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Guest Migration Revealed in CO2 Clathrate Hydrates Andrey N Salamatin, Andrzej Falenty, Thomas C Hansen, and Werner F Kuhs Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.5b01217 • Publication Date (Web): 13 Aug 2015 Downloaded from http://pubs.acs.org on August 21, 2015
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Energy & Fuels
Guest Migration Revealed in CO2 Clathrate Hydrates A.N. Salamatin1, A. Falenty 2, T.C. Hansen3, and W.F. Kuhs 2,*
2 3 4
1
Dept. of Applied Mathematics, Kazan (Volga Region) Federal University, 420008 Kazan, Russia 2
GZG Abt. Kristallographie, Georg-August-Universität Göttingen, 37077 Göttingen, Germany 3
5
Institut Laue-Langevin (ILL), 71 avenue des Martyrs, 38000 Grenoble, France
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* Werner F. Kuhs,
[email protected], tel. +49 551-39-3891, fax. +49 551-39-95-21
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RECEIVED DATE (to be automatically inserted after your manuscript is accepted if required
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according to the journal that you are submitting your paper to
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Abstract
11
The shrinking-core model of the formation of gas hydrates from ice spheres with well-defined
12
geometry gives experimental access to the gas permeation in bulk hydrates which is relevant to their
13
use as energy storage materials, their exploitation from natural resources as well as to their role in
14
flow assurance. Here we report on a new approach to model CO2 clathration experiments in the
15
temperature range from 230 to 272 K. We develop a comprehensive description of the gas
16
permeation based on the diffusion along the network of polyhedral cages, some of them being empty.
17
Following earlier molecular dynamics simulation results, the jump from a cage to one of its empty
18
neighbors is assumed to proceed via a “hole-in-cage-wall” mechanism involving water vacancies in
19
cage walls. The rate-limiting process in the investigated temperature range can be explained by the
20
creation of water-vacancy-interstitial pairs. The gas diffusion leads to a time-dependent cage filling
21
which decreases across the hydrate layer with the distance from the particle surface. The model 1 ACS Paragon Plus Environment
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allows a prediction of the time needed for a complete conversion of ice spheres into clathrate as well
2
as the time needed for a full equilibration of the cage fillings. The findings essentially support our
3
earlier results obtained in the framework of a purely phenomenological permeation model in terms of
4
the overall transformation kinetics, yet it provides for the first time insight into the cage equilibration
5
processes. The diffusion of CO2 molecules through bulk hydrate is found to be about three to four
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times faster in comparison with the CH4 case.
7 8
Keywords: gas hydrate, gas diffusion, hole-in-the-cage mechanism, water vacancies, ice, neutron
9
diffraction, Rietveld refinement
10 11
Introduction
12
Clathrate gas hydrates (GH), nonstoichiometric crystalline compounds of guest-gas molecules built
13
into small and large cages of a metastable framework of water molecules, are a widely spread matter
14
in geosystems and are important in technological applications for energy storage and flow assurance.
15
Different physical scenarios with gas/water mixtures or solutions can result in gas hydrate
16
appearance. Here we focus on the GH formation process which occurs on the ice/water-gas interfaces
17
(i.e. excluding a direct nucleation from liquid). This phenomenon has been studied under laboratory
18
conditions by exposing ice particles (e.g.1-4) or liquid-water (e.g.5-7) to clathrate-forming gas. It is
19
now well established that, in spite of obvious differences between gas-ice and water-gas interfaces,
20
in both cases the gas hydrate formation follows a similar pattern: 1) The ice/water-gas interface is,
21
first, covered by thin hydrate film composed of initially nucleated hydrate patches e.g.2, 4. The initial
22
thickness of this layer is limited to a few µm e.g.2, 8. 2) After this relatively short stage I, the next
23
stage II sets on when the further clathration reaction is maintained and, generally, limited by guest-
24
gas molecules permeation (diffusion) from the environmental gaseous atmosphere through the
25
hydrate layer towards the unreacted ice/water phase. At the same time water molecules move to the
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outer boundary of the hydrate layer to react there with the ambient gas. What drives the gas transport
27
is the supersaturation of the "gas-ice/water-hydrate" system, i.e. the difference between the chemical
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potential of the gas encaged in GH at the reaction front, at the contact with ice/water phase, and that
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on the outer GH surface at the contact with the gaseous phase.
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Powder samples of small spherical ice particles with well defined specific surface turned out to be
4
particularly helpful in accessing individual stages and establishing apparent activation energies for
5
the gas migration through the clathrate lattice. An elaborated description of the initial coating stage
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of hydrate growth on ice spheres as a stochastic “birth-and-growth” phenomenon (i.e. an interplay of
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two mechanisms: (a) creation of "two-dimensional" nucleation spots and (b) limited lateral growth of
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the hydrate patches on the ice surface) was developed and employed for interpretation of the pVT
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(in-house) and neutron diffraction hydrate formation experiments2 on the basis of the interactive
10
computer system "POWDER-4". With only small changes in resulting predictions of the
11
transformation kinetics, the stage-I sub-model is now improved to represent a more realistic scenario
12
of ice-powder structure development at microscopic randomly uniform hydrate patch nucleation2.
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As for the stage II modeling, our earlier study2-4 and the software assumed that the gas permeation
14
mechanisms were linked to polydispersity / polycrystallinity of GH build-up (particle/grain
15
boundaries, triple junctions), linear crystallographic defects, and other imperfectness of a crystalline
16
structure. A certain, though small, amount of mobile gas molecules was envisaged as an important
17
bulk property of clathrate hydrates. This pseudo-solute component could be acquired from ambient
18
atmosphere to be transported further by the quasi-stationary permeation (diffusion) process to the
19
ice-to-hydrate transformation front. Among many other works, such an approach was explored in a
20
series of our publications3,
21
modeling efforts10-12 based on formulation of the general diffusion equations in the hydrate layer do
22
not specify the gas permeation process on the microscopic level of hydrate structure and thus remain
23
essentially at the level of approximation spelled out in our earlier work3,
24
attempt to cast the model into a more physical picture of the transport phenomena.
4, 9
, but remained purely phenomenological. Unfortunately, recent
4, 9
. In the following we
25
A review of recent studies and data reconciliation2 showed that one of the most likely transport
26
mechanisms of gas molecules through the crystalline hydrate structure might be the so-called "hole-
27
in-cage-wall" migration realized via water vacancies in cage walls connecting neighboring cages.
28
Small molecules like H2 and Ne appear to move through the GH framework without water vacancies ACS Paragon Plus Environment
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for jumps between large cages (passing through the 6-membered connecting water rings), while
2
jumps involving a passage through 5-membered rings appear to involve water defects even for such
3
small molecules13. As a consequence, the migration process also involves a transient cage-filling
4
readjustment during the hydrate shell growth and some gas exchange between small and large cages.
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As for the water interstitial transport from the ice-hydrate interface to the particles’ surface, in
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accordance with the present-day understanding14, it is not the rate limiting process for the gas hydrate
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formation from ice. Here we continue our research line and complete the development of the
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generalized model for the non-equilibrium "hole-in-cage-wall" diffusion15 of the guest-gas molecules
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through the hydrate layer of structure I (sI) growing around an ice particle; this mechanism has also
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been called “water-vacancy-assisted”16 and relates to a very dynamic opening of a water ring
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structure in polyhedral cage walls of a hydrate structure by a temporary creation of a water vacancy
12
permitting the passage of guest molecules. A principal peculiarity of the sI-hydrate build-up related
13
to modeling the diffusive gas transport is that, in contrast to hydrate structure II (sII), the smaller
14
cages (SC-s) in sI are fully surrounded by larger cages (LC-s) and do not have direct "common-wall"
15
contacts with each other. Thus, in addition to the overall cage occupancy transition, a special
16
emphasis must also be paid to the gas-mass exchange between LC-s and SC-s. We, further, include
17
the diffusion theory into the macroscopic description of the single-gas sI-hydrate formation from ice
18
powders. The developed, more comprehensive and more physical (but still phenomenological) model
19
is implemented as a modified computational algorithm in the next POWDER-5 version of the
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POWDER-software series and is applied to interpret CO2-hydrate formation experiments.
21 22
Experimental
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CO2-hydrate was grown from well-defined spherical ice powders prepared in a standardized
24
procedure3, 4, 9 by spraying deuterated water into liq. N2 starting from 99.9 % purity D2O. This was
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done in a sealed glove-box under N2 saturated atmosphere in order to avoid possible contamination
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(and/or dilution) with atmospheric water. Large ice particles were removed by sieving under liq. N2
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through a 350 µm mesh, and the resulting material was stored under liq. N2. Cryo-Scanning electron
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microscopy revealed a log-normal distribution of ice spheres with a mean diameter of 52 µm. These ACS Paragon Plus Environment
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ice spheres consist of grown-together individual ice crystals of 5-15 µm in diameter. The mean
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volume-radius of 42.64 µm has been deduced from the volume distribution of ice spheres.
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Ice powders were transferred into thin walled Al vials and compacted to a remaining porosity of
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~30-40 % (estimated from volume/mass ratio). These samples containing ~ 1 g of the starting
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material were inserted in a pressure cell before CO2 gas was applied with pressures below the
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stability limit of CO2-hydrate. CO2 pressures were measured with a Ashcroft KXD linear gauge (6
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MPa). The reaction rig built around the high-flux 2-axis neutron diffractometer D20 at the Institut
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Laue-Langevin, Grenoble, France17 was used to explore the formation reaction. The excellent
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penetration power of neutrons, the high flux available on D20 instrument and its simultaneous
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readout over a 2θ range of 153.6 deg (at λ = 2.418 Å) gave the unique advantages to follow in-situ
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the nucleation and growth phase as well as the later slow transition to the diffusion limited stage
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(Fig. S1). The acquisition time was set typically to 30-60 s for the initial and 300 s for later parts of
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the formation, depending somewhat on the currently observed reaction rate. Temperature control was
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achieved with a PID regulated “orange” He-flow cryostat operating between 1.7 and 300 K with
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accuracy better than 0.1 K. The large set of diffraction data was analyzed in an automated fashion
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using the full pattern Rietveld refinement package GSAS18 (Fig S1). The phase fractions of the ice
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starting material and the newly formed CO2-hydrate were established with accuracies better than 1
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wt.%. These results present an average for the complete sample volume of ~ 1 cm3. The profile
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functions, zero point, lattice constants of GH, atomic positional and displacement parameters for
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D2O ice Ih and CO2-hydrate were kept fixed once established (Table S1). The freely refined
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parameters were the lattice constants of ice Ih, the phase fractions, five to six parameters of the
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background description by Chebyshev polynomials and occupancies of the small and large cages. It
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should be noted that the retrieval of cage fillings is a challenging task even with good quality
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diffraction data2-4 and typically requires a number of constraints. Particularly troublesome is the
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strong correlation between the cage occupancies and the atomic displacements (Uiso) that may lead to
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significant overestimations of gas occupancies in LC-s and underestimation in SC-s. Here these
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parameters have been tackled by fixing thermal displacement parameters to values derived from
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expansivity data19 cross correlated with earlier, high resolution neutron20 and X-ray powder ACS Paragon Plus Environment
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diffraction21 measurements. Correlations with instrumental parameters (zero point, instrumental
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profiles and sample position) turned out to be less troublesome and could be safely avoided by
3
careful calibration on a reference material (Na2Ca3Al2F14).
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Model Description
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Thermodynamic properties of GH
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Two basic crystallographic GH structures, the Stackelberg Structure I (sI) and Structure II (sII), are
8
distinguished22, depending on the guest-gas nature23. The unit cell characteristic of the sI clathrate
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water host lattice is summarized below in Table 1 following22, 24, 25. It is important to note that the
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density of metastable water frameworks in GH, ρw ≈ 44 kmol/m3, is essentially lower than the Ih-ice
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density ρi = 51 kmol/m3. Thus, structural changes during ice-to-hydrate transformation must proceed
12
with noticeable water mass outflow from the ice reaction front. Hereafter the two types of smaller
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and larger cavities are referred to by lower indices j =1, 2, respectively. General theoretical
14
considerations22 express the chemical potential µw of the host-water framework in clathrate hydrate
15
crystal via the guest-gas occupancies yj,
µ w = µ w ( p, T ) + RT [ν 1 ln(1 − y1 ) + ν 2 ln(1 − y 2 )] .
16
(1)
17
Here µ w is the chemical potential of the empty metastable water host lattice, being a function of
18
external pressure p and Kelvin temperature T within the GH stability region; νj is the number of
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cages of the j-th type per one water molecule; R is the gas constant. In thermodynamic equilibrium
20
with the ambient gaseous atmosphere characterized at the given p- and T- conditions by the gas
21
fugacity fa, one has
22
fa =
1 y1a 1 y 2a , = C1 1 − y1a C 2 1 − y 2 a
(2)
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where Cj are the temperature-dependent Langmuir constants of the gas in the SC and LC hydrate
24
cavities and yja are the corresponding equilibrium cage occupancies, j = 1, 2. In this work C1 and C2
25
are derived from the Langmuir isotherm fitted to cage occupancies of CO2-hydrate calculated with
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CSMGem23. Accordingly, the chemical potentials µj of the gas molecules encaged in SC-s and LC-s
2
can be expressed via their apparent fugacities fj or occupancies yj as
3
1 yj C j 1− y j
µ j = RT ln f j = RT ln
, j = 1, 2.
(3)
4
We also introduce the guest-gas density ρg in hydrate related to the molar concentration of the
5
encaged gas cg counted per a unit molar mass of host-water molecules,
6 7 8
ρ g = ρ wcg ,
c g = ν 1 y1 + ν 2 y 2 ,
(4)
Inversely, the GH-properties (1)-(4) can be rewritten in terms of the apparent gas fugacities with yj =
Cj f j 1+ C j f j
, j = 1, 2.
(5)
9 10
General description of guest-gas diffusion in sI-hydrate
11
Various natural phenomena and laboratory observations e.g.4,
25, 26
12
molecules can penetrate ("diffuse") through bulk hydrate barriers without direct recrystallization
13
and/or reformation of the lattice cages. As reviewed and discussed2, in recent years molecular
14
modeling techniques have revealed plausible mechanisms of the guest-gas permeation in single-
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crystalline hydrates. The principal conclusion is that, most probably, the gas diffusion is assisted by
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the presence of empty cages and water vacancies in the cage walls, i.e. is the so-called "hole-in-cage-
17
wall" permeation. For instance, in CO2-hydrate of structure I, around 30 % of small cages and a few
18
percent of large cages are empty at the typical conditions of laboratory experiments. However, small
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cavities in sI-hydrate are not directly linked to each other. Moreover, the activation energy to hop
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from SC-s to LC-s was found27 to be 1.5-2 times higher than that of hops between LC-s in the
21
presence of water vacancies in the hydrate framework. Therefore, one can assume that the long-range
22
transport in sI-hydrate mainly proceeds through LC-s, while SC-s work rather as sinks or sources of
23
guest-gas molecules, facilitating the gas transport by evacuating guests from LC-s, although without
24
direct participation as the permeation paths in the diffusion process.
demonstrate that the guest-gas
25
Concerning the transport of water molecules involved in the clathration process3, water interstitials
26
are suggested as a viable mechanism. Interstitial water transport is the likely mechanism for proton ACS Paragon Plus Environment
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transport in ice Ih28 and this is also a possible mechanism for hydrates29 to be even more effective
2
than a vacancy mechanism. The water mobility itself is not thought to be a rate-limiting process for
3
gas permeation14 – rather, at high temperatures, it is the creation of water vacancy-interstitial pairs15
4
that controls the guest transport rates. As shown in MD simulations14, interstitial water molecules
5
can move easily in the host framework and can interchange with host water molecules in various
6
ways and most easily when the cages are empty. Thus, we might well model the inward migration of
7
gas molecules via a hole-in-cage-wall mechanism and assume the outward migration of water
8
molecules to take place via the interstitial mechanism.
9
Let us consider the hole-in-cage-wall migration of gas molecules realized via water vacancies
10
connecting neighboring LC cavities in the crystalline hydrate structure I. This mechanism obviously
11
involves the guest-gas redistribution in the GH bulk, including gas exchange between LC-s and SC-s
12
and the cage filling transient readjustment. In accordance with the general concepts of non-
13
equilibrium thermodynamics30,
14
hydrate lattice we write after32
31
, for mass flux qg of gas molecules through larger cages in sI-
q g = − λ g ∇µ 2 ,
15
(6)
16
where λg is the mass transport (phenomenological) coefficient and ∇ is the differential Nabla-
17
operator. Hereinafter we conventionally assume that the λg-factor in Eq. (6), i.e. gas-mass transport
18
via LC-s, is proportional to the molar concentration of the guest-gas molecules in these cavities.
19
Furthermore, in contrast with the classical diffusion theory e.g.31, the potential gas mobility ϑg,
20
depending on water vacancy concentration in GH, must be additionally reduced by the probability for
21
gas molecules to meet an empty large cage. Let us also note that the probability for the guest
22
molecules to move between LC-s via intermediate jump to/from SC-s is, at least, one order of
23
magnitude less and, thus, is neglected in our further considerations. Consequently, in accordance
24
with the hole-in-cage-wall diffusion scenario, the following representation for the phenomenological
25
coefficient can be envisaged:
26 27
λ g = ρ wν 2 y 2 (1 − y 2 )ϑ g .
(7)
Combining Eqs. (3), (6), and (7), one finally arrives at the Fick's law ACS Paragon Plus Environment
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q g = − ρ wν 2 D g ∇y 2 ,
1
(8)
2
where Dg = RTϑg is, by definition, the diffusion coefficient in sI-hydrate through the LC cavities. We
3
note here that the diffusion constant is not assumed to have an explicit dependency on the cage
4
fillings; this is justified by molecular dynamics computer simulations for CH4 hydrates16, where -
5
within our experimentally established range of cage fillings - changes of only a few % are expected.
6
The next step in modeling the gas-mass transport through the newly-formed non-equilibrium
7
hydrate bulk is to describe the gas exchange between the smaller and larger cages. This is a
8
relaxation phenomenon which reveals itself as a sink/source of gas molecules in GH cavities of
9
different types, depending on their initial filling (i.e. chemical potentials of guest molecules). Hence,
10
we introduce the flux σg of gas molecules from LC-s to SC-s in a unit of GH (water framework)
11
volume per a unit of time. Formally one can present σg in two different ways, as
σ g = λ 21 ( µ 2 − µ1 ) or σ g = −λ12 ( µ1 − µ 2 ) ,
12
(9a)
13
where λ21 and λ12 are the mass transport coefficients of the gas molecules from LC-s to SC-s at
14
µ2 > µ1 and from SC-s to LC-s at µ1 > µ2, respectively.
15
Similarly to Eq. (7), we write
λ21 = ρ wν 1ν 2
16
C1 y 2 (1 − y1 ) C y (1 − y 2 ) ϑ21 and λ12 = ρ wν 1ν 2 2 1 ϑ12 . C1 + C 2 C1 + C 2
(9b)
17
Here ϑ21 and ϑ12 are the analogues of potential gas mobilities proportional to hopping frequencies of
18
gas molecules between cages of different types. In proximity to thermodynamic equilibrium
19
ϑ21 ≈ ϑ12 .
20 21 22
23 24
Accordingly, with the use of Eqs. (3), the driving force of the mass exchange process in Eqs. (9a) becomes
µ j − µ k = RT ln
C j y k (1 − y j ) , ≈ RT 1 − C y (1 − y ) fk k j k
fj
and together with Eqs. (9b) this results in the following form of the mass flux
σg =
ρ wν 1ν 2 χ g C1 + C 2
[C1 y 2 (1 − y1 ) − C 2 y1 (1 − y 2 )] ,
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(10)
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where χg = RTϑ21 ≈ RTϑ12 is the mass exchange coefficient.
2
Finally, the mass conservation laws for the guest gas molecules occluded in smaller and larger
3
cavities of non-equilibrium gas hydrate combined with constitutive equations (8) and (10) yield the
4
generalized diffusion model
5
ν 2χg ∂y1 [C1 y 2 (1 − y1 ) − C 2 y1 (1 − y 2 )] , = ∂t C1 + C 2
∂y 2 ν 1 ∂y1 + = ∇ ⋅ ( D g ∇y 2 ) . ∂t ν 2 ∂t
(11)
6 Here t is the time. 7 8
Ice-sample structure and initial stage I
9
We consider a starting ice powder as a random dense packing of spherical ice grains with the log-
10
normal size distribution characterized by initial mean radius , initial mean-volume radius
11
r i 0 = ri 0
12
follow2, 3 and use the geometrical description of the local ice-powder structure evolution during the
13
GH formation as developed33 for a random dense packing of expanding mono-disperse spheres. As
14
shown by Kuhs et al4, even in case of the polydisperse ice particle ensemble, the mono-size
15
approximation of the ice-to-hydrate transformation process remains valid up to the reaction (ice-to-
16
hydrate transformation) degree α of 40-50 %. However, the apparent initial radius of ice spheres is
17
now defined so as to represent the specific surface area (SSA) of the starting material that is the
18
initial equivalent-surface-area radius ri 0 = (1 + v0 )r i 0 . As a consequence, this will change the
19
reference particle dimensions and will affect the values of the inferable kinetic parameters, resulting
20
in more realistic estimates. On average, locally the external shape of hydrate layers formed on ice
21
particles is represented as a truncated sphere of radius rh. The ice cores shrink, their radius ri
22
decreases, due to the inward growth of the hydrate shells. But, because of the lesser density of water
23
in the hydrate phase, the excess water molecules must be transported to the outward hydrate surface
24
exposed to the ambient gas, and the hydrate layers simultaneously expand into the open space (voids)
25
between the original ice grains3. This process is governed by the hydrate volume expansion factor
26
E = ρ i ρ w − 1 , and its detailed description in terms of the relative geometric characteristics
3 1/ 3
and relative size variance ν02. As before in the POWDER-4 software model, we
2
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Ri = ri/ri0 and Rh = rh/ri0 is summarized for convenience after2,
2
Information).
3
in Appendix A (Supporting
3
The ice-particle surface fraction αS covered by hydrate film and/or occupied by developing
4
contacts of growing hydrate layers during the initial coating stage I is one of the principal overall ice-
5
to-hydrate conversion characteristics. In accordance with34, let δ0 be the thickness of the surface ice
6
layer converted to the initial hydrate shell. Further, we follow the JMAK kinetic theory35 of
7
recrystallization and employ the improved modeling approach2 presented in Appendix B (Supporting
8
Information) to simulate the hydrate film formation on ice spheres. As before, we introduce the
9
general notion of the ice-sphere coating rate ΩS, which is, by definition, the fraction of the free
10
(exposed to the ambient gas) ice surface which becomes covered by the newly nucleated hydrate
11
patches during a unit of time. The Vandermeer-Rath microstructural path methodology35 assumes
12
that the apparent radius of a hydrate patch develops with its age τ as 2Gτ m/2, where G is the growth
13
rate constant and the 2-D growth exponent is m ~ 2. Correspondingly, the nucleation rate per unit of
14
area varies with time as N0t σ−1, where N0 is the nucleation rate constant and the exponent σ ranges
15
from 0 to 1 for instantaneous and uniform nucleation, respectively.
16 17
Clathration kinetics at the recrystallization front
18
The driving force of the GH growth (crystallization) on ice spheres at the ice-hydrate interface is the
19
local difference between chemical potentials of ice (µi) and hydrate (µw) water frameworks with the
20
former one essentially higher than that of the newly formed hydrate. Based on Eq. (1), a general
21
kinetic equation for the mean ice-to-hydrate conversion rate ωV (the number of ice moles transformed
22
to hydrate on a unit area of ice-hydrate interface per a unit of time) can be formulated as
23
ωV = −
ρ i ri 0 K R ν 1 +ν 2
1 − y1i 1 − y 2i ν 1 ln + ν 2 ln 1 − y1d 1 − y 2d
.
(12)
24
Here KR is the recrystallization rate constant. The SC- and LC-occupancies y1i, y2i and respective
25
apparent gas fugacities f1i, f2i are ascribed to the clathration reaction front, and they are not
26
necessarily the equilibrium ones, i.e. f1i ≠ f2i in general. For a single-gas clathrate compound its ACS Paragon Plus Environment
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1
dissociation pressure pd(T) is an increasing function of temperature T, and the corresponding
2
apparent (equilibrium) gas fugacity f,d(T) and occupancies y1d, y2d are related by analogues of
3
Eqs. (5). The degree of cage equilibration φ1i is introduced as the ratio between the SC- and LC-
4
fugacity deviation with respect to the decomposition fugacity fd:
5
φ1i = ( f 1i − f d ) /( f 2i − f d ) ,
f 1i = f d + φ1i ( f 2i − f d ) .
(13)
6
Thus, for any fixed φ1i-value considered as a tuning parameter, the clathration reaction rate (the
7
driving force) in Eq. (12) depends only on the apparent gas fugacity f2i, i.e. on the gas occupancy y2i
8
in larger cages with the corresponding SC occupancy y1i calculated for f1i. The cage equilibration
9
degree φ1i ranges from 0, at dissociation occupancy y1d and fugacity f1i = fd in SC-s, to 1, at the cage
10
equilibrium conditions when f1i = f2i. One can speculate that φ1i is essentially less than 1 at minimum
11
occupancy level of hydrate stability in SC-s and rather high occupancy in LC-s translated through
12
larger cages from the applied environmental conditions.
13
To explore the kinetic equations (12) and (13), the gas occupancy y2i in larger cages at the reaction
14
front must be specified so as to match the clathration rate with the diffusive gas flux coming to the
15
ice-hydrate interface. This brings forward the necessity to formulate the gas diffusion model.
16 17
Hole-in-cage-wall diffusion sub-model
18
As described by Staykova et al3, a hydrate layer growing around an ice core can be divided by the
19
spherical boundary of radius rc (i.e. by the distance from the ice core centre to an average contact
20
plane) in two sub-layers: a spherical one around the ice core and a layer truncated by the interparticle
21
contacts. Accordingly, the normalized cross-flow area A of the guest-gas diffusive flux on a spherical
22
surface of relative radius R = r/ri0 in the hydrate layer is
23
R2, Ri < R < Rc , A( R) = 2 Z R c R 1 − 2 1 − R , R c < R < R h ,
2(1 − s ) , R c = R h 1 − Z
24
where R i , R c , and R h are the respective analogues of ri, rc, and rh spatially averaged over the
25
hydrate covered area of ice cores and normalized by ri0; Z is the current particle coordination number
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in the sample, and s is the fraction of the open (hydrate/ice) particle surface area exposed to the
2
ambient gas (see also Appendix A of Supporting Information).
3 4 5
At constant diffusion coefficient Dg of the guest molecules moving through water vacancies in LC walls, the general gas transport model (11) can be rewritten in one-dimensional approximation as
ν 2χg ∂y1 [C1 y 2 (1 − y1 ) − C 2 y1 (1 − y 2 )] , = ∂t C1 + C 2
D g ∂ ∂y 2 ∂y 2 ν 1 ∂y1 + = 2 A . ∂t ν 2 ∂t ri 0 A ∂R ∂R
(14)
6
Eqs (14) must be completed by the boundary condition which determines the apparent gas fugacities
7
of the newly formed hydrate so as to counterbalance the GH-growth rates by the gas flux at the ice-
8
hydrate interface, R = R i .
ωV =
9
ρ iν 2 D g ∂y 2 ri 0 c gi
∂R
,
(15)
R = Ri
10
where c gi = ν 1 y1i + ν 2 y 2i is the molar concentration of the guest-gas in the GH at the reaction front.
11
Together with Eqs. (12) and (13), the latter equation (15) predicts the clathration rate and gas
12
occupancies at the ice-core surface.
13
The physics of the initial hydrate layer formation on ice sample surface is not clear in details. The
14
ice surface coating is very difficult to measure with analytical methods, and there is no reliable
15
information on this formation step in terms of cage occupancies. Even for neutron diffraction data
16
the error bars are substantial (Fig S2). Therefore, different deviations of SC- and LC-occupancies,
17
i.e. guest gas fugacities f1h and f2h, from the equilibrium ones corresponding to the applied gas
18
fugacity fa can be envisaged. Thus, as before at the reaction front in Eq. (13), we introduce the
19
equilibration degrees φ1h and φ2h,
20 21
f 1h = f d + φ1h ( f a − f d ) ,
f 2h = f d + φ2h ( f a − f d ) ,
and calculate the corresponding occupancies y1h and y2h, using Eqs. (5).
22
Further outward growth of the hydrate shell occurs due to the outflow of the interstitial excess
23
water molecules from the ice-to-hydrate transformation front. Thus, at the contact with the gaseous
24
environment, we write after Appendix A (Supporting Information) the mass balance relation which
25
governs the outward growth of the particle outer radius R h ,
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d Rh E Ri d Ri =− , 2 dt s R h dt
1
R h t =0 = 1 ,
(16)
2
and assume that this is a slow equilibrium process proceeding at equilibrium cage filling y1a and y2a,
3
corresponding to the applied gas fugacity fa (see Eqs. (2)).
4 5 6
Eqs. (12)-(16) form a generalized microscopic shrinking-core model, based on the hole-in-cagewall transport mechanism for the GH growth from ice powders during stage II. Finally, the relationship between the clathration degree α and ice-core radius R i , R i = (1 − α / α S )1 / 3 ,
7
(17a)
8
together with the master (mass balance) equation of the ice-to-hydrate conversion on macro-scale
9
level,
10
dα 2 = S i 0 [(1 − α S ) ρ i δ 0 Ω S + α S R i ωV ] , dt
α t =0 = 0 ,
(17b)
11
complete the model of the sI-hydrate formation from ice powders. Here in Eq. (17b) S i 0 = 3 /( ri 0 ρ i )
12
is the initial specific surface area of ice spheres in the sample. Although introduced independently,
13
Eqs. (17) are closely related to the previously developed model2 and, as explained in Appendix C
14
(Supporting Information), can be directly deduced from this model.
15
In view of the general complexity of Eqs. (12)-(17), one has to think carefully about additional
16
observables that can be extracted from in-situ diffraction experiments to constrain the hole-in-cage-
17
wall diffusion model. While the fractions of the formed clathrate phase are reliably determined by
18
neutron measurements, the overall changes in SC/LC filling, being coupled with the thermal
19
displacements, can be derived with considerably lower accuracy, especially at reaction degrees lesser
20
than 10 wt.% (see Experimental). Still, the data remain useful to provide additional constraints for
21
the modeling. Ex-situ Raman or synchrotron X-ray measurements may also give further insights into
22
the composition of the newly-formed gas hydrate and could complement the information obtained
23
from neutron diffraction data.
24 25
Results and Discussion
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Three CO2-hydrate formation runs (see Table 2) starting from deuterated ice particles of 26 µm mean
2
radius at 263, 253, and 230 K and followed by neutron diffraction are considered below. They had
3
been previously simulated and interpreted on the basis of POWDER-4 program as the diffusion-
4
limited clathration process2. Here we continue this analysis with the use of the computer system
5
POWDER-5 which implements the generalized diffusion model given by Eqs. (12)-(17). Further
6
details on this implementation can be found in Appendix D (Supporting Information).
7 8
Diffusion limited ice-to-hydrate conversion
9
A series of preliminary computational experiments by POWDER-5 confirmed the conclusion2 that
10
CO2-hydrate growth on ice spheres was limited by diffusion. The best-fit equilibration factors, mass
11
exchange and diffusion coefficients χg and Dg, initial film thickness δ0, and other model parameters
12
deduced at KR → ∞ are gathered in Table 2. They allow direct estimation of the phenomenological
13
earlier introduced2 permeation coefficients,
14
D′ =
ν 2 ( y 2a − y 2d ) Dg . 2 c gi (1 + ν 0 ) ln( f a / f d )
15
The newly calculated permeation coefficients D' and initial film thickness δ0 for the diffusion-limited
16
CO2-hydrate formation (i.e. at the zero limit of the conversion-type factor F = D ′(1 + ν 0 ) /(ri 0 K R )
17
introduced in our earlier publications2-4) are compared in Table 2 to the previously inferred values
18
(given in parentheses). Close agreement between these principal parameters shows the applicability
19
of the phenomenological POWDER-2 and -4 models2-4 and their consistency with the newly
20
developed generalized theory of the guest-gas diffusion in hydrates.
21
Fig. 1 illustrates the very good match of the neutron diffraction data and performed simulations (STD
22
~0.1-0.3 wt.%). However, the model constraining procedure and sensitivity tests showed that the
23
overall kinetic curves are not noticeably affected by the finer details of the guest-gas migration such
24
as equilibration degrees and LC/SC gas exchange. With this in mind, estimates related to the SC and
25
LC filling variation during the clathration reaction (Fig. 2) have been deduced from neutron
26
measurements in the case of CO2-hydrate formation2-4. At the initial, coating phase of the formation2,
2
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1
below 5-7 wt.% of GH (after 12-18 min), we have observed increased uncertainties and larger scatter
2
of data points (see Fig. S2 for plots with error bars). A similar situation is also noticeable for the
3
whole formation run at 230 K where the conversion degree and reaction rate are low. The
4
occupancies of all investigated cases display clear trends; occupancies of LC-s rapidly settle at ~0.97
5
and SC-s show a continuous, slow readjustment during the whole reaction length. Cage occupancies
6
have allowed a first-guess constrain of all the major hole-in-cage-wall diffusion parameters,
7
including the equilibration factors and guest-gas mass exchange coefficients; the resulting best-fit
8
kinetic curves are also plotted in Fig. 2.
9
It should be noted that, in spite of the refinement constraints, the experimentally obtained absolute
10
cage occupancies show a minor overestimation with respect to the simulated curves about ~ 2 % for
11
the LC-s and ~ 5-9 % for SC-s. This is somewhat larger than the conditional 1σ error from the
12
constant Rietveld refinement (Fig. S2), yet still remains within the marginal error (i.e. including
13
uncertainties from the constrained displacement parameters). The reason for this discrepancy in
14
principle might be related either to the imperfect constraints of the data (mainly, atomic displacement
15
parameters) or to poorly defined Langmuir constants used in the model. Sensitivity tests on the first
16
possibility showed that small variation of constraints in the Rietveld refinement shifts the absolute
17
values of cage occupancies by a few % without noticeable distortion of relative changes and general
18
trends. At the same time, similar trails on the Langmuir constants allowed very little flexibility,
19
leading to unphysical diffusion rates. Hence, to reliably adjust the slopes of the simulated curves and
20
come to consistency between the data and model predictions, we simultaneously shifted the deduced
21
SC and LC measurements by the best-fit constant values (see Fig. S2).
22
For the diffusion-limited CO2-hydrate formation from ice powder, the SC and LC occupancies at
23
the reaction front are very close to the dissociation levels; the equilibration degree φ1i does not affect
24
the ice-to-hydrate conversion and has been set as φ1i = 0. Rather high guest-gas mass exchange
25
coefficients, on the order of 10-35 h−1, are estimated and reveal a fast relaxation of the depleted SC
26
occupancies to equilibrate with the local LC filling. The overall diffusion flux passing through LC-s
27
is high enough to compensate for excess or lack of guest molecules and quickly equilibrates the
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initial hydrate patches with the gaseous environment. Thus, we have had to assume φ2h = 1. Although
2
systematically small, only the initial SC equilibration degree φ1h seems to play (see Table 2) a certain
3
role as a controlling model parameter.
4 5
Reaction-controlled ice-to-hydrate conversion
6
The lowest possible values of the clathration rate constant KR and maximum bounds for the diffusion
7
coefficients Dg have been also estimated in sensitivity tests by attempts to fit the available data of the
8
three CO2-hydrate formation Runs 1-3 at highest non-zero conversion type factors F (see above)
9
which allowed the same accuracy level. The limiting best-fit model parameters have been obtained at
10
F ~ 0.06-0.07 and are summarized in Table S2; the previously deduced values2 are given in
11
parentheses. The measurements and model predictions are compared in Figs. S3 and S4. Fig. S3
12
illustrates still a good agreement between the performed simulations and the experimental values of
13
the reaction degree of a similar quality level as before in Fig. 1 for the diffusion-limited scenario.
14
The reduced reaction rate constants enhance somewhat the simulated variations of averaged SC-
15
occupancies (compare Fig. 2 and Fig. S4) but do not noticeably influence the deviations from
16
experimental LC and SC neutron diffraction measurements. Again, we have to emphasize that the
17
experimental occupancy data are not sufficiently accurate to reveal further details of the ice-to-
18
hydrate conversion especially at low reaction degree, below ~ 5-7 wt.% ( starting 12-18 min of the
19
reaction), in particular for low-temperature Run 3. Nevertheless, the performed computational
20
exercise shows that our estimates of the diffusion coefficients presented in Table 2 does not change
21
by more than a factor of 2 for the runs at 253 and 263 K and by more than a factor of 3 for the low
22
temperature run at 230 K, while all other model parameters remain practically same.
23
Although there are some uncertainties in the interpretation of the experimental data, our findings
24
suggest that the essential parameters of interest, in particular the diffusion constants, can be
25
determined from the shrinking-core modeling. Our estimates are clearly better than an order-of-
26
magnitude approximation. The work reported here is only concerned with CO2 diffusion in a hydrate
27
host lattice; yet, the results can still be compared to our earlier analysis of the CH4 diffusion, since
28
our earlier phenomenological approaches are in good agreement with the more sophisticated and 17 ACS Paragon Plus Environment
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1
physical “hole-in-cage-wall” model. It is noteworthy in this context that the former permeation
2
model was established largely considering the methane case. Thus, the ~ 3-times higher bulk
3
diffusion constants of CO2, as compared to CH4 2, would likely persist also for the CH4-case analysis
4
with our new physical model; the small influence of cage filling on the diffusion constants16 will not
5
alter this picture, despite the fact that the CO2-hydrate shows a 5% smaller total cage filling as
6
compared to CH4-hydrate at identical p-T conditions. The inferred diffusion constants could, thus, be
7
inserted into the model for gas transport phenomena in GH, e.g. in gas-exchange reaction related to
8
the replacement of CH4 for CO2, a process which would allow the concomitant sequestration of CO2
9
and recovery of CH4 from natural gas hydrates e.g.36-44.
10 11
Mid- and post-formation equilibration
12
A direct consequence of the diffusive transport of guest molecules across the GH layer is a
13
continuous readjustment of SC- and LC- occupancies. The highest difference in cage occupancies
14
between the hydrate surface (assumed to equilibrate more or less instantaneously in our experimental
15
time-resolution) and the reaction front is observed in case of the diffusion-limited scenario. The
16
compositional gradient develops after formation of the initial film of thickness δ0 (see Tables 2 and
17
S1) and remains until the equilibrium with the gas phase in whole volume of GH is reached. What
18
seems to be insufficiently appreciated in the past is the fact that the equilibration length extends
19
beyond the point of the full transformation (understood as a full conversion of ice into gas hydrate).
20
This fact is of particular importance for estimation of cage occupancies in not fully converted or not
21
fully equilibrated samples that will be smaller than predicted equilibrium values.
22
To demonstrate the duration of the readjustment process, we use the results from 263 K (Table 2,
23
Run 1) to simulate the time necessary for the full equilibration of GH grown (Fig. 3) in case of
24
26 µm-mean-radius particles. The complete transformation to CO2-hydrates takes about 470 h (i.e.
25
~19 days), after that practically uniform occupancy distributions, within 0.1‰ resolution of the
26
model, are attained in 50 and 110 h (in ~2-5 days) for LC-s and SC-s, respectively. For larger
27
particles the equilibration period increases proportionally to the time necessary for the full
28
conversion (Fig. 4). For ice spheres with the mean radius of 100 µm the formation takes nearly 290 ACS Paragon Plus Environment
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days for completion and additional 20 to 40 days for the equilibration of large and small cages. Thus,
2
the presence of large ice grains in the starting material will easily lead to an incomplete conversion
3
into GH, which is frequently observed2-4; it should be mentioned that ice grain polydispersity on its
4
own also accounts for some relic ice as a consequence of switching larger grains out of the reaction -
5
a consequence of the outward growth of GH on ice grains2, 4. Care must also be taken concerning the
6
homogeneity of GH samples prepared with the ice method. Extended reaction times are needed for a
7
full equilibration and homogenization of the cage fillings when starting from ice. Discrepancies
8
between cage fillings determined by X-ray diffraction, Raman spectroscopy or NMR on laboratory
9
specimens (obtained via the ice method) and natural samples (with presumably longer equilibration
10
times) are to be expected. Likewise, discrepancies between experimental values and theoretical
11
predictions of cage fillings may simply be due to insufficient equilibration.
12 13
Scale analysis of the phenomenological models2-4 and their consistency with the newly developed generalized “hole-in-cage-wall” diffusion model allows to introduce the dimensionless time
14
Fo =
15
6tν 2 ( y 2 a − y 2 d ) D g c g i ri 0
2
=
6tD ′ (1 + ν 0 )r i 0 2
2
ln
fa , fd
16 17
which in case of diffusion limited ice-to-hydrate conversion at relatively high SSA-equivalent
18
particle radii ri0 > 50-70 µm (i.e. mean radii > 20-30 µm) controls the hydrate formation
19
process and the post-formation equilibration. Using the deduced model parameters from Table 2, it
20
becomes possible to present all the principal simulated normalized kinetic curves (e.g. from Figs. 1
21
and 2) for Runs 1-3 at 263, 253, and 230 K versus Fo for different initial particle radii in a unified
22
practically identical form (Fig. 5); normalization of the cage filling was achieved by
23
putting y j = ( y j − y j d ) /( y j a − y j d ) , j = 1, 2. In particular, it can be concluded that the ice-to-
24
hydrate conversion time is about Fo ~ 1.08-1.13, while the equilibration is essentially finished at
25
Fo ~ 1.17-1.30. These estimates slightly vary due to differences in the deduced equilibration degrees
26
φ1h ~ 0-0.1. ACS Paragon Plus Environment
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2
CONCLUSIONS
3
1. A generalized shrinking core model for sI-hydrate formation from ice powders is developed on
4
the basis of hole-in-cage-wall gas diffusion through the hydrate layer to reaction front in mono-size
5
approximation valid up to 40-50% of ice-to-hydrate conversion.
6
2. The interactive computer system POWDER-5 is implemented to simulate the ice-to-hydrate
7
conversion in the framework of the hole-in-cage-wall diffusion concepts and to interpret the hydrate
8
formation kinetics data.
9
3. Model sensitivity analysis showed that, apart from the overall reaction degree curves, additional
10
data on spatial and temporal LC- and SC-filling variations in the growing hydrate layers would be
11
needed to reliably reveal the transient cage-occupancy readjustment and details of the inter-structural
12
gas exchange mechanisms in hydrates. Such data may be obtained from Raman spectroscopy45, 46.
13
4. The evolution of the volume-averaged LC- and SC-occupancies may be considered as one of the
14
most informative data to deeper understand and constrain the diffusion (gas mass transfer) processes
15
in gas hydrates. However, the acquisition of sufficiently accurate (neutron diffraction) data,
16
especially at reaction degrees below 5-7 wt.%-level is challenging from the instrumental point of
17
view. Weakly scattering guest molecules like CH4 or H2S will hardly be accessible by this method.
18
5. The principal model parameters inferred from the kinetic data in case of the CO2-hydrate
19
formation, such as initial hydrate-layer thickness and permeation coefficients, remain in close
20
agreement with those deduced earlier on the basis of the simplified phenomenological approach
21
employed in the POWDER-4 software.
22
6. While the modeling of the initial stages of the clathration reaction is sensitive to experimental
23
variability and bound with uncertainties of the data analysis, the later permeation-controlled stage is
24
less prone to such problems and delivers reliable kinetic parameters.
25
7. The POWDER-5 model offers a unique insight into the duration of the equilibration process of
26
the cage fillings; the time-scale of this process exceeds the time of the total ice-to-hydrate
27
conversion.
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8. The gas diffusion constants deduced from GH formation experiments are considered to be
2
relevant also to natural and technologically motivated gas exchange processes as the underlying hole-
3
in-cage-wall model is expected to be closely similar for a specific guest in a given host framework;
4
the results presented here are valid for GH crystallizing in structure type I
5
9. While it is very likely that the methane case is analogous to the CO2 “hole-in-cage-wall”
6
mechanism, the situation for larger non-polar (e.g. Xe) and for polar guest molecules in general (e.g.
7
H2S) remains to be elucidated and merits further efforts. Any H-bonding between guest and host may
8
significantly alter the concentration of water vacancies and thus affect guest permeabilities;
9
molecular dynamics simulations are likely to provide useful insights into such guest-specific
10
differences.
11 12
ACKNOWLEDGEMENTS
13
The authors thank the Institut Laue-Langevin (ILL) in Grenoble/ France for beam time and support.
14
Financial support was granted by BMBF in the framework of its SUGAR-II program (grant
15
03G0819B, TP B2-3). The technical help of Ulf Kahmann and Heiner Bartels (both Göttingen) is
16
gratefully acknowledged.
17
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1 230K 253K 263K
GH wt. fraction [α]
0.4
0.3
0.2
0.1
0.0 0
5
10
15
20
25
30
Time [h]
2 3 4 5
Figure 1 Kinetics of CO2-hydrate formation from deuterated ice powder at 263, 253, and 230 K in Runs 1, 2, and 3 (circles) of neutron diffraction experiments and the best-fit kinetic curves (solid lines) simulated by POWDER-5 system in the diffusion limited scenario.
6
230K 253K 263K
1.04
1.02
0.55
Cage occupancy (SC)
Cage occupancy (LC)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
1.00
0.98
0.96
0.50
0.45
0.40
0.94 0
5
10
15
20
25
30
0
5
10
Time [h]
15
20
25
30
Time [h]
7 8 9 10 11 12
Figure 2 Variation of occupancies versus time for LC-s and SC-s: CO2-hydrate formation from deuterated ice powder at 263, 253, and 230 K and the best-fit kinetic curves (solid lines) simulated by POWDER-5 system in the diffusion-limited scenario. LC occupancy data > 1.0 are unphysical and have been dimmed. The uncertainty of each single entry is up to 0.05 for the initial part of the reaction and becomes smaller as the conversion degree increases (see Fig. S2 for enlarged plots with error bars).
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Energy & Fuels
1 0.972
2
0.971 0.970
3
5 6
Cage occupancy
4
(2)
LC-s
0.969
(3)
(1)
0.968 0.967 0.966
0.52
(2)
SC-s
7 8
(3)
(1)
0.48
0
100
200
300
400
500
600
Time [h]
9 10 11 12 13 14
Figure 3 Simulated equilibration path of cage occupancies in GH grown from ice powders of 26-µm-mean radius at 263 K and 1 MPa: (1) during the GH formation, (2) after the complete transformation but before the equilibration; (3) The full equilibration (within 0.1‰); The readjustment is predicted to be somewhat faster for LC-s than for SC-s.
15 16 17 18
263 K 1 MPa
Full conversion & equilibraiton [days]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
300
250
200
150
100
SC LC Full conversion
50
20
40
60
80
100
Particle radius [µm]
19 20 21
Figure 4 Calculated time of the full conversion and LC/SC equilibration as a function of the mean particle radius at constant temperature of 263 K and pressure of 1 MPa.
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1 2
1.0
0.8
Cage occupancy
0.8
0.6
0.4
0.6
0.4
0.2
0.2
0.0 0.0
1.0
230 K 253 K 263 K
Ri
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.0 0.0
230 K 253 K 263 K 0.2
LC LC LC 0.4
SC SC SC
0.6
0.8
1.0
1.2
1.4
Fo
Fo
3 4 5 6 7
Figure 5 Characteristics of CO2-hydrate formation from deuterated-ice particles ( = 26 µm, r i 0 = 42.6 µm, ri0 = 69.9 µm) and post-conversion equilibration (Fo > 1.1) for Runs 1-3 at 263, 253, and 230 K (see Table 2) simulated by POWDER-5 system in diffusion limited scenario: (l.h.s) ice-core radius Ri, and (r.h.s) normalized volume-averaged SC and LC occupancies.
8 9 10 11 12 13 14 15 16 17
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Characteristics
sI
No. of H2O molecules in the unit cell
46 11.8512.05 43.8 2 5.1 1/23 6 5.8 3/23
Unit cell edge (lattice constant), Å Water density ρw, kmol/m3 Number of smaller cages (SC) SC free diameter, Å SC number per one water molecule ν1 Number of larger cages (LC) LC free diameter, Å LC number per one water molecule ν2
2 3
Table 1 The unit cell characteristics of the sI clathrate water frame.
4 5 Conditions of experiments T K 1
263
2
253
3
230
p(f) MPa 1.0 (0.93) 1.0 (0.91) 0.3 (0.29)
pd(fd) MPa 0.75 (0.71) 0.51 (0.49) 0.186 (0.183)
Deduced kinetic parameters h-1
χg
Dg m2/h
D' m2/h
0
10
2.45⋅10-10
4.26(4.3)⋅10-12
1.21⋅107
0.1
35
1.62⋅10-10
2.07(2.1)⋅10-12
2.5⋅107
0
25
2.98⋅10-11
2.71(2.7)⋅10-13
C1 MPa-1
C2 MPa-1
m
kG m/hm/2
σ
kN 1/hσm2
φ1h
1.24
36.1
2
3.5⋅10-3
0.5
1.49⋅107
1.76
61.3
2
11⋅10-3
0.6
4.5
250
1.6
9.5⋅10-4
0.4
δ0 µm 6.26 (6.7) 4.14 (4.3) 1.32 (1.33)
6 7 8 9
Table 2 Diffusion-limited scenario. Conditions of experiments and kinetic parameters of CO2-hydrate formation from ice powders /Simulations by the POWDER-5 system. Previously inferred2-4 parameters are
given in parentheses.
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