ARTICLE pubs.acs.org/Langmuir
Cavitation in Metastable Fluids Confined to Linear Mesopores Annie Grosman* and Camille Ortega Institut des NanoSciences de Paris (INSP), Universite Paris 6, UMR-CNRS 75-88, 4 Place Jussieu 75005 Paris, France ABSTRACT: We study the adsorption process of nitrogen (at 77.4 and 51.3 K) and argon (at 60 K) in porous silicon duplex layers, Si/A/B and Si/B/A, where the pores of A are on average narrower than the pores of B. We compare the experimental isotherms to that calculated from elemental isotherms measured in layers A and B supported by or detached from the silicon substrate. This allows us to confirm our previous studies which show that the relaxation of the substrate constraint modifies the adsorption strains and leads to a decrease of the adsorbed amount before condensation and consequently increases the condensation pressure. In the socalled ink-bottle Si/B/A configuration, layer B empties while layer A remains filled which proves that layer B empties via cavitation. The vapor pressure at which cavitation occurs in layer B in Si/B/A configuration is close to the pressure at which the same layer empties when it is in direct contact with the gas reservoir (Si/A/B configuration) which indicates that layer B contains all the ingredients necessary for cavitation to occur. The absolute value of the liquid pressure at which cavitation occurs is much lower than the value predicted by the theory of homogeneous nucleation. Nucleation of gas bubbles thus takes place on the surface of the pore walls. This is the crucial point of the paper. A receding meniscus with a contact angle lower than π/2 inside a pore and a gas bubble with a contact angle higher than π/2 are thus mutually exclusive. A receding meniscus cannot enter a pore. This has nothing to do with a pore-blocking effect; this is related to the physical parameters which define the contact angle inside the pores, that is, the surface energies at the solid-liquid, solid-vapor, and liquid-vapor interfaces. For argon at 60 K in the Si/B/A duplex layer, cavitation in layer B activates the emptying of a fraction of pores of layer A which constitutes a direct observation of metastable states.
I. INTRODUCTION The hysteretic behavior of fluid in porous silicon, an assembly of non-interconnected linear mesopores,1,2 is very similar to that of porous glass.3-5 The same asymmetrical hysteresis loop with a large condensation branch and a steep desorption branch (type H2 in the IUPAC classification),6 the same hysteretic behavior inside the main loop which shows that the pores interact during the emptying process. The parameter which couples the pores is not interconnectivity. Similar results have been found in SBA-15 silica7 and porous alumina.8 Two approaches were considered to explain the triangular shape of the hysteresis loop observed in porous silicon. (I) The pores are heterogeneous and can be divided into segments randomly distributed along the pore axis, the domains, which differ by the size and roughness,9,10 or by the fluidwall interaction.11,12 They are not the pores which interact since they are not connected to one another but the domains inside each pore. This is reminiscent of a pore-blocking effect in each pore. Porous silicon is assimilated to a one-dimensional network of narrow (or more attractive) sections. (II) The morphological properties of porous silicon, summarized in section II, and the adsorption experiments we performed incited us to explain the cause of the hysteresis phenomenon another way than by the presence of disorder in the pores. We try to explain the hysteretic behavior13-15 by taking into account the elastic strains undergone by the solid during the adsorption process, a phenomenon which has been known r 2011 American Chemical Society
for a long time.16-20 More recently, adsorption strains have been studied in various porous materials such as aerogel,21,22 porous silicon,23 mesoporous silica,24-27 or carbon nanotubes.28 On the other hand, the reciprocal effect of the elastic deformation on the adsorption process, predicted by Newton’s third law, is neglected in the overwhelming majority of studies. This effect has been studied using thermodynamic,13 experimental,14 or analytical,29 and Monte Carlo simulations.30,31 In the hysteresis loop region, the elastic deformation has dramatic effects on the desorption process.14,15,19 Based on experimental results obtained for porous glass18 and porous silicon,23 we have shown that the surface free energy, a physical parameter directly related to the elastic deformation of the solid, is of the same order of magnitude as the other components of the free energy and that the energy barrier to desorption depends essentially on this parameter. Neglecting the deformation cancels out the energy barrier. This is supported by recent calculations.29 The above studies should question the models and calculations for which the key parameter is the geometrical or chemical inhomogeneities. The authors of the calculations should determine the energy difference between the condensation Received: August 5, 2010 Revised: January 3, 2011 Published: February 08, 2011 2364
dx.doi.org/10.1021/la104777y | Langmuir 2011, 27, 2364–2374
Langmuir and desorption branches on the top of the plateau of the hysteresis loop. It will allow to compare their results to ours and decide which parameter is the most important, the disorder or the elastic deformation. In the present work, we pay attention to the desorption process. In principle, the pores can empty either via receding menisci or via nucleation of gas bubbles if the negative liquid pressure is sufficiently low. In complex materials composed of cavities separated from one another by constrictions, it is believed that desorption in the cavities can occur only after emptying of the constrictions via a receding meniscus. The cavities do not empty independently of each other, and the interaction mechanism is percolation. This is referred to as “pore-blocking percolation” mechanism.4,5 Now, if the size of the constriction is sufficiently low, the liquid pressure in the cavities can reach the fracture pressure of the liquid and the cavities can empty via nucleation and growth of gas bubbles in the metastable liquid while the constrictions remain filled. In this model, both desorption mechanisms, percolation and cavitation, originate from geometrical pore blocking whereas they are quite different by nature.32,33 For N2 at 77 K, the relative vapor pressure which delimits the two mechanisms is estimated to be p = 0.5. It is assumed that the nucleation occurs in the bulk liquid. The authors33 justify this assumption by invoking the presence of an adsorbed film on the pore wall surface which would “protect the interior of pores from surface pollutants and irregularities that serve as nucleation centers.” In the linear pores of porous silicon, one can reproduce artificially the so-called ink-bottle configuration by making duplex layers with different pore size distributions (PSDs). It is what Wallacher and co-workers made.34 By studying the adsorption of N2 at 77.2 K in duplex porous silicon layer Si/B/A where the pores of A are on average narrower than the pores of B, the authors have shown that layer B in configuration Si/B/A empties at a relative vapor pressure close to the pressure at which the same layer empties when it is in direct contact with the gas reservoir. The authors interpreted this result by suggesting the presence of a “quenched disorder imposed by some variation of the pore diameter along the pores.” Two other important results are shown in this paper but not discussed. Layer B empties while layer A remains filled, which proves that it empties via cavitation and, more important, the relative vapor pressure at which cavitation occurs, p = 0.6, is significantly higher than the supposed limit corresponding to homogeneous nucleation, p = 0.5. According to Naumov and co-workers,9,10 the occurrence of cavitation in porous silicon is caused by the presence of roughness and narrow sections randomly distributed in each pore of layer B which impose a negative liquid pressure sufficiently high, in absolute value, to cause the fracture of the bulk liquid. However, the authors do not discuss whether the cavitation vapor pressure value they found, ∼0.6, is compatible with homogeneous nucleation. If we refer to the literature, the fracture for liquid nitrogen at 77 K cannot be observed at pressure higher than 0.5, and the answer is no. The relatively high vapor pressure at which cavitation occurs in layer B together with the fact that, contrary to the assertion of Rasmussen and co-workers,33 we think that the large pore surface of porous materials is probably an infinite source of natural nucleation sites for gas bubbles incite us to question, in the present paper, the validity of the assumption of homogeneous cavitation. Following the work of Wallacher and co-workers,34 we study here the adsorption process of nitrogen (at 77.4 and 51.3 K) and argon (at 60 K) in porous silicon duplex layers Si/A/B and Si/B/A
ARTICLE
but where layers A and B exhibit very different PSDs so that the two desorption branches do not overlap and that desorption in layer B occurs at a relative vapor pressure as high as 0.8 that is far from the limit value for the fracture of the bulk liquid assumed to be 0.5. Unlike the previous experiment,34 the two layers were prepared on the same substrate by changing the formation conditions so that the layers have the same morphology and differ only by their PSDs and that it is possible to prepare duplex layers in Si/ A/B configuration. We compare the experimental isotherms corresponding to the Si/A/B and Si/B/A configurations to that calculated from elemental isotherms measured in layers A and B either supported by or detached from the silicon substrate. This allows us to bring additional proof of the influence on the adsorption process of the stress exerted by the substrate on the porous solid. Furthermore, this gives important information on the physical structure of the duplex layers. As already found,34 layer B full of liquid N2 empties via cavitation in configuration Si/B/A. Using the classical KelvinLaplace equation, we estimate the negative liquid pressure at which cavitation occurs in this layer and address the fundamental question of whether nucleation of gas bubbles occurs in the bulk fluid or on the pore walls. The question whether cavitation is caused by the pore-blocking effect, as proposed in the literature,9,10,33 or by another phenomenon is discussed.
II. EXPERIMENTAL SECTION A. Materials. A detailed description of the preparation and characteristics of the porous layers can be found in previous papers.1,2,35 We recall below the information required for the understanding of the paper. The mesoporous silicon layers are prepared by anodic dissolution in hydrofluoric (HF) acid solutions of highly boron doped (100) Si single crystal at constant current density. The tubular pores are perpendicular to the Si substrate and separated from each other by silicon walls of apparent constant thickness (∼5 nm). The pore length and the porosity of the layers, measured by weighing, are controlled by the conditions (HF concentration and current density) and duration of the formation. We have shown that the pores are not interconnected1 and that their average direction is parallel, to within 0.1, to the [100] crystal axis.35 The total pore volume, Vpores, deduced from N2 content measurement when the porous system is filled is proportional to the formation time which shows that there is no porosity gradient. The numerical treatment of transmission electronic microscopy (TEM) plane views of the surface of the samples, a 2D image, yields the same porosity as that obtained by weighting which corresponds to the 3D morphology.2 These 2D numerical treatments allow us to obtain a good estimation (see below) of the PSD of the layers corresponding to cylindrical pores having the same cross section area as the polygonal pores. The mean value of the distribution can be determined more precisely another way. Knowing Vpores, a macroscopic value known with a very good precision, the mean diameter is equal to ÆDæ = (4Vpores/ πNporesL)1/2, where Npores is the pore number determined by a simple counting on the 2D images which contain between 500 and 600 pores. The error on the counting, due to the fact that in some places of the micrographs the pores cannot be separated from each other precisely, does not exceed 10/500 = 2%, that on the pore volume, Vpores, is less than 1%, and that on the layer thickness, L, is about 3%. The mean value of the distributions is thus given within =3%. For samples A and B, the characteristic parameters of which are presented in Table 1, we found ÆDæA = 12.8 nm and ÆDæB = 26.6 nm. These macroscopic values are close to the values deduced directly from the numerical treatment of the 2D images, that is, 13 and 26 nm for samples A and B, respectively. 2365
dx.doi.org/10.1021/la104777y |Langmuir 2011, 27, 2364–2374
Langmuir
ARTICLE
Table 1. Formation Condition and Main Characteristic Parameters of Porous Silicon Layers A and Ba J (mA/cm2)
HF/EtOH
etching time (min)
porosity
thickness (μm)
pore density (1011 cm-2)
ÆDæ (nm)
sample A
20
3:1
20
51.4%
21.2
3.7
12.8
sample B
50
1:1
10
70.8%
18.4
1.2
26.6
porous layer
The surface electrochemically etched on the Si wafer equals 15.2 cm2. The substrate is pþ-type (3 10-3 Ω 3 cm) [100] Si. ÆDæ = (4Vpores/πNporesL)1/2, where Npores is the pore number, L is the layer thickness, and Vpores is the total pore volume deduced from N2 adsorption at 77.4 K which equals 15.5 10-3 and 18.7 10-3 cm3 for samples A and B, respectively. The surface areas deduced from the BET model analysis38 are SABET = 9.6 m2 and SBBET = 5.4 m2. a
The surface pore density, determined by counting the number of pores on the 2D images, is greater for sample A (3.7 1011 /cm2) than for sample B (1.2 1011 /cm2) which corresponds to a ratio equal to 3.08. This ratio can be determined another way from ÆDæA, ÆDæB, and the pore wall thickness (∼5 nm) values. We find (26.6 þ 5)2/(12.8 þ 5)2 = 3.15 which is very close to 3.08. We have thus complete confidence in our determination of the PSDs from TEM measurements, 13 ( 6 nm and 26 ( 14 nm. To date, in mesoporous materials, the experimental methods do not allow one to provide precise quantitative information on the variation of the pore section with depth. Nevertheless, our numerous TEM,1,2 nuclear microanalysis,35 and adsorption data studies suggest that, apart from the presence of a roughness at the atomic scale, shown by electronic paramagnetic resonance studies,36 the cross section area of each pore is rather constant with depth. Anyway, it is important to keep in mind that the distribution with depth of the pore size as well as the mean value of the pore size are not a key point for this work. As we shall see, the key parameter is the value of the liquid pressure which we determine by using the classical KelvinLaplace equation (see section IV-A) which allows us to free ourselves from the knowledge of the morphology. The consistency of the pore wall thickness can be explained by the fact that, during the pore formation, once the pore wall thickness reaches a limiting value of ∼5 nm, the pore walls become highly electrically resistive (F = 105 Ω 3 cm), which prevents any further anodic dissolution.37 This property can be used to prepare (i) porous Si membranes with tubular pores open at both ends and (ii) duplex porous layers having different porosities, that is, different PSDs, with the whole being supported by the Si substrate. (i) Porous Si membranes: Immediately after the porous layer formation, the current density is increased up to a value corresponding to the electropolishing regime, without changing the other experimental conditions. During this step, because of the high resistivity of the pore walls, the electric field lines are focused on the bottom of the pores where the Si walls are dissolved. Consequently, the porous membrane thus obtained is identical to the corresponding supported layer. This has been discussed at length in a previous paper.2 We bring further evidence of this in the present paper. (ii) Duplex porous layers: The method consists of first making a porous layer with chosen porosity and thickness values and then changing the formation conditions, that is, the HF/EtOH solution and/or the current density value to make a second porous layer of different porosity underlying the first one. Note that, as no chemical and electrochemical dissolutions of the pore walls occur once they are formed, the formation of the second porous layer does not affect the morphology of the first one. Note also that, as the two layers were prepared on the same substrate, they have the same morphology and they only differ by their pore size. Two different porous layers have been prepared, about 20 μm thick, with porosity face values of 50% (sample A) and 70% (sample B) either supported by or detached from the substrate. The layers supported by the substrate are called Asl and Bsl, and the corresponding membranes Am and Bm. Table 1 summarizes the formation condition parameters together with the main characteristic parameters of these layers. We have also made duplex porous layers noted Si/A/B and Si/B/A.
B. Measurements. The adsorption isotherms and scanning curves were measured using a Micromeritics ASAP2010 instrument equipped with pressure transducers with full scales of 1 μmHg, 10 mmHg, and 1000 mmHg, respectively. A homemade cryogenic refrigeration system with a specific analysis cell can be connected to the classical ASAP2010 apparatus. This cooling system is composed of a compressor module (helium gas at 200 bar) connected, via an expander module, to a cryostat which allows one to perform adsorption studies at constant temperature ((0.1 K) in the range 16 K ambient. A cell has been fabricated where porous samples formed on two inch Si wafers can be analyzed. The adsorption isotherms were performed at 77.4 and 51.3 K for nitrogen, and 60 K for argon. The cross section views of the interfaces between the layers A and B in the Si/A/B and Si/B/A duplex porous layers were obtained using the high spatial resolution of a field emission gun scanning electron microscope (FEG-SEM). The beam energy was 5 kV, and the magnification was 250000.
III. RESULTS AND DATA ANALYSIS Figure 1 shows the nitrogen adsorption isotherms at 77.4 K for layers Asl and Bsl supported by the substrate and for the corresponding membranes Am and Bm. The two PSDs are represented in the inset. These isotherms have been reproduced several times and discussed at length in previous papers.1,2,13 Let us have some preliminary comments about the results presented in Figure 1. (1) Whereas the supported porous layers are strictly similar to their corresponding membranes, the isotherms for these two types of layers are distinct. At a given pressure, the adsorbed amount before capillary condensation is lower for the membranes than for their corresponding supported layers and the condensation pressures are higher. We have recently explained the reason of such a behavior.13 During the adsorption process, these two types of layers react differently to the spreading pressure exerted by the adsorbed molecules. The supported layer is subjected to an additional stress, that exerted by the substrate, which constrains the planes perpendicular to the substrate to have the same interatomic spacing as that of the substrate. This external stress results in an additional extension along the [100] direction: at a given pressure, the lattice parameter parallel to the [100] direction is higher for the supported layer than for the membrane. To these different lattice parameters correspond different surface free energies and then different adsorbed amounts and condensation pressures. The lower the deformation along the [100] direction, the lower the adsorbed amount before capillary condensation and the higher the condensation pressure. This is largely corroborated by the numerous experiments presented in the following of the present paper. (2) The condensation pressure for porous silicon is low compared to that of SBA-15 or MCM-41 silica which would have similar pore size. For instance, for sample A 2366
dx.doi.org/10.1021/la104777y |Langmuir 2011, 27, 2364–2374
Langmuir
Figure 1. Nitrogen adsorption isotherms at 77.4 K for porous silicon samples A (b, O) and B (9, 0) of 50% and 70% porosity, respectively. The black symbols correspond to layers supported by the Si substrate, and the open symbols to self-supported layers, the so-called membranes. The inset shows the two PSDs, corresponding to cylindrical pores having the same cross section area as the polygonal pores, 13 ( 6 and 26 ( 14 nm for samples A and B, respectively.2 The hatched region represents the overlap of the two PSDs. The lines between the experimental points are guides for eye.
and the SBA-15 material that we have studied,7 the relative condensation pressures are similar, p = 0.7, while the mean pore size is =13 nm for sample A and =8.5 nm for SBA-15 silica. We conjecture that the gap between the condensation branches for porous silicon layers and SBA15 or MCM-41 has fundamentally the same cause as the gap between the condensation branches for the supported porous silicon layer and corresponding membrane: different adsorption strains. A start of an explanation was proposed,13 but more work, both theoretical and experimental, must be done to establish a clear picture of this phenomenon. (3) We have also shown that the isotherms for the supported porous layers shift toward that of the membrane when the layer thickness is increased from 30 to 100 μm.13 This is due to the partial relaxation of the stress exerted by the substrate as a result of the breaking of Si-Si bonds at the interface between the substrate and the porous layer. The membrane is the relaxed state of the supported layer. This probably explains why Naumov and co-workers found superimposed experimental isotherms for both types of layers.9 This is at length discussed in ref 13. (4) Let us note that if the shape of the hysteresis loop (type H2) for porous silicon has been reproduced by introducing strong disorder in the pores,9,10 nevertheless, two experimental results shown in Figure 1 cannot be explained by such models.10,12 (a) According to these models, we should observe, along the plateau of the hysteresis loop, desorption from the largest (or less attractive) segments located between the top of the pore and the narrowest (or more attractive) segments. Instead, we have shown for sample A that the amount of matter desorbed along the plateau of the hysteresis loop comes only from the nonporous surface (analysis cell and external surface of the sample) and from the formation of the menisci.2 This shows that the menisci are blocked at the pore
ARTICLE
Figure 2. (a) Nitrogen adsorption isotherms at 77.4 K, for Si/A/B (b) and Si/B/A (O) porous duplex layers, schematically represented in inset. Note that layer A in Si/B/A configuration behaves like a membrane (see Figure 1) due to the presence of dangling walls. The lines between the experimental points are guides for the eye.
ends by a phenomenon other than pore blocking. The conclusions of the present paper (section IV-C) based on experimental results independent of those presented in Figure 1 go this way and allow us to take a step forward in the understanding of this phenomenon. (b) These models which do not take into account the elastic deformation of the solid also do not reproduce the isotherms for porous silicon layers supported by and detached from the substrate. The experimental isotherms are distinct while the calculated ones are superimposable. These comments being made, let us go into the heart of the matter. Figure 2 shows the experimental nitrogen adsorption isotherms at 77.4 K corresponding to the duplex porous layers Si/ B/A and Si/A/B. These isotherms will be analyzed using the elemental ones shown in Figure 1. Let us first give some informations about the interface between the two layers. Since the pore density is greater for sample A (3.7 1011 /cm2) than for sample B (1.2 1011 /cm2), layer A has inevitably dangling walls at the interface between the two layers in the Si/B/A configuration. The inset of Figure 2 schematically represents the two configurations. After the formation of the porous layers, the Si wafers are generally cut in small pieces which can be introduced in a conventional cell. As the purpose of the paper is to study the filling and emptying of a layer through a layer full of liquid, it was important to verify that the interface between these two layers is not accessible to the gas reservoir from the sides of the layers. We have thus performed, in a homemade cell, adsorption experiments on whole Si wafers in which the porous layers are completely surrounded by compact Si so that the interface between the two layers is not in contact with the gas reservoir. As shown in Figure 3, the isotherms performed on whole wafers are identical to those performed on small pieces. We have also made scanning electron microscopy (SEM) observations of the interface in both configurations (see Figure 4). There is no visible fracture at a scale of a few tens of nanometers, that is, at a scale of some pores, which shows that even if the pores communicate with one another on short distance, the two layers have common pore walls which prevent the gas reservoir to enter the porous material from the sides of the samples. We shall bring additional proof of this in the continuation of the Article. 2367
dx.doi.org/10.1021/la104777y |Langmuir 2011, 27, 2364–2374
Langmuir
ARTICLE
Figure 3. Nitrogen adsorption isotherms at 77.4 K normalized to the same pore volume performed on Si/B/A duplex porous layers either cut into pieces before the analysis (O) or as-prepared on a whole Si wafer in which the porous layers are completely surrounded by compact Si (b) as schematically represented by insets (a) and (b), respectively. The lines between the experimental points are guides for the eye.
Duplex Layer Si/A/B. We first analyze the results concerning the duplex layer Si/A/B. Figure 5 represents a zoom on the hysteresis region of the isotherm shown in Figure 2. During the adsorption process, the pores of layer A, which are smaller than those of layer B, first fill, the end of the filling of layer A being indicated by the presence of a tipping point on the condensation branch at a relative pressure p = P/P0 = 0.8. At this pressure, the smallest pores of B are already filled. Indeed, as shown in Figure 1, the PSDs and condensation branches of samples A and B overlap one another so that the largest pores of A and the smallest pores of B capillary condense in the same pressure range. However, as the width of the intersecting region of the two PSDs is small compared to the width of each of them, the major part of the largest pores of A are not located just in front of the narrowest pores of B so that they do not fill through the pores of layer B already filled. As regards to the desorption branch, such a question does not arise since the desorption branches corresponding to samples A and B do not overlap at all, so that when layer A begins to empty layer B is already empty. In summary, in configuration Si/A/B, layers A and B should fill and empty independently of one another. It is what the results of Figure 5 show. The adsorption isotherm corresponding to this duplex layer is well fitted by the addition of isotherms corresponding to samples Asl and Bsl shown in Figure 1, that is, Si/Asl/ Bsl. For layer A, this is not surprising because it is in direct contact with the substrate. For layer B, this is less evident since the contact with the substrate is through layer A; nevertheless, layer B behaves as if it was in direct contact with the substrate. This shows that most of the pore walls of B are in contact with the substrate, that is, are common to both layers from the top to the bottom of the duplex layer making the interface between the two layers nonaccessible to the gas reservoir from the side of the layers, as in the case of Si/B/A configuration. This suggests that the pores of A are dug inside the pores of B, three pores of A in each pore of B, on average, as indicated by the pore density ratio value or equivalently by the ratio of the mean surface area of the pore sections (see section II-A). Note that, as expected (see section II-A), the formation of layer A leaves unchanged the morphology of layer B already formed, otherwise we would not have been able to fit the filling of layer B.
Figure 4. Scanning electron microscopy images of cross sections of the interfaces between layers A and B in the Si/B/A (a) and Si/A/B (b) duplex porous layers. Note the abruptness of the interface between the two layers and that there is no visible fracture. It is noteworthy that these images which were obtained by cleaving the samples before analysis do not represent plane surfaces (001) or (010). The apparent disorder is hence due to the fact that the pore walls are generally in different planes. For instance, the region in the ellipse in panel (a) which seems to be, at a first glance, a constriction is actually a piece of a pore wall in the foreground compared to the plane containing the axis of this pore. On the other hand, the region under the arrow in panel (a) with less topological defects exhibits straight and parallel pore walls.
This confirms that, once a Si porous layer is made, it is unchanged during any further anodic dissolution. This evidently holds also for the electropolishing process used to obtain self-supported membranes and brings an additional proof, if necessary, to the strict identity between the supported layer and membrane. Layers A and B in the Si/A/B configuration behave as if they were alone in contact with the gas reservoir and the substrate. This supports the fact that the elastic deformations are laterally transmitted inside each layer but not longitudinally from one layer to the other. To compress or stretch a part of a spring does not change the state of the rest of the spring. Duplex Layer Si/B/A. We consider now the duplex layer Si/ B/A, the so-called ink-bottle configuration with the narrowest pores, those of A, in contact with the gas reservoir and the largest 2368
dx.doi.org/10.1021/la104777y |Langmuir 2011, 27, 2364–2374
Langmuir
Figure 5. Zoom on the hysteresis loop regions of the nitrogen adsorption isotherm at 77.4 K shown in Figure 2 for Si/A/B porous duplex layer (b). The solid line is the sum of the isotherms corresponding to samples Asl and Bsl shown in Figure 1.
Figure 6. Zoom on the hysteresis loop regions of nitrogen adsorption isotherm at 77.4 K shown in Figure 2 for Si/B/A porous duplex layer (O). The solid line is the sum of the isotherms corresponding to samples Am and Bsl shown in Figure 1. Note that layer A, with its dangling walls at the interface between the two layers, behaves as a membrane.
pores, those of B, inside the porous duplex layer. The hysteresis region of the isotherm presented in Figure 2 is shown in Figure 6. The reversible formation of an adsorbed film on both layers, preceding capillary condensation in layer A, together with the filling and emptying of layer A are well fitted by the addition of isotherms corresponding to samples Am and Bsl shown in Figure 1, that is, Si/ Bsl/Am. This means that layer A fills and empties like a membrane (Am) while layer B in direct contact with the substrate behaves as a supported layer (Bsl) at pressures p j 0.8 corresponding to the beginning of the capillary condensation in layer B. On the other hand, the condensation and desorption of layer B cannot be fitted by this configuration. The filling of layer B through layer A full of liquid N2 takes place at vapor pressures higher than that corresponding to the filling of the same layer in direct contact with the gas reservoir. We have no explanation for such a behavior at the moment. Layer B empties while layer A remains filled. This result has been already obtained.34 This shows that nucleation of gas bubbles begins to occur in layer B at pcav v = 0.78 under the effect of the negative liquid pressure created under the concave menisci at the top of layer A, and can growth in the pores of layer B leading to the desorption of the whole layer in a narrow range of vapor pressure, 0.78-0.74. The vapor pressure at which cavitation occurs is only slightly lower
ARTICLE
Figure 7. Nitrogen adsorption isotherms at 51.3 K for Si/A/B (b) and Si/B/A (O) porous duplex layers, schematically represented in the inset. The saturation vapor pressure is P0 = 5.3 mmHg. The lines between the experimental points are guides for the eye.
= 0.84) at which layer B begins to empty that the pressure (pevap v when it is in direct contact with the gas reservoir. These results will be discussed in the next section. It is important to note that in the configuration Si/B/A, layer A has dangling walls at the interface between the two layers (see inset of Figure 2). This layer behaves like a membrane during the adsorption process. This supports our model according to which the breaking of Si-Si bonds at the interface between the substrate and the porous layer results in the relaxation of the stress exerted by the substrate on the porous layer which behaves then like a membrane, with a lower adsorbed amount and a higher condensation pressure compared to the same layer constrained by the substrate.13 Different elastic deformations, even as low as a few 10-4 result in different isotherms. Behavior of N2 and Ar at Bulk Solid Temperatures. We have checked whether solid nitrogen and argon behave like liquid N2. Figures 7 and 8 show the N2 and Ar adsorption isotherms at 51.3 and 60 K, respectively, for the duplex layers Si/A/B and Si/B/A. These isotherms exhibit essentially the same features as those obtained with nitrogen at 77.4 K. In the Si/B/A configuration, the capillary condensation is shifted toward higher pressure and cavitation occurs in layer B at a pressure slightly lower than the desorption from layer B in the Si/A/B configuration. However, there are two differences compared to liquid N2. The gap between the cavitation and desorption branches in layer B is higher, and the cavitation transition is practically vertical. Concerning solid Ar at 60 K (see Figure 8), we have obtained two specific results. (i) The first is related to the difficulties we met when making this experiment. Indeed, it was impossible to fully fill layer B in the Si/B/A configuration in a reasonable experimental time, namely, a dozen hours, and, systematically, we did not observe the saturation plateau preceding the desorption of the dense phase from layer B. This was not the case for solid N2. This may be related to the fact that the Ar-Ar interaction is higher than the N2-N2 interaction which makes more difficult the transfer of matter from layer A to layer B. Note that this transfer implies the vaporization of some Ar molecules from A to fill the pores of B, which is the opposite of what happens at the gas reservoir/layer A interface, that is, the condensation of vapor molecules. To get around this difficulty, once the duplex layer was nearly filled, we simultaneously increased the temperature of the gas 2369
dx.doi.org/10.1021/la104777y |Langmuir 2011, 27, 2364–2374
Langmuir
Figure 8. Argon adsorption isotherms at 60 K for Si/A/B (b) and Si/ B/A (O) porous duplex layers, schematically represented in inset. The saturation vapor pressure is P0 = 6.1 mmHg. The lines between the experimental points are guides for the eye.
reservoir up to 84.85 K, a value at which bulk Ar is liquid, and the vapor pressure up to the corresponding saturating value (P0 = 590 mmHg). After this step, in a reverse manner, we decreased the temperature to 60 K and the vapor pressure to the corresponding P0 = 6.1 mmHg. (ii) Figure 8 shows a surprising phenomenon which does not exist for N2. The height of the plateau corresponding to the saturation of layer A is lower in configuration Si/B/A than in configuration Si/A/B. This indicates that the emptying of layer B in the Si/B/A configuration, which occurs at a relative pressure p = 0.7, activates the emptying of a part of layer A. To confirm this idea, we have performed a primary ascending scanning curve (PASC) starting at point I on the boundary cavitation branch of layer B in configuration Si/B/A (see Figure 9). A zoom of the ascending curve, presented in the inset of Figure 9, clearly shows the presence of a tipping point at a relative pressure pJ = 0.87 which is the pressure at which a tipping point is also observed on the boundary adsorption branch of the Si/B/A duplex layer, that is, the pressure at which layer A is completely filled. Along the path IJ, the pores of layers A and B which are emptied at point I fill, and at point J layer A is again filled, then layer B continues to fill. This result shows that a fraction of the pores of layer A empties during the emptying of layer B at a pressure p = 0.7 even though the relative pressures at which these pores usually empty are around p = 0.5. The transfer of Ar molecules from layer B to A activates the desorption of the fluid contained in some pores of layer A which indicates that Ar at 60 K confined in these pores is in the metastable state at p < 0.7. This desorption at p = 0.7 is hence possible because, as shown in Figure 9, the ground states represented by the condensation branch14 for most of the pores of layer A are at pressures higher than p = 0.7. We can reasonably suppose that they are the biggest pores of layer A which empty, that is, those for which the ground states are at the largest distance, on the pressure axis, from p = 0.7. This direct observation of metastable states confirms our previous calculations which showed that, in porous materials which deform elastically during the adsorption process, such as porous silicon and porous glass, the desorption branch represents metastable states.14
ARTICLE
Figure 9. Argon adsorption isotherms, shown in Figure 8, for Si/A/B (b) and Si/B/A (O) porous duplex layers together with a PASC starting at point I on the boundary sublimation branch of layer B in the Si/B/A duplex layer. The relative pressure pJ = 0.87 is the pressure at which layer A, in the configuration Si/B/A, is completely filled. The inset shows a zoom of both the boundary sublimation branch and ascending curve for the Si/B/A duplex layer on the hysteresis loop region. The lines between the experimental points are guides for the eye.
It remains to be understood why the desorption of some pores of layer A can be triggered by the transport of Ar molecules from B to A, which is not the case for solid N2. Concerning solid Ar, our results and those obtained by Wallacher and co-workers34 seem to be contradictory at a first glance. These authors found that, in configuration Si/B/A, layer B empties at the same pressures as layer A and concluded that cavitation does not occur for solid Ar on the contrary to liquid nitrogen and that the emptying of layer B is delayed until layer A empties according to the pore-blocking model. This misinterpretation might be due to the fact that, in their experiment, the PSDs of layers A and B are too close to each other so that the gap on the pressure axis between the cavitation and desorption branches for layer B is, in a fortuitous way, equal to the gap between the desorption branches of layers A and B. In porous alumina with controlled modulations of pore diameter,39 desorption in the largest (B) and narrowest (A) sections occurs at pressures close to each other as for solid argon in ref 34. The authors concluded also that desorption in B is triggered by desorption at equilibrium (receding menisci) in A. This could be also a misinterpretation. To settle the problem, it would be preferable to redo the experiment with larger modulations and with a more simple geometry, duplex layers in both configurations with layers of comparable thicknesses.
IV. DISCUSSION The above experiments show that the pores of layer B in Si/B/A configuration empty while layer A remains filled whether the confined phase is liquid (N2 at 77.4 K) or solid (N2 at 51.3 K and Ar at 60 K). We point out that all these results were obtained with the same duplex porous layer which was successively exposed to nitrogen and argon gas reservoir at different temperatures. Moreover, the SEM analyses (see Figure 4) performed after these adsorption data do not exhibit any fracture at the interface between the two layers which would have provided direct access to the gas reservoir for layer B. This shows that nucleation and growth of gas bubbles (cavitation) occur in van der Waals liquid N2 or solid N2 and Ar confined in layer B leading to its emptying in a narrow range of relative vapor pressure. For solid N2 and Ar, this transition is practically vertical. 2370
dx.doi.org/10.1021/la104777y |Langmuir 2011, 27, 2364–2374
Langmuir
ARTICLE
It is the first time that cavitation in solid confined to mesopores is observed. While cavitation in liquid has been studied in detail by using different approaches (thermodynamics, statistics, Monte Carlo simulations), cavitation in van der Waals solids remains untouched to our knowledge. What could be the difference between cavitation in the confined fluid and confined solid? The nature of the bonds (van der Waals interactions) is the same in both phases. However, the transition from liquid to solid state results in a densification of about 10% for N2 and 14% for Ar, but these variations are similar to those observed in the liquid state when decreasing the temperature from 77.4 to 60 K for N2 or from 110 to 85 K for Ar. Thus, the passage from liquid to solid should change the nucleation rate in a similar way as the variation of temperature in the liquid sate. From these first qualitative pieces of information, we can conjecture that the cavitation phenomenon should not be qualitatively different in van der Waals liquid and solid. Nevertheless, to understand cavitation in van der Walls solids at a microscopic scale is a new field of research which requires further experimental and theoretical studies. The following discussion is hence entirely focused on the experimental results we found for liquid nitrogen which we analyze in the framework of the well established classical nucleation theory as reviewed in refs 41-44. For N2 at 77.4 K, the relative vapor pressure at which cavitation occurs, pcav v = 0.78, is high compared to other values cav reported in the literature, pcav v E 0.5 in refs 32 and 33 and pv = 0.6 in ref 34. These results raise the following fundamental questions we discuss now: (A) At what liquid pressure, Plcav, does nucleation of gas bubbles occur in sample B? (B) Does the nucleation occur in the bulk liquid (homogeneous nucleation) or on the pore walls (heterogeneous nucleation)? It is well-known that, for a given fluid, heterogeneous nucleation occurs at a negative liquid pressure lower, in absolute value, than that of homogeneous nucleation. (C) Is cavitation caused by the pore-blocking effect as proposed in the literature9,10 or by another phenomenon?
begins to empty via a receding meniscus, the contact angle of which is obtained by an additional condition, the extremalization of the grand potential (see below eq 9). For a porous material composed of pores of different sizes, at a given vapor pressure, the menisci have the same curvature radius and thus the liquid pressure has the same value in all the pores whether they are interconnected or not since μv = μl. Note that eq 1 is based on the equality μv = μl. This equality is strictly valid only under the assumption of an inert solid (see ref 40 page 33 and ref 14). The possible incidence on eq 1 of the elastic deformation of the porous solid will be addressed in a separate paper. In what follows, we will assume, as it is done in all the studies, that the chemical equilibrium between the two phases is given by the equality of their chemical potentials and we will use eq 1 to estimate the liquid pressure. Cavitation in layer B occurs at vapor pressure pcav v = 0.78. According to eq 1, Pcav l = -4.5 MPa. As it is shown in section IVB, this does not correspond to homogeneous nucleation. B. Nature of Nucleation Process. When a liquid is under tension (negative pressure), it is energetically favorable for a bubble to nucleate homogeneously. The number N of bubbles per unit time and per unit volume is related exponentially to the energy barrier ΔFmax required to form the bubble.41-44 ΔFmax N ¼ N0 exp ð2Þ kT
A. Liquid Pressure versus Vapor Pressure in Porous Materials. The capillary tension developed in fluid confined in porous
The number of bubbles with critical size R* in a pore of volume v in a time τ is thus Nvτ. The prefactor N0 can be considered as an attempt frequency per unit time multiplied by the number of sites per unit volume where nucleation can take place. The attempt frequency is generally taken to be kT/h where h is Plank’s constant. Taking for the number of sites per unit volume, the inverse of the molecular volume of liquid nitrogen, 1/vl = FNA/M, where M is the molar mass, NA is Avogadro's number, and F is the density, we find N0 = kT/hvl = 2.8 1040 m-3s -1. The number Nvτ equals 1 if
materials is given by the classical Kelvin-Laplace equation RT Pv Pl - Pv ¼ ln vl P0
ð1Þ
where Pv and Pl are the vapor and liquid pressures, respectively, vl is the molar volume of the liquid at temperature T, and R is the perfect gas constant. This equation is obtained by introducing in the Gibbs-Duhem equations for the bulk liquid and vapor, the chemical equilibrium μv = μl (μv and μl are the chemical potentials of the vapor (v) and liquid (l)), and the mechanical equilibrium at the liquid-vapor interface given by Laplace equation, Pv - Pl = 2γlv/Rm (γlv is the surface energy per unit surface area at the liquid-vapor interface and Rm is the curvature radius of the meniscus). Equation 1 means that, for a given fluid, Rm depends only on the vapor pressure and tells us how it varies from infinity at p = 1 to its lowest value, ∼RP/cos θ where θ is the equilibrium contact angle between the solid and liquid and Rp is the pore radius. It is generally admitted that once this lowest value is reached, the pore
where ΔFmax ¼
16πγlv 3 3ðδPÞ2
ð3Þ
with k being Boltzmann's constant and δP = Pv - Pl. This energy barrier corresponds to a critical nucleus of radius R ¼
2γlv δP
ΔFmax ¼ kT lnðN0 vτÞ
ð4Þ
ð5Þ
Note that the prefactor value does not significantly change that of ΔFmax due to the logarithm dependence. For layer B (see Table 1), the total pore volume, measured from adsorption data, is 18.7 10-3 cm3 and the total number of pores is 1.2 1011/cm2 15.2 cm2. Taking for v the mean value of the volume of a pore, v ∼ 10-20 m3, and τ = 1 s, we find ΔFmax ¼ 47kT 2371
ð6Þ
dx.doi.org/10.1021/la104777y |Langmuir 2011, 27, 2364–2374
Langmuir
ARTICLE
From eq 3, we deduce the negative liquid pressure at which cavitation can occur in the bulk fluid "
16π γlv 3 Plcav;bulk - Pv ¼ 3 47kT
the bulk liquid and on the pore surface is41 1=2 Plcav;surface =Plcav;bulk φðmÞ
#1=2 =Plcav;bulk
ð7Þ
For N2 at 77.4 K, we find Pcav,bulk = -15.4 MPa and R* = 1.2 nm. l ∼ -4.5 MPa that We have seen above that, for layer B, P cav l is three times lower than that of the bulk liquid. This provides evidence that, in the linear pores of porous silicon, cavitation does not occur in the bulk fluid but on the surface of the pore walls, the so-called heterogeneous nucleation which is generally observed at negative liquid pressure much lower, in absolute value, than that of the bulk. This is the crucial point of this work. We have noted above that the elastic deformation of the porous solid might have an incidence on Kelvin-Laplace equation. The question is whether this possible incidence could question our conclusion. Let us take, for instance, the case of porous silica sample SBA16-7.9 in Table 1 in ref 33. The (see eq 7) are nucleation barrier, 46.9kT, as well as Pcav,bulk l similar to that of layer B. The cavitation vapor pressure is p ∼ 0.44 which, according to Kelvin-Laplace equation, corresponds . If it to a liquid pressure Pl ∼ -15.2 MPa, a value close to Pcav,bulk l is proved true that desorption in this material occurs via homogeneous nucleation, as it is assumed by the authors, KelvinLaplace equation gives the correct values. If, on the other hand, | < |Pcav,bulk |), the Kelcavitation is heterogeneous (|Pcav,surface l l vin-Laplace equation overestimates |Pl|. In both cases, our conclusion is thus valid: |Pcav l | j 4.5 MPa. Let us give our opinion about these two hypotheses. According to the authors,33 the occurrence of homogeneous nucleation in porous silica would be due to the presence on the pore walls of a liquidlike adsorbed film which protects the interior of the pore from surface pollutants and irregularities that serve as nucleation centers for heterogeneous cavitation. We suppose that the liquidlike adsorbed film to which the authors refer is present in every porous material. If it is the case, our experimental result contradicts this assertion. As a matter of fact, we think that there are sound reasons for expecting cavitation to be heterogeneous in porous materials other than porous silicon since they are also characterized by a surface-volume ratio similar or even higher than that of porous silicon. Taking the values for sample B (see Table 1), we find SBET/Vpores = 2.9108 m-1, while for 1 g of SBA-15 silica7 SBET/Vpores = 7.5108 m-1. The problem of heterogeneous nucleation is complex. It depends on the geometry of the material which is itself complex in the case of porous materials. Even if we assimilate the actual pores to cylinders, the nucleation path does not preserve necessarily this symmetry, and the problem has no analytical solution and must be solved numerically. The fracture liquid pressure depends on the contact angle θ between the vaporliquid interface and the solid surface at the three phase line. These calculations should determine the value of θ which ∼ -4.5 MPa. corresponds to liquid pressure for fracture Pcav l Based on results found in the literature, we expect angle values much higher that π/2.41-44 If we consider, for instance, the simple case of a vapor bubble on a plane surface and if we assume that the bubble is a fraction of a sphere, the relation between the liquid pressures for fracture in
ð8Þ
where φ(m) = 1/4(2 þ m)(1 - m)2 and m ¼ cosðπ - θÞ ¼
ðγsl - γsv Þ γlv
ð9Þ
γsl, γsv, and γlv are the surface energies per unit area at the three interfaces solid-liquid, solid-vapor, and liquid-vapor. For layer B, [φ(m)]1/2 = 1/3, which corresponds to θ = 130 that is (γsl - γsv) > 0. These calculations should also explain (1) why the cavitation pressure depends on the pore size, 0.78 in the present work and 0.6 in ref 34 and (2) why the transition via cavitation is steep and even quasi vertical on the pressure axis for solid N2 and Ar while the PSD is large, which suggests that there is an interaction mechanism between the pores during the emptying via cavitation. C. Desorption Mechanism. We have seen above that the liquid pressure at which cavitation occurs in layer B in Si/B/A configuration, Pcav l ∼ -4.5 MPa, is much lower, in absolute value, than that corresponding to homogeneous nucleation, ∼ -15 MPa. Cavitation occurs via the nucleation of gas bubbles on the surface of the pore wall, and the contact angle θ between the liquid-vapor interface of the nitrogen bubble and the solid surface is higher than π/2. This indicates that, for the system porous silicon-nitrogen, the structure of the interfaces solidliquid and solid-vapor is such that γsl - γsv > 0. At this stage, it is important to distinguish between the two following situations: a meniscus at the pore end and a meniscus inside a pore. When a porous material is filled at p = 1, desorption is initiated by decreasing the vapor pressure and a concave meniscus is formed (reversibly) at the pore end. The equilibrium of the concave meniscus and vapor is achieved by the equality of the chemical potentials of the vapor and liquid and by the Laplace equation. The liquid pressure, that is, the curvature radius, is then given by Kelvin-Laplace equation. Once the curvature radius reaches a minimum value, it is generally admitted that it enters the pore and may be eventually blocked by geometric constrictions or less attractive sections of the pores. The contact angle corresponding to a meniscus inside a pore is obtained by a supplementary condition, the extremalization of the grand potential, and is also given by eq 9. A receding meniscus with a contact angle lower than π/2 (γsl - γsv < 0) inside a pore and a gas bubble with a contact angle higher than π/2 (γsl - γsv > 0) are thus mutually exclusive. Our experimental result showing that cavitation is heterogeneous indicates that a concave meniscus cannot enter a pore. This is supported by another experimental fact: for porous silicon, as already noted in section III, the amount of matter desorbed along the plateau of the hysteresis loop comes only from the nonporous surface (analysis cell and external surface of the sample) and from the formation of the menisci. The cavitation liquid pressure is hence controlled by the curvature radius of the meniscus localized at the pore end and is determined by the KelvinLaplace equation. We understand better now why layer B in cav configurations Si/B/A (pcav v = 0.78) and Si/A/B (pv = 0.84) empties at vapor pressures close to each other, since it empties in both cases via heterogeneous cavitation. These two vapor 2372
dx.doi.org/10.1021/la104777y |Langmuir 2011, 27, 2364–2374
Langmuir pressures should correspond to the same liquid pressure. This question will be addressed in a future paper. We see that our experimental results differ fundamentally from the calculations of Naumov and co-workers. Cavitation is not homogeneous but heterogeneous, a process which cannot be induced by the negative liquid pressure under a concave meniscus blocked inside a pore by narrow or more attractive sections (see Figure 4 in ref 9). The problem that we must solve is now: why, for porous silicon, it is more favorable for a gas bubble to nucleate on the pore wall than for a receding meniscus to propagate into a pore? The answer to this question will tell us if the conclusion of this paper can be extended to other porous materials.
ARTICLE
γsl - γsv > 0. A receding meniscus with a contact angle lower than π/2 (γsl - γsv < 0) inside a pore and a gas bubble with a contact angle higher than π/2 (γsl - γsv > 0) are thus mutually exclusive. A receding meniscus cannot enter the pores of porous silicon. We report, for the first time, a cavitation phenomenon in solid nitrogen (51.4 K) and Ar (60 K) confined to mesopores. In the case of Ar adsorption at 60 K in the Si/B/A duplex layer, we found in a fortuitous way a beautiful result. The emptying of a fraction of pores of layer A is triggered by the transfer of Ar molecules from B to A. This constitutes a direct observation of metastable states.
’ AUTHOR INFORMATION Corresponding Author
V. CONCLUSION We present a study of adsorption-desorption process of liquid (77.4 K) and solid (51.4 K) nitrogen and solid argon (60 K) in duplex mesoporous layers formed in silicon, Si/A/B and Si/B/A, where A and B are layers of different pore size distribution, with A having the narrowest pores on average. The mean pore diameter is two times lower for A than for B, and the number of pores is three times higher for A than for B. This means that, in the configuration Si/B/A, layer A has dangling pore walls at the interface between the two layers. For liquid nitrogen, we compare the behavior of each layer in the duplex configurations to its own behavior when it is alone in direct contact with the gas reservoir either supported or detached from the substrate. In the Si/A/B configuration, layers A and B fill and empty independently of each other as if they were in direct contact with the gas reservoir and supported by silicon substrate. This shows that layer B is attached to the substrate and that the elastic deformations undergone by each layer during adsorption are not longitudinally transmitted from one layer to the other. To compress or stretch a part of a spring does not change the state of compression or dilatation of the rest of the spring. In the Si/B/A configuration, layer A, which has dangling walls at the interface between the two layers, behaves like a membrane. The partial relaxation of the constraint of the substrate modifies the adsorption strains and leads to a decrease of the adsorbed amount before capillary condensation and consequently to an increase of the condensation pressure. This supports a previous study where we explained why the isotherms for layers supported by and detached from the substrate are distinct. In this so-called ink-bottle Si/B/A configuration, layer B empties via cavitation at a relative vapor pressure pcav v = 0.78 close to the pressure, pcav v = 0.84, at which the same layer empties when it is in direct contact with the gas reservoir (Si/A/B configuration). This indicates that the presence of layer A at the top of the duplex layer is not the determining factor for cavitation. Layer B contains the ingredients necessary for this. According to the Kelvin-Laplace equation, the liquid pressure at which cavitation occurs in layer B in the Si/B/A ∼ -4.5 MPa, is much lower, in absolute configuration, Pcav l value, than that, ∼ -15.4 MPa, predicted by the classical theory of homogeneous nucleation. Accordingly, cavitation does not occur in the bulk liquid but via nucleation of gas bubbles on the surface of the pore walls. This is the crucial point of the paper. The contact angle, θ, between the pore wall surface and the liquid-vapor interface of the bubble is higher than Π/2; that is, the structure of the solid-liquid and solid-vapor interfaces is such that
*E-mail:
[email protected].
’ ACKNOWLEDGMENT We acknowledge S. Borensztajn (LISE, UPR 15, CNRS, France) for the SEM observations. ’ REFERENCES (1) Coasne, B.; Grosman, A.; Ortega, C.; Simon, M. Phys. Rev. Lett. 2002, 88, 256102. (2) Grosman, A.; Ortega, C. Langmuir 2008, 24, 3877. (3) Brown, A. J. Thesis, Bristol, 1963. (4) Mason, J. Proc. R. Soc. London 1983, A390, 47. (5) Mason, J. Proc. R. Soc. London 1988, A415, 453. (6) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic Press: London, 1982; Chapter 3, p 111. (7) Grosman, A.; Ortega, C. Langmuir 2005, 21, 10515. (8) Bruschi, L.; Fois, G.; Mistura, G.; Sklarek, K.; Hillebrand, R.; Steinhart, M.; G€osele, U. Langmuir 2008, 24, 10936. (9) Naumov, S.; Khokhlov, A.; Valiullin, R.; K€arger, J.; Monson, P. Phys. Rev. E 2008, 78, 060601. (10) Naumov, S.; Khokhlov, A.; K€arger, J.; Monson, P. Phys. Rev. E 2009, 80, 031607. (11) Puibasset, J. J. Chem. Phys. 2007, 127, 154701. (12) Puibasset, J. Langmuir 2009, 25, 903. (13) Grosman, A.; Ortega, C. Phys. Rev. B 2008, 78, 085433. (14) Grosman, A.; Ortega, C. Langmuir 2009, 25, 8083. (15) Grosman, A.; Ortega, C. Appl. Surf. Sci. 2010, 256, 5210. (16) Bangham, D. H.; Fakhoury, N. Proc. R. Soc. London, Ser. A 1930, 130, 81. (17) Wiig, E. O.; Juhola, A. J. J. Am. Chem. Soc. 1949, 71, 561. (18) Amberg, C. H.; McIntosh, R. Can. J. Chem. 1952, 30, 1012. (19) Quinn, H. W.; McIntosh, R. Can. J. Chem. 1957, 35, 745. (20) Dash, J. G.; Suzanne, H.; Shechter, H.; Peierls, R. E. Surf. Sci. 1976, 60, 411. (21) Reichenauer, G.; Scherer, G. W. Colloids Surf., A 2001, 187188, 41. (22) Herman, T.; Day, J.; Beamish, J. Phys. Rev. B 2006, 73, 094127. (23) Dolino, G.; Bellet, D.; Faivre, C. Phys. Rev. B 1996, 54, 17919. (24) G€unther, G.; Prass, J.; Paris, O.; Schoen, M. Phys. Rev. Lett. 2008, 101, 086104. (25) Dourdain, S.; Britton, D. T.; Reichert, H.; Gibaud, A. Appl. Phys. Lett. 2008, 93, 183108. (26) Yan, M.; Dourdain, S.; Gibaud, A. Thin Solid Films 2008, 516, 7955. (27) Prass, J.; M€uter, D.; Fratzl, P.; Paris, O. Appl. Phys. Lett. 2009, 95, 083121. (28) Rossi, M. P.; Gogotsi, Y.; Kornev, K. G. Langmuir 2009, 25, 2804. (29) Kim, H. Y.; Gatica, S. M.; Stan, G.; Cole, W. M. J. Low Temp. Phys. 2009, 156, 1. 2373
dx.doi.org/10.1021/la104777y |Langmuir 2011, 27, 2364–2374
Langmuir
ARTICLE
(30) G€unther, G.; Schoen, M. Phys. Chem. Chem. Phys. 2009, 11, 9082. (31) Schoen, M.; Paris, O.; G€unther, G.; M€uter, D.; Prass, J.; Fratzl, P. Phys. Chem. Chem. Phys. 2010, 12, 11267. (32) Ravikovitch, P. I.; Neimark, A. V. Langmuir 2002, 18, 9830. (33) Rasmussen, C. J.; Vishnyakov, A.; Thommes, M.; Smarsly, B. M.; Kleitz, F.; Neimark, A. V. Langmuir 2010, 26, 10147. (34) Wallacher, D.; K€unzner, N.; Kovalev, D.; Knorr, N.; Knorr, K. Phys. Rev. Lett. 2004, 92, 19. (35) Amsel, G.; d’Artemare, E.; Battistig, G.; Morazzani, V.; Ortega, C. Nucl. Instrum. Methods Phys. Res., Sect. B 1997, 122, 99. (36) von Bardeleben, H. J.; Stievenard, D.; Grosman, A.; Ortega, C.; Siejka, J. Phys. Rev. B 1993, 47, 10899. (37) Ben Chorin, M. In Properties of Porous Silicon; Canham, L., Ed.; INSPEC Publishers: Ipswich, MA, 1997; Chapter 6, p 165. (38) Brunauer, S.; Emmett, P. H.; Teller, E. J. Am. Chem. Soc. 1938, 60, 309. (39) Bruschi, L.; Mistura, G.; Liu, L.; Lee, W.; G€osele, U.; Coasne, B. Langmuir 2010, 26, 11894. (40) Rouquerol, F.; Rouquerol, J.; Sing, K. Adsorption by powders & porous solids; Academic Press: San Diego, 1999. (41) Fisher, J. J. Appl. Phys. 1948, 19, 1062. (42) Blander, M.; Katz, J. L. AlChE J. 1975, 21, 833. (43) Maris, H. J. C. R. Phys. 2006, 7, 946. (44) Caupin, F.; Herbert, E. C. R. Phys. 2006, 7, 1000.
2374
dx.doi.org/10.1021/la104777y |Langmuir 2011, 27, 2364–2374
