Volatile Transport inside Super-Earths by Entrapment in the Water Ice Matrix
VVolatile Transport inside Super-Earths by Entrapment in theWater Ice Matrix
A. Levi , , D. Sasselov , and M. Podolak Dept. of Geophysics & Planetary Science, Tel Aviv University, Tel Aviv, Israel 69978 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138,USA
ABSTRACT
Whether volatiles can be entrapped in a background matrix composing planetary en-velopes and be dragged via convection to the surface is a key question in understandingatmospheric fluxes, cycles and composition. In this paper we consider super-Earths with anextensive water mantle (i.e. water planets), and the possibility of entrapment of methanein their extensive water ice envelopes. We adopt the theory developed by van der Waals &Platteeuw (1959) for modelling solid solutions, often used for modelling clathrate hydrates,and modify it in order to estimate the thermodynamic stability field of a new phase, calledmethane filled ice Ih. We find that in comparison to water ice VII the filled ice Ih structuremay be stable not only at the high pressures but also at the high temperatures expected atthe core-water mantle transition boundary of water planets.
1. INTRODUCTION
The discoveries and characterization of planetary systems orbiting other stars has en-tered an exciting period when we are starting to gain access to observing the atmospheres ofplanets that are essentially solid in nature - high-density rocky or icy planets of 1 to 10 Earthmasses. These planets, called collectively super-Earths, have been discovered in relativelylarge quantities, though only a handful have measured radii and masses so far (Carter et al.2012, and references therein). The mean densities derived for these exoplanets reveal a rangeof possible bulk compositions, ranging from rocky with high iron content (e.g., Kepler-10b,Batalha et al. 2011) to mini-Neptunes with high H/He fraction (e.g., Kepler-11d,e, Lissaueret al. 2011). One of these planets orbiting a nearby M-dwarf star, GJ1214b (Charbonneauet al. 2009), has been accessible to spectroscopic studies of its atmosphere with inferencesto its composition (e.g., Bean et al. 2011; Berta et al. 2012, and references therein). Corresponding author A. Levi, e-mail: [email protected] a r X i v : . [ a s t r o - ph . E P ] A p r filled ice at pressures of around 2 GPa (Loveday et al. 2001b)that can survive at even higher pressures above 86 GPa (Hirai et al. 2006). For comparison,in a typical water super-Earth Fu et al. (2010) estimate that the pressures at the bound-ary between the ice mantle and the silicate core will be of the order of 100 GPa while thetemperature will be of the order of 10 K.The basic theory of clathrates that was developed by van der Waals & Platteeuw (1959)was later applied by Lunine & Stevenson (1985) to situations of astrophysical interest. Belowwe suggest how this theory may be extended to higher pressures and temperatures. We usethis theory to estimate the stability regime of filled ice, and discuss the implications of thisfor volatile transport in super-Earths. 3 –Clathrates are crystals, whose lattice structure forms cells that act as cages for foreignmolecules. The empty cage-like structure is usually thermodynamically unstable, but thecaptured foreign molecules (i.e. guest molecules) help stabilize the clathrate crystal (vander Waals & Platteeuw 1959). Clathrate hydrate is essentially water ice. In this case aframework of groups of four coordinated water molecules creates a cage-like lattice. Thehydrogen-bonded water molecules are slightly distorted, however, from the tetrahedral angleof ordinary ice. There are two basic low pressure ( < − . − . − (b) Ice Ih(a) MH-III - - + ++- - + -+-+ - (b) Ice Ih(a) MH-III - - + ++- - + -+-+ - (b) Ice Ih(a) MH-III - - + ++- - + -+-+ - (b) Ice Ih(a) MH-III - - + ++- - + -+-+ - (b) Ice Ih(a) MH-III - - + ++- - + -+-+ - (b) Ice Ih(a) MH-III - - + ++- - + -+-+ - (b) Ice Ih(a) MH-III - - + ++- - + -+-+ - Fig. 1.—
Comparison between methane cage clathrate and methane filled ice Ih. The structure of filledice Ih is viewed down the c-axis, perpendicular to the widened channels formed in this phase. In waterice Ih every hexagon is hydrogen bonded to its neighbouring hexagons, along the c-axis, in an alternatemanner. In methane filled ice Ih three adjacent oxygens of a particular hexagon (plus signs) will bond toone neighbouring hexagon and the other three (minus signs) will bond to the other neighbouring hexagon.This difference results in a widening of the channels perpendicular to the c-axis in filled ice-Ih, thereforeallowing for the accomodation of methane. The black balls in the filled ice Ih structure (right panel) denotethe entrapped methane molecules. (Filled ice after Loveday et al. (2001b) with permission)
2. THERMODYNAMIC STABILITY FIELD
The basic theory of clathrate hydrates views their stability in terms of two phases: Theordinary water phase (e.g. ice Ih, liquid water, etc. - the α phase) and the empty clathratehydrate structure (the β phase) + a gas of guest molecules. The chemical potentials of 5 –the two phases are equal on the thermodynamic equilibrium boundary between them. Forclathrates this may be written as (van der Waals & Platteeuw 1959): µ αQ = µ βQ + kT (cid:88) i ν i ln (cid:32) − (cid:88) K y Ki (cid:33) (1) y Ki = f K C Ki (cid:80) J f J C Ji (2)Here the subindex Q refers to the molecule that makes up the host molecular network, whichin our case is H O. The second term on the RHS of Eq.(1) represents the contribution ofthe guest molecules to the clathrate hydrate chemical potential. T is the temperature, k isBoltzmann’s constant, ν i is the ratio between the number of i type cages to water moleculesper cubic unit crystal, and finally, y Ki is the probability that a guest molecule of type K occupies a clathrate cage of type i . This last function is given in Eq.(2) in terms of thevolatile fugacity ( f K ) and its Langmuir constant ( C Ki ). We will give further information onthe Langmuir constant below and refer the reader to van der Waals & Platteeuw (1959) fora more detailed derivation.Since most high pressure experimental information is for methane-filled water-ices werestrict ourselves to the case of a single species of guest molecule model, and omit thesummation over K . The equilibrium equations may therefore be written as: µ αH O kT = µ βH O kT + (cid:88) i ν i ln (1 − y CH ,i ) (3) y CH ,i = f CH C CH ,i f CH C CH ,i (4)Recalling that both the fugacity and the Langmuir constant are functions of temperatureand pressure, we differentiate and combine the last set of equations to get: ∂∂T (cid:32) µ αH O kT − µ βH O kT (cid:33) P dT + 1 kT ∂∂P (cid:16) µ αH O − µ βH O (cid:17) T dP = − (cid:88) i ν i
11 + f CH C CH ,i (cid:20) ∂∂P ( f CH C CH ,i ) T dP + ∂∂T ( f CH C CH ,i ) P dT (cid:21) (5)We introduce the thermodynamic relations: ∂∂T (cid:16) µ H O kT (cid:17) P ≡ − H H O kT (6)1 kT ∂∂P ( µ H O ) T ≡ V H O kT (7) 6 –Where H H O and V H O represent the enthalpy and volume per water molecule, respectively.Inserting these thermodynamic relations into Eq.(5) and solving for dP/dT gives: dPdT = kT (cid:16) H αH O − H βH O (cid:17) − (cid:80) i ν i fC i (cid:0) ∂f∂T C i + f ∂C i ∂T (cid:1) kT (cid:16) V αH O − V βH O (cid:17) + (cid:80) i ν i fC i (cid:0) ∂f∂P C i + f ∂C i ∂P (cid:1) (8)Here we have omitted the index CH for convenience. For a homogeneous substance(i.e. composed only of water molecules) the chemical potential equals the Gibbs free energyper particle, ˜ G , which, combined with the assumption of constant temperature gives: dµ = d ˜ G = V dP = kT d ln P (9)Here all extensive parameters are per particle. For the final equality on the RHS we haveused the equation of state for an ideal gas, so that the last relation between pressure andGibbs free energy is only valid for this case. When the gas is not ideal, we can retain thefunctional form if we replace the pressure with the fugacity, f , which acts as an effectivepressure function to correct for the effect of intermolecular interactions and which obeys thefollowing relation (Smith & Van Ness 1975): dµ = d ˜ G = V dP ≡ kT d ln f (10)From the last relation we have: (cid:18) ∂f∂P (cid:19) T = V fkT (11)Where V is the volume per methane molecule. For high pressures we can solve Eqs.(1) and(2) for the case of an SI methane clathrate hydrate numerically. We find that f C i ∼ (cid:29) dPdT = kT (cid:16) H αH O − H βH O (cid:17) − (cid:80) i ν i (cid:0) ∂∂T ln f + ∂∂T ln C i (cid:1) kT (cid:16) V αH O − V βH O (cid:17) + (cid:80) i ν i (cid:0) VkT + ∂∂P ln C i (cid:1) (12)The definition of the Langmuir constant according to van der Waals & Platteeuw (1959)is: C Ki ≡ h Ki kT ζ K ( T ) (13)where h Ki is the single cell canonical partition function for a K type guest molecule in an i type cage. ζ K is the quantum number density function for a K type molecule in an ideal 7 –gas and is independent of pressure. The cell partition function depends on pressure both viathe cell dimension and through the form of the guest-host potential, so: ∂∂P (ln C i ) T = ∂∂P (ln h i ) T (14)Here again we have omitted the index K . We are left with estimating the derivative of thesingle cell partition function, for which we give the following quasi-classical form: ∂∂P (ln h i ) T = ∂∂P ln (cid:26) (cid:126) (cid:90) d r (cid:90) d p e − /kT [ (cid:15) rot,i + (cid:15) vib,i + (cid:15) trans,i + W i ] (cid:27) (15)Here we have divided the Hamiltonian of the guest molecule into its separate kinetic andpotential contributions. As a first order approximation we assume that the kinetic degrees offreedom of the entrapped molecule are unaffected by its inclusion in the water network. Thisis a common approximation in clathrate modeling, and experiment shows that in solidifiedform methane molecules rotate freely as in an ideal gas (Hazen et al. 1980). In this ap-proximation the kinetic degrees of freedom will contribute a function that is only a functionof temperature, and the logarithm will cancel upon differentiation with respect to pressure,thus yielding a simplified form: ∂∂P (ln h i ) T ≈ ∂∂P ln (cid:90) e − W i /kT d r (16)In clathrate hydrates the cages entrapping the volatiles are assumed spherical (e.g. Sloan1998; McKoy & Sinanoˇglu 1963) so that the integration is over a spherical cage. In filledices the cages are actually cylindrical channels within the water ice structure (Loveday et al.2001b) so that we may write: ∂∂P ln (cid:90) e − W i /kT d (cid:126)r = ∂∂P ln (cid:90) z z dz (cid:90) π dφ (cid:90) a e − W i ( r,φ,z ) /kT rdr (17)Where we assume the cylindrical water-made channel has radius a and since intermolecularpotentials fall rapidly with increasing distance we limit the integration along the z coordinateto the finite values z and z .Both the limits of integration and the potential interaction of the methane molecule withits surroundings, W i , may depend on the pressure. Raman spectra of methane filled waterice shows increases in the attractive potential between methane and its water network hostwith pressure increases (Machida et al. 2007). This was also suggested by Hirai et al. (2006).The rearrangement of the water network, from cage clathrate to the filled ice structure,reduces molecular distances by ∼ . × − cm. It is also suggested, from intermolecular 8 –distance considerations, that the guest-host Lennard-Jones potential interaction estimatedfor clathrates is enhanced by weak hydrogen bonds between guest and host in the filled icestructure (Loveday et al. 2001b). Since the guest-host potential energy changes considerablywith pressure, a good first approximation will be to consider only the change of W i withpressure, therefore allowing us to insert the derivative with respect to pressure into theintegrand, giving: ∂∂P (ln C i ) T = − kT (cid:82) z z (cid:82) π (cid:82) a re − W i /kT (cid:0) ∂W i ∂P (cid:1) T dφdzdr (cid:82) z z (cid:82) π (cid:82) a re − W i /kT dφdzdr (18)Finally, taking a spatial average for the partial derivative appearing in the numerator,we have: ∂∂P (ln C i ) T = − kT (cid:28)(cid:18) ∂W i ∂P (cid:19) T (cid:29) (19)Inserting this last relation into Eq.(12) yields: dPdT = T (cid:16) H βH O − H αH O (cid:17) + kT (cid:80) i ν i (cid:0) ∂∂T ln f + ∂∂T ln C i (cid:1) V βH O − V αH O + (cid:80) i ν i (cid:10)(cid:0) ∂W i ∂P (cid:1) T (cid:11) − (cid:80) i ν i V (20)Since the pressure varies by many orders of magnitude and the temperature does not, andsince pressure has a dominant effect in determining the correction for the system non-ideality,we assume that: ∂C i ∂T (cid:29) ∂f∂T We may further approximate the following: kT (cid:88) i ν i ∂∂T ln C i = kT (cid:88) i ν i ∂∂T ln (cid:18) kT (cid:90) d (cid:126)re − W i /kT (cid:19) = kT (cid:88) i ν i (cid:34) (cid:82) d (cid:126)r W i kT e − W i /kT (cid:82) d (cid:126)re − W i /kT − T (cid:35) = (cid:88) i ν i (cid:20) T (cid:104) W i (cid:105) − k (cid:21) (21)Where in the last step we have again averaged the intermolecular interaction of the methanemolecule spatially over its surroundings. Inserting the last formula into the relation for dP/dT gives a Clausius-Clapeyron type equation of the form: dPdT = T (cid:16) H βH O − H αH O (cid:17) + (cid:80) i ν i (cid:2) T (cid:104) W i (cid:105) − k (cid:3) V βH O − V αH O + (cid:80) i ν i (cid:10)(cid:0) ∂W i ∂P (cid:1) T (cid:11) − (cid:80) i ν i V (22)This modified Clausius-Clapeyron type equation is a generalization of the equation given inLunine & Stevenson (1985) in order to explain the behaviour of clathrates. We will use this 9 –formalism to obtain the thermodynamic field of stability for the filled water ice as well. Forthis purpose we need to evaluate the different terms appearing in this equation, which wedo in the next section.
3. APPLICATION TO HIGH PRESSURE3.1. Clathrate Hydrate
Starting with clathrate hydrates, let us examine the numerator of Eq.(22). At lowpressures and temperatures the α phase is water-ice Ih whose enthalpy is taken to be equalto the enthalpy of the empty hydrate ( β ) phase (Lunine & Stevenson 1985). The potentialof interaction W i is attractive (negative) and of a Lennard-Jones type, so generally thenumerator is negative. In the denominator, at low pressures, the gaseous volatile volume( V ) is the dominant factor appearing with a minus sign. Therefore the derivative dP/dT is positive. The potential of interaction hardly changes with increasing pressure in this lowpressure regime.As we increase the temperature, the pressure increases till we reach the melting pointfor water-ice Ih and the α phase now represents liquid water. Due to the enthalpy of fusion,the enthalpy difference appearing in the numerator is no longer negligible and causes a sharpincrease in the absolute value of the numerator. This is manifested as a sharp increasein the derivative dP/dT . Every increase in temperature is now accompanied by a steeperincrease in pressure. At room temperature at 5 MPa, methane gas already deviates fromideality enough so that we need to consider a second virial correction; at 10 MPa a thirdvirial correction is required, and so on (Hirschfelder et al. 1966). This means that the volumea methane molecule occupies in the gas decreases with pressure. At high enough pressurethe volatile gas contracts enough so that the empty clathrate volume equals the liquid watervolume plus the compressed methane volume and the derivative dP/dT diverges.Any further increase in pressure brings about a situation where the volume occupied by aclathrate water molecule is larger than the sum of the volumes occupied by a water moleculein the liquid phase and a methane molecule in the gas phase weighted by the hydrationnumber ( ν ). The result is that the derivative dP/dT becomes negative and the high pressurestability limit of the clathrate hydrate is attained and the clathrate dissociates. This generaltype of clathrate behavior is shown in fig. 2, which we have derived by numerically solvingthe set of Eqs.(1) and (2).Now let us consider what happens for the case of filled water-ice. This ice is formed inthe laboratory at room temperature and at a pressure of ∼ β phase will now describe the filled ice structure while the α phase represent either water ice VII or fluid water, appropriately.To obtain the thermodynamic stability field for filled water ice we shall require thetemperature and pressure dependencies for both the different constituents’ volumes and forthe attractive potential between a methane molecule and its surroundings. In addition, theenthalpy difference between filled ice and fluid water ought be estimated for the case ofstability with respect to fluid water. We start with the general relation: dVV = χdT − κdP (23)Where χ and κ are the volumetric thermal expansivity and isothermal compressibility re-spectively. We assume the bulk modulus, B , is linearly dependent on pressure and may bewritten as: B ≡ κ = B + ˜ B P (24)Combining Eqs.(24) and (23) yields after integration: V ( T, P ) = V ( T , P ) (cid:32) B + ˜ B PB + ˜ B P (cid:33) − / ˜ B exp (cid:18)(cid:90) TT χ ( T, P ) dT (cid:19) (25)If we keep the temperature constant at T we end up with the Birch-Murnaghan equation ofstate. By setting the reference temperature ( T ) to room temperature, we can assign to eachsolid constituent a proper value for its bulk modulus ( B ) and its pressure gradient ( ˜ B ) byusing published room temperature hydrostatic compression experiments. We use the datapublished in Hemley et al. (1987) for water ice up to 128 GPa, for assigning a bulk modulusfor ice VII. For assigning a bulk modulus for solid methane we use the data published inHazen et al. (1980) for phase I and in Umemoto et al. (2002) for phases A and B up to37 GPa. Solid methane transforms at high pressures to a hexagonal close packed structure(Bini & Pratesi 1997), this phase was difficult to account for as we found no volumetric datafor it, rather we extrapolated from the phase B data in Umemoto et al. (2002) to higher 11 –pressures. For the filled ice structure we used room temperature, experimentally deduced,unit cell volumes up to 42 GPa by Hirai et al. (2003).For the thermal expansivity we have adopted the approach of Fei et al. (1993), whodetermined the thermal expansivity of water ice VII by fitting their volumetric experimentaldata to an equation of state of the form given above (Eq.25). These authors assumed thefollowing dependence for χ on the pressure: χ ( T, P ) = χ ( T, P ) (cid:32) B + ˜ B PB + ˜ B P (cid:33) − η (26)Where χ ( T, P ) is taken to be a linear function of the temperature and η is well fittedwith a numerical value of 0 .
9. For the thermal expansivity of solid methane we adopt theformalism of Eq.(26) for its dependency on pressure and set the expansivity value at P to be10 − K − , according to experimental data given by Heberlein & Adams (1970). The thermalexpansivity of filled ice is not known. As filled ice is a combination of a hydrogen bonded-network and methane molecules between which there is van der Waals type attraction, wesuggest its thermal expansivity to be intermediate between that for ice VII and for solidmethane.For the equation of state of fluid water at high pressure (up to 100 GPa) we adopt theformalism derived using molecular dynamic simulations by Belonoshko & Saxena (1991).Although this formalism is inherently dependent on the type of model used for the watermolecules’ intermolecular potential it does show good agreement with recent experimentaldata for high pressure water fluid density (Goncharov et al. 2009).We shall now turn to evaluate the energy of interaction of a methane molecule with itssurroundings in the water filled ice structure. W i In order to build the thermodynamic stability field for methane filled water ice, weneed to approximate how the potential of interaction, of the methane molecule with its sur-roundings, depends on the temperature and the pressure. As was shown by Raghavendra& Arunan (2008) a hydrogen bond may form between the water electron poor hydrogenand the center of the methane tetrahedral face which is electron rich. It was further shownby these authors that the bond energy ( E bond ) for such an interaction is − . − .For comparison a methane-methane van der Waals potential well is about − . − (Hirschfelder et al. 1966). It was also suggested by Loveday et al. (2001b) that weak hydro- 12 –gen bonds are formed in the filled water-ice structure, between the water network and thedissolved methane molecules.Let us consider a simple approximate model where with increasing pressure more ofthe tetrahedral faces, per methane molecule, create such hydrogen bonds with the waternetwork. An increase in temperature will have the opposite effect. That the number ofhydrogen bonds per molecule increases with pressure and decreases with temperature isknown for water structures (e.g. Kalinichev & Bass 1994; Pattanayak et al. 2011). We maytherefore write for the spatially averaged potential of interaction of a methane molecule withits surroundings: (cid:104) W i (cid:105) = n ( T, P ) E bond (27)Where n ( T, P ) is the number of hydrogen bonds between a methane molecule and the waternetwork, at a given pressure and temperature, and is bounded between 0 and 4, where inthe upper limit all of the methane four tetrahedral faces are hydrogen bonded to the waternetwork.We normalize n to give it a probability interpretation (i.e. What is the probability abond will form at a given pressure-temperature point). We further assume a division into atemperature dependent and pressure dependent probabilities, thus: n ( T, P ) ≡ n ( T ) n ( P ) (28)For the temperature dependent probability we assume a Boltzmann type form, of: n ( T ) = 11 + exp ( − | E bond | /kT ) (29)To obtain the form for the pressure dependent probability ( n ) we use the fact that a for-mation of a hydrogen bond is accompanied by a substantial penetration of the hydrogenbonding molecules into each other. For the case of the hydrogen bonding between water andmethane the combined van der Waals radius of 2 . × − cm, before the bonding, reducesto 2 . × − cm, after bonding (Raghavendra & Arunan 2008). This interpenetration maytherefore account for a 40% reduction in the crystal volume. This is considerable enough sothat we may relate this interpenetration (to first approximation) to the solid compressibility, κ . At low pressures only a few hydrogen bonds are formed and the solid is easily compressed,as many bonds are still ready to be formed. As the pressure increases more bonds form, permolecule, and it becomes more difficult to further compress the solid. Hypothetically, at ahigh enough pressure, all bonds per molecule are already formed and it is no longer possibleto further compress the solid via the route of hydrogen bonding molecular interpenetration.Adopting this model, to first approximation, we can write for the molecular volume, V , the 13 –following: V = n HB V HB + n nHB V nHB (30)Where n HB and n nHB are the probabilities a methane molecule is fully hydrogen bondedto the water network or not hydrogen bonded at all respectively, and V HB and V nHB arethe molecular volumes associated with these two molecular situations respectively. If thenumber of hydrogen bonds indeed determines, to first approximation, the crystal volume,we may say: κ ≡ − V (cid:18) ∂V∂P (cid:19) T = − n HB V HB + n nHB V nHB (cid:18) V HB dn HB dP + V nHB dn nHB dP (cid:19) (31)Since n HB + n nHB = 1, an integration of Eq.(31) yields: n ( P ) = n HB ( P ) = 1 − exp (cid:16) − (cid:82) P κdP (cid:17) − V HB V nHB (32)Inserting Eqs.(32) and (29) into Eq.(27) gives for the spatially averaged potential energy, ofa methane molecule with its surroundings, the following form: (cid:104) W i (cid:105) ≈
41 + e − | E bond | /kT (cid:0) − e − κ P (cid:1) E bond (33)Its derivative with respect to pressure will therefore be: (cid:28)(cid:18) ∂W i ∂P (cid:19) T (cid:29) ≈ κ E bond e − | E bond | /kT e − κ P (34)Now that we have the approximate temperature and pressure dependencies for the termsappearing in Eq.(22) we may integrate it numerically to obtain the stability regime for thefilled ice structure. The numerical integration of Eq.(22) is shown in fig. 3. As we have mentioned above, weconsider it reasonable to assume the thermal expansivity of the filled ice to be intermediate,between that for solid methane (a van der Waals solid) and water ice VII. From the laboratorydata (see subsection 3 .
2) we know the thermal expansivity for solid methane is an order ofmagnitude larger than that for ice VII. We integrate Eq.(22) assuming thermal expansivityfor the filled ice two times, three times and five times larger than the experimental value forice VII. As a limiting case we also solve assuming filled ice has a thermal expansivity equal 14 –to that of water ice VII. The four curves are given in the figure. It is appropriate to notehere that Sloan (1998) gives for a SI clathrate hydrate (also a combination of methane andwater) a thermal expansivity some five times larger than the thermal expansivity for ice Ih.More recent experiments confine the latter ratio to be between two to four (Hester et al.2007).In addition, the parameter ν , the hydration number, is important to Eq.(22), as definedearlier. Generally speaking, this parameter represents the ability to include volatiles in thewater network. As the ratio between water and methane in the filled ice is 2 : 1 (Lovedayet al. 2001b) then filled ice modeling requires a value of 1 / ν . Given that, we find thepacking efficiency for filled ice to be greater than for the case of separation to pure solidconstituents (i.e. water ice VII and pure solid methane). The contribution of the potentialof interaction (in comparison to packing efficiency considerations) is found to be small andrestricted to the lower pressure regime of the stability field.Integrating Eq.(22) and following the dissociation curve for filled-ice, with respect toice VII, a point of intersection with the melting curve for water ice VII is reached. Such anintersection is commonly referred to as a quadruple point. Up to the quadruple point theenthalpy difference between the α and β phases is relatively small, as both phases are solid.Continuing the integration beyond the quadruple point the α phase will now represent fluidwater. The enthalpy difference between the α and β phases therefore increases, and becomesdominated by the enthalpy of fusion of filled ice. Unfortunately, the enthalpy of fusion ofmethane filled-ice Ih is experimentally undetermined and we estimate its value using theenthalpy of fusion for pure solid water at high pressure.Goncharov et al. (2009) deduced experimentally the melting curves for water ice VII andfor super-ionic water. By further determining experimentally the volume difference betweensolid and melt, along the melting curve, they were able to derive the enthalpy of fusion forpure water. Their reported error is approximately 50%. As explained in Goncharov et al.(2009) the enthalpy of fusion increases with pressure since the increase in pressure meansthe melting transition is between a molecular solid and a fluid whose molecules becomeever more dissociated and ionized. At a pressure of about 47 GPa a branching occurs inthe melting curve for pure water, due to the introduction of the super-ionic phase of water(Goncharov et al. 2005). The result is a reduction in the melting curve gradient and a sharpdecrease of the enthalpy change upon melting (Goncharov et al. 2009). This behaviour isclearly seen in fig.4, where the dashed curve (red) reproduces the results from Goncharovet al. (2009) and the dashed-dotted curve (blue) is a hypothetical enthalpy of fusion, forwhich we have arbitrarily reduced the point of branching in the melting curve to 27 GPa.The reason for the introduction of this hypothetical behaviour is that the enthalpy of fusion 15 –we adopt is from experiments on a homogeneous water system and therefore it is only anapproximation for our methane filled-ice system. Molecular dynamic simulations by Iitaka &Ebisuzaki (2003) demonstrate that filled-ice exhibits behaviours similar to those appearingin homogenous water systems, but at somewhat lower pressures. A branching of the meltingcurve is therefore probable in our system of methane mixed with water as well, though,the branching may happen at a lower pressure then the 47 GPa of the pure water system.Wishing to test to what extent do our results depend on the point of branching we solve forthe hypothetical enthalpy of fusion as well.In fig.5 we solve Eq.(22) where we assume for filled-ice a thermal expansivity twice thatfor water ice VII and solve for each of the enthalpy of fusion scenarios depicted in fig.4. Inthe same figure we also show the solution of Eq.(22) with the experimental enthalpy of fusionfrom Goncharov et al. (2009) but vary it globally by 50%, which is its experimental error.From the figure we see that most of the variation in the filled-ice Ih dissociation curve dueto the changes examined in the enthalpy of fusion occur below 10 GPa.Combined with thermodynamic profiles for the interior of water exo-Earths, the ther-modynamic stability field can help us decide whether filled ice structures can form at thecore-mantle boundary and help convect methane and other hydrocarbons towards the sur-face.
4. APPLICATION TO SUPER-EARTHS
For the interior structure of water planets with masses up to 10 M ⊕ , Fu et al. (2010)consider a silicate-metal core, surrounded by a water-ice mantle. They find solid-state con-vection to prevail between the core-ice boundary and the conductive crust. Examiningdifferent surface temperatures and heat fluxes the authors find it possible for a liquid waterocean to exist beneath an ice-Ih solid surface layer. Different compounds expelled from thesilicate-metal core may be incorporated in the ice matrix and convect outward. For the2M ⊕ planet Fu et al. (2010) find the silicate core-ice boundary pressure to range from ap-proximately 50 to 90 GPa, for water mass fractions between 25% and 50% respectively. Theexpected temperatures are between 700 and 1000 K.Our extension of the theory of van der Waals & Platteeuw (1959) suggests that the filledice structure may be stable at such pressures and temperatures. In such a case any CH released from the core could be trapped in the lower part of the ice mantle. ExperimentallyLoveday et al. (2001b) inferred a 2 : 1 water-methane ratio for the filled ice phase. As themantle convects, the CH would be carried to lower pressures where the filled ice will undergo 16 –a transition to a clathrate hydrate. Using the statistical model given by van der Waals &Platteeuw (1959) we can compute the probability, y Ki , that a structure I clathrate hydratecage will be occupied. This is shown in fig. 6. It is clear that the occupancy is very close tofull occupancy (probability of unity) for which there are 5 .
75 water molecules per methanemolecule. For a structure H methane hydrate the water-methane ratio ranges from 4.25:1 to3.40:1, depending on the number of methane molecules occupying its large cage (Koh 2002).Thus as the pressure decreases in an upwelling, excess CH will be forced out of the waternetwork. This may lead to the formation of a local methane reservoir. Such reservoirs maynaturally occur on the transition from filled ice to structure H clathrate hydrate at about2 GPa, and at the transition from structure H clathrate hydrate to structure I clathratehydrate at about 1 GPa.In A. Levi et al.(2013, in preparation) we show that the introduction of filled ice as amajor constituent in an icy mantle has a large effect on the mantle thermodynamic profile.The probable higher thermal expansivity of filled ice compared with that for water ice VIIresults in a more moderate adiabatic thermal profile. The higher temperatures in the icymantle, compared with those for a pure water mantle, creates a physical route through whichsuper-Earths (objects less massive than Uranus and Neptune) may develop lower mantlesin the super-ionic and reticulating phases. Phases so far related with the interior of bodieswhose mass is equivalent to that of the icy giants of our solar system.The unique characteristics of methane clathrate hydrate, namely its low thermal con-ductivity and the topology of its melting curve, yield water planets with thin crusts ( < REFERENCES
Batalha, N. M., Borucki, W. J., Bryson, S. T., Buchhave, L. A., Caldwell, D. A., Christensen-Dalsgaard, J., Ciardi, D., Dunham, E. W., Fressin, F., Gautier, III, T. N., Gilliland,R. L., Haas, M. R., Howell, S. B., Jenkins, J. M., Kjeldsen, H., Koch, D. G., Latham,D. W., Lissauer, J. J., Marcy, G. W., Rowe, J. F., Sasselov, D. D., Seager, S., Steffen,J. H., Torres, G., Basri, G. S., Brown, T. M., Charbonneau, D., Christiansen, J.,Clarke, B., Cochran, W. D., Dupree, A., Fabrycky, D. C., Fischer, D., Ford, E. B.,Fortney, J., Girouard, F. R., Holman, M. J., Johnson, J., Isaacson, H., Klaus, T. C.,Machalek, P., Moorehead, A. V., Morehead, R. C., Ragozzine, D., Tenenbaum, P.,Twicken, J., Quinn, S., VanCleve, J., Walkowicz, L. M., Welsh, W. F., Devore, E., &Gould, A. 2011, Astrophysical J., 729, 27Bean, J. L., D´esert, J.-M., Kabath, P., Stalder, B., Seager, S., Miller-Ricci Kempton, E.,Berta, Z. K., Homeier, D., Walsh, S., & Seifahrt, A. 2011, Astrophysical J., 743, 92Belonoshko, A. & Saxena, S. K. 1991, Geochimica et Cosmochimica Acta, 55, 381 18 –Berta, Z. K., Charbonneau, D., D´esert, J.-M., Miller-Ricci Kempton, E., McCullough, P. R.,Burke, C. J., Fortney, J. J., Irwin, J., Nutzman, P., & Homeier, D. 2012, AstrophysicalJ., 747, 35Bini, R. & Pratesi, G. 1997, Phys. Rev. B, 551, 14800Carter, J. A., Agol, E., Chaplin, W. J., Basu, S., Bedding, T. R., Buchhave, L. A.,Christensen-Dalsgaard, J., Deck, K. M., Elsworth, Y., Fabrycky, D. C., Ford, E. B.,Fortney, J. J., Hale, S. J., Handberg, R., Hekker, S., Holman, M. J., Huber, D.,Karoff, C., Kawaler, S. D., Kjeldsen, H., Lissauer, J. J., Lopez, E. D., Lund, M. N.,Lundkvist, M., Metcalfe, T. S., Miglio, A., Rogers, L. A., Stello, D., Borucki, W. J.,Bryson, S., Christiansen, J. L., Cochran, W. D., Geary, J. C., Gilliland, R. L., Haas,M. R., Hall, J., Howard, A. W., Jenkins, J. M., Klaus, T., Koch, D. G., Latham,D. W., MacQueen, P. J., Sasselov, D., Steffen, J. H., Twicken, J. D., & Winn, J. N.2012, Science, 337, 556Charbonneau, D., Berta, Z. K., Irwin, J., Burke, C. J., Nutzman, P., Buchhave, L. A., Lovis,C., Bonfils, X., Latham, D. W., Udry, S., Murray-Clay, R. A., Holman, M. J., Falco,E. E., Winn, J. N., Queloz, D., Pepe, F., Mayor, M., Delfosse, X., & Forveille, T.2009, Nature, 462, 891Crowley, J. W. & O’Connell, R. J. 2012, Geophysical Journal International, 188, 61Fei, Y., Mao, H.-K., & Hemley, R. J. 1993, J. Chem. Phys., 99, 5369Fortes, A. D. & Choukroun, M. 2010, Sp. Sci. Rev., 153, 185Fu, R., O’Connell, R. J., & Sasselov, D. D. 2010, Astrophysical J., 708, 1326Goncharov, A. F., Goldman, N., Fried, L. E., Crowhurst, J. C., Kuo, I.-F. W., Mundy, C. J.,& Zaug, J. M. 2005, Physical Review Letters, 94, 125508Goncharov, A. F., Sanloup, C., Goldman, N., Crowhurst, J. C., Bastea, S., Howard, W. M.,Fried, L. E., Guignot, N., Mezouar, M., & Meng, Y. 2009, J. Chem. Phys., 130,124514Halevy, I. & Stewart, S. T. 2008, in Lunar and Planetary Inst. Technical Report, Vol. 39,Lunar and Planetary Institute Science Conference Abstracts, 1174–1175Hazen, R. M., Mao, H. K., Finger, L. W., & M., B. P. 1980, Appl. Phys. Lett., 37, 288Heberlein, D. C. & Adams, E. D. 1970, Journal of Low Temperature Physics, 3, 115 19 –Hemley, R. J., Jephcoat, A. P., Mao, H. K., Zha, C. S., Finger, L. W., & Cox, D. E. 1987,Nature, 330, 737Hester, K. C., Huo, Z., Ballard, A. L., Koh, C. A., Miller, K. T., & Sloan, E. D. 2007, J.Phys. Chem. BHirai, H., Machida, S. I., Kawamura, T., Yamamoto, Y., & Yagi, T. 2006, American Miner-alogist, 91, 826Hirai, H., Tanaka, T., Kawamura, T., Yamamoto, Y., & Yagi, T. 2003, Phys. Rev. B, 68,172102Hirai, H., Uchihara, Y., Fujihisa, K., M., S., Katoh, E., Aoki, K., Nagashima, K., Yamamoto,Y., & Yagi, T. 2001, J. Chem. Phys., 115, 7066Hirschfelder, J. O., Curtiss, J. F., & Bird, R. B. 1966, Molecular Theory of Gases andLiquids, third printing edn., Vol. 5 (John Wiley and Sons)Iitaka, T. & Ebisuzaki, T. 2003, Phys. Rev. B, 68, 172105Kalinichev, A. G. & Bass, J. D. 1994, Chem. Phys. Let., 231, 301Koh, A. C. 2002, Chem. Soc. Rev.Kuchner, M. J. 2003, Astrophysical J. Let., 596, L105L´eger, A., Selsis, F., Sotin, C., Guillot, T., Despois, D., Mawet, D., Ollivier, M., Lab`eque,A., Valette, C., Brachet, F., Chazelas, B., & Lammer, H. 2004, Icarus, 169, 499Lin, J.-F., Militzer, B., Struzhkin, V. V., Gregoryanz, E., Hemley, R. J., & Mao, H.-K. 2004,J. Chem. Phys., 121, 8423Lissauer, J. J., Fabrycky, D. C., Ford, E. B., Borucki, W. J., Fressin, F., Marcy, G. W.,Orosz, J. A., Rowe, J. F., Torres, G., Welsh, W. F., Batalha, N. M., Bryson, S. T.,Buchhave, L. A., Caldwell, D. A., Carter, J. A., Charbonneau, D., Christiansen,J. L., Cochran, W. D., Desert, J.-M., Dunham, E. W., Fanelli, M. N., Fortney, J. J.,Gautier, III, T. N., Geary, J. C., Gilliland, R. L., Haas, M. R., Hall, J. R., Holman,M. J., Koch, D. G., Latham, D. W., Lopez, E., McCauliff, S., Miller, N., Morehead,R. C., Quintana, E. V., Ragozzine, D., Sasselov, D., Short, D. R., & Steffen, J. H.2011, Nature, 470, 53Loveday, J. S., Nelmes, R. J., Guthrie, M., A., B. S., Allan, D. R., Klug, D. D., Tse, J. S.,& Handa, Y. P. 2001a, Let. Nat., 410, 661 20 –Loveday, J. S., Nelmes, R. J., Guthrie, M., D., K. D., & Tse, J. S. 2001b, Phys. Rev. Let.,87, 215501(1)Lunine, J., Choukroun, M., Stevenson, D., & Tobie, G. The Origin and Evolution of Titan,ed. Brown, R. H., Lebreton, J.-P., & Waite, J. H., 35–+Lunine, J. I. & Stevenson, D. J. 1985, A. Phys. J., 58, 493Machida, S.-I., Hirai, H., Kawamura, T., Yamamoto, Y., & Yagi, T. 2007, Physics andChemistry of Minerals, 34, 31McKoy, V. & Sinanoˇglu, O. 1963, jcp, 38, 2946Pattanayak, S. K., Prashar, N., & Chowdhuri, S. 2011, J. Chem. Phys., 134, 154506Raghavendra, B. & Arunan, E. 2008, Chem. Phys. Let., 467, 37Redmer, R., Mattsson, T. R., Nettelmann, N., & French, M. 2011, Icarus, 211, 798Schubert, G., Hussmann, H., Lainey, V., Matson, D. L., McKinnon, W. B., Sohl, F., Sotin,C., Tobie, G., Turrini, D., & van Hoolst, T. 2010, Sp. Sci. Rev., 153, 447Sloan, E. D. 1998, Clathrate Hydrates of Natural Gases, 3rd edn., Vol. 5 (New York: MarcelDekker)Smith, J. M. & Van Ness, H. C. 1975, Introduction to Chemical Engineering Thermodynam-ics, 3rd edn. (Mcgraw-Hill Book Company)Sohl, F., Choukroun, M., Kargel, J., Kimura, J., Pappalardo, R., Vance, S., & Zolotov, M.2010, Sp. Sci. Rev., 153, 485Tobie, G., Lunine, J. I., & Sotin, C. 2006, Nature, 440, 61Umemoto, S., Yoshii, T., Akahama, Y., & Kawamura, H. 2002, Journal of Physics CondensedMatter, 141, 10675van der Waals, J. H. & Platteeuw, J. C. 1959, Advances in Chemical Physics, 2, 1
This preprint was prepared with the AAS L A TEX macros v5.2.
21 – −3 l n ( d i s p r ess u r e [ b a r ]) −1 ] Fig. 2.—
This figure describes the thermodynamic stability field for CH clathrate hydrate structure I.The dotted line is water ice Ih melting curve, the thick curve profile (coloured blue in the online version) isthe dissociation pressure curve for CH clathrate hydrate structure I, which clearly follows the available datapoints (circles). The other melting curves are for water ice III, water ice V and water ice VI. The square(coloured red in the online version) marks the critical point for Methane.
22 –
Temperature [K] l og ( p r ess u r e [ G P a ] ) Fig. 3.—
The phase diagram of methane filled ice-Ih water ice crystal. The diamond is a known point ofstability (Hirai et al. 2003). The four solid curves (blue in the on-line version) are the stability boundariesfor the filled water ice, assuming for it a thermal expansivity equal to that for water ice VII and two, threeand five times larger than that for water ice VII. A Lower thermal expansivity for filled water ice increasesits stability, i.e. shifts the curve to the right to higher temperatures. The dash-dot curve is the melting curvefor molecular water ice (Lin et al. 2004) and the dashed curve is the melting curve for the super-ionic phaseof water (Goncharov et al. 2009). The square and circle data points represent the conditions at the base ofthe H/He atmosphere surrounding Neptune and Uranus, respectively (Redmer et al. 2011).
23 –
Pressure [GPa] H f u s i on [ kJ m o l − ] Fig. 4.—
Dashed curve (red in the on-line version) is the enthalpy of fusion as a function of pressure forpure water (Goncharov et al. 2009). The branching in the melting curve, due to the super-ionic phase, ishere at 47 GPa (Goncharov et al. 2005). Dashed-dotted curve (blue in the on-line version) is a hypotheticalenthalpy of fusion, where, the branching in the melting curve occurs at 27 GPa.
24 –
Temperature [K] l og ( p r ess u r e [ G P a ] ) Fig. 5.—
The solid curve (blue) is the phase diagram for filled-ice Ih assuming for it a thermal expansivitytwice as much as that of water ice VII and a branching in the melting curve at 47 GPa. The thick dashedcurve (blue) is the variation in the phase diagram if the branching in the melting curve is shifted to 27 GPa.The dashed-dotted thick curves (green) confining the thick (blue) curve from both sides represent the changein the phase diagram if the enthalpy of fusion is varied by ±
50% globally.
25 –
100 3000.950.970.981 y ij Temperature [K] y y Fig. 6.—
The probability of a methane molecule to occupy the large ( y CH ) and the small ( y CH4