Kinetics of CO2 Hydrate Formation from Water Frost at Low

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Kinetics of CO2 Hydrate Formation from Water Frost at Low Temperatures: Experimental Results and Theoretical Model Andrzej Falenty,† Georgi Genov,† Thomas C. Hansen,‡ Werner F. Kuhs,*,† and Andrey N. Salamatin§ †

GZG, Abt. Kristallographie, Universit€at G€ottingen, Goldschmidtstrasse 1, 37077 G€ottingen, Germany Institut Laue-Langevin, BP 156X, 38042 Grenoble, France § Department of Applied Mathematics, Kazan State University, Kazan 420008, Russia ‡

bS Supporting Information ABSTRACT: The gas hydrate growth from frostlike powders composed of micrometer-sized ice particles does not start with hydrate shell formation, because the initial hydrate film thickness established in earlier work exceeds the ice particle dimensions. In this limiting case, the ice grains are directly consumed by a growing nucleus created on the particle surface. The conventional Johnson-Mehl-Avrami-Kolmogorov (JMAK) model,1 which considers (re-) crystallization reactions phenomenologically in terms of the constituent nucleation and subsequent growth processes, cannot be directly applied to the hydrate formation from frost due to the assumption of an infinitely large domain of crystallization. We present here a modified approach to account for the small particle sizes of the starting material and extend the existing theory of gas hydrate formation from monodisperse ice powders3-5 to the low-temperature and low-ice-particle-size limit. This approach may also prove to be very useful for applying chemical reactions starting on the surface of nanomaterials. In situ neutron scattering was used to obtain the experimental degree of transformation as a function of temperature between 185 and 195 K. The data were analyzed with the modified JMAK model constrained by information from cryo-SEM and BET measurements. Based on the obtained activation energies for hydrate nucleation and growth, an estimate is given for the probability of formation of CO2 hydrates at conditions relevant for Mars; a direct reaction of CO2 gas with water frost is considered to be very unlikely on the Martian surface under current conditions.

1. INTRODUCTION Gas hydrates (in the following named also GH or clathrates) are crystalline, nonstoichiometric solids belonging to the clathrate hydrate structural family in which gas molecules are trapped in a 3-D hydrogen-bonded water structure forming face-sharing polyhedral cages.6 It is generally accepted that the formation of these compounds starts with hydrate nucleation7,8 usually at the interface of water/ice and gas at adequately low temperatures and/or sufficiently high gas fugacity. Nucleation, however, is not necessarily rapid, and a phenomenon called “induction time” is frequently encountered for growth from the liquid.9 For water a substantial undercooling is needed to initiate the nucleation process;10 the latter authors also found evidence for different activation energies of hydrate formation from water ice (hereinafter for simplicity termed “ice”) and water as well as a substantial decrease of sensitivity of hydrate formation to supercooling in the presence of ice. A definite molecular picture for hydrate nucleation is not yet available despite recent progress in studying this process in the water-gas system;11,12 for nucleation from ice the situation is even less clear. The subsequent growth mechanism depends on the geometry of the system and the physical state of reacting phases from which liquid water/gas and ice/gas are the most commonly studied. r 2011 American Chemical Society

For the growth of gas hydrates from relatively large ice particles exposed to a hydrate-forming gas, at least the following two stages can be distinguished:4,5,13 (1) an initial formation of a clathrate coating at the ice-gas interface (first in form of clathrate patches expanding from heterogeneously distributed nucleation centers14) and (2) a subsequent growth perpendicular to the interface (“shrinking core”) limited by the permeation of the constituents, i.e., water and gas through the hydrate layer. For isolated ice particles surrounded by a gaseous/liquid environment of the hydrate former or vice versa, the process can be mathematically treated by so-called shrinking-core models.2-4,13 Whether the permeation is limited by the water or gas transfer remains an open question and may well depend on the porosity and extent of the hydrate layer (e.g., ref 10); the microstructure of the gas hydrate undoubtedly plays a considerable role for the diffusive mass transfer between a shrinking ice core and the particle’s gas-exposed surface. Particularly low transformation rates should be expected well below the melting point of ice, where—even if the initial clathrate film can still be formed on a laboratory time scale—the following sluggish diffusion-limited Received: September 3, 2010 Revised: December 21, 2010 Published: February 23, 2011 4022

dx.doi.org/10.1021/jp1084229 | J. Phys. Chem. C 2011, 115, 4022–4032

The Journal of Physical Chemistry C reaction needs time scales beyond what can be done in the laboratory. Depending on the gas composition, particle size, microstructure, and p-T conditions, the full transformation time may vary from a few hours to billions of years.15 Consequently, it is possible that even if gas hydrates are the thermodynamically preferred phase they will be underrepresented (or not exist at all) due to the limitingly slow kinetics. This may be particularly relevant for GH from more exotic, extraterrestrial environments of the solar system like cometary ices,16-19 icy moons,20-24 or Mars25-31 where temperatures stay generally deeply below the melting point of ice. A direct condensation from a guest gaswater vapor phase seems to offer an interesting alternative32,33 to a solid-gas reaction but remains too poorly explored to draw firm conclusions. In fact, the so far observed direct growth from vapor was largely epitaxial,32 thus requiring a crystalline hydrate surface as a template. Substrates (THF-clathrate and Fe-montmorillonite clays) were also used in ref 33 for the codeposition of gas-loaded ices; by IR spectroscopy the formation of local environments with CO2 molecules surrounded by water molecules was detected, but a confirmation of the effective formation of crystalline CO2 hydrates could not be provided. The origin of the modification of the weak CO2 adsorption behavior for the clay samples remains obscure altogether and may be related to some intercalation. The stronger observed interaction with THF finds an explanation further below. In any case a straightforward low-temperature codeposition of H2O and guest gases can be expected to deliver bulk crystalline material only for gas compositions close to the one of the target solid compound, not the most likely scenario on Mars with its high CO2 content. Other low-temperature formation mechanisms of clathrate hydrates were suggested and are based on the enhancement of the molecular mobility of the constituents.34-37 Enhanced molecular mobility is the key to faster formation; this concerns both the mobilities of water as well as the guest mobilities. It is well accepted that the mobility of H-bonded water molecules is the bottleneck of any low-temperature topological rearrangements in solid water phases; it turns out that orientational defects (socalled Bjerrum defects) play an important role in this process.38 Thus, amorphous solid water phases remain metastable upon warming until the molecular mobility allows for topological rearrangements and a transformation into ice Ic somewhere in the temperature range between ∼130 and 150 K.39 A similar limitation must also be expected for topological changes involving gas hydrates at low temperature. The mechanism of water diffusion through the ice lattice was established to involve interstitials40 and is clearly an activated process with activation energy of ∼54 kJ/mol.41 There is evidence that the water transport in clathrate hydrates proceeds in an important way by a vacancy mechanism with some contribution of interstitial mechanisms, and it turns out that the guest molecules could have a significant influence:42 ether guests like THF (tetrahydrofuran) lead to a very considerable enhancement of the mobility of water molecules. It is interesting to note that Trainer et al.33 have observed an enhanced formation of CO2-filled small water cages in the presence of THF, but they did not link this observation to its catalytic enhancement of water mobilities. Water vacancies appear to be important for guest mobility; such vacancies form fairly easily in the presence of H-bonding guests like H2S or ethylene oxide.42 A pure CO2 hydrate system is known to form distinctly slower than a system with a catalyst (e.g., ethylene oxide) due to the distinctly higher activation energies for formation of a vacancy defect.42 Thus it appears that the defect

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mobilities in the water framework are crucial for the formation process of gas hydrates. The available surface area and the mobility of the constituents of gas hydrates are the limiting factors of gas hydrate growth from ice particles4,5,13,43 as described in shrinking-core models via a multistage process.3,4 The activation energies for molecular permeation are found to be similar to the self-diffusion of H2O in ice, both for CH4 and CO2 hydrate formation, suggesting (but not proving) the water mobility to be the rate-limiting step.13 In the following we will show that ice composed of extremely small particles of hundreds of nanometers in size provides an exception from the multistage formation process since the initial hydrate film thickness reaches several micrometers, thus exceeding the ice particle dimensions. Consequently, the whole ice volume is readily transformed already at the initial stage without reaching the permeation-limited regime. The time required for a full transformation of submicrometer-sized ice (“frost”) into GH can be expected to be several orders of magnitude shorter than in the case of larger particles, but more precise estimations were not available until now. The present study aims at the laboratory observations of the gas hydrate formation from frost using an extension of previously published models3-5,13 toward lower temperatures and smaller ice particle sizes based on re-examined and generalized concepts of recrystallization theory.1 As a demonstration of the predictive utility of the model, we present first results on the formation rates of CO2 hydrates at the actual conditions relevant to the Martian polar caps26 that well exceed laboratory time scales.

2. SAMPLE PREPARATION AND EXPERIMENTAL METHODS 2.1. Sample Preparation. Water frost has been produced in two separate batches using a custom-built setup based on a cold deposition technique. Deuterated water was continuously vaporized with a hot air gun and blown on a rotating copper plate submerged in liquid N2 to about 1/3 of its radius. A possible contamination with atmospheric H2O frost was considerably reduced by installing the setup in a glovebox overpressurized with N2. Layers of frost condensed on the cold copper plate were removed in regular intervals of ∼10 s down into the liquid N2 via manual scratching. The short exposition to the surrounding was maintained in order to avoid uncontrollable changes in the specific surface area (SSA) and crystallinity due to the annealing processes. It is noteworthy that the frost obtained in two batches is described crystallographically44,45 as so-called “cubic ice” (ice Ic) consisting of hexagonal and cubic stackings sequences that at temperatures between ∼160 and 240 K slowly anneal via a defective, more and more hexagonal form (Figure 1), eventually leading to well-defined hexagonal ice (ice Ih); during the annealing process the mean crystallite size evolves about fivefold between 160 and 200 K.45 From the established T dependency of the stacking fault patterns,45,46 we conclude that the effective deposition temperature of the frost on the rotating copper plate (including the very considerable local heating due to the release of the latent heat of condensation) was somewhere close to 190 K. The annealing toward ice Ih is too fast at temperatures above 195 K to be consistent with the initial diffraction patterns, but somewhat lower formation temperatures in the range of 175190 K are also compatible with our observations. Thus we may say that the frost has been formed in the range between 175 and 195 K with the most likely value of 190 K. This rather narrow 4023

dx.doi.org/10.1021/jp1084229 |J. Phys. Chem. C 2011, 115, 4022–4032

The Journal of Physical Chemistry C temperature range of formation most likely limits the varation of the particle sizes between different frost batches, in particular those two that were used in our experiments discussed below. In Figure S1 (Supporting Information), we show the diffraction pattern of the frost produced which is clearly stacking-faulty ice Ic. A preliminary characterization of the frost was done with ex situ cryo-FE-SEM imaging using a Quanta 200F and a LEO 1530 Gemini instrument equipped with Polaron and Oxford CTH1500HF cryo-stages, respectively; both instruments provide a resolution up to a few tens of nanometers. Samples (∼2 mm3) were studied at about 90 K (with cold N2 gas as coolant) and a pressure of about 0.1 Pa. In order to preserve an original morphology and avoid possible artifacts caused by a standard cold sputtering of conductive metals (e.g., Pd-Pt, Au, Au-Pd), we decided to leave the frost samples uncoated. To reduce the undesired charging effect and the surface etching of the electron beam, a fairly low acceleration voltage of up to 2.5 keV was used. Collected images show (Figure 2) irregularly shaped polycrystalline frost flakes with a rough surface composed of quasi-spherical subparticles with a radius between ∼2.5 and 7.5 μm forming a “cauliflower-like” surface structure (Figure 2B). We were not able to deduce from the electron micrographs whether the smallest visible building units are poorly developed single crystals or have a polycrystalline nature. A more detailed study on the specific surface area (SSA) of frost has been performed with a volumetric Brunauer-Emmett-Teller

Figure 1. Part of the initial neutron diffraction pattern taken at 190 K just before GH forming CO2 gas has been applied on partially annealed, stacking-faulty frost. Black crosses are observed intensities, and the red line is the expected intensities of ideal ice Ih. The defective nature of the starting material can be recognized by a larger intensity ratio of the first two ice reflections 002 and 100 from the value of ∼0.55 for ideal Ih, i.e., a relative increase of the measured 002 peak with respect to the expectation.46

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(BET) method47 based on the physisorption of gas molecules on solid surfaces where they eventually form a multilayered film bounded by van der Waals forces. This technique is particularly useful for materials with a surface roughness and inner surfaces related to open porosity. Such surfaces will also be accessible to gas leading to a hydrate formation reaction, yet may be difficult to account for in SEM images. For the BET measurements we used a custom-built setup (Figure 3) based on a device successfully used in studies of snow metamorphism.38-50 SSA was measured in a series of stepwise expansions of CH4 gas from the initial volume VI into the expansion volume VE (Figure 3). A complete description of the procedure can be found elsewhere.15 The original design of the cell used for snow studies in a large volume has been scaled down for materials of much higher SSA than snow; the modified cell could contain about ∼4.5 g of frost with an expected total surface area of several square meters. Due to lengthy equilibration periods (several to tens of minutes) between the gas admissions, a small degree of leakage—unavoidable in our system when operating at temperatures around the boiling point of N2—was lowering the calculated heat of the absorption. Additionally, some small disturbances were also caused by a fluctuating level of liquid N2 that had to be manually refilled. Consequently, the measured frost SSA of ∼1.68 m2/g (ΔQCH4 ∼1251 J/mol) was underestimated, and the true area at the ideal value of ΔQCH4 would be close to ∼1.9 m2/g (taking ΔQCH4 = 2240 J/mol48 as a reference heat of adsorption). This surface corresponds to an assumed monodispersed, spherical material with r0 =1.5 μm particle radius; this value was later adopted for the data analysis. It is noteworthy that the particle size of 2.5-7.5 μm observed by SEM (see above) would deliver a SSA of 1.2-0.4 m2/g. This

Figure 3. Scheme of the in-house BET setup. (1) He and (2) CH4 inlet, (3) vacuum pump, (4) sample cell, (A) VI—initial volume at room temperature, (B) expansion volume VE with three T zones: VER, VET, and VEL in room, transition, and liquid N2 temperatures, respectively. A more detailed description is given in ref 15.

Figure 2. Cryo-FE-SEM images of frost used as starting material for hydrate production seen under (A) irregular frost flakes and (B) close-up of a conglomerate of small, “cauliflower” ice crystallites ( ∼2.5-7.5 μm) building frost flakes. 4024


The Journal of Physical Chemistry C

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Table 1. Conditions of Experiments with the Final Transformation Degree and Cage Occupancya run

T, K

p (f), MPa

pd (fd), MPa

final degree

1

185

0.0266 (0.0262)

0.0124 (0.012)

0.2

1b/0.90 ( 0.21

25.3

2

190

0.036 (0.0357)

0.0178 (0.0177)

0.36

1b/0.91 ( 0.07

21.8

3

190

0.036 (0.0357)

0.0178 (0.0177)

0.68

1b/0.97 ( 0.04

32.7

4

195

0.0505 (0.0495)

0.0252 (0.0246)

0.80

1b/0.87 ( 0.04

15.1

cage occupancy (LC/SC)

duration (h)

LC, large cages; SC, small cages. The quality of profile fit together with reliability index and CHI2 is given in Figure S2 (Supporting Information). b Due to correlations between refined parameters, the occupancies for marked cages were fixed. a

suggests that subparticles are present in the frost sample studied, a conjecture also supported by the observed diffraction broadening44,45 strongly suggesting that the coherent crystallites are smaller than the electron-optically visible particles. 2.2. Neutron Diffraction Experiments. In situ formation experiments have been performed in a custom-built setup4,13 mounted on the high-flux two-axis neutron diffractometer D2051 at the Institut Laue-Langevin (ILL), Grenoble, France. This setup on D20 has repeatedly proven its usefulness for kinetic studies of GH52 due to its high sensitivity, the simultaneous detection in a 2θ range of 153.6°, and the high flux/short acquisition times. Temperature control was provided by a Lakeshore 340 controller attached to a so-called “orange” Heflow cryostat operating between 1.7 and 300 K to within typically (0.1 K. Samples stored under liquid N2 temperature were loaded into the pressure cell,53 transferred to the cryostat, and evacuated. Experimental runs were started at the target temperature (Table 1) by admission of CO2. It took typically on the order of a few seconds to reach a stable new pressure within the thermodynamic stability field of CO2 hydrates (Table 1); there is no evidence that the sample temperature was significantly perturbed upon gas admission. During the whole reaction both pressure and temperature were kept constant. Quantitative information on the amount of remaining gas hydrate was obtained from the Bragg diffraction data using a full pattern Rietveld refinement package—GSAS54—for every time slice with phase fraction determined typically to an accuracy of about 0.1 wt %. Refined parameters were the lattice constants of ice Ih and gas hydrate, the phase fractions, and five to six background parameters; the overall scale factor was kept fixed. The atomic positions and displacement parameters for D2O ice Ih and CO2 hydrate55 were also kept fixed. Although the experimental settings were not optimal for a detailed structural study, we also attempted to retrieve cage occupancy (Table 1), as no other data are available for the cage filling of CO2 hydrates formed at temperatures below 200 K. The weight fraction of the clathrate phase R (mole fraction of ice converted to the gas hydrate) obtained from every frame was plotted as a function of time. Experiments 1, 2, and 4 were performed with the frost samples from the same batch. Sample 3 comes from a different batch and was used in a subsequent experimental campaign.

3. THEORETICAL KINETIC MODEL The conventional Johnson-Mehl-Avrami-Kolmogorov (JMAK) model1 which considers the formal kinetics of primary (re-) crystallization phenomenologically in terms of the constituent nucleation and subsequent growth processes cannot be directly applied to the hydrate formation from highly dispersed frost material. This theory is originally based on the assumption of an infinitely large domain of crystallization. The frost medium used here, described as a packing of small ice particles of a mean

equivalent-sphere radius r0, restricts the hydrate nucleus growth to a volume of (4/3)πr03 or to an even smaller domain of a single crystallite of a lesser mean radius rc. We introduce the notion of hydrate formation rate ΩV, being, by definition, the volume fraction of nonreacted ice which is converted to hydrate during a unit of time. Accordingly, at a given moment of time t, the clathration degree R is governed by the balance relationship dR ΩV dt 1 -R and ! Z t R ¼ 1 - exp - ΩV ðτÞ dτ ð1Þ 0

The Vandermeer-Rath microstructural path methodology1 assumes that the apparent equivalent-sphere radius of a hydrate nucleus increases with its age τ as G0τm/3, where G0 is a factor proportional to the linear rate of crystal growth, i.e., interface migration, and the growth exponent m ∼ 1-3, depending on the dimensional pattern of the hydrate grain development. The nucleation rate per unit area of ice surface varies with time as N0tσ-1, where N0 is the nucleation rate factor, and the exponent σ ranges from 0 to 1 for instantaneous (site-saturated) and uniform (constant-in-time) nucleation, respectively. Correspondingly, per a unit volume of ice, · the nucleation rate is N = 3N0tσ-1/r0. Unlike the classical Johnson-Mehl-Avrami-Kolmogorov scheme, which assumes an infinitely large continuous domain of recrystallization, an important specific feature of ice-particle powders is that a hydrate nucleus formed in such a highly dispersed medium can potentially grow only to a grain of an effectively limited equivalent-sphere radius rc not exceeding the mean ice particle size r0. Although the transformation of ice to clathrate is associated with the volume increase of ∼13%, we assume this effect to be negligible for so fine powders. As discussed in the experimental section, in case of polycrystalline frost particles the hydrate nucleus development is, most probably, restricted to the subparticles. Thus, rc can be compared to the mean size of ice subparticles in the starting material to be described by the structural ratiohrc = rc/r0. In the following we assume that after essential initial annealing, for monocrystalline frost particles hrc = 1 and rc = r0 and we adopt the mean particle radius obtained from our BET analysis for further treatment. A relatively low diversity of initial crystallite volumes can be expected at stable laboratory conditions of frost production, and a monosize crystal growth domain model seems to be a plausible approximation. Consequently, at a moment t the volume V of a nucleus created at the moment τ is 4 V ðt - τÞ ¼ π min rc 3 , G0 3 ðt - τÞm 3 1 Conventionally, after, we note that the number of nuclei actually appearing in a unit volume of ice during a time interval dτ is 4025



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Table 2. Conditions of Experiments and Kinetic Parameters of CO2 Hydrate Formation from Water-Frost Powders conditions of experiments

deduced kinetic parameters

run

T, K

r0, μm

ln(f/fd)

duration, h

m

kN, 1/(h m2)

kG, m/hm/3

ωV, 1/h

t0, h

SD 3 102

1

185

1.5

0.779

25.3

1.28

6.13 108

5.85 10-7

0.0132

16.3

0.21

9

6.69 10-7

0.0323

14.6

0.24

0.0730

30.1

0.34

0.137

4.8

0.70

2

190

1.5

0.702

21.8

1.3

1.63 10

3

190

2.25

0.702

32.7

1.28

1.63 109

7.5 10-7

4

195

1.5

0.699

15.1

1.14

6.93 109

1.185 10-6

· less than N dτ, because hydrate nuclei cannot form in those parts of the surface (of fraction R) which have already been converted to hydrate. Therefore this quantity includes the so-called “phantom” nuclei RN_ dτ which would form if the ice surface was completely free. The fraction Aex of the ice powder volume which would be consumed by the hydrate nuclei at the moment t if the phantom nuclei were real is termed1 the extended hydrate · volume. In accordance with the above definitions, Aex = ΩV, and the generalized JMAK approach yields the following parametrization: Z t Z t _ ΩV ðτÞ dτ ¼ V ðt - τÞNðτÞ dτ Aex ¼ 0

¼

4πN0 r0

Z

0 t

min rc 3 , G0 3 ðt - τÞm τσ

-1

dτ

0

This finally specifies the balance eq 1 as

8 Rt τ m > > > ðt - τÞσ - 1 dτ, t > 0 > ωV ðt - τÞσ - 1 dτ þ ðt - t0 Þσ , : t σ 0 0

Z

t >t0

ð2Þ Here t0 = (rc/G0)3/m is the typical time of the hydrate grain growth, and the factor ωV is the nucleation-limited rate of hydrate formation, ωV = 4πrc3N0/r0. At a fixed temperature T (in K), the driving force of the hydrate formation process is the excess of the chemical potential of the ambient gaseous phase characterized by the gas fugacity f over the equilibrium (dissociation) conditions determined by fugacity fd. Hence, we write " t0 ¼

r 0 kG ln f = fd

#3=m ,

ωV ¼ 4πr0 2 kN ln

f fd

ð3Þ

where kG and kN are the apparent growth and nucleation rate constants dependent on temperature. It should be emphasized that the poorly known structural factors (rhc)-1 and (rhc)3 are directly included into these characteristics, rendering them oppositely affected by the maximum relative size of the hydrate grain growth (i.e., by the frost particle polycrystallinity). Differentiation of eq 2 with respect to t yields the relationship for ΩV. The substage solely controlled by nucleation sets on at t > t0, and for high rates of hydrate nuclei growth, when t0 f 0, one obtains ΩV ≈ ωVtσ-1. For the constant nucleation rate at σ = 1, this results in the limiting approximation employed earlier in refs 3, 4, and 13 for modeling ice surface coating by initial hydrate film. A more general situation with predominant hydrate nucleation in cracks considered in earlier work5 corresponds to σ < 1. On the other hand, for low hydrate grain growth rates, at large t0

and t < t0, eqs 1 and 2 represent a classical sigmoidal curve of JMAK type which is sensitive only to the size of frost particles (hence the SSA), since for t < t0 the ratio ωV/t0m, by definition, does not depend on hrc, being inversely proportional to r0.

4. RESULTS AND DISCUSSION The modified JMAK model presented above has been implemented as an interactive computer system “FROST” capable of simulating the hydrate formation from ice frost. Built-in routines allow also for comparison and interpretation of theoretical curves with the hydrate growth kinetics measured in neutron diffraction experiments if a set of initial parameters (r0, ln(f/fd), σ, m, kN, kG) is provided. Typically three of them, m and kN, kG (i.e., ωV, t0) were kept free for tuning. The particle size estimates and the thermodynamic conditions of the experimental runs are presented in Tables 1 and 2. The nucleation exponent σ = 1 has been derived from an earlier formation experiment performed on ice spheres at 193 K and 80 kPa14 where clathrates were seen to nucleate uniformly in time until the whole available surface was consumed. The dimensionality of growth described by the exponent m is difficult to determine precisely, but with the guidance of SEM images taken at the initial stage of a reaction on sprayed ice spheres14consistent with images obtained on largely transformed frost spheres (Figure 4)- we were able to narrow down possible choices to values between 1 and 2 indicating a dendritic/2D growth. The best-fit model parameters m, kN, kG together with ωV, t0 are given in Table 2. The simulated kinetic curves plotted against the experimental data can be seen in Figure 5. In particular, to deduce the values of kN and kG for runs 1, 2, and 4, we have adopted the typical (mean) ice particle size r0 of 1.5 μm (1.9 m2/g) taken directly from combined cryo-SEM images and BET measurements. Although the experimental run 3 from another batch was conducted at thermodynamic conditions (190 K and 0.036 MPa) identical to run 2 and very close growth dimensionalities m were deduced, the inferred nucleation rate factors, ωV, which in accordance with eq 3 essentially depend on the particle size, appeared to be more than two times different (see Table 2). In spite of the fact that the frost preparation technique has repetitively proven to return a material of comparable characteristics, small differences in the SSA (hence particle size) can occur, for instance, because the particle size is influenced by the exact deposition temperature.45 Consequently, we have assumed the kN factor in run 3 to be equal to that found for run 2 and, using eq 3, calculated r0 of 2.25 μm from ωV deduced for sample 3 of the other batch. Very similar kG factors obtained in these two experiments provide an additional and independent confirmation for validity of such interpretation. As can be seen from Tables 1 and 2, for both batches of ice, the total duration of the experiments noticeably exceeds the time t0 when the reaction switches from the grain-growth-limited phase to the nucleation-limited phase. In accordance with the modified 4026



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Figure 4. SEM images taken at ∼90K and 0.1Pa from a partly transformed frost sample (run 3) recovered after 32.7h at 190K and 36kPa (final GH fraction 0.68). (A,B) The underlying typical frost appearance (Figure 2) is still seen with its typically 1-2 μm-sized single crystalline GH particles and some larger ones (compare with Figure 2). (C) A close-up on an aggregate of small GH crystallites with distinguishable crystal boundaries between individual single crystals. (D) To the left, a fragment of a larger rounded particle with a well visible sub-μ porosity characteristic for GH.4 To the right, one can see well exposed, connected single crystals of GH. Some fresh frost particles deposited in the SEM during handling are also seen loosely attached to the GH surface.

Figure 5. CO2 hydrate formation kinetic curves (Table 1) fitted with the modified JMAK model using the FROST program (left) and Avrami plot (right).

JMAK model 1 and 2, the slope of the kinetic curve in logarithmic scales ln(ln(1 - R)-1) and ln(t) decreases around this moment from m þ 1 ∼ 2.15-2.3 to σ ∼ 1, not being constant as in the classical theory. A certain discrepancy between the theoretical curve and the data can be observed only at the very end of the experimental run 4 (Figure 5) at the limiting values of the reaction degree R ∼ 0.8 when the reaction may be noticeably influenced by the polydispersity effects due to two minor fractions of remaining utmost small and large ice particles. As discussed in more detail below, the first group has the lowest probability of hydrate nucleation because of their small surface area in combination with the stoichiastic nature of a nucleation event, while the transformation rates of shrinking ice cores in the largest particles can be essentially suppressed by the diffusive resistance of the surrounding hydrate shell. This likely scenario is additionally

supported by SEM images (Figure 4) where one can clearly distinguish two suits of differently developed particles. The vast majority forms small 1-2 μm-sized single crystals that with some exceptions (Figure 4D) occupy the whole volume of a former frost particle (Figure 4C). Some larger 15-20 μm particles are covered by a clathrate film commonly without clearly recognizable crystal boundaries. The deduced values of the hydrate growth rate exponent m are also comparable and range from 1.14 to 1.3, corresponding to a situation between a dendritic 1D type of growth of hydrate crystals and a two-dimensional one of the hydrate crystals. Finally, the deduced hydrate nucleation and growth rate constants can be plotted versus inverse temperature to determine the respective activation energies (Figure 6) that can be later used for an extrapolation on lower temperatures exceeding laboratory time scales. The respective activation 4027



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Figure 6. Hydrate nucleation rate constant and growth rate constant versus inverse temperature (solid squares) and the least-squares approximation (solid line).

energies are inferred as QN = 72.5 kJ/mol and QG = 20.9 kJ/mol. Considerable differences between both values show that it is much more difficult for a GH embryo to cross its critical size and form a nucleus than later to grow. The crucial importance of nucleation for clathrate formation in small-grained systems is highlighted in additional observations made by SEM on ice spheres of the mean radius r0 ∼ 27.5 μm at 193 K and 0.080 MPa in one of our previous experiments.14 As seen in SEM images, the maximum site density at a nearly full coverage of accessible ice surface by hydrate patches was on the order of 108 nuclei/m2 and was achieved in 15 min of the clathration reaction. In case of the frost material with the particle size of r0 ∼ 1.5 μm, the derived model parameters allow us to predict that about 1011 nuclei/m2 are needed to transform the same volume of ice to hydrate at identical thermodynamic conditions. The respective nucleation rates can be estimated as 4 108 nuclei/(m2 h) on the ice spheres and 6 109 nuclei/(m2 h) on the frost particles. Thus, even much higher imperfectness (expected for the frost grain surfaces) resulting in more than one order higher nucleation rates does not compensate for the restricted GH growth. In other words, this suggests that after the formation of a nucleus in a small-grained system, GH growth quickly consumes the ice grain on which the nucleation has occurred but does not spread to neighboring particles. As a result, more nucleation events are needed to transform a small-grained ice system to hydrate in comparison with a coarse ice powder of the same total volume. Another evidence supporting this conclusion comes from frost SEM images (Figure 4 D, E) where in most cases individual frost particles seem to be substituted by a single clathrate crystal. Moreover, as it was shown in ref 14, the average surface (∼ 615 μm2) occupied by one clathrate crystallite on the ice spheres greatly exceeds the averaged surface of the frost particles (28.27 μm2). Consequently, at the investigated as well as at lower temperatures the main limiting factor for forming CO2 clathrates from water ice and volatile CO2 may well be the geometry of the starting material as well as the stochastic nature of the nucleation process. Additionally, the observations show the importance of discontinuities (cracks, structural defects, grain boundaries) within an isolated ice particle; these discontinuities may also create a considerable hindrance for the clathrate hydrate spreading. The overall speed of transformation then depends on the initial effective particle size (considering also the particles’ internal discontinuities). Each of these effective particles needs to experience a nucleation event before it can be converted to GH. Thus, in general, due to the high activation energy of nucleation, the stochastic nature of the nucleation event, and the low

activation energy of growth, smaller particles in frost will tend to transform more slowly, while larger particles with larger surface areas have a higher chance to experience a nucleation event on their surfaces and would then transform faster by the GH spreading growth. Further on, quite the reverse tendency will take place when ice particle size exceeds 20-30 μm, allowing for hydrate shell formation around the remaining ice core. We speculate that this scenario may well be common for all clathrates formed from ice in a gaseous guest-gas phase (except for those with guests of small radius, e.g., H243). At very low temperatures, the GH formation in the small-grained systems may become effectively impossible due to the very low probability of nucleation events. It is also noteworthy that, in accordance with explanations given in the Introduction, the formation of GH through a condensation process, sometimes invoked as a faster alternative pathway, is not a realistic scenario for a pure system. The limitations of growth and nucleation observed in our study are fully compatible with the expectations from other low-temperature kinetic measurements as well as molecular dynamics simulations on a variety of clathrate hydrate systems.42

5. APPLICATION TO GH FORMATION ON MARS In the following we exemplify the functionality and usefulness of the JMAK model applied to the formation experiments of CO2 hydrates at the conditions relevant to the surface and close subsurface of the Martian polar regions. Although numerous researchers attempted to explain geomorphological features or chemical and isotopic composition of atmospheric gases on Mars (see Introduction) in terms of clathrate hydrates, any positive evidence for the existence of these compounds is still missing. Despite some effort in applying spectroscopic methods56,57 to the search for GH, large similarities to water ice hinder a successful detection from the orbit. With the information obtained from fitting our modified JMAK model to our experimental GH formation data, one can now start to extrapolate the formation processes to lower temperatures relevant for the present Mars surface. Using the deduced activation energies (Figure 6), we have extrapolated the nucleation (kN) and growth (kG) rate constants down to 150 K (the lowest possible temperatures on the Martian surface where CO2 clathrates might be formed28) for selected pressures (Figure 7). m and σ have been assumed to be constant. As a reference ice particle radius we chose r0 = 1.5 μm (within the size limit suggested for Martian frost58). This value is close to the 4028


The Journal of Physical Chemistry C particle radius of the frost produced in our laboratory, an observation which is not that surprising as our frost was produced at around 190 K (see above), i.e., at typical Martian surface conditions. The calculated values were reintroduced into the model and used to obtain kinetic curves for temperatures between 150 and 170 K (Figure 8A). In addition, we also demonstrate the influence of the particle radius on formation rates at 150 K (Figure 8B). As expected, the formation rates are very sensitive to temperature (Figure 8A), decreasing rapidly within lowering by 20 K from time scales of years to hundreds of years (Table 3). An equally interesting result has been returned from the isothermal modeling on four selected monodispersed powders at 150 K (Figure 8B). At this temperature the probability for the nucleation is already very low, and even if a clathrate crystal succeeds to nucleate on a single particle the reaction will come to a halt due to very restricted volume of ice. Consequently, the model predicts diminishing transformation rates along with the decreasing particle radius. Within the suite of the modeled particle radii, the time necessary for the comparable degree of conversion was changing up to 2 orders of magnitude (Table 4). A less pronounced but nevertheless still considerable impact has also the excess fugacity. The transformation time difference between

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the modeled conditions (Max) and (Mid) reaches a factor of 2-3 (Tables 3, 4). Although the model presented above is capable of fitting the experimental data very well, it is important to note that the extrapolation toward lower temperatures should be taken with some precaution due to a number of facts: (1) Below 130-150 K ice frost formed by precipitation in the atmosphere would likely result in an amorphous form of ice with different microstructure and surface reactivity. (2) To our knowledge there is no reliable source of fugacities of CO2 to precisely constrain the driving force at the modeled p-T conditions. As the discrepancy between f and p is small at low pressures (roughly below 10-2 MPa), we assumed both parameters to be equal down to 150 K. It remains an open question, however, whether there is any driving force to transform ice into CO2 hydrate at all at very low temperatures; instead, a mixture of volatile CO2 þ N2 as well as ice þ solid CO2 phases may be present leaving no stability region for CO2 hydrate28 (3) The model treats samples as a monodisperse powder that is a reasonable but not perfect approximation to the real, polydisperse frost (Figure 2). Consequently, part of particles that exceed the thickness of the initial film (∼2.5 μm) is likely to need considerably longer reaction times due to the rate-limiting diffusive transport from the surface to the reaction front inside.4,5,13 (4) The model does not explicitly account for the defective nature of ice Ic frost and its potential influence on the nucleation rate. This defective nature, however, has been implicitly introduced by the experimental determination of the nucleation rate constant on such defective ice samples. Yet, with decreasing ice “quality” for ice formed at Table 3. Excess Fugacity for Mid and Max Pressures (Figure 7) at Selected Temperatures with Resulting Time Scales (Figure 8) for 50 and 90 wt % Transformation of r0 = 1.5 μm Ice Powder excess fugacity ln(f/fd)

Figure 7. Part of the CO2-H2O phase diagram with marked conditions (Max and Mid) used in the modeling. The Max line (red) marks a phase boundary between volatile (light gray) and solid (dark gray) CO2,59 and Min shows the low-pressure limit for the CO2 clathrate hydrate (GH) stability field.60 The Mid line marks a pressure line along which the intermediate excess fugacity was calculated.

time scales for r0 = 1.5 μm powder

T, K

max (mid)

170

1.004 (0.502)

0.34 (0.73)

1.11 (2.28)

165

0.916 (0.458)

1.8 (5.8)

3.6 (11.8)

160

0.824 (0.412)

10 (21)

34 (66)

155

0.724 (0.362)

68 (134)

221 (445)

150

0.688 (0.344)

470 (950)

1540 (3090)

50 wt %, years

90 wt %, years

Figure 8. (A) Predictive kinetic curves for CO2 clathrate formation between 150 and 170 K (Table 3) within the stability field of clathrate grown from ice and vapor phase (see Figure 7). (B) Kinetic curves for CO2 clathrate formation for selected monodisperse powders at 150 K. Solid and open symbols correspond to the modeling at Max and Mid excess fugacity, respectively (Table 3). 4029



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Table 4. Formation Time to 50 and 90 wt % of Transformation for Four Selected Particle Radii Calculated at Max and Mid (in Brackets) Pressures (See Figure 7) time scales for the formation at 150 K r0, μm

50 wt %, years

90 wt %, years

0.25

17326 (33440)

54994 (109988)

0.5

4119 (8359)

13749 (27497)

1.5

470 (950)

1540 (3090)

2.5

180 (350)

558 (1158)

44,45

lower temperatures, imperfections of various nature (e.g., point defects, stacking faults, grain boundaries, Bjerrum defects, or dangling OH bonds) may facilitate (or also hamper) the nucleation process by changing the activation energy at defective sites. Some support for this hypothesis may be found in our earlier studies on ice spheres14 where clathrates were indeed preferentially growing on cracks, surface imperfections, and grain boundaries. (5) The model assumes constant p-T conditions over the whole calculated time period (Figure 7) that might not always be realistic for natural systems, particularly over geological time scales. Notwithstanding these disclaimers, it is clear from the introductory remarks and our experimental results that it will be difficult to form CO2 hydrates on the surface of Mars from ice frost or, even, from CO2-water vapor phase without the presence of help molecules or catalysts. The existence of such species (e.g., THF or ethylene oxide) on Mars is highly speculative and certainly unproven to date. On thermodynamic grounds a formation can presently occur only in the polar regions of Mars; however, due to the tilt of the rotation axis the temperatures in polar summer61 exceed the stability limit of CO2 hydrate. Thus, if the formation of detectable amounts of GH takes more than 1 year (roughly corresponding to one Martian winter), any possibly formed small amount of GH is very likely to be destroyed in the Martian summer (NB: the unit year quoted in the text, Figure 8, as well as in Tables 3 and 4, relates to a terrestrial year; a Martian “year” corresponds to 687 terrestrial days, and a Martian “day” corresponds to 24.6 h). With very low winter temperatures in the polar regions, staying below 150 K, it seems unlikely that detectable amounts of CO2 hydrates are formed on the Mars surface according at least to present day knowledge as the known mechanisms for faster formation processes do not operate at the relevant temperatures in the pure H2O-CO2 system. As we have shown above, even if the frost particle sizes are considerably smaller or larger, the reaction rates would not become substantially faster as for smaller particles we run into nucleation problems and for larger particles we run into permeation problems, both limiting the transformation rates. Only if near subsurface parts of the regolith could be sealed off from the atmosphere (e.g., by water ice formation blocking circulation of CO2 and thereby increasing its chemical activity), there remains a chance to form CO2 hydrates a few meters below the surface in the regolith; from our model we can estimate that the kinetics would be sufficiently fast to form significant amounts of CO2 clathrate in one season at temperatures of above ∼165 K.

6. CONCLUSION We have presented a phenomenological modified JMAK model capable of predicting the formation rates of gas hydrates from (sub)micrometer-sized monodispersed ice powders. The

model has been constrained with help of specific surface area measurements using the BET method, a series of in situ neutron diffraction runs to determine the degree of GH formation, and cryoSEM imagery on laboratory-formed ice frost. The activation energies for the nucleation and growth obtained from the model are used to calculate formation rates at conditions exceeding laboratory capabilities due to extreme p-T conditions and/or extended time scales. It turns out that the stochastic nucleation process of gas hydrates on ice frost particles at low temperatures is so infrequent that nucleation becomes the bottleneck of gas hydrate formation in such small-grained systems, at least for pure systems. As the known catalysts for the enhancement of molecular mobilities and clathrate formation (e.g., ethers) are unlikely to be present on Mars, we predict that the formation of CO2 hydrates on the Martian (sub)surface at the prevailing low winter temperatures is not possible. On more general terms we may also state that when going from coarse-grained systems to fine-grained systems or nanomaterials any surface reactivity involving a nucleation step may become slower despite the larger specific surface area; in such cases intermediate sizes may show enhanced reactivity.

’ ASSOCIATED CONTENT

bS

Supporting Information. Diffraction pattern of unreacted frost (Figure S1) and last neutron diffraction data set for each of the four formation experiments fitted with GSAS (Figure S2). This material is available free of charge via the Internet at http:// pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*Phone: þ49 551-39-3891. Fax: þ49 551-39-95-21. E-mail: [email protected].

’ ACKNOWLEDGMENT We thank the Deutsche Forschungsgemeinschaft for financial support (Grant Ku 920/11), the Institut Laue-Langevin (ILL) in Grenoble for beam time and support, and Dr. Kirsten Techmer (G€ottingen) for her help in the SEM sessions. The technical help of Heiner Bartels and Eberhard Hensel in setting up the BET apparatus as well as the reaction cells is very gratefully acknowledged. ’ REFERENCES (1) Humphreys, F. J.; Hatherly, M. Recrystallization and Related Annealing Phenomena, 2nd ed.; Elsevier: Oxford, U.K., 2004; p 628. (2) Salamatin, A. N.; Hondoh, T.; Uchida, T.; Lipenkov, V. Ya. Postnucleation conversion of an air bubble to clathrate air-hydrate crystal in ice. J. Cryst. Growth 1998, 193 (1-2), 197–218. (3) Salamatin, A. N.; Kuhs, W. F. Formation of porous gas hydrates. Proceedings of the Fourth International Conference on Gas Hydrates, Yokohama, May 19-23, 2002; pp 766-770. (4) Staykova, D. K.; Kuhs, W. F.; Salamatin, A. N.; Hansen, T. Formation of porous gas hydrates from ice powders: Diffraction experiments and multistage model. J. Phys. Chem. B 2003, 107 (37), 10299–10311. (5) Genov, G.; Kuhs, W. F.; Staykova, D. K.; Goreshnik, E.; Salamatin, A. N. Experimental studies on the formation of porous gas hydrates. Am. Mineral. 2004, 89, 1228–1239. (6) Sloan, E. D.; Koh, C. A. Clathrate Hydrates of Natural Gases, 3rd ed.; Taylor & Francis Group, LLC: Boca Raton, FL, 2008. (7) Kashchiev, D.; Firoozabadi, A. Driving force for crystallization of gas hydrates. J. Cryst. Growth 2002a, 241 (1-2), 220–230. 4030


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