RESEARCH PAPER
a
b
PCCP
B. Kvamme,a A. Graue,a E. Aspenes,a T. Kuznetsova,a L. Gránásy,b G. Tóth,b T. Pusztaib and
G. Tegzeb
www.rsc.org/pccp
Kinetics of solid hydrate formation by carbon dioxide: Phase field
theory of hydrate nucleation and magnetic resonance imagingy
Department of Physics, University of Bergen, Allégt. 55, 5007 Bergen, Norway
Research Institute for Solid State Physics and Optics, H-1525 Budapest, POB 49, Hungary
Received 15th September 2003, Accepted 14th November 2003
F|rst published as an Advance Article on the web 8th December 2003
In the course of developing a general kinetic model of hydrate formation/reaction that can be used to establish/
optimize technologies for the exploitation of hydrate reservoirs, two aspects of CO2 hydrate formation have
been studied. (i) We developed a phase field theory for describing the nucleation of CO2 hydrate in aqueous
solutions. The accuracy of the model has been demonstrated on the hard-sphere model system, for which all
information needed to calculate the height of the nucleation barrier is known accurately. It has been shown that
the phase field theory is considerably more accurate than the sharp-interface droplet model of the classical
nucleation theory. Starting from realistic estimates for the thermodynamic and interfacial properties, we have
shown that under typical conditions of CO2 formation, the size of the critical fluctuations (nuclei) is comparable
to the interface thickness, implying that the droplet model should be rather inaccurate. Indeed the phase field
theory predicts considerably smaller height for the nucleation barrier than the classical approach. (ii) In order to
provide accurate transformation rates to test the kinetic model under development, we applied magnetic
resonance imaging to monitor hydrate phase transitions in porous media under realistic conditions. The
mechanism of natural gas hydrate conversion to CO2-hydrate implies storage potential for CO2 in natural gas
hydrate reservoirs, with the additional benefit of methane production. We present the transformation rates for
the relevant processes (hydrate formation, dissociation and recovery).
DOI: 10.1039/b311202k
I.
Introduction
The worldwide amounts of carbon bound in gas hydrates are
conservatively estimated to total twice the amount of carbon
to be found in all known fossil fuels on Earth.1 Assuming
the lattice is filled to the maximum capacity, dissociation of
1 m3 gas hydrate may release about 164 m3 methane under
standard temperature and pressure (STP) condition.2
CO2 hydrate is significantly more stable thermodynamically
than methane hydrate. Storage of CO2 in hydrate reservoirs
through replacement of natural gas is therefore considered as
an interesting option for safe long terms storage of CO2 , which
can be economically feasible due to the produced natural gas.
The efficiency of any exploitation strategy based on CO2
depends on the composite dynamics of the system, where
knowledge of the kinetics of hydrate reformation is crucial.
Storage of CO2 in aquifers is another option for reducing
CO2 emissions to the atmosphere. This option is already in
use outside the coast of Norway, where CO2 from the Sleipner
field is being injected into the Utsira formation. Similar storage
option may be used outside the northern part of Norway. In
these regions the seabed temperature may reach as low as
1 C and there may be regions of the reservoirs beneath that
are inside the hydrate stability zones. Therefore, hydrate may
form homogeneously from dissolved CO2 or at the interface
between free CO2 and water.
Development/optimization of the related technologies
requires a detailed understanding of the processes involved, a
knowledge that can be gained by combined experimenting
and modeling. Two essential ingredients of such an approach
are the collection of reliable experimental information on
the transition rates, and the development of appropriate
y Presented at the 3rd International Workshop on Global Phase
Diagrams, Odessa, Ukraine, September 14–19, 2003.
theoretical tools that can describe all stages of the processes
involved. For example, one of the least understood steps of
hydrate formation is the early nucleation stage, in which nanometer sized hydrate crystallites form via thermal fluctuations.
Accordingly, in this paper, we apply two modern
approaches to study the hydrate phase transitions involved
in the storage of CO2 in reservoirs. First, we present a model
for the nucleation of CO2 hydrate under conditions specific
to such reservoirs. In this we rely on the phase field theory
(PFT); one of the most potent methods developed recently to
model solidification in binary, ternary and multi-component
melts. Over the past decade it has demonstrated its ability to
describe complex solidification morphologies3 including thermal and solutal dendrites4–7 and eutectic/peritectic fronts.8–12
In addition, we demonstrate the accuracy of our approach on
the hard-sphere system, for which all essential properties are
well known, making it an ideal testing ground for theory.
Secondly, we apply magnetic resonance imaging13 (MRI) to
monitor hydrate phase transitions. We present experimental
results for the formation and dissociation of CO2 hydrate in
real porous media.
II. Nucleation theory
The freezing of homogeneous undercooled liquids starts with
the formation of heterophase fluctuations whose central part
shows crystal-like atomic arrangement. Those heterophase
fluctuations that exceed a critical size (determined by the interplay of the interfacial and volumetric contributions to the cluster free energy) have a good chance to reach macroscopic
dimensions, while the smaller ones decay with a high probability. (Heterophase fluctuations of the critical size are called
nuclei.) The description of the near-critical fluctuations is problematic even in single component systems. The main difficulty
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Phys. Chem. Chem. Phys., 2004, 6, 2327–2334
2327
is that the typical size of the critical fluctuations, forming on
the human time scale, is about 1 nm, comparable to the thickness of the crystal-liquid interface, which in turn extends to
several molecular layers.14 Therefore, the droplet model of
the classical nucleation theory, which relies on a sharp interface and bulk crystal properties, is inappropriate for such fluctuations. Field theoretic models, that predict a diffuse
interface, offer a natural way to handle such a situation.15
For example, in recent works, the phase field theory has been
shown to describe such fluctuations quantitatively.16,17
center of the fluctuations. Under such conditions, the
Lagrange multiplier can be identified as l ¼ (@ c/@ w)r!1 .
Assuming spherical symmetry, which is reasonable considering the low anisotropy of the crystal– liquid interface at small
undercoolings, the EL equations take the following form:
(
)
d2 m 2 dm
2
e T
¼ wTg0 ðmÞ þ p0 ðmÞf fL fS g;
þ
ð3aÞ
dr2
r dr
and
dfS
dfL dfL þ pðmÞ
:
0 ¼ wTgðmÞ þ ½1 pðmÞ
dw
dw
dw r!1
ð3bÞ
II.1 Phase field theory of nuclei
Our starting point is the standard phase field theory of binary
alloys as developed by several authors.3,18 In the present
approach, the local state of the matter is characterized by
two fields; a structural order parameter, m, called the phase
field, that describes the transition between the disordered
liquid and ordered crystalline structures, and a conserved field,
w, which may be either the coarse-grained density, r, or the
solute concentration, c.
The structural order parameter can be viewed as the Fourier
amplitude of the dominant density wave of the time averaged
singlet density in the solid. As pointed out by Shen and
Oxtoby19 if the density peaks in the solid can be well approximated by Gaussians placed at the atomic sites, all Fourier
amplitudes can be expressed uniquely in terms of the amplitude of the dominant wave, thus a single structural order parameter suffices. Here we take m ¼ 0 in the solid and m ¼ 1 in
the liquid. We assume mass conservation, which implies that
the integral of the conservative fields over volume is a
constant.
The Helmholtz free energy of the system is a functional of
these fields:
Z
F ¼ d3 rf12 e2 TðHmÞ2 þ f ðm; wÞg
ð1Þ
where e is a constant, T is the temperature, and f (m,w)
is the local free energy density. The first term on the right
hand side is responsible for the appearance of the diffuse
interface. The local free energy density has the form
f (m,w) ¼ wTg(m) þ [1 p(m)] fS(w) þ p(m) fL(w), where the
‘‘ double well ’’ and ‘‘ interpolation ’’ functions have the forms
g(m) ¼ 14 m2(1 m)2 and p(m) ¼ m3(10 15m þ 6m2), respectively, that emerge from the thermodynamically consistent formulation of the PFT,20 w is the free energy scale, while the
Helmholtz free energy densities of the homogeneous solid
and liquid, fS and fL , depend on the local value of the conservative field. These relationships result in a free energy surface
that has two minima, whose relative depth depends on the
deviation from equilibrium.
Being in unstable equilibrium, the critical fluctuation (the
nucleus) can be found as an extremum of this free energy functional,18–20 subject to the solute conservation constraint discussed above. To impose this constraint one adds the volume
integral over the conserved
field times a Lagrange multiplier,
R
l, to the free energy: l d3r w(r). The field distributions, that
extremize the free energy, have to obey the appropriate
Euler–Lagrange (EL) equations, which in the case of such a
local functional take the form
dF dc
dc
¼
¼ H
¼ 0;
dw
dw
dHw
ð2Þ
where dF/dw stands for the first functional derivative of the
Helmholtz free energy with respect to the field w, while c is
the total free energy density. The EL equations have to be
solved assuming that unperturbed liquid exists in the far field,
while, for symmetry reasons, zero field gradients exist at the
2328
Phys. Chem. Chem. Phys., 2004, 6, 2327–2334
0
Here stands for differentiation with respect to the argument of
the function. The last term in eqn. (3b) originates from the
Lagrange multiplier. Since the right hand side of eqn. (3b) is
a function of fields w and m, it provides the implicit relationship w ¼ w(m). Accordingly, eqn. (3a) is an ordinary differential equation for m(r). This equation has been solved here
numerically using a fourth order Runge–Kutta method. Since
m and dm/dr are fixed at different locations, the central value
of m that satisfies m ! m1 ¼ 1 for r ! 1, has been determined iteratively. Having determined the solutions m(r) and
w(r), the work of formation of the nucleus W* can be obtained
by inserting the solution into the free energy functional. Provided that the bulk free energy densities, fS(w) and fL(w), are
known, the only model parameters, we need to fix to evaluate
W*, are w and e. These model parameters are related to the
interface thickness and the interfacial free energy,6,21 thus these
quantities can be used to determine w and e, and calculate W*
without adjustable parameters.
The steady state nucleation rate (number of nuclei formed in
unit volume and time), JSS , can be calculated as
J SS ¼ J 0 expfW =kTg;
using the classical nucleation prefactor,
mentally on oxide glasses.23
II.2
22
ð4Þ
J0 , verified experi-
Classical droplet model (CDM)
For comparison with our non-classical approach, we calculate
the height of the nucleation barrier using the sharp interface
droplet model of the classical nucleation theory. In this
approach, the free energy of heterophase fluctuations of radius
R is given as WCDM ¼ (4p/3)R3Dg þ 4pR2G1 , where
Dg ¼ r(mL mS) is the volumetric Gibbs free energy difference
between the bulk liquid and solid, mL and mS are the chemical
potential of the liquid and the solid, and G1 is the free
energy of the equilibrium planar interface. Then, the free
energy of the critical fluctuation of radius R*CDM ¼ 2G1/Dg
is WCDM ¼ (16p/3)G31 /Dg2. This approach is expected to be
accurate if only the interface thickness is negligible with respect
to the size of the critical fluctuation.
II.3
Material properties
We apply these models for crystal nucleation in the hardsphere system, and for CO2 hydrate nucleation in aqueous
CO2 solutions.
In the case of the hard-sphere system, the virtually exact
equations of state for the solid and liquid phases24 deduced
from molecular dynamics (MD) simulations have been integrated to obtain fS(r) and fL(r). Owing to the complex form
of these functions, eqn. (3b) could only be inverted numerically. The interface thickness for the phase field was evaluated
from the cross-interfacial variation of the height of the singlet
density peaks determined by MD simulations.25,26 The model
parameters were fixed in equilibrium so that the free energy27
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and thickness of the (111), (110), and (100) interfaces from
molecular dynamics were recovered. Considering that due to
the small anisotropy the equilibrium shape is expected to be
nearly spherical, we also performed calculations with an average of the interface free energy over the known orientations.
In the case of hydrate nucleation, the molar Gibbs free
energy of the aqueous CO2 solution has been calculated as
GL ¼ (1 c)GL,W þ cGL,CO2 , where c is the mole fraction of
CO2 . The partial molar Gibbs free energy of water in solution
has been obtained as GL,W ¼ G0L;W þ RT ln[(1 c)gL,W(c)],
where the free energy of pure water has been calculated as
0
¼
GL;W
3
X
ki
i¼0
Ti
;
ð5Þ
with coefficients ki taken from ref. 28. Here R is the gas constant and gL,W is the activity coefficient of water in solution.
The partial molar free energy of CO2 in solution is
GL,CO2 ¼ G1
L;CO2 þ RT ln[cgL,CO2(c)], where the molar free
energy of CO2 at infinite dilution, G1
L;CO2 ¼ 19.67 kJ/mol,
has been taken from molecular dynamics simulations.29 The
temperature and pressure dependent activity coefficient of
CO2 in aqueous solution deduced from CO2 solubility experiments of Stewart and Munjal30 have been fitted using the form
X5
ln gL;CO2 ¼
a ðTÞ½ln xi ;
ð6Þ
i¼0 i
with ai(T ) given by third order polynomials. The activity coefficient of water, gL,W in aqueous solution has been obtained
from eqn. (6) via the Gibbs–Duhem relationship.
The Gibbs free energy of the hydrate is given by
GS ¼ (1 c)GS,W þ cGS,CO2 . Owing to the lack of experimental information, the partial molar quantities have been calculated using the model described in ref. 28. For water and
CO2 we use the relationships GS,W ¼ G0S;W þ RT(3/23)ln(1 y),
), and GS,CO2 ¼ Ginc
S;CO þ RT ln[y/(1 y)], respectively, where the
hole occupancy is y ¼ c/(3/23). Here, the partial molar Gibbs
free energies of the empty clathrate, G0S;W , and that of guest
inclusion, Ginc
S;CO , are given by eqn. (6), with the appropriate
ki taken from ref. 28.
Following other authors,31 we approximate the free energy
of the hydrate–solution interface by that of the ice–water interface, taken from the work of Hardy, 29.1 0.8 mJ m2.32
Owing to a lack of information on the CO2 hydrate/aqueous
solution interface, we use the 10%–90% interface thickness, d,
(the distance on which the phase field changes between 0.1
and 0.9) as an adjustable parameter in the calculations. Molecular dynamics simulations on other clathrate hydrates indicate
that the full interface thickness is about 2–3 nm,33 that corresponds to roughly d 1–1.5 nm. Assuming that similar values
apply for the CO2 hydrate, we vary d in the 0.5–1.5 nm range.
The computations were performed under conditions typical
for the seabed reservoirs, i.e. T ¼ 274 K, p ¼ 15 MPa ( 1500
m depth), furthermore, we assume that water has been saturated by CO2 (c ¼ 0.033, obtained by extrapolating the relevant data by Teng and Yamasaki34). These experimental
data are for synthetic average seawater. The salinity of groundwater in reservoirs may vary from close to zero up to seawater
salinity in regions where the penetration of seawater dominates
the salinity.
III. Experimental
Monitoring of hydrate phase transitions in real porous media
(a sandstone core plug) has been performed using Magnetic
Resonance Imaging.13 The experimental setup is shown schematically in Fig. 1. The core length is 9.4 cm and the core diameter is 1.5 cm. Porosity was measured at 22.1% and the brine
permeability at 1050 mD.
Fig. 1 The core plug inside the MRI tomograph (left) has been
cooled using a coolant fluid (right) that controls the temperature with
an accuracy of 0.1 C. The arrows show the flow of the fluids (coolant,
confinement fluid for the pressure vessel and CO2).
The core was saturated with brine, placed inside the core
holder in the MRI tomograph and pressurized applying 80
bars confinement pressure. In an attempt to reproduce conditions that are, or would be, present in nature, i.e. gas with
water present, a gas-water displacement process was initiated.
A liquid gas phase, CO2 at 63 bars and 25 C, was introduced
at one end of the 100% brine saturated core sample. The liquid
gas-water displacement was imaged. The core holder was then
cooled to 2.0 C and hydrate began to form. Imaging the
change in the free water saturation as function time provides
an indirect measure of the hydrate formation and disassociation during temperature variations.
IV. Results and discussion
IV.1 Nucleation
A. Hard-sphere system. Fixing the model parameters e and
w so that the free energy and the 10%–90% thickness of the
equilibrium interfaces are recovered, the grand potential surface (Fig. 2), and the phase field and coarse-grained density
profiles for the equilibrium solid-liquid interface have been
determined. The profiles predicted for the (111) interface are
Fig. 2 Grand potential density surface (o ¼ f mLr) for the hardsphere system in equilibrium, evaluated with the free energy scale w
corresponding to the (111) interface. (m–phase field, f–volume fraction.) Note, that the minima at (f ¼ 0.494, m ¼ 1) and (f ¼ 0.546,
m ¼ 0) correspond to the equilibrium liquid and solid (fcc) phases.
The calculation has been performed for a colloidal suspension of
hard-sphere diameter s ¼ 890 nm, a particle size comparable to that
in many experiments. The small value of the grand potential density
follows from the large ‘‘ molecular ’’ diameter. (T ¼ 293 K.)
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Phys. Chem. Chem. Phys., 2004, 6, 2327–2334
2329
compared with their counterparts from MD simulations in
Figs. 3. For the sake of this comparison, all quantities are presented in a normalized form, (X Xmin)/(Xmax Xmin), where
X is either the density peak height, filtered density, 1 m, or
the coarse-grained density, while Xmin and Xmax are the minimum and maximum values of these quantities. For the (111)
and (100) interfaces, and to a smaller extent for the (110) interface, the structural order parameter profiles are in a good
agreement with the cross-interfacial variation of the density
peak height from MD. Note that while d has been fitted, the
agreement depends also on the shape/symmetry of the predicted profile. It appears that the nearly symmetric structural
order parameter profile of the PFT approximates reasonably
well the behavior seen in the simulations. Remarkably, the
density functional theories by Shen and Oxtoby,19 and
Gránásy and Pusztai35 derived for the fcc structure predict significantly more asymmetric structural order parameter profiles
for the crystal-liquid interfaces, which are sharper on the crystal side and have a long tail on the liquid side. Therefore, their
match to the simulated profiles is apparently less satisfactory.
It is an intriguing question why the quartic form of g(m), that
is consistent with the symmetries of the base centered cubic
(bcc) structure,36 leads to a better fit to the simulation profile
then those obtained by using free energy functionals consistent
with the fcc symmetries. A distinct possibility is the presence of
a bcc-like layer at the interface as reported in the case of the
Lennard-Jones system.37
Summarizing, with model parameters fixed as described
above, the PFT reproduces exactly the thermodynamics of
the bulk phases, the interface free energy, and quite reasonably
both the structural order parameter profile and the density
profile. Therefore, we may calculate the height of the nucleation barrier with some confidence.
The reduced nucleation barrier heights, W*/kT, calculated
with e and w that fit to the properties of the (111) and (110)
interfaces and to the average interface properties are shown
in Fig. 4 for the phase field theory. For comparison, the predictions of the classical nucleation theory and the exact results
from Monte Carlo simulations38 are also presented. The
W*/kT vs. initial liquid volume fraction curves predicted by
the PFT using the properties of the (111) and (110) interfaces
envelope the MC simulations, while the predictions with the
average interface properties fall close to the MC results. (The
initial volume fraction of the liquid is defined as
f1 ¼ ps3r1/6, where r1 is the initial density of the liquid.)
Fig. 3 Reduced interfacial profiles for the (111) interface. (Solid
lines: predictions of the phase field theory; heavy solid line: structural
order parameter, 1 m; light solid line: density. Dashed lines: results
from molecular dynamics;25,26 heavy dashed line: height of singlet
density peaks; light dashed line: filtered density profile. Note that
the heavy lines should be compared to each other, as the light ones.)
2330
Phys. Chem. Chem. Phys., 2004, 6, 2327–2334
Fig. 4 Reduced nucleation barrier height vs. volume fraction of the
initial liquid. [Upper three lines (solid): phase field theory (PFT); lower
three lines (dashed/dotted): classical droplet model–CDM.] Within
these triplets of lines for PFT and CDM, the upper and lower curves
were calculated using the physical properties of the (110) and (111)
interfaces, respectively (as indicated by caption in the figure). The prediction for the (100) interface (not shown here) falls slightly below the
results obtained with the average interface properties (heavy lines). The
results of Monte Carlo simulations38 are denoted by triangles.
The radial order parameter and density profiles for the critical fluctuations (nuclei) corresponding to the volume fractions
used in MC simulations of ref. 38 are shown in Fig. 5. Where
available (f1 ¼ 0.5207) these profiles are in a reasonable
agreement with the size of the critical fluctuation from MC
simulation (cf. our Fig. 5 and Fig. 3 of ref. 38). Note the lack
of bulk crystalline properties even at the centers of the critical
fluctuations. As a result, the classical droplet model significantly underestimates the height of the nucleation barrier for
all orientations. Since the nucleation rate is proportional to
exp(W*/kT ), these differences amount to orders of magnitude. For example, the CDM calculation made with the average interface free energy overestimates the nucleation rate by
three to five orders of magnitude.
These findings for the hard-sphere model system indicate
that our non-classical approach is considerably more accurate
Fig. 5 Radial 1 m profiles (solid lines) and reduced volume fraction
profiles (dashed lines) for critical fluctuations that correspond to the
initial liquid volume fractions f1 ¼ 0.5207, 0.5277, and 0.5343.
The PFT calculations were performed with e and w corresponding to
the average interface properties (G1s2/kT ¼ 0.6133 and d/s ¼ 5.410).
Here fS ( ¼ 0.5796, 0.5875, and 0.5948, respectively) is the volume
fraction of the solid that is in mechanical equilibrium with the initial
liquid (the crystal that has the same pressure as the liquid of volume
fraction f1).
This journal is Q The Owner Societies 2004
than the classical droplet model, a conclusion supported
by earlier evidence on the Lennard-Jones and ice–water
systems.16,17
B. Formation of CO2 hydrate in aqueous solution. The free
energy surface, corresponding to e and w that reproduce the
ice-water interface free energy and a 10%–90% thickness of 1
nm at T ¼ 274 K, is shown in Fig. 6. The radial structural
order parameter and concentration profiles of the critical fluctuations forming in saturated aqueous CO2 solution
(c ¼ 0.033) at the same temperature are shown in Fig. 7 as a
function of the interface thickness, d. Remarkably, at the realistic interface thickness (d ¼ 1.0 to 1.5 nm) the bulk crystalline
structure is not yet established even at the center of the
nucleus, indicating that the nucleus is softer (the molecules
have larger amplitude of oscillations around the crystal sites)
than the bulk crystal. Despite these, we have almost full
hole-occupancy in the central part, c ¼ 0.1235 or y ¼ 0.946.
Furthermore, the interface thickness for the concentration field
is far sharper than for structure. It extends to only a few Å,
which is consistent with the picture that the nucleus is a small
piece of hydrate crystal embedded into the solution, however,
built of somewhat distorted H2O cages seen at the interface in
MD simulations with realistic potentials39 (Fig. 8). It is
remarkable, that the interface for the solute falls close to the
classical radius RCDM ¼ 1.76 nm.
In agreement with previous studies on other systems,16,17 a
substantial difference can be seen between predictions made
for the height of the nucleation barrier by the phase field theory and the classical droplet model (Table 1). This difference
emerges from the fact that the interface thickness is comparable to the size of the critical fluctuation. As one should expect,
for decreasing interface thickness, the PFT prediction for the
height of the nucleation barrier converges to that by the sharp
interface CDM (WPFT ! WCDM for d ! 0; see Fig. 9), indicating the coherency of our results. These findings imply that,
similarly to many other systems, in hydrate forming systems
the classical droplet model of crystal nuclei is rather inaccurate
and should be replaced by more advanced approaches such as
the phase field theory.
We note finally, that the predicted W* is rather sensitive to
the values of G1 and d used to fix the model parameters e and
w. Thus, accurate results may only be expected if these input
properties are known with a high accuracy. Unfortunately,
none of these input properties is available for the CO2
hydrate/solution interface (the values used here and in other
Fig. 6 Helmoltz free energy density surface for the CO2 hydrate system at 274 K, evaluated with free energy scale w corresponding d ¼ 1
nm and G1 ¼ 29.1 0.8 mJ m2. (m: phase field, c: mole fraction of
CO2 .)
Fig. 7 Radial 1 m profiles (solid lines) and CO2 concentration profiles (dashed lines) for critical fluctuations calculated at T ¼ 274 K, for
10%–90% interface thickness d ¼ (a) 0.5, (b) 1.0 and (c) 1.5 nm. For
comparison the critical radius from the CDM is also shown (dotted
lines). Note that with increasing interface thickness the difference
between the PFT and CDM predictions for the height of the nucleation
barrier increases (see Table 1).
work31 are only rough estimates). Therefore, further effort is
needed to establish accurate nucleation rates.
The experimental determination of the input properties is
susceptible to errors. In the case of transparent substances,
meticulous experiments using the grain boundary groove technique may reduce the error of the interfacial free energy to
3%.32 In ideal case, a comparable error can be established
for the thermodynamic properties. These add up to 15%
error in W*, corresponding to about 2.5 orders of magnitude
in the nucleation rate at W*/kT ¼ 40. Another possibility is
to evaluate the interfacial and thermodynamic properties from
MD simulations25,27,40 with realistic interaction potentials.
Then, in principle, virtually exact thermodynamic properties
can be used (for example, for the hard sphere fluid the first
eight virial coefficients have been determined), and the interface free energy can be evaluated with an error as low as
1.5 2%.27 Accordingly, the overall uncertainty of W* is
5 6%, which corresponds to about an order of magnitude
uncertainty in the nucleation rate at W*/kT ¼ 40. This
appears to be the maximum precision one may achieve for
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Phys. Chem. Chem. Phys., 2004, 6, 2327–2334
2331
Fig. 8 Snapshot of the water–CO2 hydrate interface in molecular dynamics simulation (top). Note the regular clathrate cages inside the solid (on
the right) and the distorted cages at the interface (on the left). For comparison, the ideal crystal structure (bottom left), and the real crystal with
thermal fluctuations (bottom right) from MD simulations are also shown. We used SPC water and EPM2 CO2 potentials.39 The diameter of the
H2O cages of circular projection (tetrakaidecahedra) is 0.866 nm. The rods at the center of the cages denote the CO2 molecules. [These illustrations
have been made using the Visual Molecular Dynamics package.41]
model potentials. The accuracy, however, will depend on how
realistic the applied intermolecular potential is. Work is underway to determine the thermodynamic and interfacial properties of CO2 hydrate from MD simulations with realistic
intermolecular potentials.
IV.2 Magnetic resonance imaging
The MRI intensity profile of water along the length of the partially water saturated sandstone core plug is shown in Fig. 10
as a function of time. Liquid CO2has been injected into the
Table 1 Free energy of critical fluctuations (W*) vs. the interface
thickness in the CO2 hydrate system, as predicted by the phase field
theory (PFT) and the classical droplet model (CDM) at 274 K and
c ¼ 0.033
2332
d (nm)
1019W PFT /J
1019W CDM /J
0.5
1.0
1.5
3.18
2.66
2.23
3.76
Phys. Chem. Chem. Phys., 2004, 6, 2327–2334
Fig. 9 Height of the nucleation barrier for CO2 hydrate formation as
a function of interface thickness in the phase field theory (squares). For
comparison, the prediction of the classical droplet model that assumes
a sharp interface (d ¼ 0) is also presented (circle). Note that for d ! 0
the PFT results converge to the value predicted by the CDM (dashed
line: second order polynomial fitted to the PFT results).
This journal is Q The Owner Societies 2004
Fig. 10 Formation of CO2 hydrate along the core as function of
time.
plug at the face being at the left-hand side of this plot. The
water saturation distribution, after 0.5 PV of liquid CO2 displaced the water is shown by the upper saturation profile. At
static conditions, the temperature was then lowered to 2 C.
The downward arrow shows the trend of intensity variation
with time. Decreasing intensity indicates the formation of
hydrate since the MRI parameters were adjusted to image only
the liquid hydrate precursors. Hydrate formation was indicated by a measurable decrease in the amount of free water
and the simultaneous decrease in permeability with time. The
noticeable difference in the change of the MRI intensity along
the length of the plug suggests that the hydrate formation was
primarily limited by the amount of CO2 available, i.e. the
greatest change was near the CO2 inlet face and the smallest
of change was at the far end of the plug. The permeability
decreased to zero as hydrate formed, implying that selected,
but not readily detected, areas were completely filled with
hydrate before all of the pore volume was filled.
A sequence of two-dimensional (2D) longitudinal images of
a 4 mm thick slice along the center of the core plug during
hydrate formation is shown in Fig. 11. These images imply that
the hydrate primarily formed (areas of reduced intensity) near
the CO2 inlet end of the plug, the right-hand side of these
images. The hydrate formation process continued over a relatively long period of time, and the growth from right to the left
is consistent with the initial nucleation of hydrate followed by
the growth of hydrate crystals. These 2D images show that
hydrate formed as a uniform band across the plug, rather than
as separate, discrete clumps, again consistent with nucleationinitiated hydrate growth from the right-hand side of the plug.
V. Summary
We reported on advances made in two research areas related
to the formation/dissolution of CO2 hydrate in aqueous
solutions:
A theoretical development by us, based on the phase field
theory of Gránásy et al.,16,17 addresses the nucleation of CO2
hydrate in aqueous solution. First, we demonstrated for the
hard-sphere system that this model is able to predict the
nucleation rate quantitatively without adjustable parameters.
In contrast, the classical droplet model, used widely, predicts
nucleation rates that are orders of magnitude too high. Therefore, we adapted the phase field theory to describe homogeneous nucleation of hydrate from dissolved CO2 in aqueous
solution. It has been demonstrated under typical conditions
that the thickness of the hydrate-solution interface is comparable with the size of nuclei, implying that the classical droplet
model might be inaccurate. Indeed, as found for other systems,
the nucleation barriers predicted by the phase field theory and
Fig. 11 A sequence of two-dimensional images that shows the formation of CO2 hydrate as function of time. The vertical axis displays position along the core height in cm. The horizontal axis is the length of the
core plug. Top: when cooled to 2 C. Center: After 30 min at 2 C.
Lower: After 8 h at 2 C. Gray scale indicates fraction of water not
bound in hydrate.
the classical droplet model differ considerably. Apparently,
advanced models are needed to evaluate the rate of hydrate
nucleation accurately. To improve the accuracy of predictions,
work has been started to estimate the interface free energy and
the width of the interface from molecular dynamics simulations for this system.
On the experimental side, we demonstrated the feasibility of
magnetic resonance imaging as a tool for monitoring phase
transitions involving hydrate in real porous media.
Acknowledgements
We thank R. L. Davidchack and B. B. Laird for communicating us their new high-resolution density data for the (111),
This journal is Q The Owner Societies 2004
Phys. Chem. Chem. Phys., 2004, 6, 2327–2334
2333
(110), and (100) interfaces in the hard-sphere system prior to
publication. This work has been supported by the Norwegian
Research Council under project Nos. 153213/432 and 151400/
210, by the ESA under Prodex Contract No. 14613/00/NL/
SFe, and by the Hungarian Academy of Sciences under contract No. OTKA-T-037323. T. P. acknowledges support by
the Bolyai János Scholarship of the Hungarian Academy of
Sciences.
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