Intrusion and Retraction of Fluids in Nanopores: Effect of

J. Phys. Chem. C 2009, 113, 1953–1962

1953

Intrusion and Retraction of Fluids in Nanopores: Effect of Morphological Heterogeneity Benoit Coasne,*,† Anne Galarneau,† Francesco Di Renzo,† and R. J. M. Pellenq‡ Institut Charles Gerhardt Montpellier, UMR 5253 CNRS, UniVersite´ Montpellier 2, ENSCM, Place Euge`ne Bataillon, 34095 Montpellier Cedex 05, France, Department of CiVil and EnVironmental Engineering, Massachusetts Institute of Technology, 77 Massachusetts AVenue, Cambridge, Massachusetts 02139, and Centre Interdisciplinaire des Nanosciences de Marseille, UPR 7251 CNRS, Campus de Luminy, 13288 Marseille Cedex 09, France ReceiVed: September 3, 2008; ReVised Manuscript ReceiVed: October 27, 2008

This paper reports on a molecular simulation study of intrusion and retraction of a nonwetting fluid in silica nanopores without or with morphological defects (constrictions or undulations) to mimic mercury porosimetry in porous materials. All the pores considered in this work are of a finite length and are connected to bulk reservoirs so that they mimic real materials for which the confined fluid is always in contact with the external phase. Our results for pores with constant or variable cross-section are in qualitative agreement with experimental data: (1) the intrusion and retraction pressures are larger than the bulk saturating vapor pressures, (2) the intrusion and retraction pressures are decreasing functions of the pore size, and (3) entrapment of the confined fluid leads to irreversible intrusion/retraction isotherms (except for the smallest pore, for which intrusion is reversible). Our data show that the intrusion pressure varies linearly with the pore diameter when both quantities are plotted on a logarithmic scale; the slope, -1.1 ( 0.1, is close to the theoretical value -1 given by the Washburn-Laplace equation (the contact angle θa is found to be about 102° ( 15°). In contrast, the retraction pressure varies with a larger slope, -1.6 ( 0.1, due to the fact that retraction consists of the nucleation of a gas bubble within the pore. We find that the intrusion chemical potential for the large cavities of the constricted pores corresponds to that for a regular cylindrical nanopore having the same diameter as the constriction, i.e. intrusion in the constricted pores occurs when the nonwetting fluid invades the constrictions that isolate the large cavity from the bulk external phase. On the other hand, the retraction chemical potentials for pores with uneven diameter underestimates the value found for a regular cylindrical pore having the pore diameter of the wider parts of the variable-diameter pore. These results suggest that intrusion and retraction experiments (porosimetry) can be used to assess and characterize morphological defects in nanopores, provided that the pore size is known from other independent measurements. The present work also provides a theoretical frame to explain some discrepancies observed between properties assessed by mercury porosimetry and nitrogen adsorption. 1. Introduction The behavior of fluids confined within nanometric pores (size of a few molecular diameters) significantly differs from that of the bulk.1,2 In particular, confinement and surface forces affect the phase transitions (condensation, freezing, etc.). Significant shifts in phase transitions (e.g., pressure, temperature) are observed and, in some cases, new types of phase transitions (layering, wetting, etc.) can also be found for these inhomogeneous systems. Understanding the effects of confinement and surface on the thermodynamics of fluids is of crucial interest for both fundamental research and potential applications. Among nanoporous solids, siliceous MCM-413 and SBA-154 are important materials because of their possible applications for gas adsorption, phase separation, catalysis, preparation of nanostructured materials, drug delivery, etc.5-7 These materials are obtained by a template mechanism involving the formation of surfactant or block copolymer micelles in a mixture composed of a solvent and a silica source. Polymerization of the silica * To whom correspondence should be addressed. Phone: +33 4 67 14 33 78. Fax: +33 4 67 14 42 90. E-mail: [email protected]. † Institut Charles Gerhardt Montpellier. ‡ Massachusetts Institute of Technology and Centre de Recherche en Matie`re Condense´e et Nanosciences.

and removal of the organic micelles allows one to obtain a material made up of an array of regular pores. The pore diameter distribution is narrow with an average value that can be varied from 2 up to 20 nm, depending on the synthesis conditions.7 From a fundamental point of view, MCM-41 and SBA-15 are considered as model materials to investigate the effect of nanoconfinement on the thermodynamic properties of fluids. In particular, the cylindrical geometry of the pores in these materials makes it possible to address in a simple way the effect of confinement on the adsorption or intrusion of fluids in nanopores. As a result, many experimental, theoretical, and molecular simulation studies have been reported on the thermodynamics of fluids confined in these materials (for reviews, see refs 1 and 2). Properties of MCM-41 and SBA-15 have been extensively studied by combining transmission electronic microscopy (TEM), X-ray diffraction, and adsorption experiments.5,6 For instance, it has been shown that the cylindrical nanopores in SBA-15 materials are connected through transversal nanoporous channels.8-12 On the other hand, it is generally agreed that MCM-41 is made up of unconnected nanopores.9,13 Despite the significant amount of information gained on the properties of these materials, some uncertainties remain regarding their surface chemistry (presence of impurities, defects, etc.) as well

10.1021/jp807828a CCC: $40.75  2009 American Chemical Society Published on Web 01/13/2009

1954 J. Phys. Chem. C, Vol. 113, No. 5, 2009

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TABLE 1: Morphological Properties of the Regular (pores A1, A2, A3, A4, and A5) and Constricted (pores B1 and B2) Nanopores Considered in This Worka porous material regular pores pore A1 pore A2 pore A3 pore A4 pore A5 constricted pores pore B1 pore B2

diameter D in nm (length L in nm) 1.6 (25.6) 2.4 (25.6) 3.2 (25.6) 4.8 (25.6) 6.4 (25.6) 3.2 (1.1) 4.8 (1.1)

6.4 (6.4) 6.4 (6.4)

3.2 (2.1) 4.8 (2.1)

6.4 (6.4) 6.4 (6.4)

3.2 (2.1) 4.8 (2.1)

6.4 (6.4) 6.4 (6.4)

3.2 (1.1) 4.8 (1.1)

a The numbers provide the sequence of cavity and constriction diameters D starting from one of the pore openings. For each cavity or constriction, the number in parentheses corresponds to its length L. For example, Dp(Lp)-Dc(Lc) corresponds to a cavity of a diameter Dp and a length Lp followed by a constriction of a diameter Dc and a length Lc. For all systems, the total length of the pore is 25.6 nm.

as their surface texture (microporosity, surface roughness, existence of defects such as constrictions).14-16 Adsorption and intrusion of fluids in nanoporous materials still attract a great deal of attention, since they are involved in many industrial processes (catalysis, phase separation, etc.). Adsorption and porosimetry experiments are also routinely used for the characterization of porous solids.17 Among these characterization techniques, porosimetry is an important one in which the intrusion pressure of a nonwetting fluid (usually mercury) is related to the pore size using the Washburn-Laplace equation. In contrast to adsorption of wetting fluids, considerably less attention has been paid to the simulation of intrusion and retraction of nonwetting fluids in porous materials. Pioneering works have mainly focused on the nucleation of the liquid or gas phases in pores using the density functional theory,18 molecular simulations,19-21 or macroscopic approaches.22-24 More recently, Monson and co-workers have developed a lattice model that allows studying the intrusion and retraction of nonwetting fluids in ordered pores of a simple geometry or disordered porous matrices such as Vycor or controlled pore glass.25-27 As mentioned above, intrusion experiments are usually interpreted using the Washburn-Laplace equation for the intrusion of nonwetting fluids:17,28,29

∆P ) -

4γ cos θ D

(1)

where D is the pore diameter and ∆P ) Pl - Pg is the presure difference across the curved interface between the liquid and gas phases. γ and θ are the gas-liquid surface tension and contact angle, respectively. If the saturating vapor pressure Pl of the liquid is neglected, we obtained that the external pressure Pg on intrusion is equal to ∆P. Porosimetry experiments usually exhibit a hysteresis loop that reveals the irreversibility of the intrusion/retraction mechanisms. A common explanation for such hysteresis loops is that the receding contact angle θr upon retraction is different from the advancing contact angle θa upon intrusion. For a pore with constriction or disordered porous materials, another common interpretation for the irreversibility of the intrusion/retraction cycles is that the retraction process is driven by the nucleation of a vapor bubble (for a detailed discussion on the possible retraction mechanisms, see ref 23). The aim of the present work is to investigate by means of grand canonical Monte Carlo (GCMC) molecular simulation the behavior of a simple nonwetting fluid confined within atomistic silica nanopores. Two models of cylindrical silica nanopores are considered. Model A is a regular cylindrical pore of a finite length that is opened at both ends toward an external bulk reservoir. For this reference system, five different pore diameters D ) 1.6, 2.4, 3.2, 4.8, and 6.4 nm have been

considered. Model B is a cylindrical pore of a finite length having constrictions. We address the effect of the size of these constrictions on intrusion and retraction of the nonwetting fluid by varying their diameter. The comparison between models without (model A) and with (model B) constrictions allows us to discuss the effect of such morphological defects on the intrusion and retraction mechanisms. We also discuss the ability of the Washburn-Laplace equation to describe intrusion and retraction of fluids in nanopores. This paper complements our previous work in which we studied the effect of constrictions on the adsorption and desorption of a wetting fluid in cylindrical silica nanopores.30 The remainder of the paper is organized as follows. In section 2, we present the models of MCM-41 nanopores used in this work and briefly discuss the details of the simulation technique. We report results for intrusion and retraction of a nonwetting Lennard-Jones fluid confined in these silica nanopores in section 3 and discuss the results in section 4. Finally, section 5 contains concluding remarks and suggestions for future work. 2. Methods 2.1. Pore Models. A porous material can be defined using a mathematical function η(x,y,z) that equals 1 if the coordinates (x, y, z) belong to the pore wall and 0 if the coordinates (x, y, z) belong to the pore void. In this work, two models of cylindrical silica nanopores were considered. Model A is simply a regular cylindrical pore, while model B is a cylindrical pore having constrictions. The properties of the different pores considered in this work for models A and B are reported in Table 1. We also report in Figures 1 and 2 transversal views of the silica pores without and with constrictions. All of the pores considered in this work are of a finite length and are connected to bulk reservoirs so that they mimic real materials for which the confined fluid is always in contact with the external phase. For all pores, the total pore length is Lp ) 25.6 nm and the size of the reservoirs at the top and bottom of the pore is 3.2 nm. For model A, five pore diameters have been considered: D ) 1.6, 2.4, 3.2, 4.8, and 6.4 nm (pores A1, A2, A3, A4, and A5, respectively). The porous void for model A is defined as

{

0 if |z| > Lp/2

η(x, y, z) ) 0 if √x2 + y2 < D/2 1 otherwise

(2)

For model B, we prepared two pores of a diameter D ) 6.4 nm with different constriction diameters Dc (pores B1 and B2). Pore B1 has constrictions with a diameter Dc ) 3.2 nm, while pore B2 has constrictions with a diameter Dc ) 4.8 nm. The

Intrusion and Retraction of Fluids in Nanopores

J. Phys. Chem. C, Vol. 113, No. 5, 2009 1955

Figure 2. Transversal views of the MCM-41 silica nanopores having a constriction: (top) pore B1 and (bottom) pore B2 (see Table 1). White and gray spheres are oxygen and silicon atoms, respectively. Black spheres correspond to hydrogen atoms, which delimit the pore surface.

Figure 1. Transversal views of MCM-41 silica nanopores having a regular cylindrical diameter. From top to bottom: D ) 2.4 nm (pore A2), D ) 4.8 nm (pore A4), and D ) 6.4 nm (pore A5). White and gray spheres are oxygen and silicon atoms, respectively. Black spheres correspond to hydrogen atoms, which delimit the pore surface.

constricted pores are composed of three different regions: R1, where the pore section is such that the diameter is equal to D (the largest cavity); R2, where the pore section is such that the diameter is equal to Dc (the largest cavity); and R3, where the pore section is such that the diameter varies linearly from D to Dc or from Dc to D (see Table 1 and Figure 2 for a clearer visualization of the constricted nanopores considered in this work). The porous void for model B is defined by:

{

0 if |z| > Lp/2

η(x, y, z) ) 0 if √x2 + y2 < f(z) 1 otherwise with

{

D/2 if z ∈ R1 f(z) ) Dc/2 if z ∈ R2 az + b if z ∈ R3

(3)

where a and b are scalar values that describe the linear change in the pore section diameter as it varies from the large cavity to

the constriction diameters or the constriction to the large cavity diameters. The atomistic pores used in this work were generated according to the method proposed by Pellenq and Levitz to prepare numerical Vycor samples.31 Coasne and Pellenq have shown that this technique can be used to prepare pores of various morphologies and/or topologies, such as cylindrical, hexagonal, ellipsoidal, and constricted pores.32-34 The pore models used in this work were obtained by carving out of an atomistic block of cristobalite (cristalline silica), the porous network corresponding to η(x,y,z) ) 0 (see eqs 2 and 3). In order to mimic the pore surface in a realistic way, we removed in a second step the Si atoms that are in an incomplete tetrahedral environment. We then removed all oxygen atoms that are nonbonded. This procedure ensures that the remaining silicon atoms have no dangling bonds and the remaining oxygen atoms have at least one saturated bond with a Si atom. Then, the electroneutrality of the simulation box was ensured by saturating all oxygen dangling bonds with hydrogen atoms. The latter are placed in the pore void, perpendicularly to the pore surface, at a distance of 1 Å from the closest unsaturated oxygen atom. Then, we displace slightly and randomly all the O, Si, and H atoms in order to mimic an amorphous silica surface (the maximum displacement in each direction x, y, and z is 0.7 Å). It has been shown31 that the density of OH groups obtained using such a procedure (7 OH per nm2) is close to that obtained experimentally for porous silica glasses such as Vycor (5-7 OH per nm2).35,36 On the other hand, the density of OH groups at the surface of real MCM-41 is usually smaller (2 or 3 OH per nm2).37-41 Despite this quantitative difference in the surface chemistry of the numerical and real samples, it has been shown that the simulated materials obtained using the procedure above are able to describe adsorption and condensation of fluids in nanoporous silicas.31,33,34 Moreover, the discussion on the effect of the existence of morphological defects by comparing the


Coasne et al.

Figure 3. Intrusion isotherm for a nonwetting fluid in a MCM-41 nanopore with D ) 1.6 nm (pore A1). µ0 is the chemical potential corresponding to the bulk saturating vapor pressure.

simulations for our models of silica pores is relevant since the surface chemistry is the same for all samples. 2.2. Grand Canonical Monte Carlo. We performed GCMC simulations of intrusion in the atomistic models of silica nanopores with and without constrictions. The GCMC technique is a stochastic method that simulates a system having a constant volume V (the pore with the adsorbed phase), in equilibrium with an infinite reservoir of particles imposing its chemical potential µ and temperature T.42-44 The absolute intrusion/ retraction isotherm is given by the ensemble average of the number of adsorbed atoms as a function of the chemical potential of the gas reservoir µ. The fluid/fluid interaction was calculated using a Lennard-Jones potential with ε ) 120 K and σ ) 0.3405 nm, which corresponds to the usual parameters for argon.45 In order to model the intrusion of a nonwetting fluid in silica nanopores, the interactions between the fluid and substrate atoms were calculated using the PN-TraZ potential31,46 developed for the adsorption of rare gases, but only the repulsive contribution of the potential was considered. Although this system does not correspond to any realistic fluid adsorbed in silica pores, it allows studying the intrusion and retraction of a nonwetting fluid in a porous matrix. This simple repulsive potential was used for both the pores of models A and B. We calculated the adsorbate/substrate interaction using an energy grid;47 the potential energy is calculated at each corner of each elementary cube (about 1 Å3). An accurate estimate of the energy is then obtained by a linear interpolation of the grid values. Such a procedure enables simulation of adsorption in mesoporous media of complex morphology and/or topology without a direct summation over matrix species in the course of GCMC runs.34,48-50 3. Results 3.1. Regular Cylindrical Pores. The intrusion isotherms obtained for the regular cylindrical nanopores with D ) 1.6, 2.4, 3.2, 4.8, and 6.4 nm (pores A1, A2, A3, A4, and A5, respectively) are shown in Figures 3 and 4. The data are reported as a function of the excess chemical potential β(µ - µ0), where β ) 1/kBT is the reciprocal temperature and µ0 ) kBT ln(P0) ) 6.48 kJ/mol is the chemical potential corresponding to the bulk saturating vapor pressure. Adsorbed amounts are reported as the density of fluid in the system, i.e. the pore and the two reservoirs at the bottom and top of the pore. The density is in reduced unit with respect to σ, the Lennard-Jones parameter for the fluid-fluid interaction, F ) Ν/Vσ3. We emphasize that this density is not universal, as it depends on the sizes of the pore and the reservoirs. On the one hand, for a given pore size and length, if one increases the size of the reservoirs (bulk external part), the variation corresponding to the intrusion/ retraction of fluid in the pore will appear smaller. On the other

Figure 4. Intrusion isotherm for a nonwetting fluid in MCM-41 nanopores: (from bottom to top) pore A2 (D ) 2.4 nm), pore A3 (D ) 3.2 nm), pore A4 (D ) 4.8 nm), and pore A5 (D ) 6.4 nm). Open and closed symbols correspond to the intrusion and retraction data, respectively. µ0 is the chemical potential corresponding to the bulk saturating vapor pressure.

hand, for a given size of the reservoirs, if one increases the pore length or size, the variation corresponding to the intrusion/ retraction of fluid in the pore will appear larger. As a result, the density reported in our data is not representative of the pore volume or surface area. It is worth noting that we did not correct our results for the amount of fluid located outside the pore, as it corresponds to the experimental situation of mercury porosimetry, in which one measures upon intrusion/retraction cycles the total variation of mercury volume in the system (i.e., the fluid confined in the porosity of the sample and the bulk-like fluid located between or outside the grains that constitute the system). Then, the porosity of the sample is assessed by measuring the sharp variations that occur as the fluid invades (or retracts from) the sample. For pores with D g 2.4 nm, the adsorbed amount increases until intrusion of the fluid within the pore occurs. As expected for a nonwetting fluid, the chemical potential at which intrusion occurs is larger than that corresponding to the bulk saturating vapor pressure of the fluid. The intrusion chemical potentials, i.e. β(µ - µ0) ) 1.13, 0.79, 0.53, 0.38 for D ) 2.4, 3.2, 4.8, 6.4 nm, respectively, decreases as the pore size increases. This result can be explained as follows. The surface to volume ratio for a cylindrical pore increases upon decreasing the pore size, so the effect of the nonwetting wall/ fluid interaction becomes more important as D decreases. As a result, the chemical potential needed for the liquid to invade the pore increases as the pore size decreases. The retraction chemical potentials β(µ - µ0) ) 0.86, 0.53, 0.32, 0.22 for D ) 2.4, 3.2, 4.8, 6.4 nm, respectively, are significantly lower than the intrusion chemical potentials, so large hysteresis loops are


Figure 5. Typical molecular configurations upon intrusion of a nonwetting fluid in a MCM-41 nanopore with D ) 1.6 nm (pore A1): β(µ - µ0) ) 2.01 (top) and β(µ - µ0) ) 2.10 (bottom). White spheres are fluid atoms, while gray spheres are hydrogen atoms at the pore surface.

observed. These results are in qualitative agreement with experimental data23,24,28,29 which show that (1) the intrusion and retraction chemical potentials are larger than that corresponding to the bulk saturating vapor pressure, (2) the intrusion and retraction chemical potentials are decreasing functions of the pore size, and (3) entrapment of the confined fluid leads to irreversible intrusion/retraction isotherms. In a more evident way than usually observed in mercury porosimetry experiments, the amount of fluid intruded in the system after pore filling keeps increasing with increasing pressure (Figures 3 and 4). This is due to the compressibility of argon (used in this work) being significantly higher than the compressibility of mercury. In contrast to the pores with D g 2.4 nm, the intrusion at β(µ µ0) ) 2.04 for the pore with D ) 1.6 nm is reversible and continuous. Interestingly, the critical diameter below which the intrusion/retraction becomes reversible is the same as that found for capillary condensation of wetting fluids (1.6 nm < Dc < 2.4 nm for argon).30 This result suggests that the so-called critical capillary condensation temperature, Tcc,1,51 above which capillary condensation in nanoporous solids becomes reversible, is independent of the couple adsorbate/adsorbent.52 This result is in agreement with our recent work53 showing that Tcc depends only on the reduced pore size, defined as the pore size D divided by the kinetic diameter of the adsorbate σ. We report in Figure 5 typical molecular configurations upon intrusion of the nonwetting fluid in the pore with D ) 1.6 nm (pore A1). We also report in Figure 6 molecular configurations of the intrusion and retraction of the nonwetting fluid in the pore with D ) 4.8 nm (pore A4). The filling occurs in a continuous way for the pore D ) 1.6 nm. The invasion of the porous space by the fluid starts from the external liquid phase that penetrates the nanopore. In a similar way, the intrusion mechanism for the pore with D ) 4.8 nm starts with the formation at the pore entrance of a convex hemispherical meniscus (due to the fact that the pressure in the liquid phase is larger than that in the gas phase). As the chemical potential increases, the meniscus enters the pore and its radius decreases; there is coexistence across the meniscus between the gas inside the pore and the bulk liquid located in the external reservoir (Figure 6). At the intrusion chemical potential, the fluid invades the pore through the displacement of the hemispherical meniscus along the pore axis; this filling mechanism corresponds to the

J. Phys. Chem. C, Vol. 113, No. 5, 2009 1957 transition at equilibrium between the empty pore and the filled pore configurations. Starting with a completely filled pore, the retraction in the case where a hysteresis loop is observed occurs through the cavitation of the gas phase within the pore (Figure 6). In contrast to the intrusion mechanism, such a retraction is a transition from a metastable state to a stable state. These results show that intrusion and retraction of nonwetting fluids are symmetrical with respect to desorption and adsorption of wetting fluids (adsorption is a transition from a metastable state to a stable state, while desorption occurs through the displacement at equilibrium of a hemispherical meniscus along the pore axis30,50,54). This result is in agreement with previous gas-lattice calculations by Monson and co-workers on intrusion of fluids in nanoporous solids.25-27 The nature and transition chemical potentials of the intrusion and retraction processes are summarized in Tables 2 and 3 for the different pores considered in this work. 3.2. Cylindrical Pores with Constrictions. We discuss in this section the results obtained for the two cylindrical pores having constrictions (see Table 1). As for the nanopores without constrictions, the results (nature and transition chemical potentials) are summarized in Tables 2 and 3. We note that recent gas-lattice calculations of intrusion and retraction in constricted pores and disordered porous materials have been reported in the literature.25-27,55 The intrusion isotherm for the silica nanopore of a diameter D ) 6.4 nm and a constriction with Dc ) 3.2 nm (pore B1) is shown in Figure 7. Again, adsorbed amounts are reported as the density of fluid in the system, i.e., the pore and the two reservoirs at the bottom and top of the pore. We also present in Figure 8 a series of molecular configurations in this pore obtained at different chemical potentials upon intrusion and retraction. As for the regular cylindrical pores, there is at low pressures coexistence across a hemispherical convex meniscus between the gas within the pore and the bulk liquid located in the external reservoir. As the chemical potential increases, the liquid first invades the cavities located near the pore openings at β(µ - µ0) ) 0.75 (Figure 8). This intrusion chemical potential is close to that found above for the regular pore with D ) 3.2 nm, β(µ - µ0) ) 0.79. This result suggests that intrusion in the main cavities of the pore with constriction is driven by intrusion of the fluid in the constriction that isolates the cavities from the external reservoir. Such an intrusion mechanism is symmetrical to the so-called pore-blocking effect that can be observed in adsorption/ desorption of wetting fluids in a constricted pore; the confined liquid only evaporates when the constriction, which isolates the pore from the gas, empties.56,57 This result suggests that intrusion in the large cavity D ) 6.4 nm for this constricted pore is a transition from a metastable to a stable state as the intrusion is delayed compared to what is expected for a regular cylindrical pore having the same diameter. After intrusion in the cavities located near the pore openings and the constrictions, the cavity with D ) 6.4 nm in the pore center remains empty at this chemical potential. This result can be explained by the fact that the liquid located near the pore openings is more stable than in the pore center as it interacts with a larger number of liquid atoms (favorable interactions) and a smaller number of wall atoms (nonfavorable interactions). Finally, the intrusion mechanism for this pore ends with the invasion at β(µ - µ0) ) 0.80 of atoms in the cavity in the pore center. Such a two-step intrusion mechanism shows that the pore external surfaces introduce some heterogeneity in the overall invasion process (in the sense that the intrusion chemical potential of the fluid near the pore opening is different from that of the fluid in the


Coasne et al.

Figure 6. Typical molecular configurations upon intrusion and retraction of a nonwetting fluid in a MCM-41 nanopore with D ) 4.8 nm (pore A4). (Left) Intrusion: β(µ - µ0) ) 0.51 (top) and β(µ - µ0) ) 0.53 (bottom). (Right) Retraction: β(µ - µ0) ) 0.34 (top) and β(µ - µ0) ) 0.32 (bottom). White spheres are fluid atoms, while gray spheres are hydrogen atoms at the pore surface.

TABLE 2: Intrusion and Retraction Chemical Potentials β(µ - µ0) for the Different Silica Nanopores Considered in This Work porous material regular pores pore A1 (1.6 nm) pore A2 (2.4 nm) pore A3 (3.2 nm) pore A4 (4.8 nm) pore A5 (6.4 nm) constricted pores pore B1 (3.2-6.4 nm) pore B2 (4.8-6.4 nm)

intrusion

retraction

2.04 1.13 0.79 0.53 0.38

2.04 0.86 0.53 0.32 0.22

0.75 (cavities) 0.80 (constrictions) 0.51

0.37 0.28

TABLE 3: Nature of the intrusion and Retraction Mechanisms for the Different Silica Nanopores Considered in This Worka porous material regular pores pore A1 (1.6 nm) pore A2 (2.4 nm) pore A3 (3.2 nm) pore A4 (4.8 nm) pore A5 (6.4 nm) constricted pores pore B1 (3.2-6.4 nm) pore B2 (4.8-6.4 nm)

intrusion

retraction

equilibrium equilibrium equilibrium equilibrium equilibrium

equilibrium metastable (cavitation) metastable (cavitation) metastable (cavitation) metastable (cavitation)

metastable metastable

metastable (cavitation) metastable (cavitation)

a Quasiequilibrium is used when the path is made of both at equilibrium (continuous, reversible) and metastable (discontinuous, irreversible) parts.

pore center). Starting with a completely filled pore, the retraction for the constricted pore occurs at β(µ - µ0) ) 0.37 through the cavitation of the gas phase within the pore (Figure 8). Such a retraction mechanism is a transition from a metastable state to a stable state, during which all the liquid in the pore center and near the pore opening evaporates. In contrast to intrusion, retraction for this pore is not a two-step process, as the cavities near the pore openings and that in the pore center empty at the same retraction chemical potential. Such a behavior can be explained as follows. Given that the wall-fluid interaction is purely repulsive, the stability of the confined liquid decreases as the contact surface with the pore wall increases. As a result, the liquid confined in the constrictions is less stable than in the

Figure 7. Intrusion isotherm for a nonwetting fluid in silica nanopores: (triangles) cylindrical pore of a diameter D ) 6.4 nm and a constriction Dc ) 3.2 nm (pore B1) and (circles) cylindrical pore of a diameter D ) 6.4 nm and a constriction Dc ) 4.8 nm (pore B2). Open and closed symbols correspond to the intrusion and retraction data, respectively. µ0 is the chemical potential corresponding to the bulk saturating vapor pressure.

pore cavities, as it interacts with a larger number of wall atoms. Hence, starting with a completely filled pore, retraction first occurs through cavitation in the constrictions as this corresponds to the region where the confined liquid is the least stable. Once the constrictions are empty, the liquid in the pore cavities finds itself in equilibrium with the gas phase and spontaneously retracts if the chemical potential is below that at which retraction should occur at equilibrium. The two-step intrusion and one-step retraction for this constricted pore is to be compared with our previous results30 on adsorption/desorption of argon (wetting fluid) for the same system. In this previous work, filling and emptying of argon were found to be one-step and two-step mechanisms, in qualitative agreement with the experiments by Ravikovitch and Neimark58 on N2, Ar, and Kr adsorption in ordered cagelike mesoporous silicas. Such a similarity between intrusion and desorption of fluids in constricted pores is another illustration of the symmetry between intrusion/retraction of nonwetting fluids and adsorption/desorption of wetting fluids.25-27 We now discuss the intrusion isotherm for the silica nanopore of a diameter D ) 6.4 nm and a constriction with Dc ) 4.8 nm (pore B2) shown in Figure 7. We report in Figure 9 typical molecular configurations in this pore obtained at different chemical potentials upon intrusion and retraction. Again, at low pressures, there is coexistence across a hemispherical convex meniscus between the gas within the pore and the bulk liquid located in the external reservoir. Intrusion for this pore occurs


J. Phys. Chem. C, Vol. 113, No. 5, 2009 1959

Figure 8. Typical molecular configurations upon intrusion and retraction of a nonwetting fluid in a silica nanopore of a diameter D ) 6.4 nm and a constriction Dc ) 3.2 nm (pore B1). (Left) Intrusion: β(µ - µ0) ) 0.75 (top) and β(µ - µ0) ) 0.80 (bottom). (Right) Retraction: β(µ - µ0) ) 0.38 (top) and β(µ - µ0) ) 0.36 (bottom). White spheres are fluid atoms, while gray spheres are hydrogen atoms at the pore surface.

Figure 9. Typical molecular configurations upon intrusion and retraction of a nonwetting fluid in a silica nanopore of a diameter D ) 6.4 nm and a constriction Dc ) 4.8 nm (pore B2). (Left) Intrusion: β(µ - µ0) ) 0.38 (top) and β(µ - µ0) ) 0.51 (bottom). (Right) Retraction: β(µ - µ0) ) 0.28 (top) and β(µ - µ0) ) 0.26 (bottom). White spheres are fluid atoms, while gray spheres are hydrogen atoms at the pore surface.

at β(µ - µ0) ) 0.51 with the invasion of the whole pore by the liquid. This intrusion chemical potential is close to that found for the regular cylindrical pore with D ) 4.8 nm, β(µ - µ0) ) 0.53. This result suggests that the intrusion in a constricted pore is driven by the intrusion of the fluid in the constriction that isolates the cavities from the external reservoir. This conclusion is also supported by the data reported above for pore B1. Again, such an intrusion mechanism is a transition from a metastable to a stable state, as intrusion is delayed compared to what is expected for a regular cylindrical pore having the same diameter. In contrast to the two-step intrusion mechanism observed for pore B1 (the liquid invades the pore center at a different chemical potential than the region near the pore openings), intrusion for pore B2 is a one-stage mechanism. This result shows that the difference between the constriction diameter and the cavity diameter in pore B2 is not large enough to introduce some heterogeneity in the intrusion process. Retraction of the nonwetting fluid for pore B2 follows the same pattern observed for pore B1; i.e., starting with a completely filled pore, the retraction consists of the nucleation at β(µ - µ0) ) 0.28 of the gas phase within the pore (cavitation). Again, this retraction mechanism is a transition from a metastable state to a stable state, during which all the liquid in the pore center and near the pore opening evaporates.

4. Discussion In this paper, we report molecular simulations of intrusion and retraction of a nonwetting fluid confined in silica nanopores without or with morphological defects, which can be interpreted as constrictions or enlargments of the pore according to the choice of the reference diameter. From a fundamental point of view, these morphological defects are equivalent to chemical heterogeneity along the pore axis in the sense that they introduce some dispersion in the potential energy along the pore surface (see refs 59-63 and 64 on adsorption and desorption in chemically heterogeneous nanopores). All the pores considered in this work are of a finite length and are connected to bulk reservoirs, so they mimic real materials for which the confined fluid is always in contact with the external phase (this departs from infinitely long pores for which the intrusion necessarily occurs through the nucleation of the liquid phase as there is no interface with the external phase). For the pores without constrictions, we considered five pore diameters, D ) 1.6, 2.4, 3.2, 4.8, and 6.4 nm, in order to address the effect of the pore size. For pores with constrictions, we address the effect of the size of these constrictions by varying their diameter. All our results in terms of nature of the mechanism and transition


Figure 10. Chemical potential of gas and liquid (supercooled) argon at 77 K as a function of pressure. The chemical potential µ is given with respect to the reference state corresponding to the chemical potential at the liquid/gas coexistence, µ0 ) 6.48 kJ/mol. The error bars are smaller than the size of the symbols.

chemical potentials for the intrusion and retraction mechanisms are reported in Tables 2 and 3. Our results for the regular and constricted pores are in qualitative agreement with experimental data: (1) the intrusion and retraction chemical potentials are larger than that corresponding to the bulk saturating vapor pressure, (2) the intrusion and retraction chemical potentials are decreasing functions of the pore size, and (3) entrapment of the confined fluid leads to irreversible intrusion/retraction isotherms. Intrusion and retraction experiments of nonwetting fluids in nanopores are usually interpreted on the basis of the WashburnLaplace equation (eq 1). The common explanation for the difference between the intrusion and retraction pressures (hysteresis loops) is that the receding contact angle θr upon retraction is different from the advancing contact angle θa upon intrusion.65 In this situation, according to eq 1 and assuming that the pressure of the vapor is negligible, the same slope ∼ -1 for ln(P) as a function of ln(D) is expected for the intrusion and retraction mechanisms. For a pore with constriction or disordered porous materials, another common interpretation for the irreversibility of the intrusion/retraction cycles is that retraction is driven by the nucleation of a vapor bubble.66 In this case, the absolute value of the slope for ln(P) as a function of ln(D) for the retraction is larger than that for the intrusion (see refs 23 and,24 for a detailed discussion). The pressure of the liquid phase is not used as an input parameter in our simulations of the intrusion of nonwetting fluids (only the temperature and chemical potential are fixed in a GCMC simulation). Consequently, in order to compare our data with the predictions of the Washburn-Laplace equation, we determined the relationship between pressure and chemical potential for supercooled liquid argon at 77 K using the following method proposed by Desbiens et al.67 We first performed GCMC simulations of bulk argon in order to obtain the relationship between chemical potential and density, µ(F). Then, we computedusingMonteCarlosimulationsintheisobaric-isothermal ensemble (NPT) the relationship between pressure and density, P(F). Gathering all these data together, one obtains the relationship µ(P) shown in Figure 10. The linear region in the lowpressure range corresponds to the gas/liquid equilibrium, while the weak pressure dependence of the chemical potential above the saturating vapor pressure illustrates the low compressibility of the liquid. The intrusion and retraction pressures calculated for the regular pores A1, A2, A3, A4, and A5 are reported in Figure 11 as a function of the pore diameter, D. A straight line is obtained when both the pressure and pore diameter are plotted in a logarithmic scale; the slope, -1.1 ( 0.1, is close to the theoretical value -1 given by the Washburn-Laplace equation

Coasne et al.

Figure 11. Intrusion (open symbols) and retraction (closed symbols) pressure as a function of the pore diameter. The circles are for the regular cylindrical pores (pores A1, A2, A3, A4, and A4) while the triangles are for the pores with constrictions (pores B1 and B2). The size of the smallest cavity (constriction) was used for the data corresponding to pores B1 and B2. The error bars are smaller than the size of the symbols.

(eq 1). We note that the fit was done without the data for the smallest pore, D ) 1.6 nm, as intrusion and retraction for this pore are not supposed to obey the Washburn-Laplace equation, in the sense that they are reversible continuous processes that do not involve any gas/liquid interface (meniscus). Knowing the gas/liquid surface tension for bulk argon at 77 K (γlv ) 15.2 mN/m), we also estimated the angle θa from the fit of our data against eq 1. We found that θa is 102° (15°, which is a reasonable value for a nonwetting fluid, although the theoretical nature of the nonwetting argon used in this work prevents any meaningful comparison with actual contact angles. We note that these values for the slope and contact angle are almost unchanged if the surface tension is corrected for the curvature effect using the Tolman equation.68 The results above are in qualitative agreement with previous experiments.23,69 in which the Washburn-Laplace equation was found to successfully describe the intrusion pressure of a nonwetting fluid in cylindrical nanopores. The logarithm of the retraction pressure also varies linearly with the logarithm of the pore diameter. However, in contrast to the intrusion pressure that varies with a slope of about -1 (Washburn-Laplace equation), the retraction pressure varies with a larger slope, -1.6 ( 0.1. This result is due to the fact that retraction consists of the nucleation (out-of-equilibrium) of a gas bubble within the pore, which is not supposed to obey the Washburn-Laplace equation describing the mechanical stability of a gas/liquid interface. Finally, our results are in agreement with several sets of experimental data23,69-71 showing that the intrusion and retraction pressures do not follow the same law when plotted as a function of the pore diameter. This suggests that the common explanation for the hysteresis loop assuming two different contact angles upon intrusion and retraction is not valid; the retraction pressure should vary in the same way as the intrusion pressure if retraction occurred through the displacement of a meniscus with a receding contact angle θr. We also report in Figure 11 the intrusion and retraction chemical potentials for the constricted pores (pores B1 and B2) as a function of the smaller size of the pore (constriction). On the one hand, we find that the intrusion chemical potential in the large cavity of the constricted pores corresponds to that for a regular cylindrical nanopore having the same diameter as the constriction; the open triangles (constricted pores) in Figure 11 fall on the same curve as that for the regular pores when plotted as a function of the constriction size. This result suggests that intrusion in the constricted pores occurs when the nonwetting fluid invades (i.e., becomes stable within) the constrictions that

Intrusion and Retraction of Fluids in Nanopores isolate the large cavity from the bulk external phase. On the other hand, the retraction chemical potentials for the constricted pores (filled triangles in Figure 11) are lower than the value found for a regular cylindrical pore having the pore diameter of the constriction. This result, which is more pronounced for pore B1, as this pore exhibits a smaller constriction to cavity size ratio and a lower tapering angle (compared to pore B2), is an illustration of the entrapment effect of nonwetting fluids confined in nanoporous materials. This result is of crucial interest for characterization of porous solids using mercury porosimetry and/or nitrogen adsorption. 5. Conclusion Our findings can be summarized as follows. The intrusion isotherms for the regular cylindrical pores conform to the typical experimental behavior observed for MCM-41, as both the intrusion and retraction chemical potentials increase when the pore size decreases. The intrusion mechanism corresponds to the displacement at equilibrium of a gas/liquid curved interface (convex meniscus) within the pore. The contact angle θa is found to be about 102° ( 15°. Our data also show that the intrusion pressure varies linearly with the pore diameter when both quantities are plotted in a logarithmic scale; the slope, -1.1 ( 0.1, is close to the theoretical value -1 expected on the basis of the Washburn-Laplace equation. This qualitative agreement with the behavior predicted by the Washburn-Laplace equation together with the observation of a hemispherical meniscus upon intrusion suggest that, even for a fluid confined at the nanoscale, the concept of surface tension and contact angle remains meaningful. In particular, this suggests that the use of macroscopic equations such as the Washburn-Laplace equation to relate the pore size and intrusion pressure remains, at least qualitatively, valid (nevertheless, macroscopic approaches need to be modified to account for curvature or nanoconfinement effects using corrections such as the Tolman law, etc.). This conclusion is consistent with our previous work in which it was found that adsorption/desorption of a wetting fluid in pores proceed through the formation of curved interfaces that are, at least qualitatively, described by the Kelvin equation. In contrast to intrusion, retraction does not proceed through the displacement of a curved interface, as the confined liquid is not in equilibrium with the gas phase prior to retraction. As a result, retraction necessarily occurs through the metastable nucleation of a gas bubble within the confined liquid (cavitation). This absence of gas/liquid meniscus upon retraction of the confined liquid suggests that the Washburn-Laplace equation is not suitable to describe the emptying process, in contrast to the intrusion mechanism. This is confirmed by the fact the logarithm of the retraction pressure varies with the logarithm of the pore diameter with a steeper slope, -1.6 ( 0.1, compared to the value -1 expected on the basis of the Washburn-Laplace equation. This departure from the behavior predicted by the Washburn-Laplace equation could be expected as the latter assumes the formation of a hemispherical meniscus that is not observed upon retraction. In particular, these results suggest that the common explanation for the hysteresis loop assuming two different contact angles upon intrusion and retraction is not valid (otherwise the same slope would be observed for the intrusion and retraction data). The intrusion mechanism for the pores with constrictions departs from what is observed for regular nanopores. We find that the intrusion chemical potential in the constricted pore corresponds to that for a regular cylindrical nanopore having the same diameter as the constriction. Depending on the relative

J. Phys. Chem. C, Vol. 113, No. 5, 2009 1961 size of the cavity and constriction, the intrusion process for the constricted pore is a one- or two-step process (i.e., in the latter case the cavities near the pore openings are filled at a different chemical potential than the cavities in the pore center). On the other hand, retraction for the constricted pores is of the same nature as that observed for the regular pores (i.e., cavitation), but occurs at a pressure lower than that expected for a regular pore having the size of the constrictions and larger than that expected for a regular pore having the size of the cavites. From a general point of view, our results suggest that intrusion and retraction experiments (porosimetry) can be used to assess and characterize morphological defects in nanopores, provided that the pore size is known from other independent measurements. References and Notes (1) Gelb, L. D.; Gubbins, K. E.; Radhakrishnan, R.; SliwinskaBartkowiak, M. Rep. Prog. Phys. 1999, 62, 1573. (2) Alba-Simionesco, C.; Coasne, B.; Dosseh, G.; Dudziak, G.; Gubbins, K. E.; Radhakrishnan, R.; Sliwinska-Bartkowiak, M. J. Phys.: Condens. Matter 2006, 18, R15. (3) Beck, J. S.; Vartulli, J. C.; Roth, W. J.; Leonowicz, M. E.; Kresge, C. T.; Schmitt, K. D.; Chu, C. T.-W.; Olson, D. H.; Sheppard, E. W.; McCullen, S. B.; Higgins, J. B.; Schlenker, J. L. J. Am. Chem. Soc. 1992, 114, 10834. (4) Zhao, D.; Feng, J.; Huo, Q.; Melosh, N.; Fredrickson, G. H.; Chmelka, B. F.; Stucky, G. D. Science 1998, 279, 548. (5) Corma, A. Chem. ReV. 1997, 97, 2373. (6) Ciesla, U.; Schu¨th, F. Microporous Mesoporous Mater. 1999, 27, 131. (7) Soler-Illia, G. J. de A. A.; Sanchez, C.; Lebeau, B.; Patarin, J. Chem. ReV. 2002, 102, 4093. (8) Imperor-Clerc, M.; Davidson, P.; Davidson, A. J. Am. Chem. Soc. 2000, 122, 11925. (9) Ryoo, R.; Ko, C. H.; Kruk, M.; Antochshuk, V.; Jaroniec, M. J. Phys. Chem. B 2000, 104, 11465. (10) Liu, Z.; Terasaki, O.; Ohsuna, T.; Hiraga, K.; Shin, H. J.; Ryoo, R. Chem. Phys. Chem 2001, 2, 229. (11) Galarneau, A.; Cambon, H.; Di Renzo, F.; Fajula, F. Langmuir 2001, 17, 8328. (12) Galarneau, A.; Cambon, H.; Di Renzo, F.; Ryoo, R.; Choi, M.; Fajula, F. New J. Chem. 2003, 27, 73. (13) Jun, S.; Joo, S. H.; Ryoo, R.; Kruk, M.; Jaroniec, M.; Liu, Z.; Ohsuna, T.; Terasaki, O. J. Am. Chem. Soc. 2000, 122, 10712. (14) Sonwane, C. G.; Bhatia, S. K.; Calos, N. J. Langmuir 1999, 15, 4603. (15) Berenguer-Murcia, A.; Garcia-Martinez, J.; Cazorla-Amoros, D.; Martinez-Alonso, A.; Tascon, J. M. D.; Linares-Solano, A. In Studies in Surface Science and Catalysis 144; Rodriguez-Reinoso, F., McEnaney, B., Rouquerol, J., Unger, K., Eds.; Elsevier Science: New York, 2002; p 83. (16) Edler, K. J.; Reynolds, P. A.; White, J. W. J. Phys. Chem. B 1998, 102, 3676. (17) Rouquerol, F.; Rouquerol, J.; Sing, K. S. W. Adsorption by Powders and Porous Solids; Academic Press: London, 1999. (18) Talanquer, V.; Oxtoby, D. W. J. Chem. Phys. 2001, 114, 2793. (19) Lum, K.; Chandler, D.; Weeks, J. D. J. Phys.Chem. B 1999, 103, 4750. (20) Bolhuis, P. G.; Chandler, D. J. Chem. Phys. 2000, 113, 8154. (21) Cottin-Bizonne, C.; Barrat, J. L.; Bocquet, L.; Charlaix, E. J. Chem. Phys. 2003, 2, 237. (22) Restagno, F.; Bocquet, L.; Biben, T. Phys. ReV. Lett. 2000, 84, 2433. (23) Lefevre, B.; Saugey, A.; Barrat, J. L.; Boquet, L.; Charlaix, E.; Gobin, P. F.; Vigier, G. J. Chem. Phys. 2004, 120, 4927. (24) Lefevre, B.; Saugey, A.; Barrat, J. L.; Boquet, L.; Charlaix, E.; Gobin, P. F.; Vigier, G. Colloids Surf. A 2004, 241, 265. (25) Porcheron, F.; Monson, P. A.; Thommes, M. Langmuir 2004, 20, 6482. (26) Porcheron, F.; Monson, P. A.; Thommes, M. Adsorption 2005, 11, 325. (27) Porcheron, F.; Monson, P. A. Langmuir 2005, 21, 3179. (28) Gomez, F.; Denoyel, R.; Rouquerol, J. Langmuir 2000, 16, 4374. (29) Martin, T.; Lefevre, B.; Brunel, D.; Galarneau, A.; Di Renzo, F.; Fajula, F.; Gobin, P. F.; Quinson, J. F.; Vigier, G. Chem. Commun. 2002, 24–25. (30) Coasne, B.; Galarneau, A.; Di Renzo, F.; Pellenq, R. J.-M. J. Phys. Chem. C 2007, 111, 15759–15770. (31) Pellenq, R. J.-M.; Levitz, P. E. Mol. Phys. 2002, 100, 2059.

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Intrusion and Retraction of Fluids in Nanopores: Effect of

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