Equilibrium and out-of-equilibrium (hysteretic) behavior of Ñuids in
disordered porous materials : Theoretical predictions
E. Kierlik, M. L. Rosinberg, G. Tarjus and P. Viot
L aboratoire de Physique T he orique des L iquides,¤ Universite Pierre et Marie Curie,
4 Place Jussieu, 75252 Paris Cedex 05, France
Received 26th October 2000, Accepted 19th January 2001
First published as an Advance Article on the web 2nd March 2001
We study the relation between out-of-equilibrium (hysteretic) and equilibrium behavior in the capillary
condensation of Ñuids in disordered mesoporous solids. Using mean-Ðeld density functional theory, we show
that a simple lattice-gas model can reproduce the major experimental observations and that the classical van
der Waals picture of metastability fails due the appearance of many metastable states. We Ðnd that (i) a true
equilibrium capillary phase transition may occur when the perturbation induced by the solid is sufficiently
small ; (ii) hysteresis does not necessarily imply the existence of this phase transition ; (iii) the disappearance of
the hysteresis loop is not associated with capillary criticality ; and (iv) thermodynamic consistency is violated
along the adsorptionÈdesorption isotherms.
I. Introduction
Low-temperature adsorption isotherms of gases in mesoporous adsorbents are often characterized by a steep increase
in the mass adsorbed at some pressure below the saturated
vapor pressure of the bulk gas. This phenomenon, known as
capillary condensation, is usually accompanied by a pronounced hysteresis loop that shrinks and eventually disappears at some temperature T less than T , the bulk critical
h
c
temperature. Hysteresis loops are reproducible, with a shape
that depends on the speciÐcity of the pore structure, and they
are widely used for characterizing porous solids.1 The origin
of this phenomenon, however, remains controversial, and
neither theories that relate hysteresis to the existence of metastable states in a single pore nor those which focus on
network e†ects (e.g., pore blocking) can account for the experimentally observed behavior.2 Actually, the very concept of
capillary condensation is ambiguous as it is used to describe
both the steep but continuous Ðlling of the pore space that is
observed in real amorphous solids like Vycor, controlled pore
glasses or silica gels, and the discontinuous jump that is predicted to occur in a single pore of simple and regular shape
below the so-called capillary critical temperature T . In the
cc
latter case, capillary condensation is a genuine Ðrst-order
phase transition that corresponds to a shift of the bulk gasÈ
liquid transition induced by conÐnement, and the adsorbed
Ñuid has a well deÐned coexistence curve below T .3 Morecc
over, in a single pore, hysteresis can occur only below T , as
cc
described by a classical (mean-Ðeld) scenario in which the
metastable portions of the adsorption and desorption
branches correspond to the two local minima of the grand
potential. In real systems, the relation between T and T is
h
cc
more elusive because one cannot identify T with the disapcc
pearance of a vertical jump in the adsorption isotherm.
Although one can try to use another criterion to extract some
““ crossover ÏÏ temperature from the experimental data,4h6 the
existence of a true capillary critical temperature is still an
open question that theory and/or simulation should help to
¤ The Laboratoire de Physique Theorique des Liquides is the UMR
7600 of the CNRS.
DOI : 10.1039/b008636n
elucidate. In other words, one would like to answer the following two questions : (i) Is there a genuine thermodynamic
(i.e., equilibrium) phase transition underlying capillary condensation in real porous solids (so that notions like capillary
phase diagram, capillary critical temperature, etc., take a
precise meaning) ? (ii) Is the hysteresis loop a signature of this
transition ?
In this paper, we try to shed some light on the problem by
studying a lattice-gas model that has been introduced some
time ago as a simple model of a Ñuid conÐned in a disordered
matrix.7 The use of a lattice model allows us to perform a
detailed investigation of the hysteretic behavior that would
not be feasible for the more realistic molecular models that
have been proposed in the recent literature and that focus on
speciÐc systems (see ref. 8 for a review). We believe, however,
that our model incorporates some of the essential physical
ingredients that characterize real systems, and this is conÐrmed a posteriori by the results presented here which exhibit
a number of important features observed in experiments. The
method we use is the mean-Ðeld density functional theory (in
its lattice version) that has proved a valuable technique for
studying adsorption phenomena in simple systems.3,8
II. Model and density functional theory
We consider a three-dimensional lattice where the N sites
(i \ 1. . .N) may be either empty or occupied by one matrix or
one Ñuid particle (this is of course a coarse-grained model and
the term ““ particle ÏÏ must not be taken literally). Matrix particles are distributed at random with an average density o
m
whereas Ñuid particles are allowed to thermalize in contact
with an external reservoir that Ðxes the temperature T and the
chemical potential k. To keep things simple, ÑuidÈÑuid and
matrixÈÑuid interactions act only between nearest neighbor
(n.n.) pairs and have strength [w and [w , respectively
ff
mf
(we only consider the case of attractive interactions). This
deÐnes the following Hamiltonian :
H \ [w
;qq gg
i j i j
WijX
[ w ; [q g (1 [ g ) ] q g (1 [ g )]
mf
i i
j
j j
i
WijX
ff
Phys. Chem. Chem. Phys., 2001, 3, 1201È1206
This journal is ( The Owner Societies 2001
(1)
1201
where q \ 1 if site i is occupied by a Ñuid particle and q \ 0 if
i
i
it is not, g \ 0 if site i is occupied by a matrix particle and
i
g \ 1 if it is not (for historical reasons related to the study of
i
the site-diluted Ising model,9 g is used in place of (1 [ g ), the
i
i
occupancy variable for matrix particles). The notation SijT
indicates that the sums are restricted to distinct n.n. pairs. In
this model, the properties of the adsorbentÈadsorbate system
are thus speciÐed by two parameters : the density o which
m
Ðxes the pore space accessible to the Ñuid (conÐnement is
replaced here by dilution) and the interaction ratio y \
w /w which determines the ““ wettability ÏÏ of the matrix.
mf ff
(There is also a dependence on the lattice structure ; the results
presented here correspond to the bcc lattice.) Increasing the
size of the matrix particles would certainly add more realism
to the model. Such a modiÐed version has been proposed
recently.10
In a previous study,11 the equilibrium properties of the
model were investigated in the framework of the meanspherical approximation, by using the replica method to
perform the average over the (quenched) matrix variables Mg N
i
and by choosing the energy route to calculate the thermodynamics. This study showed that the conÐned lattice Ñuid has a
Ðnite (i.e. non zero) capillary critical temperature T when o
cc
m
and y are not too large. Below T , there is coexistence
cc
between a gas-like and a liquid-like phase, and the capillary
phase diagram becomes more and more asymmetric and
shifted to higher densities on the vapor branch as y increases.
These results are conÐrmed by recent Monte-Carlo simulations : 12 for illustration see Fig. 1. This previous study,
however, does not provide insight into the out-of-equilibrium
(hysteretic) behavior of the model because such information is
lost in the average over disorder. SpeciÐcally, the metastable
states corresponding to the (two or three) minima of the
average Ñuid grand potential within the mean spherical
approximation are not related to the metastable states corresponding to the (numerous) local minima of the grand potential for a particular matrix realization. As we shall see below,
the free-energy landscape of the system is very complex and
the hysteretic behavior cannot be explained by the usual
single loop picture à la van der Waals.
In order to investigate this question, we propose a microscopic analysis based on the mean-Ðeld density functional
theory. For a given matrix realization deÐned by the set Mg N,
i
the grand potential function X (Mo N is expressed as
i
X(Mo N) \ k T ; [o ln o ] (g [ o )ln(g [ o )]
i
B
i
i
i
i
i
i
i
[ w ; o o [ w ; [o (1 [ g ) ] o (1 [ g )]
ff
i j
mf
i
j
j
i
WijX
WijX
[k ; o
(2)
i
i
where o \ Sq g T is the average Ñuid density at site i (note
i
i i
that the usual factor 1/2 in the mean Ðeld term w ;
oo
ff WijX i j
is absent because the sum runs over distinct n.n. pairs). Minimization with respect to the local Ñuid densities gives a set of
N nonlinear coupled equations
g
i
o\
i 1 ] e~bKk`&j@i*wffoj`wmf(1~gj)+L
(3)
where b \ 1/(k T ) and ; indicates that the sum over j is
B
j@i
restricted to n.n. sites of i. The overall density of the conÐned
Ñuid is then o \ (1/N) ; o . Inserting the solution of eqn. (3)
f
i i
into X(Mo N) yields the corresponding grand potential,
i
o
(4)
X \ k T ; g ln 1 [ i ] w ; o o
ff
i j
B
i
g
i
WijX
i
where we have used the property ; g ln g \ 0. These local
i i
i
mean-Ðeld equations can be also derived directly from the
original Hamiltonian, eqn. (1), by neglecting Ñuid density Ñuctuations beyond the linear term, as is done in ref. 10.
The conÐned Ñuid is in equilibrium with a bulk Ñuid whose
uniform density obulk satisÐes the mean-Ðeld equation of state
f
obulk
f
k \ k T ln
[ cw obulk
(5)
B
ff f
1 [ obulk
f
where c is the lattice coordination number (here c \ 8). Eqn.
(5) implies that the bulk gasÈliquid transition occurs on the
line k /w \ [c/2 with a critical point located at k T /w \
sat ff
B c ff
c/4 and oc \ 1/2.
f
Starting from an initial distribution of the o Ïs, the coupled
i
EulerÈLagrange equations, eqn. (3) have been solved by
simple iteration, o(t`1) \ f (Mo(t)N), with the strict convergence
i
j
requirement (1/N) ; (o(t`1) [ o(t))2 \ 10~14. As we are
i i
i
dealing with a system with quenched disorder, one should in
principle consider a large number of di†erent matrix realizations (the smaller the lattice size, the larger the number of
realizations) and average the results over these realizations. In
the present work, however, we have studied only a few typical
realizations, using a periodic lattice of size L \ 48 (N \
2L 3 \ 221 184). We expect that the main conclusions of this
paper will not be a†ected by the average over disorder. This
important issue is currently under investigation.13
A
B
III. Results and discussion
Fig. 1 Capillary phase diagram of the conÐned Ñuid for o \ 0.1
m
and y \ 0.6 calculated via the replica/mean-spherical approximation
(solid curve) and via Monte-Carlo (MC) simulations for 100 matrix
realizations on a bcc lattice of linear size L \ 16 (open symbols). The
MC estimate for the capillary critical temperature of the inÐnite
system is T \ 1.43 (the critical temperature of the bulk Ñuid is T \
cc
c
1.589).
1202
Phys. Chem. Chem. Phys., 2001, 3, 1201È1206
The results we shall present have been obtained for a matrix
density o \ 0.25 (i.e., a ““ porosity ÏÏ 1 [ o \ 0.75) and two
m
m
values of the interaction ratio, y \ 1 and y \ 1.5. It would be
worth performing a more systematic study of the inÑuence of
o and y on the behavior of the conÐned Ñuid, but these
m
typical cases will do for our present purpose.
Fig. 2 shows typical adsorptionÈdesorption isotherms
obtained by progressively increasing (respectively, decreasing)
the Ñuid activity j \ ebk from a state of low activity
(respectively, high) and using the Ðnal values of the local Ñuid
densities obtained as solution of eqn. (3) at each state as the
initial values for the next state. This procedure mimics what is
done experimentally. The theoretical hysteresis loops have the
same shape as those observed in experiments on porous
Fig. 2 Theoretical adsorption and desorption isotherms for y \ 1 at
T /T \ 0.60, 0.65, 0.70, 0.75 and for y \ 1.5 at T /T \ 0.50, 0.55, 0.6,
c
c
0.65. These isotherms are of type V and IV, respectively, in the
IUPAC classiÐcation.
see that the two diagrams have roughly the same asymmetric
shape (with a density on the ““ gas ÏÏ branch much higher than
in the bulk), that is again very similar to the one observed
experimentally. By reducing both the temperature and the
Ñuid density by their values at the maximum of the hysteresis
diagram, the points for y \ 1 and y \ 1.5 fall approximately
on a common curve (see Fig. 4), as it is the case with the real
data for CO and Xe adsorbed in Vycor.14
2
As pointed out by Ball and Evans,2 ““ any proposal for the
mechanism of hysteresis should account for the scanning
behavior observed experimentally ÏÏ. The scanning curves are
obtained by changing the sign of the evolution of k at di†erent
points along the adsorption and desorption branches. The
theoretical scanning curves shown in Fig. 5 bear a close
resemblance to those observed experimentally.20 In particular,
for a given k, the desorption scanning curves are steeper when
starting at lower points on the adsorption branch. Interestingly, the independent pore model, which is widely used to
measure pore size distribution in porous materials,1 incorrectly predicts the opposite trend.2,21
The results presented above show that the present approach
reproduces, at least qualitatively, the main experimental
glasses and silica gels (see for instance ref. 6, 14 and 15), with a
steep desorption branch and a smoothly increasing adsorption
branch : these loops belong to the H2 category in the 1985
IUPAC classiÐcation.16 Similar hysteresis loops have been
obtained in recent simulation studies of model heterogeneous
adsorbents.17h19 As the temperature is raised, the loops shrink
in size, and the shrinkage involves a decrease in both the
height and the width, in agreement with experimental observations. Fig. 2 also illustrates the inÑuence of a change in the
solidÈÑuid interaction on the adsorption isotherms : at a given
temperature, increasing y induces a narrowing of the hysteresis region. This implies that the temperature T at which the
h
hysteresis loop disappears is depressed as y increases. There is,
however, some ““ universality ÏÏ in this phenomenon. This can
be seen by constructing the so-called hysteresis phase diagram
of the conÐned Ñuid14 which is the locus of the upper and
lower limits of the hysteresis loops as a function of temperature. Fig. 3 shows the diagrams for y \ 1 and y \ 1.5,
together with the coexistence curve of the bulk Ñuid. One can
Fig. 4 Comparison of theoretical hysteresis phase diagrams for
y \ 1 and y \ 1.5 in reduced units.
Fig. 3 Theoretical hysteresis phase diagrams for y \ 1 and y \ 1.5.
(The apparent small ““ jumps ÏÏ in the curves are due to the uncertainty
in determining the upper and lower limits of the hysteresis loops.) The
top curve is the mean-Ðeld coexistence curve of the bulk Ñuid.
Fig. 5 Theoretical desorption (a) and adsorption (b) scanning curves
for y \ 1.5 at T /T \ 0.4.
c
Phys. Chem. Chem. Phys., 2001, 3, 1201È1206
1203
observations on capillary condensation in disordered porous
solids. This indicates that our simple lattice-gas model properly incorporates the main physical ingredients that characterize these systems : limitation of the space accessible to the
Ñuid, ““ wettability ÏÏ of the solid surface, connectivity of the
pore space, geometric and energetic disorder. Of course, activated processes due to thermal Ñuctuations are not taken into
account in the mean-Ðeld description, and in comparing theoretical hysteresis loops and scanning curves (which do not represent equilibrium states) to the experimental ones, we
implicitly assume that the system does not have time to equilibrate at constant T and k : adsorption and desorption are
then only driven by changes in the chemical potential and, on
practical time scales, the system moves from one metastable
local free-energy minimum to the nearest available one.
Whether or not this is true for all experimental systems and
existing observation techniques is still an open question. Let
us recall that these many metastable states are also
responsible for the hysteretic behavior observed in Monte
Carlo simulations which are carried out by the conventional
Metropolis method.17h19 When one uses more sophisticated
algorithms to overcome the local free-energy barriers at constant T and k, the hysteresis can be eliminated, or at least
signiÐcantly reduced, as illustrated in Fig. 6 for the present
model.12
To gain a better perspective on the Ñuid behavior in the
hysteresis region, we have looked for other solutions of the
mean-Ðeld equations, starting the iteration procedure for a
given k with di†erent initial conditions. For simplicity, we
have only considered initial conditions corresponding to
uniform Ðllings of the lattice, o(0) \ o(0) with 0 O o(0) O 1 (an
i
exhaustive quest would require to consider also non-uniform
Ðllings such as checkerboard and random conÐgurations, as
has been partly done in the case of the random Ðeld Ising
model,22 a model somewhat related to the one studied here).
Fig. 7(a) and (b) show the results of this search for y \ 1.5,
T /T \ 0.4 and y \ 1, T /T \ 0.5. One can see that there are
c
c
many solutions inside the hysteresis loops. (Note that the
loops are not Ðlled uniformly ; although we have considered
more than 100 di†erent values of o(0), some of them give the
same Ðnal conÐgurations.) The adsorption and desorption isotherms are found to coincide with the solutions obtained from
the two extreme initial conditions o(0) \ 0 and o(0) \ 1,
i
i
Fig. 6 Sorption curves for y \ 1 at w /(k T ) \ 1.3 (T /T \ 0.484)
ff a Bsingle matrix realization
c
obtained via Monte Carlo simulations for
on a lattice of size L \ 24, using the grand canonical Metropolis algorithm (crosses and circles) and a grand canonical ““ tempering ÏÏ method
which samples several chemical potentials at the same time (squares).
The latter method allows to reduce dramatically the relaxation time27
and then to equilibrate the system.
1204
Phys. Chem. Chem. Phys., 2001, 3, 1201È1206
Fig. 7 Multiple solutions obtained by solving eqn. (3) with di†erent
initial conditions o \ o(0) for each value of k : (a) y \ 1.5 and T /T \
c
0.4, (b) y \ 1 and Ti /T \ 0.5. The solid lines represent the equilibrium
c
isotherms obtained by connecting the states of lowest grand potential.
respectively (it is easy to show that all other solutions are
bound by these two extremal curves). The existence of many
solutions shows that the free-energy landscape of the system is
very complex, as it is often the case when disorder is present,
with a large number of metastable states inside the hysteresis
loops. The very existence of the scanning curves is a manifestation of these metastable states.
Within the (limited) set of solutions that have been
obtained, there is one solution which gives the lowest value of
X for given T and k. This may be not the global minimum
that corresponds to the true thermodynamic equilibrium, but
we think that it is a good approximation (see below). A preliminary study of size e†ects has shown that this minimum is
separated from the closest other minima by a di†erence in X
that increases with system size. Therefore, at the temperatures
studied, only one solution contributes to equilibrium properties. (On the other hand, the numerical study of the same
equations in the case of the random Ðeld Ising model22 suggests a more complicated behavior in the vicinity of the capillary critical point.) The resulting ““ equilibrium ÏÏ isotherms are
shown in Fig. 7(a) and (b). Two di†erent cases may occur
depending on the value of y : (i) the equilibrium isotherm is
smooth, and it remains smooth at all temperatures, which
means that there is no equilibrium phase transition (this is
what happens when the matrixÈÑuid interaction is too large as
is the case for y \ 1.5) ; (ii) a jump is present below some capillary critical temperature, indicating a true phase coexistence
between gas-like and liquid-like states, as is the case for y \ 1.
One thus concludes that hysteresis in the present model is not
a signature of an underlying phase transition, as it is often
stated. One may observe hysteresis with or without a phase
transition. Accordingly, the hysteresis phase diagram, which
describes out-of-equilibrium situations, has nothing to do with
the true capillary phase diagram, when the latter exists. In
particular, the disappearance of the hysteresis loop is not
associated with capillary criticality. Clearly, the classical
picture of a van der Waals loop with only two possible metastable states is irrelevant here. The independent-pore model is
also disqualiÐed as it cannot predict the existence of a sharp
equilibrium transition which is a collective e†ect involving
Ñuid molecules in a large number of interconnected pores.
More striking evidence for the failure of the van der Waals
picture of metastability in the present problem can be seen in
Fig. 8(a) and (b) where we show the violation of the Gibbs
adsorption isotherm (dX/dk) \ [No along the adsorption
T
f
Fig. 8 Check of thermodynamic consistency ((dX/dk) \ [No )
T
f
along the adsorption (a) and desorption (b) branches for y \ 1.5 at
T /T \ 0.4. Filled symbols : average Ñuid density obtained from the
c
solutions of eqn. (3). Open symbols : quantity obtained by di†erentiating the corresponding grand potentials.
Fig. 9 Same as Fig. 8 but for the equilibrium isotherm.
and desorption branches for y \ 1.5 at T /T \ 0.4. The
c
numerical derivative has been calculated with a step
*(k/w ) \ 0.02. One can see that there is a loss of thermodyff
namical consistency which comes from the fact that the
system often jumps from one metastable local minimum of the
grand potential to another along the two branches. In other
words, the function X(k) is not continuous everywhere and its
derivative is not well deÐned. As a result, one cannot use thermodynamic integration to locate the equilibrium capillary
transition, as is done in a single pore of ideal geometry.8 For
instance, thermodynamic integration predicts a capillary
coexistence curve for y \ 1.5 whereas the above analysis
shows that no equilibrium transition takes place in this case.
This questions the use of thermodynamic integration in recent
work on model heterogeneous adsorbents.23,24 On the other
hand, one expects to recover thermodynamic consistency
along the true equilibrium isotherm, as illustrated in Fig. 9
(this indicates a posteriori that our estimate of the global
minimum was reasonably accurate in spite of the use of a
limited set of solutions of the local mean-Ðeld equations).
Another important consequence of the above calculations is
that the capillary critical point, when it exists, is located at a
lower temperature than T , the temperature at which the hysh
teresis Ðrst appears (T \ T ). This result is in agreement with
cc
h
the experimental observations for magnetic systems in a
random Ðeld25 but it contradicts recent statements in the literature.4,6,8 We think that this is because the capillary critical
temperature empirically determined in disordered porous
materials represents a crossover point at which steep capillary
condensation ceases to exist rather than a true critical point.
This is illustrated in Fig. 10 where we use the procedure suggested in ref. 4 and 6 to analyze the theoretical sorption isotherms for y \ 1. On a logarithmic scale, we observe that the
positive deviation from a power law behavior disappears
around T \ 0.8 T , a temperature which is indeed higher than
c
T . But this is not the true capillary critical temperature at
h
which the jump in the equilibrium isotherm disappears. This
latter temperature is below T . Note however that the present
h
discussion may not apply to Ñuids adsorbed in disordered
solids of very high porosity like aerogels. In this case, the perturbation induced by the matrix may be small enough to
allow the system to reach equilibrium on the experimental
time scale. Indeed, there is evidence of a vaporÈliquid phase
separation that terminates at a genuine critical point for 4He
and N in aerogel of 95% porosity, but no sign of hysteresis.26
2
IV. Conclusion
The work presented in this paper is a Ðrst step towards understanding the relation between out-of-equilibrium (hysteretic)
and equilibrium behavior in the capillary condensation of
Ñuids in real disordered mesoporous solids. Using mean-Ðeld
density functional theory, we have shown that a simple latticegas model can reproduce the essential experimental observations and that the classical picture of metastability fails due
to the appearance of many metastable states. Our main Ðndings are that (i) a true equilibrium capillary phase transition
may occur when the perturbation induced by the solid is sufficiently small ; (ii) hysteresis does not necessarily imply the existence of this phase transition ; (iii) the disappearance of the
hysteresis loop is not associated with capillary criticality
(speciÐcally, one has T \ T when T exists) ; and (iv) thercc
h
cc
modynamic consistency is violated along the adsorptionÈ
desorption isotherms, which forbids the use of thermodynamic
integration to map out the capillary phase diagram.
Fig. 10 Logarithmic plot of the theoretical isotherms for y \ 1 at
T /T \ 0.8, 0.75, 0.70 and 0.65, (the curves are o†set for clarity). The
solidc lines represent a Ðt to a power law behavior in the intermediate
density region.
Acknowledgements
We are grateful to P. Monson for useful discussions and for
communicating his preprint10 prior to publication.
Phys. Chem. Chem. Phys., 2001, 3, 1201È1206
1205
References
1
2
3
4
5
6
7
8
9
10
11
12
13
14
S. J. Gregg and K. S. W. Sing, Adsorption, Surface Area and Porosity, Academic Press, London, 1982.
P. C. Ball and R. Evans, L angmuir, 1989, 5, 714.
R. Evans, J. Phys. : Condens. Matter, 1990, 2, 8989.
M. Thommes and G. H. Findenegg, L angmuir, 1994, 10, 4277.
K. Morishige and M. Shikimi, J. Chem. Phys., 1998, 108, 7821.
S. Gross, T. Michalski, A. Schreiber, M. Thommes and G. H.
Findenegg, Phys. Chem. Chem. Phys., 2001, 3 (DOI : 10.1039/
b010086m).
E. Pitard, M. L. Rosinberg, G. Stell and G. Tarjus, Phys. Rev.
L ett., 1995, 74, 4361.
L. D. Gelb, K. E. Gubbins, R. Radhakrishnan and M. SliwinskaBartkowiak, Rep. Prog. Phys., 1999, 62, 1573.
R. B. Stinchcombe, in Phase T ransitions and Critical Phenomena,
ed. C. Domb and J. L. Lebowitz, Academic Press, London, 1983,
vol. 7, p. 151.
L. Sarkisov and P. A. Monson, preprint, 2000.
E. Kierlik, M. L. Rosinberg, G. Tarjus and E. Pitard, Mol. Phys.,
1998, 95, 341.
P. Viot, unpublished results.
E. Kierlik, P. A. Monson, M. L. Rosinberg, L. Sarkisov and G.
Tarjus, in preparation.
C. G. V. Burgess, D. H. Everett and S. Nuttall, Pure Appl. Chem.,
1989, 61, 1845.
1206
Phys. Chem. Chem. Phys., 2001, 3, 1201È1206
15 W. D. Machin, L angmuir, 1994, 10, 1235.
16 K. S. W. Sing, D. H. Everett, R. A. Haul, L. Moscou, R. A.
Pierotti, J. Rouquerol and T. Siemieniewska, Pure Appl. Chem.,
1985, 57, 603.
17 L. Sarkisov, S. K. Page and P. A. Monson, in Fundamentals of
Adsorption 6, ed. F. Meunier, Elsevier, 1998, p. 847.
18 L. D. Gelb and K. E. Gubbins, L angmuir, 1998, 14, 2097.
19 R. Pellenq, S. Rodts, V. Pasquier, A. Delville and P. Levitz,
Adsorption, 2000, 6, 241.
20 A. J. Brown, PhD Thesis, University of Bristol, 1963.
21 D. H. Everett, in T he Solid-Gas Interface, ed. E. A. Flood, Marcel
Dekker, New York, 1967, vol. 2, p. 1055.
22 D. Lancaster, E. Marinari and G. Parisi, J. Phys. A, 1995, 28,
3959.
23 K. S. Page and P. A. Monson, Phys. Rev. E, 1996, 54, 6557.
24 R. Salazar, R. Toral and A. Chakrabarti, J. Sol Gel Sci. T echnol.,
1999, 15, 175.
25 D. S. Fisher, G. M. Grinstein and A. Khurana, Phys. T oday,
1988, 41(12), 58.
26 A. P. Y. Wong and M. H. W. Chan, Phys. Rev. L ett., 1990, 65,
2567 ; A. P. Y. Wong, S. B. Kim, W. I. Goldburg and M. H. W.
Chan, Phys. Rev. L ett., 1993, 70, 954.
27 Q. Yan and J. J. de Pablo, J. Chem. Phys., 1999, 111, 9509.