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Langmuir 2007, 23, 3372-3380
Mercury Porosimetry in Mesoporous Glasses: A Comparison of Experiments with Results from a Molecular Model F. Porcheron,† M. Thommes,‡ R. Ahmad,‡ and P. A. Monson*,† Department of Chemical Engineering, UniVersity of Massachusetts, Amherst, Massachusetts 01003-9303, and Quantachrome Instruments, 1900 Corporate DriVe, Boynton Beach, Florida 33426 ReceiVed October 19, 2006. In Final Form: December 16, 2006 We present results from experiments and molecular modeling of mercury porosimetry into mesoporous Vycor and controlled pore glass (CPG) solid materials. The experimental intrusion/extrusion curves show a transition from a type H2 hysteresis for the Vycor glass to a type H1 hysteresis for the CPG. Mercury entrapment is observed in both materials, but we find that the amount of entrapped mercury depends on the chosen experimental relaxation time. No additional entrapment is found in a second intrusion/extrusion cycle, but hysteresis is still observed. This indicates that hysteresis and entrapment are of different origin. The experimental observations are qualitatively reproduced in theoretical calculations based on lattice models, which provide significant insights of the molecular mechanisms occurring during mercury porosimetry experiments in these porous glasses.
I. Introduction Gas adsorption and mercury porosimetry are widely accepted methods to characterize porous solids with respect to their pore size, pore size distribution, pore volume, and surface area. In contrast to gas adsorption and pore condensation, where the pore fluid wets the pore walls (contact angle 90°) and (hydraulic) pressure must be applied to force mercury into the pores. The attraction of mercury porosimetry is based on its ability to perform pore size analysis over a wide range of pore sizes from about 0.003 µm up to about 400 µm. In addition, mercury porosimetry data have been interpreted in terms of particle size distributions, tortuosity factors, permeability, fractal dimension, and compressibility, as well as to account for pore shapes and network effects in any porous solid.1-3 Information about skeletal and bulk density of the material can also be obtained. Hence, over the course of several decades, mercury porosimetry has proven to be of substantial value to industries dealing for instance with catalysts, ceramics, minerals, coals, soils, and pharmaceuticals. In most of the applications, pore size distribution curves are simply obtained by monitoring the volume of mercury intruded into pores as a function of applied pressure. The estimation of the pore size distribution curves from these data is traditionally based on the use of the Washburn equation4
Pr ) -2γ cos θ
(1.1)
where P is the hydraulic pressure that must be applied to a nonwetting liquid mercury to penetrate cylindrical pores of radius r and γ and θ are the mercury surface tension and contact angle, respectively. A significant feature of mercury porosimetry curves is the occurrence of hysteresis between the intrusion and extrusion branches. In addition, entrapment is observed, i.e., mercury * Corresponding author:
[email protected]. † University of Massachusetts. ‡ Quantachrome Instruments. (1) Lowell, S.; Shields, J.; Thomas, M. A.; Thommes, M. Characterization of Porous Solids and Powders: Surface Area, Pore Size and Density; Kluwer Academic Publisher: Dordrecht, Boston, London, 2004. (2) Leo´n y Leo´n, C. AdV. Colloid Interface Sci. 1998, 341, 76-77. (3) Moscou, L.; Lub, S. Powder Technol. 1981, 29, 45. (4) Washburn, E. W. Phys. ReV. 1921, 17, 273.
remains contained in the porous network after extrusion. A detailed understanding of these phenomena is most important in order to be able to obtain a more accurate mesopore size analysis. Both phenomena, i.e., intrusion/extrusion hysteresis and entrapment, can be observed in the first mercury intrusion/extrusion cycle. However, entrapment does usually not occur in the second intrusion/extrusion cycle performed on the same material, but the hysteresis loop is still observed.1-3 This indicates that entrapment and hysteresis have different origins. There is extensive literature on hysteresis and entrapment phenomena in mercury porosimetry.1-11 Different models (mostly based on mesoscopic and macroscopic concepts) are discussed in the literature, and among them are (i) contact angle hysteresis,10-15 (ii) the energy barrier model,16 and (iii) network models.17-21 Within the contact angle hysteresis model,10-15 hysteresis is attributed to different contact angles during intrusion and extrusion. The energy barrier model takes into account that during (5) Pirard, R.; Blacher, S.; Brouers, F.; Pirard, J. P. J. Mater. Res. 1995, 10, 2114. (6) Pismen, L. M. Dokl. Akad. Nauk SSSR 1973, 211, 1398. Neimark, A. V. Colloid J. USSR 1984, 46, 639. (7) Neimark, A. V. Colloid J. USSR 1985, 47, 67. (8) Day, M.; Parker, I. B; Bell, J.; Thomas, M.; Fletcher, R.; Duffie, J. In Characterization of Porous Solids II; Rouquerol, J., Rodriguez-Reinoso, F., Sing, K. S. W., Unger, K. K., Eds.; Elsevier: Amsterdam, 1991; p 75. (9) Tsakiroglou, C. D.; Payatakes, A. C. AdV. Colloid Interface Sci. 1998, 75, 215. (10) Rigby, S. Stud. Surf. Sci. Catal. 2002, 144, 185. (11) Rigby, S. P.; Fletcher, R. S.; Riley, S. N. J. Colloid Interface Sci. 2001, 240, 190. (12) Lowell, S. Powder Technol. 1980, 25, 37. Lowell, S.; Shields, J. E. J. Colloid Interface Sci. 1982, 80, 192. Lowell, S.; Shields, J. E. J. Colloid Interface Sci. 1981, 83, 271. (13) Kloubek, J. Powder Technol. 1981, 29, 63. (14) Salmas, C.; Androutsopoulos, G. J. Colloid Interface Sci. 2001, 239, 178. (15) Rigby, S. P.; Edler, K. J. J. Colloid Interface Sci. 2002, 250, 175. (16) Giesche, H. Mater. Res. Soc. Symp. Proc. 1982, 80, 192. (17) Zgrablich, G.; Mendioroz, S.; Daza, L.; Pajares, J.; Mayagoitia, V.; Rojas, F.; Conner, W. C. Langmuir 1991, 7, 779. (18) Day, M.; Parker, I. B.; Bell, J.; Fletcher, R.; Duffe, J.; Sing, K. S. W.; Nicholson, D. In Characterization of Porous Solids III; Rouquerol, J., RodriguezReinoso, F., Sing, K. S. W., Unger, K. K., Eds.; Elsevier: Amsterdam, 1994; p 225. (19) Matthews, G. P.; Ridgway, C. J.; Spearing, M. C. J. Colloid Interface Sci. 1995, 171, 8. (20) Kornhauser, I.; Cordero, S.; Felipe, C.; Rojas, F.; Ramirez-Cuesta, A. J.; Riccardo, J. L. Proceedings of Fundamenals of Adsorption 7; Kaneko, K., Ed.; IK International, Ltd.: Chiba City, Japan, 2002; p 1030. (21) Felipe, C.; Cordero, S.; Kornhauser, I.; Zgrablich, G; Lopez, R; Rojas, F. Part. Part. Syst. Charact., in press.
10.1021/la063080e CCC: $37.00 © 2007 American Chemical Society Published on Web 02/17/2007
Mercury Porosimetry in Glass
extrusion the liquid/vapor interface has to be newly created (and thus needs an extra amount of energy). Similarly, as in gas adsorption, network models describe hysteresis as a consequence of the distribution of pore sizes and their connectivity in the porous material. Woo and Monson22 have presented evidence (based on a lattice gas model) that the hysteresis loops observed during gas adsorption experiments for fluids confined to disordered porous materials (such as Vycor) involve two differential dynamical regimes. The first regime (the so-called “transport regime”) is associated with mass transfer to and from the external surface and can be thought as diffusional relaxation. Typically the experimental sample time, τ, is longer than the time scale covering this regime. Of course, if the sample time is too short, the width of the hysteresis would change significantly with time. The second regime is the so-called “quasi-equilibrium regime”, which corresponds to the redistribution of fluid inside the material. Equilibration is here so slow that the associated changes are not detectable on an experimental time scale. This explains why experimentally observed gas adsorption hysteresis loops are reproducible. In recent papers23,24 we have considered mercury porosimetry from the perpective of statistical thermodynamics, mean field density functional theory (MF-DFT), and Monte Carlo (MC) simulations, applied to a lattice gas model of penetration of a nonwetting liquid into a porous material. Application was made to mercury in the porous glass Vycor. The lattice model exhibits a symmetry that provides a direct relationship between intrusion/extrusion curves for a nonwetting fluid and adsorption/ desorption isotherms for a wetting fluid. On the basis of these results, we could conclude that in the first instance hysteresis in mercury porosimetry for disordered porous materials has the same origin as hysteresis in gas adsorption/desorption experiments in the same material. Hence, the symmetry of the lattice model means that the arguments of Woo and Monson with regard to the two dynamical regimes (associated with the occurrence of hysteresis) carry over into mercury porosimetry. Entrapment is generally believed to be associated with kinetic effects during mercury extrusion, coupled with the tortuosity of the disordered pore network and the surface chemistry of the material (reflected in the contact angle).1-3 Porcheron and Monson24 performed dynamic Monte Carlo simulations using both Glauber dynamics and Kawasaki dynamics for a lattice gas model to study the behavior of a nonwetting fluid confined in a porous material. They studied intrusion and extrusion into slit and ink-bottle pore geometries as well as into the porous Vycor glass. Their results suggest that mercury entrapment is caused by a decrease in the rate of mass transfer associated with the fragmentation of liquid during extrusion. Under such conditions extrusion involves vapor-phase transport with an intrinsically low flux, a mechanism representing a slowing down of the dynamics in the “transport” regime. This is also in agreement with the findings of Lowell and Shields1 who noted that often at the completion of an intrusion/extrusion cycle mercury would slowly continue to extrude for hours. The purpose of the present paper is to present a comparison of the theoretical predictions with new experimental results for intrusion/extrusion dynamics of mercury in Vycor and CPG. In the experiments intrusion/extrusion results are obtained for different experimental sample times and studies of the systems during second intrusion/extrusion cycles are made. This makes it possible to investigate the relationship between hysteresis and (22) Woo, H. J.; Monson, P. A. Phys. ReV. E 2003, 67, 041207. (23) Porcheron, F.; Monson, P. A.; Thommes, M. Langmuir 2004, 20, 6482. (24) Porcheron, F.; Monson, P. A. Langmuir 2005, 21, 3179.
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entrapment. Our paper is organized as follows. In section II we describe important details about the experiments while section III gives a review of the theory based on the lattice model. Section IV is devoted to the description of and comparison between the experimental and the simulation results. Finally, section V gives a summary of our results and conclusions. II. Experiments Mercury intrusion and extrusion experiments on Vycor and CPGs were performed over a wide range of pressures starting from vacuum to over 400 MPa (using a Quantachrome Poremaster 60 instrument). Vycor (Corning 7930) was obtained from Corning (via Advanced Glass and Ceramics, 108 Valley Hill Dr., Holden, MA), and CPG was obtained from Millipore (290 Concord Rd, Billerica, MA). Prior to the analysis the samples were outgassed overnight at 150 °C. Data acquisition was performed in both the so-called continuous scanning and stepwise modes, in order to evaluate the effect of experimental sample time on the hysteresis and entrapment. In the scanning mode the rate of pressurization is controlled by the motor speed of the pressure generator system. The motor speed can be set to a fixed, constant value; however, it is also possible (with the help of a computer) to adjust the pressurization and depressurization rate in inverse proportion to the rate of intrusion or extrusion, respectively. Thus, the porosimeter provides maximum speed in the absence of intrusion or extrusion and maximum resolution and in most cases sufficient relaxation time (sampling time) when most required, that is, when intrusion or extrusion is occurring rapidly with changing pressure. The use of this scanning mode allows us to obtain highresolution intrusion/extrusion curves. In the stepwise data acquisition mode the pressure can be incremented or decremented by programmed amounts between each measurement. The user has to program the pressure points to be analyzed and a proper equilibration period (relaxation time) (for more details, see refs 1 and 2). Nitrogen (77.35 K) adsorption/desorption isotherm measurements were also performed using an Autosorb-I-MP sorption instrument (Quantachrome Instruments, Boynton Beach, FL) over a range of relative pressure (P/P0) between 10-6 and 1. As in the case of the mercury porosimetry samples, the samples were outgassed overnight at 150 °C prior to the adsorption analysis.25
III. Model and Methods A. Lattice Hamiltonian. In order to model the mercury porosimetry experiments in porous glasses, we discretize the space using a simple cubic lattice.23 Each lattice site has therefore Z ) 6 nearest neighbors and can either be occupied by a solid particle (ζ ) 0) or available for a fluid occupation (ζ ) 1). Each fluid site can then either be occupied, with fluid occupation variable ni ) 1, or be vacant, with ni ) 0. The Hamiltonian of the lattice model can be written as
H ) -
∑ ζiζjninj - µ ∑i ζini - R ∑ [niζi(1 - ζj) + 〈ij〉
〈ij〉
njζj(1 - ζi)] (1) where is the fluid-fluid interaction parameter, R is the ratio of the wall-fluid to the fluid-fluid interaction parameter, and (25) The analysis station of the volumetric sorption apparatus is equipped with, in addition to the obligatory pressure transducers in the dosing volume (manifold) of the apparatus, high precision pressure transducers (Baratron MKS) dedicated to read the pressure just in the sample cell. Hence, the sample cell is isolated during equilibration, which ensures a very small effective void volume and therefore a highly accurate determination of the adsorbed amount. The saturation pressure P0 is measured throughout the entire analysis by means of a dedicated saturation pressure transducer, which allows the vapor pressure to be monitored for each data point. This leads to high accuracy and precision in the determination of P/P0 and thus in the determination of the adsorption isotherm and the pore size distribution.
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Porcheron et al.
µ is the chemical potential.26 The double summations in the first and third terms of eq 2 run over all the distinct nearest neighbor site pairs. This lattice Hamiltonian exhibits a symmetry with respect to bulk liquid-vapor coexistence chemical potential µ0 and with respect to R ) 1/2. We have that
H′(1/2 - δR, µ0 + δµ) ) H′(1/2 + δR,µ0 - δµ)
(2)
that is, the Hamiltonian of a state of a fluid where δµ < 0 and δR > 0 (i.e., gas adsorption/desorption of a wetting fluid) has the same value as that for the fluid with δµ > 0 and δR < 0 (i.e., liquid intrusion/extrusion of a nonwetting fluid) and all the fluid occupancies (denoted by s) reversed. In our previous paper we showed that this symmetry contained in the lattice Hamiltonian leads to a symmetry between gas adsorption isotherms and mercury porosimetry curves. The configuration of the solid sites {ζi} that we use is built to model the mesoporous structure of Vycor and CPG. Each sample of a glass material is obtained with the Gaussian random field method where an auxiliary random field Φ(r) is used to generate the physical density of the solid via the level cut22,27,28
Fs(r) ) Θ[Φ(r) - Φc]
(3)
where Θ(x) is the Heaviside step function. The lattice variable ζi is obtained via the discretization ζi ) 1 - Fs(ri). The field cutoff and the spatial correlation of the field are fixed so that spatial fluctuations of the solid density match known experimental properties of the solid. For a mesoporous glass, those properties are the porosity and the structure factor I(q) which has been measured for Vycor.29 The model is then characterized by the porosity p ) 1 - Ns/Ntot, where Ns is the number of solid sites and Ntot is the total number of lattice sites, and by a lattice constant a setting the scale of the resolution. The CPG samples are obtained by scaling the experimental structure factor of the Vycor to obtain the desired porosity. In our simulations we consider a solid material geometry of volume V ) 2 N × N × N in contact with bulk reservoirs at its extremities along the x direction,22,28 and to avoid interfacial effect due to the finite size of the material, averages of thermodynamic quantities are performed only over the central portion of the sample (i.e., N/2 < x < 3N/2 and 0 < y,z < N). B. Mean-Field Density Functional Theory. Mean-field density functional theory (MF-DFT) calculations have been widely used for the modeling of gas adsorption in disordered mesoporous materials.28,30-32 For our lattice model, the meanfield expression of the grand potential is
βΩ[ζi] )
∑i [Fi ln Fi + (ζi - Fi) ln(ζi - Fi) - βµFi] β ∑ [FiFj + RFi(1 - ζj) + RFj(1 - ζi)]
Figure 1. Nitrogen adsorption/desorption isotherms at 77.4 K from experiment for Vycor (porosity ca. 32%, b) and CPG 75 (porosity ca. 55%, 0).
chemical potential yields a set of nonlinear algebraic equations for the average local densities at sites i, Fi. We have
Fi )
ζi
1 + exp[-βµ - β
∑j∈i [Fj + R(1 - ζj)]]
(5)
The equations are solved by an iteration scheme with convergence assumed when for an iteration k + 1 we obtain [∑i(Fk+1 - Fki )2]1/2 i -4 < 10 . C. Monte Carlo Simulations. To study the dynamic behavior of fluids penetrating porous glasses, we have performed Monte Carlo simulations using both Glauber and Kawasaki algorithms.22,35 The straightforward implementation of the Metropolis algorithm in the Grand Canonical ensemble for the lattice model generates the Glauber dynamics, where the creation and destruction of particles are possible throughout the system regardless of the relative position to the bulk interface. This method ignores the fact that density fluctuations inside the material can only occur via diffusion or flow. We therefore also consider a more realistic but more computationally intensive method where the particles are treated via a blend of Glauber and Kawasaki dynamics. The Kawasaki dynamics is generated by modeling a diffusive process where an occupied lattice site i can swap its occupancy only with one of its nearest unoccupied neighbors. In this composite algorithm, a Monte Carlo step then consists of treating the bulk reservoirs with Glauber dynamics and then performing a series of Kawasaki steps for each occupied fluid site in the system.
IV. Results (4)
〈ij〉
where β ) 1/kbT and Fi is the average density of the site i. The minimization of the grand potential under the constraint of fixed (26) We include the term involving the chemical potential since the lattice gas Hamiltonian is then isomorphic with that of an Ising type model of a magnet and it permits us to exhibit the symmetry of the model more easily. (27) Levitz, P. AdV. Colloid Interface Sci. 1998, 77, 71. (28) Woo, H. J.; Sarkisov, L.; Monson, P. A. Langmuir 2001, 17, 7472. (29) Wiltzius, P.; Bates, F. S.; Dierker, S. B.; Wignall, G. D. Phys. ReV. A 1987, 36, 2991. (30) Sarkisov, L.; Monson, P. A. Phys. ReV. E 2001, 65, 011202. (31) Woo, H. J.; Monson, P. A. Phys. ReV. E 2003, 67, 041207. (32) Kierlik, E.; Monson, P. A.; Sarkisov, L.; Tarjus, G. Phys. ReV. Lett. 2001, 87, 055701.
A. Experimental Results. 1. Mercury Intrusion/Extrusion and Nitrogen Sorption. Figure 1 shows nitrogen adsorption isotherms at 77.4 K on both porous Vycor glass and CPG 75. The adsorbed amount of gas (in Volume STP) is given as a function of relative pressure (P/P0, where P0 is the saturation vapor pressure). Figure 2 shows the mercury intrusion/extrusion curves obtained on the same materials; here the intruded/extruded (liquid) volume of mercury is shown as a function of hydraulic pressure. The nitrogen sorption isotherms clearly reveal a type H1 hysteresis loop for the CPG sample and a type H2 hysteresis loop for the porous Vycor glass. The mercury intrusion/extrusion curves also show well-defined hysteresis loops, but of course entrapment was also observed. Nevertheless, the shape of the intrusion/extrusion hysteresis loops agrees with the hysteresis observed in gas adsorption; i.e., CPG shows type H1 hysteresis,
Mercury Porosimetry in Glass
Figure 2. Mercury porosimetry intrusion/extrusion isotherms at 298 K in Vycor (porosity ca. 32%; b, τ ) 400 s; ), τ ) 300 s) and CPG 75 (porosity ca. 55%, 0).
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Figure 4. Pore size distribution from nitrogen adsorption (b) and mercury porosimetry (0) for CPG 75.
whereas a type H2 hysteresis is observed for porous Vycor glass. From an analysis of the sorption isotherms as well from the mercury intrusion/extrusion curves, the pore volumes of CPG 75 and Vycor glass differ significantly, i.e., one obtains from gas adsorption a porosity of ca. 32% for the porous Vycor glass; however a value of 55% is obtained for this CPG sample. It is important to note that in contrast to the situation in CPG, where the system can be completely filled with mercury, a complete filling of the Vycor system is not possible since it would require hydraulic pressures larger than the 60000 psi limit of the porosimetry instrument. On the basis of the Washburn equation this corresponds to a pore size of ca. 40 Å. A clear high-pressure plateau seen in the case of the CPG experiments can therefore not be observed in the case of Vycor. On the basis of a comparison with gas adsorption experiments, at least 0.05 cm3/g of the total Vycor pore volume remains unfilled, which distorts the shape of the hysteresis loop. This may also contribute to the fact that the experimentally observed mercury intrusion/extrusion hysteresis loop on Vycor is truncated and does not reveal the same clear type H2 behavior as we would expect based on the gas sorption experiments. Figure 3 shows pore size distribution curves obtained by applying the BJH method, which is based on the Kelvin equation.33
It can be seen that the maximum in the CPG pore size distribution curve is shifted to larger pore sizes as compared to the Vycor glass and that the pore size distribution is narrower. For Vycor glass the pore size distribution extends over a wide range, from below 2 nm up to 12 nm, whereas in the case of CPG the pore size distribution extends from ca. 7 to 12 nm. The difference in the width of the pore size distributions reveals differences in the texture of the material, indicating that the pore network of Vycor glass can be considered as to be more heterogeneous as compared to CPG. We can compare the pore size distributions obtained from mercury intrusion to those from the BJH analysis of gas desorption. In order to obtain the pore size distribution from the mercury porosimetry isotherm, we apply the Washburn equation, with an assumed contact angle of 145°, based on measurements for mercury on amorphous silica.34 The resultant pore size distribution, as shown in Figure 4, is in good agreement with that obtained from N2 desorption. This good agreement lends support to the idea that that mercury intrusion and desorption are closely related phenomena. 2. Entrapment and Hysteresis. As already discussed in the introduction a significant difference between gas adsorption and mercury porosimetry for these mesoporous materials is the occurrence of entrapment in mercury porosimetry. According to the work presented in our previous papers23,24 we expect that entrapment has a dynamic origin. In order to test this prediction, we performed experiments in which we studied the mercury intrusion/extrusion behavior into Vycor glass as a function of the sample time, i.e., the time allowed for relaxation of the system after each step change in the pressure. The experiments were performed in the stepwise experimental mode, as described in the experimental section, with various sample times ranging from 10 to 300s at each value of the pressure. We find that both hysteresis and entrapment are sensitive to the sample time. This is illustrated in Figure 5, which shows intrusion/extrusion isotherms for three values of the sample time. For shorter sample times we find that intrusion curve is shifted to higher pressure. The shape of the extrusion branch is less sensitive to the sample time except at lower pressures, indicating a difference in mechanism between intrusion and extrusion. At lower pressures, we see that the amount of entrapped mercury increases substantially as the sample time is decreased. For CPG 75 the intrusion/extrusion curves are substantially less sensitive to the sample time used even in the so-called scanning mode, where
(33) Barrett, E. P.; Joyner, L. G.; Halenda, P. P. J. Am. Chem. Soc. 1951, 73, 373.
(34) Simon, J.; Saffer, S.; and Kim, C. J. J. Microelectromech. Syst. 1997, 6, 208.
Figure 3. BJH pore size distribution for CPG 75 (0) and Vycor (b) obtained from gas adsorption data.
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Figure 5. Mercury intrusion/extrusion into Vycor for different sample times (b, τ ) 300 s; 0, τ ) 50 s; (, τ ) 10 s).
Figure 6. Primary (0) and secondary (b) intrusion/extrusion cycles into Vycor glass.
the complete intrusion/extrusion experiment is performed in less than 2 h. The apparent crossover between results for intrusion for different sample times at around 40 kPa and the differences between the extrusion curves for different sample times at high pressures are likely a consequence of variations between porous material samples between experiments. In order to investigate the relationship between entrapment and hysteresis, we have performed experiments where an additional intrusion/extrusion cycle was carried out on the system immediately after the first cycle so that the second cycle begins with mercury entrapped in the system. Figures 6 and 7 show results of these experiments for CPG and Vycor, respectively. We observe that in both cases a hysteresis loop is present in the second cycle but that there is no additional entrapment, an indication that hysteresis and entrapment have different origins. In the case of CPG the second intrusion/extrusion curves follow closely the first intrusion/extrusion curves, i.e., the width and shape, and position of the hysteresis loop remains unchanged. For Vycor glass, however, the shape and width of the hysteresis loop changes because the second intrusion curve is shifted significantly to lower pressures and appears to be less steep as compared to the first intrusion/extrusion cycle. The position and shape of the extrusion branch on the other hand are unaffected. We note that it should be expected that the extrusion curves for the first and second intrusion/extrusion cycles would be identical given that in each case the extrusion starts from the same state (i.e., the material mostly filled with mercury). How can we explain these differences in the intrusion/extrusion cycles between CPG and Vycor? One possible explanation is
Porcheron et al.
Figure 7. Primary (0) and secondary (b) intrusion/extrusion cycles into CPG 75.
that the entrapment of mercury in CPG does not change the overall character of the pore network since the pores filled with entrapped mercury represent only about 10% of the complete pore volume and the pore size is relatively uniform in comparison with that of Vycor, as indicated in Figure 3. For Vycor the fraction of the pore space with entrapped mercury after the first intrusion/ extrusion cycle is significantly larger, approaching 25% in the experiments with the longest sample times and much greater for shorter sample times (see Figure 5). Also the distribution of pore size is more nonuniform so that at the beginning of the second cycle the available void space can be expected to have a different topology. For Vycor the shift of the intrusion branch to lower pressures in the second cycle is consistent with the assumption that mercury rentrapment occurs in the smaller pores. However, the volume of entrapped mercury is rather large, and GCMC simulations of mercury extrusion within a model of Vycor indicate that large pores can remain filled after fragmentation of the mercury during extrusion.23,24 The kinetics of extrusion of this entrapped mercury is extremely slow, because it can only occur via a process of vapor transfer from isolated droplets of entrapped mercury.1,23,24 Such an interpretation is somewhat similar to a point of view presented by Moscou and Lub.3 They argued that pore shape determines the mercury retraction because mercury that has been penetrated into ink-bottle-shape pores will not leave these pores through smaller pore entrances during extrusion. In conclusion, based on the experimental findings one could assume that differences in the reintrusion/extrusion behavior between Vycor and CPG are due to the fact that Vycor exhibits a much more pronounced textural heterogeneity as compared to CPG 75 (this is also reflected in the much wider pore size distribution as shown in Figure 3). However, more comprehensive insight into the microscopic mechanism of intrusion/extrusion can be obtained by utilizing molecular simulations based on lattice models. These results are discussed in the next section. B. Results from the Lattice Model. 1. Structural Characterization of Lattice Models of Porous Glasses. We now consider the modeling of mercury porosimetry experiments using a lattice model. A porous glass sample with lattice of volume V ) 180 × 180 × 180 sites is built using the Gaussian random field method with a lattice parameter a ) 15 Å. The Vycor glass model is generated with a porosity p ) 0.3 while we consider p ) 0.6 for the CPG. In order to characterize the porous structure of the materials, we plot in Figure 8 the pore size distribution of the Vycor and CPG models. These distributions are obtained by computing the largest sphere passing through each fluid lattice
Mercury Porosimetry in Glass
Figure 8. Pore size distributions for the lattice models of Vycor (porosity ) 0.30, b) and CPG (porosity ) 0.60, 0).
site before encountering a wall site.36 The pore size distribution of the Vycor displays a wide peak centered around 90 Å, while the CPG 75 has a characteristic size of about 140 Å. The qualitative behavior is in very good agreement with the observed experimental pore size distributions obtained from the BHJ method (see Figure 3). The peak in the pore size distribution for CPG 75 based on the BJH analysis of the experimental adsorption isotherms occurs at somewhat lower pore diameters. This can in part be attributed to a lower porosity (p ) 0.55) of the experimental sample compared to the model (p ) 0.6). Additionally, it is known that the BJH approach may underestimate the pore size,37 and the application of microscopic approaches such as nonlocal density functional theory (NLDFT) can yield a more accurate pore size distribution.37,38 If we apply such a NLDFT method to CPG 75, assuming a distribution of independent cylindrical silica pores, then the peak of the experimental pore size distribution curve for CPG shifts to ca. 130 Å, significantly closer to the value for our CPG model. One potential weakness of the current implementation of the model is that we do not have the structure factor for CPG. Instead we are using the structure factor for Vycor rescaled to the porosity of CPG, assuming that the structures are self-similar. This assumption is a reasonable first approximation given that both materials are generated by a spinodal decomposition process. Comparisons of the pore size distributions in Figures 3, 4, and 8 show that our model CPG has a wider distribution of pore sizes than that estimated for the real material from gas adsorption and mercury porosimetry. 2. Mean-Field Theory Results. In Figure 9 we plot adsorption/ desorption isotherms obtained from MF-DFT for our models of Vycor and CPG. We set the wall-fluid interaction parameter R ) 1 and the temperature to T ) 0.9165 so that T/Tc ) 0.611 (where Tc is the bulk critical temperature), similar to the experimental conditions. The isotherms for both systems display hysteresis with that for CPG appearing at higher relative pressures. The Vycor hysteresis is of type H2 while we observe type H1 hysteresis for the CPG. The isotherm shapes and hysteresis positions of the sorption isotherms obtained from the CPG and Vycor glass models (Figure 9) are in good agreement with the (35) Newman, M. E. J.; Barkema, G. T. Monte Carlo Methods in Statistical Physics; Clarendon Press: Oxford, 1999. (36) Gelb, L. D.; Gubbins, K. E. Langmuir 1999, 15, 305. (37) Neimark A. V.; Ravikovitch P. I.; Vishnyakov A. J. Phys. Condens. Matter. 2003, 1, 347. (38) Thommes, M. In Nanoporous Materials: Science and Engineering; Lu,G. Q., Zhao, X. S., Eds.; Imperial College Press, 2004; Chapter 11.
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Figure 9. Gas adsorption isotherms for Vycor (b) and CPG (0) from MF-DFT for the lattice model.
Figure 10. Mercury porosimetry isotherms for Vycor (b) and CPG (0) from MF-DFT for the lattice model.
experimental results (see Figure 1), supporting the qualitative accuracy of our approach. In Figure 10 we report the MF-DFT intrusion/extrusion curves obtained in the Vycor and the CPG for T/Tc ) 0.5 and a repulsive wall-fluid interaction parameter R ) 0. We have chosen a ratio T/Tc larger than that obtained in experiments, where T/Tc ) 0.17, since this facilitates our study of the dynamics for longer times that are not accessible at the lower temperature. The intrusion of mercury into the mesoporous solids occurs at pressure above P0 and is shifted to larger pressures for the Vycor glass due to the stronger confinement of the liquid in the Vycor pore network. We observe intrusion/extrusion hysteresis for both materials and there is clearly a transition from type H2 hysteresis for the Vycor glass to type H1 hysteresis for the CPG. Again the qualitative agreement with experiment (see Figure 2) is good. The difference between the extrusion curves for CPG from theory and experiment (where the onset of retraction is more sudden) reflect, we believe, either the effect of the difference in the T/Tc value used or the use of the rescaled Vycor structure factor in building our model of CPG. Inspection of computer graphics visualizations of the liquid distribution (see Figure 11 and 12) during intrusion reveals a inhomogeneous intrusion of the mercury into the Vycor, whereas the penetration into the CPG is more percolation-like. Both extrusion processes seem to have a similar overall mechanism as the mercury fragments in the porous network ultimately lead to scattered droplets of liquid surrounded by a vapor phase. However, this fragmentation is more heterogeneous for the Vycor
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Figure 11. Computer graphics visualizations of the liquid distribution during intrusion (left) and extrusion (right) in Vycor from MF-DFT calculations for the lattice model. The liquid is shown in blue, the solid in gray, and the void in white. Figure 13. Mercury porosimetry isotherms for Vycor from Glauber dynamics for the lattice model with variable sample time (b, τ ) 1 MCS; 0, τ ) 4 MCS; (, τ ) 20 MCS).
Figure 12. Computer graphics visualizations of the liquid distribution during intrusion (left) and extrusion (right) in CPG from MF-DFT calculations. The liquid is shown in blue, the solid in gray, and the void in white.
case, and this correlates with the difference in steepness between the extrusion curves in the two cases. Calculations performed in our previous work23,24 suggest that this fragmentation is the microscopic origin of mercury entrapment. However, since the dynamics of intrusion/extrusion is not included in the MF-DFT, the extrusion branch closes the intrusion one at µ0. Calculations taking into account the dynamics of the system are therefore necessary to investigate the mechanisms encountered in experiments. 3. Monte Carlo Simulations. We first consider Monte Carlo simulations using Glauber dynamics. In Figures 13 and 14 we plot the intrusion/extrusion curves obtained with different sample times in the Monte Carlo simulations, measured in Monte Carlo steps (MCS). As the sample time decreases, the width of the hysteresis loop increases and can ultimately reach a point where the density in the mesoporous material is different than zero at the end of the extrusion branch (when µ ) µ0). This is in qualitative agreement with the experimental results obtained on Vycor glass as shown in Figure 5. Using the composite algorithm described in section 3.3, involving Kawasaki dynamics to model the fluid relaxation inside the porous material, Woo and Monson obtained results from the lattice model indicating that the adsorption of a gas into a mesoporous material is subject to two different regimes as a function of time. The first or “transport” regime occurs at short times and corresponds to the diffusion of molecules from the bulk reservoirs into the mesoporous material. During this regime the density inside the material changes relatively quickly as a function of time. The second regime happens at much longer times and corresponds to a “quasi-equilibrium” regime where the density varies much more slowly as a function of time. In this regime the relaxation process corresponds to
Figure 14. Mercury porosimetry isotherms for CPG from Glauber dynamics with variable sample time (b:, τ ) 4 MCS; 0, τ ) 20 MCS; (, τ ) 40 MCS).
transitions between the multiple local minima of the free energy of the confined fluid. Woo and Monson then suggested that the appearance of hysteresis in adsorption/desorption experiments originates from this regime. In previous work we have shown that this idea can be applied to the dynamics of extrusion of mercury from Vycor24 and that mercury entrapment observed in experiments is a consequence of the transport regime. As we observe here the entrapment occurs for very small relaxation times in the Glauber dynamics which corresponds to the latter part of the transport regime. We now focus on the nature of reintrusion into the Vycor and CPG. To study this we also use the composite dynamics algorithm described in section 3.3. Due to the computational expense of the composite dynamics, it was necessary to use a smaller sample of porous glass of volume V ) 128 × 64 × 64 lattice sites. Even with this reduced system size it was still not feasible to run an intrusion/extrusion cycle except with a very large amount of entrapment. We typically found 70-80% of the void space filled with liquid at the end of the extrusion branch after using the sample times that were accessible in our calculations. For this reason the initial configuration for the reintrusion was obtained by performing a Glauber simulation on a completely filled material at µ0. The number of steps was chosen so that the final density in the material is F = 0.15. Then to avoid a quick draining of the sample, we start the sequential simulations of reintrusion slightly above µ0. For each simulation we perform 2 × 105 Monte
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Figure 15. Mercury porosimetry intrusion (0) and reintrusion (b) isotherms for Vycor for the lattice model from the composite Kawasaki/Glauber dynamics. The reintrusion curve has been shifted down by an amount equal to the entrapped density.
Figure 17. Computer graphics visualizations of the liquid distribution during intrusion and reintrusion for Vycor glass from Kawasaki dynamics for the lattice model.
Figure 16. Mercury porosimetry intrusion (0) and reintrusion (b) isotherms for CPG for the lattice model from the composite Kawasaki/ Glauber dynamics. The reintrusion curve has been shifted down by an amount equal to the entrapped density.
Carlo steps where the first 105 steps are used to relax the system and the last 105 are used to perform averages. A comparison of the intrusion and reintrusion curves from these calculations is given in Figures 15 and 16. To aid in the comparison, the reintrusion curves have been shifted down by an amount equal to the entrapped density at the end of the first cycle. As in the experiments we see that the reintrusion and intrusion curves for CPG are quite similar but those for Vycor differ significantly. In discussing the experimental results, we suggested that this difference in behavior for CPG and Vycor indicated a difference in the pore structure of the materials. To understand this difference we plot in Figures 17 and 18 computer graphics visualizations of the liquid distribution observed during intrusion and reintrusion in Vycor and CPG. The primary intrusion in Vycor and CPG proceed through the formation and the progression of a liquid front starting from the external surface of the material. The progression in Vycor is more heterogeneous than that in CPG, and simulations on a larger sample revealed that droplets of liquid can form in pores ahead of the progressing liquid front via vapor transfer.24 This mechanism is more pronounced during the reintrusion in the Vycor glass where the droplets located in the middle of the sample act as an additional attractive field enhancing the vapor transport from the liquid front. Therefore during reintrusion, we observe that while the
Figure 18. Computer graphics visualizations of the liquid distribution during intrusion and reintrusion for CPG from Kawasaki dynamics for the lattice model.
liquid front is progressing into the Vycor, liquid droplets located deep into the material grow in size. The difference in density between reintrusion curve and the primary intrusion curve at the same pressure is larger than that represented by the amount of entrapped liquid at the beginning of the reintrusion. The reintrusion in the CPG proceeds with a different mechanism as the larger pores of the material sustain the liquid front geometry while the droplets located deep into the material are not affected. Therefore at the same chemical potential, the only difference between the intrusion and the reintrusion is the liquid density coming from the liquid initially present in the material at the start of reintrusion.
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V. Summary and Conclusions We have performed experiments of mercury porosimetry on mesoporous Vycor and CPG and compared the results with those from the lattice fluid models of these systems. The experimental intrusion/extrusion curves reveal a transition from type H2 hysteresis for the Vycor to type H1 for the CPG. The inversion of the curves using the Washburn equation gives a very good agreement with the pore size distribution obtained from gas sorption. At the end of the extrusion curve we observe that the density in the material is different than zero and therefore some mercury has been entrapped in the mesoporous material. When we perform a secondary cycle of intrusion/extrusion, the intrusion curve into the Vycor glass differs from the primary intrusion while the one in the CPG remains largely the same. The results from our lattice model calculations provide insight into the experimental results. In order to understand the mechanisms related to these observations, we use a lattice model framework which has been successfully used to model gas sorption30-32 and mercury porosimetry23,24 into mesoporous materials. Most of the features observed in the experiments also
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appear in our model. For instance MF-DFT calculations shows the transition from type H2 for the Vycor to type HI hysteresis for the CPG. The explanation of entrapment requires us to take into account the dynamics of the system, and therefore we performed dynamic Monte Carlo simulations using both Glauber and Kawasaki dynamics. The origin of entrapment is the slowdown of the dynamics associated with the fragmentation of the liquid in the void space that makes vapor transfer an important part of the extrusion process. The difference obtained between the primary and the secondary intrusion curves in Vycor also appears in our calculations. This behavior arises from vapor transfer ahead of the front of liquid percolating in from the external surfaces. This permits the growth of the droplets within the material. On the other hand, the intrusion into CPG is predominantly a percolation process and therefore the secondary intrusion curve is similar to the primary one. Acknowledgment. This work was supported by the National Science Foundation (CTS-0220835). LA063080E
