminated Self-Assembled Monolayer with OH, O3, and HO2[END_REF].

Conclusions

In this paper, the adsorption of ethylene, propanol and hexanal molecules on crystalline ice has been characterised, for the first time, by computer simulations performed at 236 K, a temperature which is typical of Enceladus' environments. The results of the simulations turned out to be in good agreement with the major features provided by the available experiments on similar systems, especially at low coverage when the lateral interactions within the adsorbed layer do not play any significant role.

In analysing the properties of the adsorption layer formed by propanol and hexanal adsorbates at the surface of ice, we have found that their molecularlevel structures change upon saturation. Indeed, at low surface coverage, the adsorption is driven by the interaction of the organic molecules with the ice phase and, as a consequence, the adsorbed propanol and hexanal molecules lie more or less parallel to the ice surface. On the other hand, upon saturation, the adsorbate-adsorbate interactions become more and more important and the molecular axis tend to become tilted with respect to the surface normal by about 25 -30 • . Nevertheless, the polar head of these molecules always stay attached to the ice surface, irrespective of the coverage.

More interestingly, the results of the present simulations confirm that the binding energy to ice of propanol and hexanal has values falling either in (for hexanal) or at the limit of (in the case of propanol) the intermediate energy range (between 0.5 and 0.7 eV) for which the model of Bouquet et al. [START_REF] Bouquet | How Adsorption Affects the Gas-Ice Partitioning of Organics Erupted from Enceladus[END_REF] indicates that the organic compounds may be present on the ice surface only if the competition to occupy adsorption sites allows. In addition, considering that the saturation vapour pressure of water ice at the wall's temperature of the ice cracks has been estimated to be about 17 Pa at 236 K [START_REF] Bouquet | How Adsorption Affects the Gas-Ice Partitioning of Organics Erupted from Enceladus[END_REF] and that the organic compounds accounts for a few tenth of percent of the gas phase composition [START_REF] Khawaja | Low-mass nitrogen-, oxygenbearing, and aromatic compounds in Enceladean ice grains[END_REF], we can infer that only the first part of the propanol and hexanal adsorption isotherms, calculated at low fugacity, likely fits the real conditions of the ice cracks on Enceladus. Consequently, if more quantitative data on the possible competitive adsorption process on ice are required, further studies should focus especially on the very low coverage stage of the adsorption process. GCMC simulations taking into account a gas phase containing various mixtures of both propanol and hexanal molecules could be thus envisaged, following the approach recently developed to characterise the selective trapping of CO and N 2 by clathrate hydrates and various ice surfaces [START_REF] Patt | A grand canonical monte carlo study of the N2, CO, and mixed N2-CO clathrate hydrates[END_REF][START_REF] Petuya | Molecular Selectivity of CO-N2 Mixed Hydrates: Raman Spectroscopy and GCMC Studies[END_REF][START_REF] Patt | Adsorption of CO and N2 molecules at the surface of solid water. A grand canonical Monte Carlo study[END_REF]. Work in that direction is currently in progress.

Figure 1 :

 1 Figure 1: Visualisation of the crystalline I h structure of ice used in the present work. The geometry of the ethylene, propanol and hexanal molecules considered as adsorbates in the GCMC simulations is also shown on the right hand side of the figure, with the corresponding atom numbering. Oxygen, carbon and hydrogen atoms are represented as red, black, and white balls, respectively.

Figure 2 :

 2 Figure 2: Average number of ethylene molecules in the basic simulation box as a function of the fugacity, as calculated from GCMC simulations performed on the ethylene/I h ice system at 236 K. The arrows indicate the systems used in the detailed analysis. Error bars are also indicated. The line connecting the points issued from the simulations only serves as a guide to the eye.

Figure 3 :

 3 Figure 3: (a) Number density profile of the ethylene centres of mass in Systems 1 (circles), 2 (triangles), 3 (squares) and 4 (crosses), as obtained from the simulations. For reference, the outer tail of the number density profile of the water centres of mass in System 1 is also shown (dashed blue line). All profiles shown are symmetrised over the two surfaces present in the basic simulation box. (b) Distribution of the interaction energy of an adsorbed ethylene molecule with the ice phase and (c) with the other adsorbed molecules (circle, triangle and square symbols for Systems 1, 2 and 3, respectively). A snapshot issued from the simulations of System 4 is also given in panel (a) to illustrate the geometry of the ethylene molecules at the surface of ice.

Figure 4 :

 4 Figure 4: (a) Average number of propanol (red circles) and hexanal (black triangles) molecules in the basic simulation box as a function of the fugacity (errors bars are also indicated), as calculated from the present GCMC simulations. The arrows indicate the systems used in the detailed analyses. Lines are guide for eyes, only. Panels (b) and (c) show the corresponding curves fitted by the Langmuir equation to these data (best fit) for propanol (red) and hexanal (black), respectively. In these panels (b) and (c), inserts show the range of relative pressure in which the best Langmuir fit can be achieved from the simulation data.

Figure 5 :

 5 Figure 5: Number density profile of the propanol (a) centres of mass, (b) C 3 (c) O and (d) H atoms, in Systems 1 (circles), 2 (triangles), 3 (squares) and 4 (crosses), as obtained from the simulations. For reference, the outer tail of the number density profile of the water centres of mass in System 1 is also given (dashed blue line). All profiles shown are symmetrised over the two surfaces present in the basic simulation box. In addition, snapshots issued from the simulations of Systems 1 (low coverage) and 3 (high coverage) are given in panel (a) and (b), respectively, to illustrate the adsorption geometry of the propanol molecules at various coverages of the ice surface.

Figure 6 :

 6 Figure 6: Number density profile of the hexanal (a) centres of mass, (b) C 6 (c) O and (d) C 1 atoms, in Systems 1 (circles), 2 (triangles), 3 (squares) and 4 (crosses), as obtained from the simulations. For reference, the outer tail of the number density profile of the water centres of mass in System 1 is also given (dashed blue line). All profiles shown are symmetrised over the two surfaces present in the basic simulation box. In addition, snapshots issued from the simulations of Systems 1 (low coverage) and 3 (high coverage) are given in panel (a) and (b), respectively, to illustrate the adsorption geometry of the hexanal molecules at various coverages of the ice surface.

Figure 7 :

 7 Figure 7: Distribution of the interaction energy of an adsorbed molecule with the ice phase (P(U ads-w ), top row) and with the other adsorbed molecules (P(U ads-ads ), bottom row) when considering propanol (left hand side, red curves) and hexanal (right hand side, black curves) as adsorbates in the GCMC simulations, for Systems 1 (circles), 2 (triangles) and 3 (squares).

Figure 8 :

 8 Figure8: Distribution of the molecular orientations for propanol (red curves) and hexanal (black curves) molecules adsorbed on ice, for Systems 1 (circles), 2 (triangles), and 3 (squares). Top row: θ 1 is the angle formed between the surface normal to the ice surface (z) and the C-C vector joining the C atom (C 1 , see Figure1) of the terminal aliphatic group (CH 3 ) to the C atom bearing the alcohol (C 3 , see Figure1) or the aldehyde function (C 6 , see Figure1). Middle row: θ 2 is the angle formed between z and the CO vector; bottom row (for propanol, only), θ 3 is the angle formed between z and the OH vector.

Acknowledgements

The authors gratefully acknowledge financial support from the Region Bourgogne Franche-Comté and from the Observatoire des Sciences de l'Univers OSU THETA (France). Computations have been performed on the supercomputer facilities of the Mésocentre de calcul de Franche-Comté.