The Ins and Outs of Capillary Condensation in
Cylindrical Pores
Lev D. Gelb
Florida State University,
Department of Chemistry,
Tallahassee, FL 32306-4390 USA
January 6, 2002
1
Abstract
We have performed a series of simulations of adsorption and desorption of a simple
model of xenon in cylindrical pores of a silica-like material. Closed-ended, open-ended,
and single-ended pores of either 3 nm or 4 nm diameter are considered, and the pore
length is varied between 8 nm and 108 nm. This study exposes some of the possible
mechanisms of pore filling and emptying, and demonstrates that hysteresis can be
almost entirely suppressed in certain pore geometries. The effects of pore length are
considered, and the thermodynamics of one-dimensional systems and the nature of
“capillary critical points” are discussed.
2
1
Introduction
The adsorption and desorption of gases in cylindrical pores has been extensively studied using
computer simulations [1–10], theoretical treatments such as classical fluid density functional
theory [2, 4, 11–18] and other methods [19, 20], and experimentally, using materials such as
MCM-41 [16, 21–28] and carbon nanotubes [29]. This literature is also reviewed in [30]. In
the present work we attempt to address some of the aspects of these systems that have
not been considered in great depth. Specifically, we simulate capillary phenomena in openended and single-ended pores, and consider the effects of the length of the simulation cell
in periodically-bound systems. Very long pores have not been widely studied largely due
to prior lack of computational power. Also, few studies of closed-ended and open-ended
pores have been made, partly because little information is available on the structure of pore
mouths and ends, and partly because comparisons between simulation results and mean-field
theories have been quite fruitful in other work (cited above), but computational treatment
of open-ended pores with density functional theories would be non-trivial [31].
Much of the literature on this subject has focused on identification and characterization of
capillary critical points, which are postulated in analogy with higher-dimensional systems,
and hysteresis critical points, the temperatures at which adsorption/desorption hysteresis
disappears. As is well known from classical statistical mechanics, a one-dimensional system 1
cannot exhibit a first-order phase transition or associated critical point. The fact that
cylindrical mesopores are not strictly one-dimensional but three-dimensional is irrelevant,
as the classical proof applies as long as the system is infinite in only one dimension. [The
possibility of substantial interactions between fluid particles in different pores increases the
effective dimensionality of the system, but this is rarely considered in theory or simulation
[32], and the inter-pore interactions in real materials are difficult to characterize.] The
argument [33], cast in the thermodynamics of capillary phenomena, is as follows. The
1
More strictly, a one-dimensional system that possesses only short-ranged interactions.
3
pore fluid, locally, exhibits two stable “phases”, corresponding to multilayer adsorption (low
density) and a filled pore (high density). At the chemical potential corresponding to the
coexistence of these “phases”, these have the same grand free energy per unit length; Ω h = Ωl .
For a two-domain system with a single interface, the free energy of the system becomes
Ω = Lh Ωh + Ll Ωl + γ, where Lh and Ll are the lengths of the two phases, and γ is the free
energy of the interface between the two phases, which is an approximately hemispherical
meniscus. This clearly reduces to Ω = LΩ + γ at coexistence, where L is the total pore
length. This expression is independent of the location of the interface, so we must then
add a third term corresponding to the entropy associated with movement of the interface,
S = kB ln(L/D), so Ω = LΩ − kB T ln(L/D) + γ; the pore diameter D is taken as the natural
unit of length in this expression, making the argument of the logarithm dimensionless. If n
interfaces are introduced, then the total free energy of the system is increased by a factor
nγ, and the entropy associated with these interfaces is S = kB ln(Ln /D n n!), recognizing
that the interfaces are indistinguishable. Applying Stirling’s approximation for the factorial,
we obtain Ω(n) = LΩ + n[γ − kB T ln(eL/Dn)]. For sufficiently large L/Dn the term in
brackets will be negative, and the most stable state of the system will involve an equilibrium
concentration of interfaces; at fixed L/D, the minimum free energy satisfies dΩ(n)/dn = 0,
and occurs at Dn/L = e−γ/kB T .
As mentioned above, the interface between the two phases is observed to be approximately
hemispherical. For a 5 nm pore, with a 0.5 nm thick adsorbed layer in the low-density
phase, the area of this interface is approximately 25 nm2 . For nitrogen adsorption at 77 K,
the surface tension is, approximately, 0.009 J/m2 [34], leading to a (dimensionless) interface
concentration of approximately 10−92 . This is an extremely low concentration, and for
pores of experimentally reasonable length (say, one micron or less) totally negligible. As
the temperature is raised, the adsorbed-layer thickness in the low-density phase increases,
reducing the surface area of the meniscus, and the surface tension of the fluid is reduced. The
4
concentration of interfaces therefore increases exponentially with increasing temperature. We
note that the case of binary liquid phase separation in cylindrical pores is formally equivalent
to the liquid-vapor problem, in that the same statistical mechanics can be applied and both
systems can be mapped onto a simple Ising model [35]. In previous work on liquid-liquid
phase separation in cylindrical pores, we have demonstrated that the mean distance between
interfaces eventually becomes comparable with the pore diameter, and that the “critical”
state of the fluid mixture consists of a domain structure characterized by strictly microscopic
length scales, rather than anything resembling a bulk or 2D critical point [36–38].
The effect, then, of inhomogeneity along the pore axis is negligible under most conditions.
In systems where there is no hysteresis, it will lead to a small rounding-off of the otherwise
sharp phase transition, similar to the effects of the use of periodic boundary conditions in
the simulation of bulk phase equilibria [39]. It is therefore reasonable to speak of the phase
transition in nano-scale capillaries as effectively first order, and apply standard thermodynamics to it. Furthermore, the scaling relations and thermodynamic expressions used to
characterize phase transitions [27, 28] will be applicable to these systems as long as they are
not used at temperatures very close to the “critical point”. However, inhomogeneity along
the pore axis does preclude the existence of a true critical point (in the statistical mechanical
sense [40, 41]) or any of the interesting properties normally associated with critical points,
including long wavelength fluctuations, divergences in susceptibilities, etc. Much of the
discussion of capillary critical points as “critical points”, therefore, is somewhat misleading.
Thus, for studies of capillary phenomena at low temperatures, the use of short, periodic,
simulation cells is thermodynamically acceptable because the length-scales necessary to see
inhomogeneity along the pore axis are macroscopic. The usual application of classical fluid
density functional theory to the system, which assumes translational invariance along the
pore axis, is also justified. [We note that such mean field theories incorrectly predict phase
transitions for one-dimensional systems [42] precisely because of this translational invariance.]
5
Likewise, at very high temperatures only a single “phase” exists at a given pressure, and short
simulation cells are also acceptable. In the region of the putative “capillary critical point”,
however, the pore length could have a substantial effect, interfering with the formation and
motion of domain boundaries, and artificially stabilizing single-phase states.
Hysteresis is present in experiments, molecular simulations, and mean-field treatments.
Generally, desorption hysteresis in simulations and numerical treatments is more pronounced
than in experiments. In Monte Carlo simulations, this is often blamed on the relatively short
effective time-scale that can be accessed. For comparison between open-ended experimental pore systems and periodic-cell molecular simulations, it is also often argued that the
presence of the pore mouth provides an easy nucleation of the low-density phase, and that
the experimental desorption curve corresponds to the simulated thermodynamic coexistence
point, and that experimental hysteresis occurs mainly on the adsorption branch. Likewise,
it is sensible to expect larger density fluctuations in simulations of long pores than in short
ones, and thus differences in nucleation limits and hysteresis loops.
In pores closed at one end , the closed end could serve as an already nucleated dense
phase, and so there might be no adsorption hysteresis either! This situation is considered in
the simulations presented below.
A unified, unambiguous picture of what really happens in these systems can be obtained
using simulations which dispense with the periodic-cell approximation. Direct treatment of
open-ended, infinitely periodic, or single-ended pores in excess of 100 nm length is possible
with modern equipment, and may serve to resolve many of the remaining questions in this
area.
2
Glass models and potentials
A simplified model for xenon adsorption on silica was used throughout this study. The pore
material was modeled by an amorphous configuration of particles representing the oxygen
6
atoms in silica. Silicon atoms are omitted, as they are not found on the surface and, being
not very polarizable, exhibit only weak dispersion interactions. An amorphous configuration
was generated by simulating a Lennard-Jones liquid at the density corresponding to that of
oxygen atoms in silica glass, and then filling the few holes in this configuration with additional
Lennard-Jones particles. To prepare pore models, cylindrical holes were then “drilled” out of
these configurations; no surface relaxation was applied, and pore-wall atoms were kept frozen
throughout the adsorption simulations. We have used such methods in previous studies [43]
to prepare idealized pore models which have pore-wall properties similar to those of our
models of Vycor and controlled-pore glass [43–45]; such models have also been used by
others [9]. The potential parameters used for xenon were σ = 0.391 nm, = 227 kB , and for
the pore particles, σ = 0.27 nm, and = 230 kB , as in earlier work [46, 47]. The potential
interactions between fluid particles were truncated (but not shifted) at 2.5σXe , and potential
interactions between fluid and wall particles were cut-and-shifted at the same distance. A
short-ranged potential is desirable for computational efficiency, and the shifting of the fluidsolid potential prevents artifacts in the shape of the multilayer adsorption isotherm.
The pore models used in this study, therefore, have surfaces which are rough on an
atomic scale, though quite smooth on larger scales. The use of atomically rough surfaces
instead of the smooth cylindrical potentials widely applied in such simulations [2,37] increases
somewhat the cost of the computation, but removes the perfect translational symmetry of
the smooth-walled models. This symmetry is not present in real materials, and is thought
to lead to unphysical stabilization of layered structures at low temperatures [30]. Atomistic
models of the pore material also allow for straightforward treatment of open-ended pores,
constrictions, “dead-end” pores, and any variations on these geometries, without losing a
reasonable description of the surface.
Three lengths of pore have been used: 8 nm, 16 nm and 108 nm. The first two are
comparable with most work in this area; we are only aware of a few quench Molecular
7
Dynamics simulations in very large-scale models [4, 36, 38]. These cylindrical pores were
made by removing all the particles within a cylinder from simulation cells of the appropriate
length and 5.4 nm (20σO ) square cross-section. For open-ended pores, the periodic simulation
cell length was then increased without moving any pore particles, introducing a 10 nm void
space with two planar surfaces. To generate single-ended pores, the cylinder was simply
made slightly shorter than the cell length.
In all the results that follow, data are given in Lennard-Jones reduced units [48], reduced
by the xenon parameters given above. For the potential cutoff used, the critical temperature
of this fluid is known to be Tc = 1.1876 [41].
3
Simulation protocols
“Normal” Grand Canonical Monte Carlo simulations have been used in this work, rather than
expanded-ensemble [49] methods or histogram-reweighting [41, 50]. The standard GCMC
method has recently been argued [51, 52] to provide a qualitatively reasonable picture of
the process of adsorption and desorption, as well as sampling the appropriate statisticalmechanical ensemble once equilibrium is reached. The basis of this argument is that one
observes most of the successful particle insertions and deletions at the liquid-vapor interface,
just as condensation and evaporation in an experimental system proceeds from the interface.
GCMC does not accurately account for the transport of fluid or vapor to and from the
interface, of course. However, as shown below, visualizations of the system moving towards
equilibrium in GCMC calculations can provide a microscopic, qualitative view of the pore
filling and emptying processes. In what follows, the traditional recipe of equal numbers of
insertion, deletion, and displacement moves has been used, with the maximum displacement
size adjusted during the run to give a 50% acceptance ratio. In the isotherms presented here,
each point was run for at least 30 million moves; points that were not equilibrated in this
time, or exhibited large fluctuations, were run for (much) longer.
8
4
Simulation results
4.1
Pore filling and emptying mechanisms
Early in this work it was noted that in the long-pore systems, equilibration at the top of the
capillary rise and the bottom of the desorption drop was very slow; as many as thirty times
as many moves as usual were required at some points. The configurations output during
these equilibrations nicely illustrate the mechanism by which fluid addition and removal
occurs in these simulations. In Figure 1 are shown several snapshots from the pore emptying
process observed in a 4 nm diameter, 108 nm long, open-ended pore. Only the last 30 nm
of the system is shown. As the simulation moves towards equilibrium at constant chemical
potential, the liquid-vapor meniscus moves into the pore mouth at a constant “velocity”.
The simulation never nucleates a “bubble” further inside the pore. In closed-ended long
pores, desorption occurs through the nucleation and growth of bubbles of vapor within the
pore; once bubbles are present, interfacial motions at constant velocity are observed in these
systems as well. One expects, then, that the chemical potential at which desorption occurs
in the open-ended pore is mediated by the structure of the pore mouth and its interaction
with the liquid-vapor meniscus.
In Figure 2 the opposite situation is presented, being the filling-up of a “single-ended”
pore, with one closed end and one open mouth. The high-density phase nucleates at the pore
end, and moves, again at constant velocity, towards the pore mouth. The open pore mouth
does not appear to play any role in the early stages of this process. The last configuration
shown in Figure 2 is typical of the equilibrium state at this chemical potential, with a roughly
hemispherical meniscus clearly visible at the pore mouth. In the periodic and open-ended
pores we have simulated, which have no “end” at which to start this process, the capillary
rise occurs only at higher chemical potential, through the spontaneous formation of multiple
“droplets” of the dense phase.
9
In Figure 3 are shown the adsorption and desorption isotherms measured in the singleended pore model, along with isotherms measured in a 4 nm infinite (periodic) pore of similar
cell size. It is apparent that the presence of both a pore opening and a pore end greatly
reduces the size of the hysteresis loop and the temperature at which hysteresis disappears.
This seems to occur because both high and low density phases are easily nucleated in this
system, which is not the case in the periodic pore. We have observed in the equilibration of
this system that at points on or near the capillary transition, large, long-lived fluctuations
in density occur, indicating that the system is near to a coexistence point and can explore
a large range of density. These fluctuations should not be compared with those of a bulk
or 2D critical state; rather, this is an unusual system in which kinetic barriers to density
fluctuations have been removed and large fluctuations occur frequently. A series of snapshots
taken at one pressure in this system is shown in Figure 4.
The most striking feature of Figure 3 is that the non-hysteretic capillary rise in the
single-ended pores occurs at a chemical potential roughly in the center of the hysteresis loop
of the periodic system, indicating that the periodic system exhibits roughly equal amounts
of adsorption and desorption hysteresis. The small hysteresis loop observed in the singleended pore at the lowest temperature does appear to be stable, exhibiting the large density
fluctuations discussed in the preceding paragraph. In this case, weak repulsive interactions
between the liquid-vapor meniscus and the ends of the pore may be responsible for the
stability of points halfway down the desorption drop or halfway up the capillary rise. By
moving near to either the closed end of the pore or its open mouth, the structure of the
interface is disturbed and the free energy of the system would rise, leading to a preference
for the center of the pore, as in Figure 4(a) and (e).
10
4.2
Open vs. infinite pores
Open-ended and infinite (periodic) pores were compared using model pores of 108 nm length,
which is comparable with the lengths one might expect to see in a typical experimental system. Figure 5 and Figure 6 show the results for 3 nm diameter pores and 4 nm diameter
pores, respectively. In both models, the presence of the pore ends somewhat reduces the
width of the hysteresis loop, from the desorption side. In the 3 nm pores, hysteresis disappears below T∗ =0.890 in the open-ended pore, and between T∗ =0.890 and T∗ =0.927 in the
periodic pore. In the 4 nm diameter pores, hysteresis probably disappears in both systems
between T∗ =0.974 and T∗ =1.025, and isotherms will have to be obtained at more temperatures to determine in which system it disappears first. At higher temperatures there does
not seem to be a substantial effect due to the opening of the pore ends.
The effects of open pore mouths are more dramatic in the 4 nm pore than in the 3 nm
pore, which is consistent with the picture of fluctuation-induced nucleation of the favored
phase. In 4 nm pores such fluctuations would have to be much larger than in 3 nm pores, and
thus the presence of the open pore mouth is more important in the 4 nm system. The very
small effect of the open pore mouth on hysteresis in the 3 nm system is difficult to reconcile
with fluctuation-induced nucleation. If the pore mouth does not contribute substantially to
reducing hysteresis on the desorption branch, we hypothesize that the periodic 3 nm system
doesn’t exhibit desorption hysteresis, and that the hysteresis observed at T∗ =0.801 is entirely
adsorption hysteresis.
In several systems the capillary rise actually occurs at slightly higher chemical potential
in the open-ended system then the infinite pore. This may occur because the pore mouths,
with their reduced adsorbed layer thickness (this is evident in the simulation snapshots),
serve to slightly stabilize the low-density phase at the ends of the pore.
11
4.3
Pore length effects (periodic systems)
There are two reasons why one might expect to see different behavior in long periodic pores
and short periodic pores. The first is the domain-structure argument given above, which
states that in long systems it becomes entropically favorable to see heterogeneous states,
which has the effect of smoothing out the capillary transition. Secondly, in systems which
exhibit substantial hysteresis, the width of the hysteresis loop can change. In longer pores,
larger fluctuations in the structure of either phase are possible, which should make nucleation
of the other phase more likely, shrinking the hysteresis loop. One might thus expect hysteresis
to disappear at a lower temperature in long pores than in short pores.
Figure 7 shows the adsorption and desorption isotherms measured in 4 nm diameter
infinitely-periodic pores of lengths 8 nm, 16 nm, and 108 nm, at several temperatures. At
the lowest temperature, T∗ =0.801, the 108 nm and 16 nm systems exhibit slightly narrower
hysteresis loops than the 8 nm pore. At the next higher temperature, T∗ =0.890, a similar
but more dramatic effect is observed, with the hysteresis loop in the shortest pore clearly
bracketing those of the two longer systems. At the third temperature, T∗ =0.927, the hysteresis loop in all three systems is quite narrow, and a clear trend is difficult to observe, though
the shortest pore still has a slightly broader loop. At the fourth temperature, T∗ =0.974,
no hysteresis is present in the two longer systems, with a very narrow loop observed in the
8 nm pore. At the highest temperature, T∗ =1.025, all three systems are reversible, and superimpose to within the resolution of these simulations. The influence of pore length on the
width of the hysteresis loop appears more pronounced at T∗ =0.890 than at either T∗ =0.801
or T∗ =0.927, which is interesting but probably not significant, as this appearance is caused
by only two data points in the desorption isotherm in the 8 nm system, and may thus be a
poor-sampling artifact due to finite run-time.
These results are reasonable within the thermodynamic framework presented above.
However, we have not observed multi-domain states in any of the simulations performed
12
here. Only within the very longest pore, at temperatures above the disappearance of hysteresis, would there be any possibility of observing heterogeneous multi-domain states at
equilibrium, and doing so would require fixing the chemical potential precisely at its coexistence value, since small shifts in chemical potential away from coexistence would favor the
presence of one phase over the other, breaking the translational invariance associated with
the domain boundaries. We also note that the time-scale involved in seeing these fluctuations is likely to be inaccessible even if the chemical potential could be specified to sufficient
precision; as demonstrated above in the discussion of pore filling and emptying mechanisms,
even fast motions of the liquid-vapor meniscus along the pore axis occurs on a scale of many
hundreds of millions of Monte Carlo moves.
As to the width of the hysteresis loop, the dependence of the loop size on pore length
is consistent with the hypothesis that fluctuations in density along the pore axis lead to
nucleation of each phase. Longer pores would permit larger fluctuations, and would thus
exhibit narrower hysteresis loops. Furthermore, the near-agreement of the 16 nm and 108 nm
pores indicates that pore lengths much longer than the pore diameter do not contribute
further to such fluctuations, which is reasonable.
5
Discussion
We have simulated a number of systems which are particularly relevant for developing a
better understanding of capillary phenomena in cylindrical pores. While the statistical mechanical argument for the absence of real critical behavior in such systems is certainly applicable in these systems, we believe that it is only important under very precise and particular
conditions. Nonetheless, the constant references in the literature to capillary critical points
in these systems are misleading, in that the states being described bear little resemblance to
bulk, 2D, or other critical phenomena. Pore length effects on hysteresis loops are found to
be small, though definitely present. The effects of open pore mouths on adsorption in very
13
long pores are found to be quite small in 3 nm pore systems, but substantial in 4 nm pores,
suggesting that 3 nm pore systems naturally exhibit much less desorption hysteresis than do
4 nm pores. Lastly, the mechanisms of pore filling and emptying have been visualized, agree
well with expectations, and suggest a number of related studies, as discussed below.
Multi-domain states were not observed in the long pores used in these studies. Further
investigation of the presence or absence of axial heterogeneity in long pores will require
a number of developments. The first is the precise location of the coexistence chemical
potential, which can be accomplished with standard thermodynamic integration methods [2].
A study of the effects of the pore length on the probability histogram of the system density,
as conducted in our previous work on liquid-liquid transitions [36–38], will also be useful, and
can be accomplished using umbrella-sampling methods or histogram-reweighting. Lastly, in
order to directly observe the domain structure in such systems, longer time-scales must be
accessed, which could be accomplished through the development of a Monte Carlo scheme
which preferentially samples particles near to the hemispherical menisci in such systems.
The short, single-ended pores considered here suggest a possible route for further research
in both simulations and experiments. These systems exhibit two unusual properties: a neartotal lack of hysteresis, and large density fluctuations at pressures near to condensation.
Should an experimental realization of such a material be accomplished, it could provide a
direct experimental confirmation of the hypothesis that the capillary rise is nucleated by the
pore end. Observation of large-scale density fluctuations would be difficult since the densities
in different pores would be uncoupled, but the presence of weak pore-pore correlations [32] in
an array of such pores could give rise to a novel type of phase transition in two dimensions.
The preparation of an array of short pores might be accomplished either through templating
using thin films of the MCM-41 mesophase system, somehow oriented perpendicular to a
silica surface, or by nanofabrication techniques.
Lastly, a more thorough investigation of the effects of the shapes of both pore mouths
14
and pore ends is clearly required in order to catalog the possible behavior observed in these
systems, and to place the observations made here in their proper context. Little is known
about the structure of pore mouths and pore ends in real mesoporous materials, though this
information might be obtained via AFM or similar methods.
6
Acknowledgments
Thanks are due to the Editors of this special issue, for inviting the submission of this paper,
and, of course, to Keith Gubbins, for many useful discussions, guidance and advice, and for
introducing me to this subject.
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20
(a)
(b)
(c)
(d)
(e)
Figure 1: Desorption in an open-ended, 4 nm diameter pore, at T∗ =0.801 and µ∗ =-4.33. This
chemical potential is just below the desorption shoulder, as shown in Figure 6. Equilibration
of the Monte Carlo simulation is observed for 300 million moves. Snapshots are shown in
which the simulation cell has been cut in half along the pore axis; pore material is dark
gray, and particles are light gray. “Cut” particles are white, showing the pore density in
cross-section. Snapshot (a) is taken at 30 million moves, (b) at 90 million, (c) at 150 million,
(d) at 210 million, and (e) at 270 million. The liquid-vapor meniscus moves into the pore
at a constant rate.
21
(a)
(b)
(c)
(d)
(e)
(f)
Figure 2: Adsorption in a single-ended, 4 nm diameter pore at T∗ =0.801 and µ∗ =-4.18.
This chemical potential is just above the capillary rise (see Figure 3). Equilibration of the
Monte Carlo simulation is observed for 200 million moves. Snapshots are shown in which
the simulation cell has been cut in half along the pore axis; pore material is dark gray, and
particles are light gray. “Cut” particles are white, showing the pore density in cross-section.
Snapshot (a) is taken at 15 million moves, (b) at 30 million, (c) at 45 million, (d) at 60
million, (e) at 75 million, and (f ) at 180 million. The liquid-vapor meniscus at the mouth
of the pore in (f ) appears to be stable.
22
density
0.9
single−ended
periodic
(a)
(b)
(c)
(d)
(e)
0.7
0.5
0.3
−4.75
−4.25
−3.75
chemical potential
−3.25
Figure 3: Adsorption and desorption isotherms in the single-ended, 4 nm diameter pore
system and the 4 nm diameter, 16 nm long periodic pore system, shown for comparison.
Reduced temperatures are (a) 0.801, (b) 0.890, (c) 0.927, (d) 0.974, and (e) 1.025. Isotherms
(b)–(e) have been shifted 0.1, 0.2, 0.3 and 0.4 reduced chemical potential units to the right,
respectively. The densities reported are the number of particles in the simulation cell divided
by the volume of the cylindrical pore. In the open-ended pore, the density data have been
uniformly shifted downwards by 0.21 density units so that the hysteresis loops approximately
superimpose; this is necessary because of the additional adsorption in this system on the two
planar surfaces visible in Figure 2.
23
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4: Large density fluctuations in the single-ended 4 nm diameter pore system. (a)-(f )
are sequential snapshots from a single long simulation at constant T∗ =0.801 and µ∗ =-4.21,
at the top of the desorption shoulder shown in Figure 3. The liquid-vapor meniscus exhibits
slow, diffusion-like motion. The spacing between snapshots is 60 million Monte Carlo moves.
24
periodic
open−ended
0.9
(a)
(b)
(c)
density
(d)
0.7
(e)
0.5
0.3
−5
−4.5
−4
chemical potential
−3.5
Figure 5: Adsorption and desorption isotherms at five different temperatures, in open-ended
and infinite (periodic) pores of 3 nm diameter and 108 nm length. The temperatures (a)–
(e) are the same as in Figure 3. Isotherms (b)–(e) have been shifted 0.1, 0.2, 0.3 and 0.4
reduced chemical potential units to the right, respectively. The density is normalized by
the cylindrical pore volume, as in Figure 3. Adsorption on the surfaces at the ends of
the open-ended pore causes a small vertical shift in these isotherms. Open-ended pores
have marginally smaller hysteresis loops in this system, and hysteresis disappears at a lower
temperature in the open-ended system.
25
0.9
periodic
open−ended
(a)
(b)
density
(c)
(d)
0.7
0.5
(e)
0.3
−4.75
−4.25
−3.75
chemical potential
−3.25
Figure 6: Adsorption and desorption isotherms at five different temperatures, in open-ended
and infinite (periodic) pores of 4 nm diameter and 108 nm length. The temperatures (a)–
(e) are the same as in Figure 3. Isotherms (b)–(e) have been shifted 0.1, 0.2, 0.3 and 0.4
reduced chemical potential units to the right, respectively. The density is normalized by the
cylindrical pore volume; adsorption on the surfaces at the ends of the open-ended pore causes
a small vertical shift in these isotherms. Open-ended pores have slightly smaller hysteresis
loops in this system, and hysteresis disappears at a lower temperature in the open-ended
system.
26
density
0.9
108 nm
16 nm
8 nm
(a)
(b)
(c)
(d)
0.7
(e)
0.5
0.3
−4.75
−4.25
−3.75
chemical potential
−3.25
Figure 7: Adsorption and desorption isotherms at five different temperatures, in infinite
(periodic) 4 nm diameter pores of lengths 108 nm, 16 nm, and 8 nm. The temperatures
(a)–(e) are the same as in Figure 3. Isotherms (b)–(e) have been shifted 0.1, 0.2, 0.3 and 0.4
reduced chemical potential units to the right, respectively.
27