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Condensation Model for Cylindrical Nanopores Applied to Realistic Porous Glass Generated by Molecular Simulation Hideki Kanda,† Minoru Miyahara,* and Ko Higashitani Department of Chemical Engineering, Kyoto University, Kyoto 606-8501, Japan Received December 1, 1999. In Final Form: April 3, 2000
1. Introduction Pore-size distributions from adsorption isotherms of simple molecules such as nitrogen are nowadays measured almost routinely in response to the development of automated sorption apparatuses. The pressing need in this field, however, would be an accurate and handy model to describe condensation in pores of a few to several nanometers in size. Many researchers including the authors reported that the Kelvin model underestimates pore sizes in the order of nanometers,1-6 yet it still receives widespread use. Recently, we developed a new condensation model within nanopores7,8 as a possible replacement for the Kelvin model. The concept is that the meniscus of a condensate need not have the extent of curvature as expected by the Kelvin model because the attractive potential from pore walls also stabilizes the condensed phase. Thus the meniscus exhibits nonuniform curvature. Surface tension deviation from that of a flat interface9,10 is also taken into account. The model was compared with molecular dynamics (MD) simulations of a Lennard-Jones (LJ) fluid as an ideal experimental system and proved its reliability in predicting capillary coexistence relation for slit-shaped nanopores7 and cylindrical ones.8 However, this means that the model was verified only for pores with simplest geometries. Examinations of the model in a more realistic system would be desired to strengthen reliability in applying it to real porous solids, which often have random nature. Gelb and Gubbins quite recently developed a realistic model for studying adsorption and condensation in porous glasses by a quench MD procedure that mimics the processes by which controlled-pore glasses and Vycor glasses are produced.11 Then they obtained isotherms for N2-like Lennard-Jones (LJ) fluid in the glasses by Grand * To whom correspondence should be addressed. Tel: +81-75753-5582. Fax: +81-75-753-5913. E-mail: miyahara@ cheme.kyoto-u.ac.jp † Present address: Department of Chemical Energy Engineering, Central Research Institute of Electric Power Industry, Yokosuka City, Kanagawa Prefecture 240-0196, Japan. (1) Evans, R.; Marconi, U. M. B.; Tarazona, P. J. Chem. Phys. 1986, 84, 2376. (2) Heffelfinger, G. S.; Swol, F. van; Gubbins, K. E. Mol. Phys. 1987, 61, 1381. (3) Peterson, B. K.; Gubbins, K. E. Mol. Phys. 1987, 62, 215. (4) Seaton, N. A.; Walton, J. P. R. B.; Quirke, N. Carbon 1989, 27, 853. (5) Saito, A.; Foley, H. C. AIChE J. 1991, 37, 429. (6) Miyahara, M.; Yoshioka T.; Okazaki, M. J. Chem. Phys. 1997, 106, 8124. (7) Yoshioka, T.; Miyahara, M.; Okazaki, M. J. Chem. Eng. Jpn. 1997, 30, 274. (8) Miyahara, M.; Kanda, H.; Yoshioka, T.; Okazaki, M. Langmuir 2000, 16, 4293. (9) Tolman, R. C. J. Chem. Phys. 1949, 17, 333. (10) Melrose, J. C. Ind. Eng. Chem. 1968, 60, 53. (11) Gelb, L. D.; Gubbins, K. E. Langmuir 1998, 14, 2097
Canonical Monte Carlo (GCMC) simulations.12 Their isotherms and pore size distributions (PSD) determined geometrically are quite useful for our purpose to test the condensation model. In this study, we apply our condensation model to their isotherms to estimate PSD and examine the reliability thorough comparison with true PSD, or the one they determined geometrically. 2. Concept and Outline of Condensation Model The model treats the condensate as continuum throughout, in order not to lose simplicity. Since the detail of the model is given elsewhere,8 only the concept and outline of the model are briefly explained below, comparing it with the Kelvin model. The Kelvin equation describes the effect of curved surface for condensation in the following form for cylindrical pores with zero contact angle
kT ln
pg 2γVp )psat R-t
(1)
where k is Boltzmann’s constant, p is vapor pressure, and the subscript g indicates gas phase and sat indicates saturation. γ is surface tension of liquid, Vp the volume per molecule of liquid, R the pore radius, and t the statistical thickness of adsorption film on a pore wall. In nanopores, however, attractive potential from pore walls should contribute to the condensation equilibrium. Also, the surface tension deviates from that of a flat surface. Including the above two effects, the basic equation to describe the condensation is
kT ln
pg γ(F) ) ∆ψ(r) - Vp psat F(r)
(2)
where
1 1 1 ) + F(r) F1(r) F2(r) F1(r) and F2(r) are local radii of two principal curvatures of the gas-condensate interface at radial position r. ∆ψ(r) is the contribution of the attractive potential energy from the pore wall, which must be expressed as an “excess” amount compared with a potential energy that a molecule would feel if the pore wall consists of the same molecules as adsorbate. The surface tension is treated as a function of the curvature, and the relation given by the GibbsTolman-Koenig-Buff equation9 is adopted for the dependence. The contribution of the potential energy, together with the curvature-dependent surface tension, gives the two different principle radii of curvature in eq 2, instead of a single term of 2γ/(R - t) seen in the Kelvin model: The meniscus is not hemispherical in the proposed model. Geometrical integration of eq 2 will determine the shape of the interface, which will give pore size if summed with the thickness of the adsorbed film on the interior surface of the pore, t ()R - r0). The thickness of the adsorbed film is given by eq 3 since the radii of the two principal curvatures will be zero and r0, respectively, at the surface of the adsorption film in a cylindrical pore. (12) Gelb, L. D.; Gubbins, K. E. Langmuir 1999, 15, 305
10.1021/la9915769 CCC: $19.00 © 2000 American Chemical Society Published on Web 06/08/2000
Notes
Langmuir, Vol. 16, No. 14, 2000 6065
kT ln
pg γ(r0) ) ∆ψ(r0) - Vp psat r0
(3)
Summarizing, eqs 2 and 3, together with the GibbsTolman-Koenig-Buff equation, should be solved simultaneously to find R, t, F1(r), and F2(r) for a given relative pressure. 3. Examination of the Model 3.1. Isotherms in Realistic Model Pores. In Gelb and Gubbins’s GCMC simulation for nitrogen adsorption on the model porous glasses,12 which was generated by a quench MD simulation technique, the LJ parameters gg/k and σgg for the adsorbate were 95.2 K and 0.375 nm, respectively. The pore wall was set as a silica-like solid. Interactions with silicon atoms were neglected as in many of the previous works for this kind of material.13 To represent bridging oxygen in silica, the parameters for the adsorbent atoms were set as ss/k ) 230 K and σss ) 0.27 nm. The Lorentz-Berthelot combining rules were applied to give parameters for the nitrogen-oxygen interactions. The cutoff distance of adsorbent and adsorbate was 3.5σss ()2.5σgg) to simulate large-scale systems easily. Other constants are the reduced temperature, T* ) Tk/gg ) 0.80, and the reduced density of oxygen in silica, Fs* ) Fsσss3 ) 0.868. They determined “nitrogen” isotherms for each of four model glasses with various pore sizes. A model glass they call “sample A”, which has the smallest pore size among the four, exhibits only a small hysteresis loop, while “sample D” with the largest pore shows quite pronounced ranging from a relative pressure of ca. 0.5 to 0.8. For the other two samples, desorption branches are not shown and the extent of the hysteresis cannot be known. Because of the clarity in small hysteresis, which is explained further below, we make the examination on sample A here. There is still argument about which branch to take for pore-size determination, though adsorption branch is often recommended. Analysis of an isotherm with large hysteresis confuses the purpose of the present examination. Another argument would arise concerning a difficulty with GCMC simulation. It has often been pointed out that the GCMC method suffers from difficulty in determining true equilibrium between two metastable branches near the critical condensation pressure and that an artificial hysteresis may arise even for ideal straight pores (e.g., refs 3 and 14). This may be, or may not be, the case for the adsorption/condensation simulations in the model glasses: The apparent adsorption branches may, to some extent, be shifted from true adsorption branches. Employing “sample A” with only a narrow hysteresis loop, the examination can be free from the above two uncertainties. Then we analyze the adsorption isotherms of the smallest pore, which is plotted in Figure 1 using the data given by them. 3.2. Application of the Condensation Model to Nitrogen Isotherms. To calculate a capillary coexistence relation with the authors’ model, some physical properties for the adsobate in the above computer experiment must be known: the volume per molecule of liquid and the gasliquid surface tension of the adsorbate particles for bulk liquid. According to literature on simulations of the LJ liquid,15,16 in which the cutoff distance was 2.5σgg, Vp/σgg3 (13) Heuchel, M.; Snurr, R. Q.; Buss, E. Langmuir 1997, 13, 6795. (14) Papadopoulou, A.; Swol, F. von; Marconi, U. M. B. J. Chem. Phys. 1992, 97, 6942. (15) Nijmeijer, M. J. P.; Bakker, A. F.; Bruin, C. J. Chem. Phys. 1988, 89, 3789.
Figure 1. Adsorption isotherm of N2-like LJ fluid on realistic porous glass of “sample A” obtained by GCMC simulations.12
and γσgg2/gg were determined to be 1.367 and 0.389 at T* ) 0.809, respectively. As for the potential function included in eq 2, employment of a complicated functional formula would ruin the handiness of the model, and we propose to use the most simple concept of a structureless cylindrical wall made of LJ solid given by Peterson et al.,17 which is analogous to the LJ 9-3 potential for a planar surface. The interaction of one adsorbate particle and semi-infinite pore wall is given by
ψ(r,R) ) πgsFs
[
]
7σ12 gs K (r,R) - σ6gsK3(r,R) 32 9
(4)
where
∫0π dΘ[- Rr cos Θ +
Kn(r,R) ) R-n
(1 - (Rr )
2
sin2 Θ
1/2 -n
) ]
gs and σgs are the energy and size parameter between the LJ particle and solid wall, and those employed in the simulations on the model glass, as described in section 3.1, are to be used. However, in the model glass system, the adsorbate-solid interaction had a cutoff distance of 2.5σgg. Thus the corresponding part of the infinite solid must be subtracted. Further, only the attractive term of the potential is enough to be considered. The potential field exerted by the pore wall of the model glass ψgs is then
ψgs ) -πgsFsσgs6K3(r,R) (-πgsFsσgs6K3(r,r + 2.5σgg)) (5) If r + 2.5σgg < R, the above equation formally gives positive value of potential. In that case ψgs was set to be zero since this condition corresponds to a situation that the location does not feel any potential from the wall. The excess potential ∆ψ should be obtained by subtracting a corresponding potential energy for the adsorbate’s liquid state from the potential of the pore wall. Considering again the cutoff distance between the adsorbate particles, the potential energy that a molecule would feel if the pore wall consisted of liquid of adsorbate molecules, ψgg, is expressed as follows (16) Holcomb, C. D.; Clancy, P.; Zollweg, J. A. Mol. Phys. 1993, 78, 437. (17) Peterson, B. K.; Walton, J. P. R. B.; Gubbins, K. E. J. Chem. Soc., Faraday Trans. 2 1986, 82, 1789.
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Notes
ψgg ) -πggFgσgg6K3(r,R) (-πggFgσgg6K3(r,r + 2.5σgg)) (6) Thus the function ∆ψ is
∆ψ ) ψgs - ψgg ) -π(gsFsσgs6 - ggFgσgg6)(K3(r,R) K3(r,r + 2.5σgg)) (7) Here, if r + 2.5σgg < R, ψgg and ψgs are set to be zero. The rightmost term with the cutoff distance comes only for the accord with the model fluid employed in the simulation. When applying the condensation model to a real experimental system, one need not include the term. Using the excess potential of eq 7 and the “physical properties”, the relation between condensation pressure and pore size is calculated by the present model as shown in Figure 2. Also shown in the figure is the one given by the Kelvin model for which the “physical properties” of the LJ particles as explained above are commonly used. The raw value of pore diameter expresses the distance between the center of surface atoms of pore walls. Then the actual pore diameter is smaller than the value by about the size of the surface atoms. The LJ diameter of a bridged oxygen atom was subtracted from the raw value: Thus the obtained value expresses the size of accessible pore space, which corresponds to the geometrical pore size determined in ref 12. On the basis of these coexistence curves, we calculated corresponding pore-size distributions for the realistic porous glass. The scheme of calculation was similar to that proposed by Dollimore and Heal.18 Figure 3 shows the results, where the solid line gives the “true” pore size determined geometrically.12 As pointed out by Gelb and Gubbins, significant underestimation in pore size by the Kelvin model (broken line) is recognizable: The peak pore size lies around 2 nm, while the geometrical one is around 3 nm. On the other hand the PSD calculated by the proposed model (dotted line) agrees almost perfectly with the “true” distribution. The realistic porous glass provides many trial grounds, such as atomic scale of surface roughness, noncircular cross section, and meandering connection of pores with varying pore size, for applicability of the condensation model. After the examination, and showing the agreement, the model now has strengthen reliability in application to real existing porous solids, which are often of random nature. The present model is based on the simple concept of continuum assumption, similar to the Kelvin model, and does not need much computation. The key factor to improve the underestimation of the conventional model is the introduction of the excess potential of pore walls. Some technical matters have arisen, in this test, to obtain ∆ψ, but it originates from the setting in the molecular simulation for “nitrogen adsorption”. In a real experimental system, ∆ψ can easily be obtained from a standard adsorption isotherm, and it would not bring much trouble in applying the model. Considering handiness with the simple concept, accuracy proved in an ideal system,8 and (18) Dollimore, D.; Heal, G. R. J. Appl. Chem. 1964, 14, 109.
Figure 2. Capillary coexistence curves: solid line, proposed model; dashed line, the Kelvin model.
Figure 3. Pore size distribution (PSD) calculated from adsorption isotherm based on capillary coexistence curves of Figure 2: solid line, “true” PSD determined geometrically;12 dashed line, prediction by the Kelvin model; dotted line, prediction by our model.
applicability to a realistic system examined here, we expect the model to be a replacement for the Kelvin model for pore characterization on the order of nanometers. 4. Conclusion We have examined our condensation model for cylindrical nanopores, which takes account of the effect of the attractive potential energy from pore walls and the surface tension deviated from that of a flat interface, employing LJ nitrogen isotherms for the LJ porous glasses that was generated by Gelb and Gubbins. The model, with a simple concept and handy calculation, successfully describes a “true” pore size distribution that was determined geometrically and proves its reliability not only in simple pores but also in a realistic pore with random nature. Acknowledgment. We thank Dr. Lev D. Gelb for generosity in providing isotherm data. This research was supported in part by the Grant-in-Aid for Scientific Research, No. 09750827, provided by the Ministry of Education of Japan. LA9915769
