On the Cavitation and Pore Blocking in Slit-Shaped Ink-Bottle Pores

ARTICLE pubs.acs.org/Langmuir

On the Cavitation and Pore Blocking in Slit-Shaped Ink-Bottle Pores Chunyan Fan, D. D. Do,* and D. Nicholson School of Chemical Engineering, University of Queensland, St. Lucia, Qld 4072 Australia ABSTRACT: We present GCMC simulations of argon adsorption in slit pores of different channel geometry. We show that the isotherm for an ink-bottle pore can be reconstructed as a linear combination of the local isotherms of appropriately chosen independent unit cells. Second, depending on the system parameters and operating conditions, the phenomena of cavitation and pore blocking can occur for a given configuration of the ink-bottle pore by varying the geometrical aspect ratio. Although it has been argued in the literature that the geometrical aspects of the system govern the evaporation mechanism (either cavitation or pore blocking), we here put forward an argument that the local compressibility in different parts of the ink-bottle pore is the deciding factor for evaporation. When the fluid in the small neck is strongly bound, cavitation is the governing process, and molecules in the cavity evaporate to the surrounding bulk gas via a mass transfer mechanism through the pore neck. When the pore neck is sufficiently large, the system of neck and cavity evaporates at the same pressure, which is a consequence of the comparable compressibility between the fluid in the neck and that in the cavity. This suggests that local compressibility is the measure of cohesiveness of the fluid prior to evaporation. One consequence that we derive from the analysis of isotherms of a number of connected pores is that by analyzing the adsorption branch or the desorption branch of an experimental isotherm may not lead to the correct pore sizes and the correct pore volume distribution.

1. INTRODUCTION The phenomena associated with adsorption and desorption in porous materials are well-known.1,2 The basic observations have been documented in hundreds of experimental papers over more than a century. The essential features can be summarized as follows: When the pores are sufficiently large (usually referred to as mesopores) and the temperature is low enough (and below the critical temperature of the adsorptive) capillary condensation takes place and is accompanied by hysteresis. When the pore cross section, relative to the adsorbate molecule size, is below a certain limit, condensation is replaced by a direct filling process, and there is no longer hysteresis. The hysteresis loop size also decreases as temperature is raised toward the critical temperature.3 The main reason for the enduring interest and investigation of these phenomena rests in their relationship to the character of the porous materials in which they occur. Descriptions of the basic physics of capillary condensation frequently invoke the Laplace equation, which expresses the pressure difference across a curved interface, and the Kelvin equation, which follows from this and gives the relation between critical vapor phase pressure at which condensation occurs and the size of the pore. The existence of hysteresis is always a challenge to physics because strict thermodynamics insists that only one equilibrium state can exist for a given thermodynamic state. Moreover, it gradually emerged from adsorption studies that not only were the loops stable over very long times but that quite characteristic loop r 2011 American Chemical Society

shapes could occur depending on the material examined. Early attempts were made to relate these shapes to pore shapes,4 and the loop shapes have been classified into types H1 to H4.5,6 Two simple explanations for the phenomena observed in single pores were advanced several decades ago. The first, the Kelvin-Cohan hypothesis, proposed that in cylindrical pores adsorption proceeds in a layer-wise fashion, building up a cylindrical meniscus that eventually becomes critical (according to the Kelvin-Laplace equation) when condensation occurs.7,8 The resulting filled pore now has a hemispherical meniscus in contact with the adsorptive vapor, and this becomes unstable at a lower pressure than the cylindrical meniscus. The second explanation was first proposed by McBain and is known as the inkbottle theory.9 In this scenario, the pore fills by condensation into the large and small sections, but when the desorption pressure for the large pore is reached, the liquid remains trapped because the small pore has not yet reached its desorption-pressure, and emptying is blocked by the adsorbate fluid held in this pore. This idea has been extended by modeling the porous material as a network of interconnected pores of different size. Emptying of the network then becomes a percolation problem in which a

Received: October 25, 2010 Revised: January 23, 2011 Published: March 03, 2011 3511

dx.doi.org/10.1021/la104279v | Langmuir 2011, 27, 3511–3526

Langmuir critical pathway to the trapped fluid must be established before this can occur.10-12 Computer simulation of adsorption started in the 70s and was applied to this problem by many workers in the 80s.13-21 Initially it was surprising to find that hysteresis was observed in grand canonical simulations and perhaps even more so that the loops corresponded quite well to experimental studies. However, no immediate explanation for its existence was forthcoming. Indeed, the best thermodynamic argument remained that was first proposed by Hill as far back as 1947,22 namely, that there are two free energy minima accessible from either the adsorption or the desorption direction and that these are separated by an energy barrier so high that it cannot be crossed on any reasonable experimental time scale. Later work has established these transitions as being closely related to experimental observations, although the question remains as to whether they occur at precisely the same pressures as experimental transitions.23 Furthermore, it became increasingly clear that hysteresis can occur in slit pores (which rules out a strict application of the Cohan-Kelvin hypothesis), and when the more recently developed DCV24-26 simulation technique was applied to an ink-bottle model, the long-held suspicion was confirmed that a pressure drop can drive adsorbate fluid out of a trapped large pore, thereby severely damaging the network or ink-bottle proposals.27-31 However, later simulations have shown the difficulty of making lasting generalizations in this matter because pore blocking effects have been found in ink-bottle models of sphere-cylinder geometry when the trapping pores are large but not for smaller trapping pores.32-34 Pore blocking effects were not always found, however, in similar slit pore models.35 In reconstructed networks, adsorbed fluid can be trapped by adsorbate in molecular sized spaces.36 The possibility that both pore blocking and cavitation mechanisms operate in conjunction has been broached elsewhere.37 None of these advances has been able to explain comprehensively the different shapes of observed hysteresis loops and do not support a general explanation of the observations. However, molecular fluctuation theory, as first proposed by Hill,38 can provide a feasible mechanism, at least in general terms.39-41 The filling of pores by condensation is expected to occur when the growing adsorbate layers form bridges that increase in size until a pressure where fluid fills the vapor space.31,42 The nucleation processes in the adsorbate fluid that precede bridging would be controlled to a large extent by the adsorbent potential energy field that tends to order the layers.43,44 When pressure is decreased during the desorption stage, fluctuations in the filled pore will create intermolecular spaces that increase in size as the pressure is lowered; eventually, these spaces will cluster into cavities that become unstable, and emptying occurs. Discussions of nucleation and cavitation in liquids are widespread in the scientific literature.45 However, it is important to appreciate that adsorbed fluids are highly nonuniform and that the magnitude of the fluctuations is consequently strongly localized. To a first approximation, this magnitude would be expected to depend on the local density and temperature. The spatial identification of locality, as always in nonuniform fluids, presents problems. Because the nucleation processes underlying emptying and filling of the capillaries are activated processes, kinetics also play an important role in the overall phenomena.46 Condensation and evaporation, like all adsorption phenomena, are modified by changes in the adsorption potential field. There is a number of ways in which the potential field can be

ARTICLE

altered, and recent computer simulations have investigated several instances. For example, the cross sectional shape of a cylindrical pore can be varied from circular (an aspect ratio of one) through elliptical to open slit-shaped (an aspect ratio of infinity). The field can also be modified by introducing defects, surface roughness, or heterogeneities along the length of a single pore. More subtle effects can be introduced by using finite length pores rather than the usual periodic boundary conditions.47 Here the atoms close to the ends of the pores will not experience all adsorbent atom interactions in the same way as adsorbate atoms toward the center of the long axis. As a very rough rule, the potential field decreases as the cube of the distance so that atoms more than about three atomic diameters from the pore end will be subject to approximately the same field as those in an infinitely long pore. In other words, the adsorption field in pores of less than about six diameters in length will appear to be quite different from that in an infinite pore. Similarly, finite length pores incorporated into a network will generate a potential field with similar properties to an isolated finite length pore, but in this case, the field will be modified by fields from the surrounding solid. Coasne, Pellenq, and coworkers have carried out a number of simulation studies of a variety of topologies of these types48-51 and find several variations in the character of the hysteresis loops. An important advance, first noted by Sarkisov and Monson,28 is the discovery that a dynamical simulation in which gas was only allowed to enter the pores through spaces open to the external phase gave the same results as the GCMC simulation. In this article, we embark on a systematic GCMC computer simulation of slit-shaped ink-bottle pores to demonstrate cavitation and pore blocking in model slit-shaped pores and to investigate the role of compressibility as a measure to identify these two phenomena. First, we argue that the adsorption and desorption in a connected pore is merely the combination of local isotherms of some appropriately chosen independent unit cells. Definitions of these unit cells will be given in the Theory section. Next, we illustrate the concept of a column of liquid condensate by carrying out a simulation of pores of uniform size but different surface strengths along the pore. With this configuration, we show that adsorption is a sequence of filling the sections having the highest surface strength, followed by filling sections of lower surface strength. Desorption from this type of pore follows a different mechanism from that in a pore with a uniform surface; here the column of liquid in the core of this pore is stretched, as the pressure is decreased, to the extent that particles in the liquid column desorb simultaneously, giving rise to the hysteresis, thereby separating the adsorption and desorption branches. This analysis provides information about the critical bubble size prior to condensation and the critical cavity size just after evaporation.

2. THEORY To simulate the adsorption isotherms, we use the grand canonical Monte Carlo simulation (GCMC), details of which can be found in ref 52. We briefly summarize below the essential points used in our simulations. The adsorbate is argon represented by a Lennard-Jones potential with parameters εff/kB = 119.8 K and σff = 0.3405 nm.53,54 2.1. Model Solids with Connectivity. The four solid models studied are illustrated in Figure 1. The walls are made up by flat strips, which are finite in the y direction and infinite in the x direction. The interaction of a fluid particle with each of the strips is modeled by the Bojan-Steele potential (Section 2.2). 3512

dx.doi.org/10.1021/la104279v |Langmuir 2011, 27, 3511–3526

Langmuir

ARTICLE

The origin of the y coordinate is positioned at the center of the strip. The repulsive and attractive functions on the RHS of eq 1a are given by y 1 σ sf 10 1 σ10 sf þ jrep ðz, yÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi½ 10 z8 ðy2 þ z2 Þ y2 þ z2 5 z 3 σ 10 1 σ10 sf sf þ 2 40 z6 ðy2 þ z2 Þ 16 z4 ðy2 þ z2 Þ3 7 σ10 sf þ 128 z2 ðy2 þ z2 Þ4

þ

" # σ 4sf y 1 σsf 4 1 jatt ðz, yÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 2 4 z ðy þ z2 Þ y2 þ z 2 2 z Figure 1. Configurations of the four solid models: (a) finite slit pore, (b) finite slit pore with one close end, (c) two connected pores exposed to bulk, and (d) cavity connected to smaller necks.

1 The first model is a finite slit pore, whose solid-fluid interaction is modeled by the Bojan-Steele potential (Figure 1a). Two ends are open to the gas phase. 2 The second model has Bojan-Steele surfaces, but one end is closed (Figure 1b). 3 The third model is a connected pore, formed by joining a small pore to a larger pore. The two ends of this connected pore system are exposed to the adsorptive gas. (See Figure 1c.) 4 The fourth model is an ink-bottle pore, formed by joining a larger pore (cavity) to two smaller pores at the two ends (necks) (Figure 1d). Only the smaller pores are exposed to the adsorptive gas. The first model is chosen to show the behavior of hysteresis in finite single slit pores and to explore how it changes with the system parameters. The second one is chosen to show the effects of closing one end of the pore. The third model is to study the effects of connectivity, which gives rise to the competition of two types of interfaces (two flat interfaces and one cylindrical interface), whereas the fourth model is an ink-bottle type of pore, in which we investigate whether pore blocking is relevant in hysteresis. 2.2. Solid-Fluid Potential. We model a finite sized surface of constant surface density with a Bojan-Steele surface. The equation describing the solid-fluid interaction of this surface is55-58 jf , s ¼ 2πðFs σ2sf Þεsf f½jrep ðz, yþ Þ - jrep ðz, y- Þ - ½jatt ðz, yþ Þ - jatt ðz, y- Þg

ð1aÞ

where z is the shortest distance between a particle and the surface, Fs is the surface density (taken as 38.2 nm-2 in this work), σsf and εsf are the cross collision diameter and welldepth of the interaction energy of the fluid particle with the surface, and they are calculated from the Lorentz-Berthelot rule: σsf = (σss þ σff)/2 and εsf = (εssεff)1/2, with σss and εss taking the values of carbon atom in a graphene layer, εss/kB = 28 K and σss = 0.34 nm. The other variables in the BojanSteele (BS) equation are yþ and y-, which are the y coordinates of the right-hand edge and the left-hand edge relative to the fluid particle yþ ¼

W W - y; y- ¼ - - y 2 2

ð1bÞ

ð2aÞ

ð2bÞ

The potential equation in eqs 1a and 1b is valid for any particle around the strip. In the case of a particle positioned exactly in the same level as the strip (i.e., z = 0), eqs 2a and 2b become undefined. In that case, we apply the Taylor series expansion to eqs 2a and 2b, and the Taylor series of the solid-fluid potential for small values of z is given in Appendix 1. 2.3. Simulation. A cutoff radius of half of the box length was used, and 30 000 cycles were run for equilibration and for sampling, where one cycle consists of 1000 displacement moves and attempts of either insertion or deletion with equal probability. The initial maximum displacement step length was half of the box length and was adjusted during the equilibration stage such that the acceptance ratio of displacement fell between 25 and 75%. The maximum displacement length was then kept constant during the sampling stage and taken as the value achieved at the end of the equilibrium stage. Periodic boundary conditions were applied in the x-direction to simulate the infinite extent of the surface in that direction. The results are presented as (i) absolute pore density based on the accessible volume versus pressure and (ii) absolute density in different sections of the pore. The absolute pore density is defined as F¼

ÆNæ - Fgas Vacc Vacc

ð3Þ

where Fgas is the bulk molecular density, ÆNæ is the ensemble average of the number of particle in the simulation box, and Vacc is the accessible pore volume, which is defined as volume where the solid-fluid potential is nonpositive. The local absolute density is defined in a similar manner as in eq 3 with the parameters being replaced by those relevant for the local section.

3. RESULTS AND DISCUSSION 3.1. Model 1: Finite Slit Pores. The adsorption isotherms for argon adsorption in finite length slit pores of width H = 3 and 4 nm are shown in Figure 2a,b, respectively. To avoid cluttering, the isotherms for L = 2, 3, and 6 nm are shifted up by 90, 60, and 30 kmol/m3, respectively. Also shown in this graph are the isotherms for pores of infinite length with the same pore widths. The solid-fluid interaction in infinitely long pores is described by the Steele 10-4-3 potential equation. From Figure 2, we make the following observations: (i) An H1 type hysteresis loop5 is observed for both infinitely long and finite pores. A similar observation was made by Coasne et al.59 in their simulations of pores with elliptical and circular cross section. 3513


Langmuir

ARTICLE

Figure 3. Local isotherms for argon adsorption at 87.3 K in a finite slit pore of width H = 3 nm and L = 6 nm.

Figure 4. Adsorption isotherms of argon at 87.3 K in finite slit pores with one end closed. The pore width is H = 3 nm, and the lengths are 2, 6, and 10 nm.

this by dividing the pore into two sections, one at the pore mouth and the other for inside the pore. The local isotherms from these two sections are shown in Figure 3 for a finite slit pore of width H = 3 nm and length of 6 nm. The demarcation of these sections is based on the configuration of particles in the pore just before the evaporation occurs. It can be seen that in the region of the knee, the local density in the pore mouth decreases gradually because of the receding of the meniscus, whereas the local isotherm of the interior of the pore is nearly constant. As pointed out by Monson,31 in an ideal open pore, the desorption happens when the system is very close to the equilibrium transition. On the other hand, no such knee is observed with infinitely long pores because there is no meniscus, and evaporation results purely from the evaporation of liquid condensate beyond the liquid spinodal point. (iv) For finite pores, the condensation and evaporation pressures increase with pore size, and the extent of the increase in the condensation pressure is greater than that

Figure 2. Adsorption isotherms for argon at 87.3 K in finite slit pores of (a) H = 3 nm and (b) H = 4 nm with different lengths compared with infinite pores of the same width.

(ii) The condensation pressure is higher for the shorter pore length. When the length is sufficiently small, the hysteresis loop disappears, that is, the adsorption isotherm is reversible because the adsorption and desorption follow the same mechanism; during adsorption, a liquid bridge is formed at the middle of the pore, and the two resulting cylindrical menisci advance to the pore mouths as pressure is increased; on desorption, evaporation occurs at the pore mouth via the cylindrical interfaces, which recede into the center of the pore as pressure is decreased. (iii) Along the desorption path, the isotherms for finite pores show a knee just before the evaporation. The knee results from the recession of the menisci into the pore. We prove 3514


Langmuir

ARTICLE

Figure 5. Schematic diagram showing the advance of different interfaces in (a) a slit pore with two ends open to the bulk phase and (b) a slit pore with one end closed. Pressure increases from left to right.

of the evaporation pressure, making the hysteresis loop larger with pore size. 3.2. Model 2: Finite Slit Pores with One End Closed. Figure 4 shows the adsorption isotherm for argon in finite pores of width 3 nm and different lengths. For short pores (L = 2 and 6 nm), the adsorption isotherm is reversible, in contrast with the finite pores with open ends considered in Section 3.1. This reversibility is due to the formation of a cylindrical meniscus at the closed end of the pore. In the initial stages of the adsorption, adsorbate particles fill the closed end of the pore because there is a stronger potential field due to the presence of three adsorbent walls. The potential of a fluid particle near the cylindrical meniscus is stronger than that at other points in the pore, resulting in a favorable adsorption on the cylindrical meniscus, which advances steadily with pressure toward the pore mouth. The schematic diagram of the advance of different interfaces in the finite slit pore with two open ends and in that with one closed end is shown in Figure 5. Upon desorption, the mechanism is exactly the reverse of that just described, resulting in a reversible adsorption isotherm. However, when the pore length is increased to 10 nm, we observe a hysteresis loop with condensation and evaporation pressures smaller than those of the corresponding pore with two open ends. The loop is smaller in size and the slopes of the adsorption, and desorption branches are less steep. So what is the mechanism of hysteresis in long closed end pore? Along the adsorption path, we have adsorption layers on the walls and on the closed end. As pressure is increased, even though adsorption is favored at the cylindrical meniscus and accounts for the advance of this meniscus, adsorption also occurs to a lesser extent on the two flat interfaces. Because the pore is very long, the two growing interfaces exert increasingly stronger forces on each other and can approach close enough for attraction between molecules from opposite walls to induce a capillary condensation. The molecules at the advancing cylindrical meniscus are in a weaker field because the enhancement from the pore end decreases rapidly with separation. Therefore, it is possible for condensation to occur before the cylindrical interface reaches the pore mouth. Therefore, the presence or absence of hysteresis depends on the relative advance of the cylindrical interface and the flat interfaces on the pore walls. If the cylindrical interface wins, we have a reversible isotherm (like the ones with shorter length, 2 and 6 nm). However, if the flat interfaces win, we have hysteresis, which is the case for a very long pore (for example, the pore of length 10 nm). Upon desorption from a very long pore, the cylindrical interface recedes at the pore mouth to the extent that

Figure 6. Adsorption isotherm of argon in a connected pore joining a narrower (width 1.7 nm) and a wider (width 3 nm) pore. The length of the narrow pore is 2 nm and the lengths of the wider pore are 2, 6, and 15 nm.

the liquid condensate becomes metastable and evaporates, resulting in two flat interfaces and one cylindrical interface. The desorption path is not the same as the adsorption path, giving rise to a hysteresis loop. Loops with similar general characteristics were observed by Sarkisov and Monson28 for similar models. 3.3. Model 3: Connected Pore with Two Open Ends. We consider two cases. In the first, the width of the narrower pore (H1) is less than the critical hysteresis pore width, and the adsorption isotherm in this pore is therefore reversible; in the second case, H1 is greater than the critical hysteresis pore width. This means that in the first case we expect reversible pore-filling, followed by a hysteresis loop associated with the larger pore (if this is chosen to be long enough). If this pore is too short, then we expect reversible adsorption in the wider pore as well. The discrimination between hysteresis and reversibility in the wider pore is the competition between the advance of the two interfaces: a flat interface at the layers building on the planar walls and the cylindrical interface, as demonstrated in the model of the closed end pore in Section 3.2. In other words, once the narrower pore is filled, its combination with the junction of the two pores acts like a closed end for the wider pore. These points are illustrated with specific examples in Figures 6-12. In Figure 6 , we show isotherms for a model with 3515


Langmuir

ARTICLE

Figure 7. Schematic diagram of the advancing of interfaces in short and longer pores: (a) dominated by a semicylindrical interface in the short large pore and (b) dominated by a flat interface in the long large pore.

Figure 8. Adsorption isotherms for argon at 87.3 K in a connected pore joining a narrow pore (width 3 nm) and wider pores (width 3.68 nm). The lengths of the narrow and wide pores are 6 and 10 nm, respectively.

H1 = 1.7 nm and H2 = 3 nm and different lengths of the wider pore. The width of the narrower pore is less than the critical pore width, and the existence of hysteresis in the larger pore is dictated by the two interfaces (cylindrical or flat) in this pore. A schematic picture of the advancing interfaces is shown in Figure 7. If the wider pore is very long (L2 = 15 nm), then we expect that condensation is promoted by the approach of the two interfaces of adsorbate from opposite pore walls (Figure 7b). Upon desorption, particles evaporate from the semicylindrical interface at the pore mouth of the larger pore, and this interface recedes toward the junction. As the pressure is further reduced, the fluid in the inner core of the larger pore is stretched in a metastable state and eventually evaporates, leaving adsorbed layers on the two walls. Because the mechanism of condensation is different from that of evaporation, we observe hysteresis in the long, wide pore. If the wider pore is short (L2 = 2 or 6 nm), then the advance of the semicylindrical interface governs adsorption, and this interface reaches the pore mouth before the flat interfaces are close enough to induce condensation (Figure 7a). This results in a reversible adsorption of the larger pore because upon desorption the process is simply the reverse of adsorption. Next, we show the second case where the width of the smaller pore is greater than the critical hysteresis pore width. Two examples are given to show the effects of the pore length and width on adsorption behavior. In the first example, we choose H1 = 3 nm and H2 = 3.68 nm with different lengths to show the

adsorption behavior when the difference in pore width is small. Both pores exhibit hysteresis if they are independent (i.e., nonconnected). Figure 8 shows the adsorption isotherms of the connected pore with L1 = L2 = 6 nm and L1 = L2 = 10 nm. If the pore is short (L1 = L2 = 6 nm), then, because the pore widths of these two pores are not very different, we see a sharp increase in adsorption initiated in the narrower pore, followed immediately by the condensation in the wider pore (a responsive effect). However, upon desorption, we observe a two-stage decrease in the adsorbed density; the first decrease is due to evaporation from the wider pore, whereas the second is from the narrower pore. As can be seen, both of these stages of desorption are initiated when the semicylindrical interfaces recede, either from the pore mouths, or from the junction. This process is reflected in the knees preceding the sharp decrease in the adsorbed density. This case is interesting because adsorption isotherms exhibited by other pore configurations usually show a gradually increasing adsorption branch and a sharp desorption branch (H2 hysteresis). In the longer pore length, L1 = L2 = 10 nm, the responsive effect observed in the shorter pore is not seen but is replaced by a two-stage uptake. This is due to the advance of the cylindrical interface toward the pore mouth. In shorter length (L2 = 6 nm), the advance gives rise to a responsive branch, whereas in the case of longer length (L2 = 10 nm), the advance gives rise to a gradual but sharp adsorption branch C0 D0 . We substantiate this with the snapshots of particles for short and long pores in Figure 9. We see that the responsive effect is observed when the widths of the two sections of the connected pore are almost the same. We now address this situation where the width of the wider section is increased. The simulated isotherms are shown in Figure 10 for H2 = 3.68, 4.3, 5, and 6.4 nm. Because the difference in the pore size is greater, we see a deconvolution of the single hysteresis loop (for H1 = 3 nm and H2 = 3.68 nm) into two distinct hysteresis loops. The second loop is larger and more distinct from the first loop as the width of the larger section increases. The shape of the hysteresis loop for comparable widths of the narrow and wide pores is quite distinct from many other shapes of hysteresis. We will summarize all hysteresis shapes in Section 3.5 after we have discussed all other pores. We next show that the isotherm of a connected pore can be derived from the linear combination of two independent pores, which we refer to as unit cells. A somewhat similar approach to complex materials was proposed by Puibasset,60 who studied a multiscale model that combined pores of different sizes and surface heterogeneity. Consider the example where the width of the smaller pore is less than the critical hysteresis pore width. The first unit cell is the narrower pore exposed at two ends, and the 3516


Langmuir

ARTICLE

Figure 9. Snapshots during the adsorption process for the connected pore joining a narrow pore (3 nm) and a wider pore (3.68 nm); the lengths of two sections are 6 and 10 nm. The points A-D and A0 -D0 are as labeled in Figure 8.

second is the wider pore with one end open to the bulk gas and the other end closed, with surface and frozen argon particles, as shown in Figure 11a, to mimic the junction of the two pores and liquid condensate that has been formed in the smaller pore. The local isotherms of these two unit cells are shown in Figure 11a. For ease of comparison, we present these isotherms as the number of particles as a function of pressure. We see that the local isotherm of the narrower unit cell is reversible because the pore size is less than the pore size for critical hysteresis, whereas that of the wider unit cell has a hysteresis loop. We assume that the connected pore of Figure 6 is a combination of the two unit cells that we just described, and the sum of their local isotherms is shown in Figure 11b. Also plotted in the same figure is the isotherm obtained directly from the GCMC simulation of the connected pore. These isotherms agree very well with each other, suggesting that an isotherm of a connected pore can be constructed from a linear combination of isotherms of appropriately chosen unit cells. In the second case, the width of the smaller pore is greater than the critical hysteresis pore width, and the widths of the small and large pores are not very different. The two unit cells have the same respective widths, and the two ends of the small pore are open to the bulk gas, whereas two different unit cells are chosen for the larger pore here: one with two open ends and the other one closed at one end. Figure 12a shows the local adsorption isotherms of the unit cells, and their linear combination is shown in Figure 12b. Also plotted in Figure 12b is the isotherm obtained directly from the GCMC simulation of the connected pore. The ordinate scale of the combined isotherm for the case of unit cells with the wider pore having two open ends is shifted by 1500, whereas that for the unit cells with wider pore having one end closed is shifted by 3000. It is interesting but not too surprising that the isotherm obtained through the combination of the two unit cells, both of which have two ends open to the bulk phase, does not agree with that of the connected pore. However, when the unit cell of the larger pore is closed at one end, the combined isotherm agrees better with the isotherm of

the connected pore, supporting the responsive theory. One significant implication from the isotherm for the connected pore (Figure 12) is that by analyzing the adsorption branch for the determination of a pore size distribution (PSD) using a kernel of unit cells with two open ends (which is a common practice in the literature) we obtain one pore size that is the size of the smaller pore. If we use the desorption branch of the connected pore to analyze for the PSD, then we will get a pore size that is smaller than the two actual pore sizes because the desorption branch of the connected pore occurs at a lower pressure than those of the two local isotherms. This implication calls for caution about the characterization for PSD using just one adsorption isotherm. It might be better to use isotherms at various temperatures and different adsorbates to determine PSD. This will be the subject of a future publication. 3.4. Model 4: Ink-Bottle Pore with Two Open Ends. We next consider the case of an ink-bottle pore with a wider pore in the middle connected to two narrower pores (necks) that are exposed to the bulk gas (Figure 1d). This ink-bottle pore is the classic example for the discussion of the phenomena of cavitation and pore blocking. We consider two cases: (i) the widths of the necks are less than the critical hysteresis pore size Hc and that of the wider pore is greater than Hc and (ii) the widths of both pores are greater than Hc. For the first case, two examples are chosen: In the first, H1 = 0.68 nm (small neck), H2 = 3.36 nm, and the length of the wider pore is 4 nm and the length of the neck is either 2 nm or 4 nm. The isotherms in Figure 13a show that the isotherm is not affected by the length of the neck. In the adsorption branch, there are two stages in the uptake, corresponding to condensation in the neck, followed by condensation in the wider pore. Because this pore is long enough (L2 = 4 nm), the condensation is due to the approach of the two flat interfaces, advancing from the flat walls, followed by an instant filling of the core with liquid condensate. If the wider pore is short, then the condensation is due to the approach of the two cylindrical interfaces, advancing from the junctions between the different pores. Upon desorption, hysteresis is observed due to the evaporation from the wide 3517


Langmuir

Figure 10. Adsorption isotherms of argon in a connected pore joining a small pore (3 nm) and larger pores (3.68, 4.3, 5, and 6.4 nm). The lengths of the smaller and the larger pores are as shown for each case.

ARTICLE

pore, whereas the smaller necks remain filled. This is the phenomenon of cavitation. To confirm these features further, we choose another example with a larger neck size (H1 = 1.7 nm) but still smaller than the critical hysteresis pore width (Hc). The larger pore width is H2 = 3 nm. The isotherms for this pore model are shown in Figure 13b for three different lengths: L2 = 2, 4, 6 nm, to study the effects of the length of the central pore. Our first observation is that a hysteresis loop is observed, no matter how long this section is. The distinction, however, is that for a short section L2 = 2 nm the hysteresis loop is not only smaller than for the two cases when L2 = 4 or 6 nm but is markedly shifted to lower pressures. This is due to the different mechanisms of condensation in the central section. When this is short, the condensation is due to the advance of the cylindrical interfaces, starting from the junctions between the pore sections, followed by merging of these two cylindrical interfaces with liquid condensate. (See the double arrow in Figure 14a.) When the central section is longer, condensation is due to the advance of the two flat interfaces, starting from the walls of the wider section, followed by the bridging and then filling of these two flat interfaces with the adsorbate fluid. (See Figure 14b.) To illustrate further the microscopic details of the condensation and evaporation, snapshots for the example with a central section of 2 nm are shown in Figure 13c. (The points A-F are labeled in the isotherm of Figure 13b.) At very low pressures, we see layers of molecules in both parts of the model. As the pressure is increased, the necks are filled with adsorbate molecules. Further increase in pressure results in capillary condensation in the center. Upon desorption, we observe evaporation of argon from the center, whereas the necks remain filled (cavitation). Any further reduction in pressure will result in desorption of argon from the necks. This process can be simply viewed as desorption from the center section with appropriate constraints. We illustrate this by considering two independent unit cells. One is a small pore with two open ends having the same width as the neck and the other is a closed pore having the same width as the center pore. They are shown in Figure 15a together with their local isotherms. By summing these two local isotherms, we see that this sum is almost the same as the isotherm obtained directly from the GCMC simulation of the ink-bottle pore (Figure 15b). The difference between the two, at pressures close to the vapor pressure, is because in the independent unit cell of a narrow pore we have two menisci connecting the pore fluid and the gas phase, whereas in the of ink-bottle pore we have only one meniscus, resulting in a lower adsorbed amount for the summed local isotherms. We now address the question of why the central pore empties while the necks remain filled. Our hypothesis is that the difference is in the degree of cohesiveness of the liquid condensate in the two sections of the model. We use local compressibility as a measure of cohesiveness and show that when the local compressibility of the neck is much smaller than that of the center section, cavitation will occur because the liquid condensate in here is fully stretched and this leads to evaporation, whereas the liquid condensate in the neck remains under-stretched and therefore remains until the pressure is further reduced to the point where the evaporation of the neck occurs. 3.4.1. Local Compressibility. Compressibility is a measure of the cohesiveness of a fluid. From the definition of the compressibility as the relative change of the volume with pressure at 3518


Langmuir

ARTICLE

Figure 11. (a) Configuration of the two unit cells for the case when the width of the smaller pore is smaller than the critical hysteresis pore width and their corresponding adsorption isotherms. (b) Adsorption isotherm of the connected pore and the isotherm obtained from the linear combination of two unit cells’ isotherms.

constant temperature k¼ -

1 DV V Dp

! ¼ T

1 f ðN, NÞ kTÆFæ ÆNæ

ð4Þ

in which we can used the fluctuation variable61 for a grand canonical ensemble. We can write the same compressibility equation for a local section K in the simulation box as kk ¼

1 f ðNk , Nk Þ Fk ¼ kTÆFk æ ÆNk æ kTÆFk æ

ð5Þ

because each section in the simulation box behaves like a subsystem subjected to the same chemical potential (i.e., in the grand canonical ensemble). Because the variable Fk is on the order of unity (except when the pressure is very high), the local compressibility is proportional to the inverse of the local density. For the ink-bottle pore, dealt with in Figure 13, we compute the compressibility of the liquid condensate in the neck (H1 = 1.7 nm) and that in the center section (H2 = 3 nm); when this has a length of 2 nm, the ratio of the compressibility in the two sections of the model is 3:08 10-9 Pa-1 ¼ 0:3 1:01 10-8 Pa-1 whereas when the length of the center section is 6 nm, this ratio is 5:75 10-10 Pa-1 ¼ 0:16 4:7 10-9 Pa-1

In both cases, the compressibility of the neck is smaller than that in the center, making the liquid condensate here more susceptible to evaporation when the pressure is reduced. Using this argument, we can deduce that pore blocking will occur when the compressibility of the neck is either greater than or comparable to that in the wider section. One example of this is an ink-bottle pore where the neck and the cavity are not very different in size and their widths are greater than the critical hysteresis pore size. We show in Figure 16 the GCMCadsorption isotherms for the ink-bottle pores with the neck having width of 3 nm (H1) and center sections of widths 3.68 and 5 nm. In the first case, where the pore widths of the two sections are comparable, we see a reasonably sharp increase in the amount adsorbed as the pressure is increased. This is due to the filling of the neck, followed immediately by the filling in the center. Upon desorption, we see a simultaneous evaporation from both sections. At the pressure just before the evaporation, we find that the compressibility in the center section is 3.50 10-9 Pa-1, which is smaller than that of the neck (1.97 10-8 Pa-1). This is opposite to what we have observed for cavitation. Therefore, the molecules in the neck evaporate, resulting in the exposure of the cavity to the bulk gas, and this then leads to an immediate evaporation from the center. This is the process of pore blocking. When H2 is increased from 3.68 to 5 nm, we observe the same phenomenon of pore blocking, the isotherm is shown in the bottom panel of Figure 16. As can be seen, compared with the former case (H2 = 3.68 nm), the condensation is more gradual, and the condensation pressure shifts to a 3519


Langmuir

ARTICLE

Figure 12. (a) Configuration of the two unit cells for the case when the width of the smaller pore is greater than the critical hysteresis pore width. (b) Adsorption isotherms of the two unit cells and its combined isotherm.

higher pressure, whereas the evaporation occurs at the same pressure. The compressibilities just before evaporation are 3.06 10-8 and 4.05 10-9 Pa-1 for the neck and the center section, respectively. Once again, it supports our hypothesis that: (i) if the compressibility of the neck is lower, then we have cavitation, and (ii) if the compressibility of the neck is higher, then we have pore blocking. To understand better the mechanisms of condensation and evaporation, we argue that if there is a critical bubble size (in our systems, the bubble is in the form of an infinitely long cylinder) condensation will occur, and there will be a critical cavity size just after evaporation. This critical cavity size is not the same as the critical bubble size in the case of adsorption. We study this with the solid model, as shown in the inset of the top panel of Figure 17. The pore width is uniform, and the surface strength is much reduced in the middle section of the pore to simulate the region for bubble formation. We choose a surface strength of εss/ k = 28 K for the sections close to the opening and a surface strength of 1 K for the middle section. To study the critical bubble size just before condensation, we vary the length of the

middle section. The adsorption isotherms for various lengths of the middle section are shown in Figure 17. We see from Figure 17 that the isotherm shows a two-stage uptake when the length of the middle part is varied from 2 to 4 nm. The first is due to the filling of the strong sections, and the second results from the filling of the weaker middle section. Because the length of the middle section is shorter, we see that the second stage uptake shifts to the left and approaches the firststage uptake. When the length is equal to the critical length (L2 = 1.5 nm), we see the filling of the stronger and weaker sections occurring simultaneously (a single-stage uptake). This critical length represents the critical bubble size (1.5 nm) prior to the condensation. For the desorption branch, we see that when the length of the middle section is greater than a critical length for desorption, the evaporation occurs at a higher pressure than that for an independent slit pore with two open ends (H = 3 nm and L = 6 nm). The surface strength of this independent pore is the same as the strong section of the heterogeneous pore. If the length of the middle section is much longer than the critical length of 3520


Langmuir

ARTICLE

Figure 13. Isotherm in ink-bottle pores open at both ends. When the cavity is short, the two cylindrical interfaces merge via a liquid bridge; when the cavity is long, the liquid bridge joins the two flat interfaces. 3521


Langmuir

ARTICLE

Figure 14. Schematic diagram of different interfaces in an Ink-Bottle pore with two open ends when the larger pore has a different length: (a) Dominated by a semicylindrical interface in the short large pore. (b) Dominated by a flat interface in the longer large pore.

Figure 15. (a) Configuration of the two unit cells and their adsorption isotherms. (b) Combined isotherm of the two unit cells and the isotherm of the ink-bottle pore.

desorption, then a two-stage desorption is observed (L2 = 4 nm). As the length of the middle pore is reduced, we see that the evaporation of the heterogeneous pore occurs at the same pressure as the independent pore when the length of the middle section is 2 nm. This is the critical length for desorption, compared with the critical length for adsorption of 1.5 nm. This finding suggests that in adsorption in a slit pore, when the core size is ∼1.5 nm, condensation will occur, and when evaporation has just occurred, the core size is ∼2 nm. These detailed calculations of compressibility confirm that the hypothesis advanced previously29 that fluctuations of the liquid in the expanded state lead to cavitation and destabilization. 3.5. Shape of the Hysteresis Loops. Having studied a number of model pore configurations, we see a range of isotherms with different shapes and sizes of hysteresis loop. Using the IUPAC classification, we list in Table 1 the common shapes

of the hysteresis loop and the possible pore configurations that could give rise to these shapes. Type H1 occurs when the adsorption and desorption branches are both (nearly) vertical, and is observed with the following pore configurations: 1 Slit pores of infinite extent. This pore gives a vertical loop. 2 Slit pores with finite length. If they are long enough, then they show a vertical loop, but the desorption branch, prior to evaporation, has a distinct knee. When the pore length is sufficiently small, the loop completely disappears. 3 Slit pore with one end closed. Type H1 loop is observed for this pore, except when the pore is sufficiently short when we have a reversible isotherm. 4 Connected system with two pores where the width of the narrower pore is less than the pore width for critical hysteresis. 3522


Langmuir

Figure 16. Adsorption isotherms of ink-bottle pores where the smaller size is greater than the critical hysteresis and close to the larger size.

5 Ink-bottle pores with two open ends. The loop is associated with the wider section. For the case of the small neck, the adsorption branch increase gradually because adsorption proceeds with the advance of the cylindrical interfaces from the junctions of the neck and the center and the advance of flat interfaces from the walls of the cavity. Desorption is the instant evaporation of the stretched (metastable) liquid in the center section. The two mechanisms are different; therefore, we have hysteresis, and this is the classic cavitation mechanism. When the neck size is greater than the critical width, the adsorption branch is sharper, and evaporation occurs simultaneously in the cavity and the neck, which is the pore-blocking mechanism. There are two other types, observed in this work, but not included in the four loops classified by the IUPAC. We shall denote these loop shapes as Types H5 and H6, respectively. For Type H5, the adsorption branch is very steep, whereas the desorption branch is shallower and occurs in two stages. The possible candidates for this hysteresis loop shape are a connected system of two pores with similar widths, both greater than the critical hysteresis size. The last loop-type (H6) is an overlapping hysteresis loop. This is observed for a connected system with two pores whose widths are greater than the hysteresis critical size when the difference between the widths is large enough. It is notable that several of the loop shapes identified in the IUPAC classification were not found for these pore geometries; in particular, the common H2 loop.

ARTICLE

Figure 17. Adsorption isotherm of pore having a weak surface strength in the middle for all curves shown in this Figure. H = 3 nm; L1 = 3 nm (the length of the high surface strength section); L2 is the length of the weak affinity section.

4. CONCLUSIONS We have presented a detailed computer simulation study of adsorption of argon in connected graphitic slit pores to study the capillary condensation and evaporation, with particular attention on the pore size and the pore length. For a simple connectivity of two pores of different sizes, we found that the adsorption and desorption behavior of the smaller pore is unaffected by the adjacent larger pore, but the adsorption behavior of the larger pore is affected, and how significant this is depends on the relative pore sizes of these pores and the length of the larger pore. In general, the capillary condensation and evaporation of the larger pore depend on the relative build-up of materials on the flat interfaces coming from the walls of the larger pore and the cylindrical interface coming from the junction of the two connected pores. If the latter is dominating, then we have a reversibility of the larger pore. If the former wins, we have a hysteresis contributed by the larger pore. We have found an interesting case where the sizes of the wide and narrow pores are comparable and are greater than the critical pore width when the condensation is vertical (filling of two pores simultaneously; an responsive effect) and the desorption is gradual in two stages, one from the wider pore, followed by one from the narrower pore. In the case of connected pores where the wider pore is bound by the two narrower pores, we have found that the cavitation and the pore blocking depend on the geometrical aspects of the connected 3523


Langmuir

ARTICLE

Table 1

pore. If the size of the narrow neck is smaller than some threshold value (Hc), then we have a cavitation in the wider pore. The pore blocking occurs when the size of the narrower necks is greater than Hc. One significant finding from this simulation study is that by using either the adsorption branch or the desorption branch we might not be able to derive correctly the pore size distribution. To provide a reason for

the distinction between the cavitation and the pore blocking, we use the local compressibility as a measure. If the local compressibility of the narrower neck is smaller than that of the middle wider pore, then we have cavitation; otherwise, we have pore blocking. Finally, we have summarized the various pore configurations that give rise to different shapes of hysteresis loop. 3524


Langmuir

’ APPENDIX 1 The Taylor series expansion of the Bojan-Steele solid-fluid potential energy equation is ( " 10 # φf , S 63 σsf 10 σ sf ¼ - þ 1280 yΨ2πðFs σ 2sf Þεsf y " #) " 4 2 ( 3 σsf 4 σ sf z 231 σ sf 12 16 yσ sf 1024 yyþ " 12 6 #) σsf 5 σsf 6 σsf - þ - þ 16 y y y " " 4 ( 14 # 14 z 1287 σ sf σsf 105 σsf 8 þ - þ σ sf 2048 y256 yy #) ( " 8 6 16 # σ sf z 45045 σsf 16 σsf - þ - þ σsf 32768 yy y " #) ( " 10 8 63 σsf 10 σ sf z 85085 σ sf 18 - þ þ 128 yσsf 32768 yy " #) 18 12 σ sf 1155 σ sf 12 σsf - þ - þ þ Oðz=σsf Þ10 2048 yy y where Ψ = þ1 for positive yþ and y- and Ψ = -1 for negative yþ and y-. We see that there is no singularity in the solid-fluid potential energy, as one would physically expect, and this is important in the GCMC calculations when the z coordinate of a particle is very close to zero.

’ AUTHOR INFORMATION Corresponding Author

*Fax: þ61-7-3365-2789. E-mail: [email protected].

’ ACKNOWLEDGMENT This project is funded by the Australian Research Council. ’ REFERENCES (1) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area, and Porosity, 2nd ed.; Academic Press: London, 1982. (2) Rouquerol, F.; Rouquerol, J.; Sing, K. Adsorption by Powders and Porous Solids; Academic Press: London, 1999. (3) Everett, D. H. Adsorption Hysteresis; Marcel Dekker: New York, 1967; Vol. 2, pp 1055-1110. (4) De Boer, J. H. The structure and texture of a physical adsorbent. Colloq. Int. C.N.R.S. 1972, 201, 407. (5) Sing, K. S. W.; Everett, D. H.; Haul, R. A. W.; Moscou, L.; Pierotti, R. A.; Rouquerol, J.; Siemieniewska, T. Reporting physisorption data for gas/solid systems with special reference to the determination of surface area and porosity (Recommendations 1984). Pure Appl. Chem. 1985, 57, 603–619. (6) Sing, K. S. W.; Williams, R. T. Physisorption hysteresis loops and the characterization of nanoporous materials. Adsorpt. Sci. Technol. 2004, 22, 773–782. (7) Cohan, L. H. Sorption hysteresis and the vapor pressure of concave surfaces. J. Am. Chem. Soc. 1938, 60, 433–435. (8) Cohan, L. H. Hysteresis and the capillary theory of adsorption of vapors. J. Am. Chem. Soc. 1944, 66, 98–105. (9) McBain, J. W. An explanation of hysteresis in the hydration and dehydration of gels. J. Am. Chem. Soc. 1935, 57, 699–700.

ARTICLE

(10) Mason, G. Model of adsorption desorption hysteresis in which hysteresis is primarily developed by the interconnections in a network of pores. Proc. R. Soc. London, Ser. A 1983, 390, 47. (11) Mason, G. The effect of pore-space connectivity on the hysteresis of capillary condensation in adsorption desorption isotherms. J. Colloid Interface Sci. 1982, 88, 36. (12) Mayagoitia, V. R.; Kornhauser, F.; Perez-Aguilar, I. Modeling of porous media and surface structures: their true essence as networks. Langmuir 1997, 13, 1327. (13) Heffelfinger, G. S.; van Swol, F.; Gubbins, K. E. Liquid-vapour coexistence in a cylindrical pore. Mol. Phys. 1987, 61, 1381–1390. (14) Heffelfinger, G. S.; van Swol, F.; Gubbins, K. E. Adsorption hysteresis in narrow pores. J. Chem. Phys. 1988, 89, 5202–5205. (15) Heffelfinger, G. S.; Tan, Z.; Gubbins, K. E.; Marconi, U. M. B.; Vanswol, F. Fluid mixtures in narrow cylindrical pores - computer simulation and theory. Int. J. Thermophys. 1988, 9, 1051–1060. (16) Peterson, B. K.; Gubbins, K. E. Phase transitions in a cylindrical pore - grand canonical Monte Carlo, mean field theory and the Kelvin equation. Mol. Phys. 1987, 62, 215–226. (17) Peterson, B. K.; Gubbins, K. E.; Heffelfinger, G. S.; Marconi, U. M. B.; van Swol, F. Lennard-Jones fluids in cylindrical pores: nonlocal theory and computer simulation. J. Chem. Phys. 1988, 88, 6487–6500. (18) Peterson, B. K.; Heffelfinger, G. S.; Gubbins, K. E.; van Swol, F. Layering transitions in cylindrical pores. J. Chem. Phys. 1990, 93, 679– 685. (19) Peterson, B. K.; Waltont, J. P. R. B.; Gubbins, K. E. Fluid behaviour in narrow pores. J. Chem. Soc., Faraday Trans. 1986, 2, 1789–1800. (20) Schoen, M.; Rhykerd, C. L.; Cushman, J. H.; Diestler, D. J. Slitpore sorption isotherms by the grand-canonical Monte-Carlo method manifestations of hysteresis. Mol. Phys. 1989, 66, 1171–1182. (21) Marconi, U. M. B.; van Swol, F. A model of hysteresis in narrow pores. Europhys. Lett. 1989, 8, 531–535. (22) Hill, T. L. Statistical mechanics of multimolecular adsorption. 0.3. Introductory treatment of horizontal interactions - capillary condensation and hysteresis. J. Chem. Phys. 1947, 15, 767. (23) Rasmussen, C. J.; Vishnyakov, A.; Thommes, M.; Smarsly, B. M.; Kleitz, F.; Neimark, A. V. Cavitation in metastable liquid nitrogen confined to nanoscale pores. Langmuir 2010, 26, 10147–10157. (24) Heffelfinger, G.; van Swol, F. Diffusion in Lennard-Jones fluids using dual control volume grand canonical molecular dynamics simulation (DCV-GCMD). J. Chem. Phys. 1994, 100, 7548–7552. (25) MacElroy, J. M. D. Nonequilibrium molecular dynamics simulation of diffusion and flow in thin microporous membranes. J. Chem. Phys. 1994, 101, 5274–5280. (26) Cracknell, R. F.; Nicholson, D.; Quirke, N. Direct moleculardynamics simulation of flow down a chemical-potential gradient in a slitshaped micropore. Phys. Rev. Lett. 1995, 74, 2463–2466. (27) Libby, B.; Monson, P. A. Adsorption/desorption hysteresis in inkbottle pores: a density functional theory and Monte Carlo simulation study. Langmuir 2004, 20, 4289–4294. (28) Sarkisov, L.; Monson, P. A. Hysteresis in Monte Carlo and molecular dynamics simulations of adsorption in porous materials. Langmuir 2000, 16, 9857–9860. (29) Sarkisov, L.; Monson, P. A. Modeling of adsorption and desorption in pores of simple geometry using molecular dynamics. Langmuir 2001, 17, 7600–7604. (30) Marconi, U. M. B.; Vanswol, F. Microscopic model for hysteresis and phase-equilibria of fluids confined between parallel plates. Phys. Rev. A 1989, 39, 4109–4116. (31) Monson, P. A. Mean field kinetic theory for a lattice gas model of fluids confined in porous materials. J. Chem. Phys. 2008, 128, 084701. (32) Vishnyakov, A.; Neimark, A. V. Monte Carlo simulation test of pore blocking effects. Langmuir 2003, 19, 3240–3247. (33) Ravikovitch, P. I.; Neimark, A. V. Experimental confirmation of different mechanisms of evaporation from ink-bottle type pores: equilibrium, pore blocking, and cavitation. Langmuir 2002, 18, 9830– 9837. 3525


Langmuir (34) Ravikovitch, P. I.; Neimark, A. V. Density functional theory of adsorption in spherical cavities and pore size characterization of templated nanoporous silicas with cubic and three-dimensional hexagonal structures. Langmuir 2002, 18, 1550–1560. (35) Wongkoblap, A.; Do, D. D. Effect of pore constriction on adsorption behaviour in nanoporous carbon slit pore: a computer simulation. Mol. Simul. 2006, 32, 539–549. (36) Nguyen, T. B., S. Accessibility of simple gases in disordered carbons: theory and simulation. Asia-Pacific Journal of Chemical Engineering 2009, 4, 557. (37) Kikkinides, E. S.; Kainourgiakis, M. E.; Stubos, A. K. Origin of hysteresis of gas adsorption in disordered porous media: lattice gas model versus percolation theory. Langmuir 2003, 19, 3338–3344. (38) Hill, T. L. Note on the physical adsorption of gases in capillaries and on small particles (Nucleation of Condensation). J. Phys. Chem. 1950, AA3, 1186. (39) Neimark, A. V.; Vishnyakov, A. The birth of a bubble: a molecular simulation study. J. Chem. Phys. 2005, 122, 5. (40) Neimark, A. V.; Vishnyakov, A. Phase transitions and criticality in small systems: Vapor-liquid transition in nanoscale spherical cavities. J. Phys. Chem. B 2006, 110, 9403–9412. (41) Ustinov, E. A.; Do, D. D. Modeling of adsorption in finite cylindrical pores by means of density functional theory. Adsorption 2005, 11, 455–477. (42) Ustinov, E. D.; D. D. Capillary phenomena in the framework of the two-dimensional density functional theory. Adsorption 2005, 11, 133. (43) Husowitz, B. T., V Nucleation in cylindrical capillaries. J. Chem. Phys. 2004, 121, 8021. (44) Coasne, B.; Galarneau, A.; Di Renzo, F.; Pellenq, R. Molecular simulation of adsorption and intrusion in nanopores. Adsorption 2008, 14, 215–221. (45) Young, D. M. Cavitation; McGraw-Hill: New York, 1989. (46) Valiullin, R.; Naumov, S.; Galvosas, P.; Karger, J.; Woo, H. J.; Porcheron, F.; Monson, P. A. Exploration of molecular dynamics during transient sorption of fluids in mesoporous materials. Nature 2006, 443, 965–968. (47) Gelb, L. D. The ins and outs of capillary condensation in cylindrical pores. Mol. Phys. 2002, 100, 2049–2057. (48) Coasne, B. G., K. E.; Pellenq, R. J. M A grand canonical Monte Carlo study of adsorption and capillary phenomena in nanopores of various morphologies and topologies: testing the BET and BJH characterization methods. Part. Part. Syst. Charact. 2004, 21, 149. (49) Coasne, B.; Hung, F. R.; Pellenq, R. J. M.; Siperstein, F. R.; Gubbins, K. E. Adsorption of simple gases in MCM-41 materials: the role of surface roughness. Langmuir 2006, 22, 194–202. (50) Coasne, B.; Galarneau, A.; Di Renzo, F.; Pellenq, R. J. M. Effect of morphological defects on gas adsorption in nanoporous silicas. J. Phys. Chem. C 2007, 111, 15759–15770. (51) Pellenq, R. J. M.; Coasne, B.; Denoyel, R. O.; Coussy, O. Simple phenomenological model for phase transitions in confined geometry. 2. Capillary condensation/evaporation in cylindrical mesopores. Langmuir 2009, 25, 1393–1402. (52) Frenkel, D.; Smit, B. Understanding Molecular Simulation: From Algorithms to Applications; Academic Press: San Diego, 1996; pp 17, 443. (53) Hauschild, T.; Prausnitz, J. M. Monte Carlo calculations for methane and argon over a wide range of density and temperature, including the two-phase vapor-liquid region. Mol. Phys. 1993, 11, 177. (54) Do, D. D.; Do, H. D. Effects of potential models in the vaporliquid equilibria and adsorption of simple gases on graphitized thermal carbon black. Fluid Phase Equilib. 2005, 236, 169–177. (55) Bojan, M. J.; Steele, W. A. Computer simulation of physisorption on a heterogeneous surface. Surf. Sci. 1988, 199, L395–L402. (56) Bojan, M. J.; Steele, W. A. Computer simulations of the adsorption of xenon on stepped surfaces. Mol. Phys. 1998, 95, 431–437. (57) Bojan, M. J.; Steele, W. A. Computer simulation of physisorbed krypton on a heterogeneous surface. Langmuir 1989, 5, 625–633. (58) Bojan, M. J.; Steele, W. A. Computer simulation of physical adsorption on stepped surfaces. Langmuir 1992, 9, 2569–2575.

ARTICLE

(59) Coasne, B.; RJ-M, P. A grand canonical Monte Carlo study of capillary condensation in mesoporous media: effect of the pore morphology and topology. J. Chem. Phys. 2004, 121, 3767–3774. (60) Puibasset, J. l. Monte-Carlo multiscale simulation study of argon adsorption/desorption hysteresis in mesoporous heterogeneous tubular pores like MCM-41 or oxidized porous silicon. Langmuir 2009, 25, 903–911. (61) Nicholson, D.; Parsonage, G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: London, 1982.

3526


On the Cavitation and Pore Blocking in Slit-Shaped Ink-Bottle Pores

Recommend Documents