Langmuir 2008, 24, 12295-12302
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Contact Angles, Pore Condensation, and Hysteresis: Insights from a Simple Molecular Model P. A. Monson* Department of Chemical Engineering, UniVersity of Massachusetts, Amherst, Massachusetts 01003-9303 ReceiVed June 23, 2008. ReVised Manuscript ReceiVed August 19, 2008 We discuss the thermodynamics of adsorption of fluids in pores when the solid-fluid interactions lead to partial wetting of the pore walls, a situation encountered, for example, in water adsorption in porous carbons. Our discussion is based on calculations for a lattice gas model of a fluid in a slit pore treated via mean field density functional theory (MFDFT). We calculate contact angles for pore walls as a function of solid-fluid interaction parameter, R, in the model, using Young’s equation and the interfacial tensions calculated in MFDFT. We consider adsorption and desorption in both infinite pores and in finite length pores in contact with the bulk. In the latter case, contact with the bulk can promote evaporation or condensation, thereby dramatically reducing the width of hysteresis loops. We show how the observed behavior changes with R. By using a value of R that yields a contact angle of about 85° and maintaining the bulk fluid in a supersaturated vapor state on adsorption, we find an adsorption/desorption isotherm qualitatively similar to those for graphitized carbon black where pore condensation occurs at supersaturated bulk vapor states in the spaces between the primary particles of the adsorbent.
I. Introduction This paper was partly inspired by an experimental study of adsorption of water in graphitized carbon black by Easton and Machin.1 Some of their adsorption/desorption data is illustrated in Figure 1. A striking feature of this data is the continuation of the adsorption isotherm to supersaturated states of water vapor, up to the point where capillary condensation of water takes place in the spaces between the primary particles of the carbon black. The Easton and Machin work provides a nice illustration of capillary condensation for a case where the liquid only partially wets the surface and, in particular, the interplay between contact angle (wettability), pore condensation, and hysteresis. This is important in understanding water adsorption in porous carbons especially materials with graphite-like surfaces such as carbon nanotubessthrough experiments2-9 and modeling.10-19 †
Figure 1. Adsorption/desorption isotherm for water in graphitized carbon black at 280.15 K. The filled symbols denote adsorption data and the open symbols desorption data. The data are from Easton and Machin.1
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(1) Easton, E. B.; Machin, W. D. J. Colloid Interface Sci. 2000, 231(1), 204– 206. (2) Ghosh, S.; Ramanathan, K. V.; Sood, A. K. Europhys. Lett. 2004, 65(5), 678–684. (3) Hanasaki, I.; Nakatani, A. J. Chem. Phys. 2006, 124(17), xx–xx. (4) Kaneko, K.; Hanzawa, Y.; Iiyama, T.; Kanda, T.; Suzuki, T. Adsorpt.sJ. Int. Adsorpt. Soc. 1999, 5(1), 7–13. (5) Maniwa, Y.; Kataura, H.; Abe, M.; Suzuki, S.; Achiba, Y.; Kira, H.; Matsuda, K. J. Phys. Soc. Jpn. 2002, 71(12), 2863–2866. (6) Mao, S. H.; Kleinhammes, A.; Wu, Y. Chem. Phys. Lett. 2006, 421(4-6), 513–517. (7) Cossarutto, L.; Zimny, T.; Kaczmarczyk, J.; Siemieniewska, T.; Bimer, J.; Weber, J. V. Carbon 2001, 39(15), 2339–2346. (8) Rudisill, E. N.; Hacskaylo, J. J.; Levan, M. D. Ind. Eng. Chem. Res. 1992, 31, 1122–1130. (9) Salame, I.; Bandosz, T. J. Langmuir 1999, 15, 587–593. (10) Brennan, J. K.; Bandosz, T. J.; Thomson, K. T.; Gubbins, K. E. Colloids Surf. AsPhysicochem. Eng. Aspects 2001, 187, 539–568. (11) Alcaniz-Monge, J.; Linares-Solano, A.; Rand, B. J. Phys. Chem. B 2001, 105, 7998–8006. (12) Slasli, A. M.; Jorge, M.; Stoeckli, F.; Seaton, N. A. Carbon 2004, 42(10), 1947–1952. (13) Slasli, A. M.; Jorge, M.; Stoeckli, F.; Seaton, N. A. Carbon 2003, 41(3), 479–486. (14) Striolo, A.; Chialvo, A. A.; Gubbins, K. E.; Cummings, P. T. J. Chem. Phys. 2005, 122(23), (15) Striolo, A.; Chialvo, A. A.; Cummings, P. T.; Gubbins, K. E. Langmuir 2003, 19, 8583–8591.
The link between wetting and capillary condensation is described at the simplest level by the Kelvin equation, which for a slit pore can be expressed as
RT ln
()
-2σVL cos θ Pc ) P0 (nL - nV)H
(1.1)
where Pc is the pressure at which pore condensation occurs, P0 is the bulk saturated vapor pressure, T the absolute temperature, R the gas constant, σVL the vapor-liquid surface tension, θ is the contact angle, and nL and nV are the molar densities of the bulk liquid and bulk vapor phase, respectively, while H is the pore width. Contact angles less than 90° lead to capillary condensation at P < P0 while contact angles greater than 90° lead to capillary condensation at P > P0. Since the contact angle of water on graphite is less than 90°,20 we should expect water to condense in a graphite pore at P < P0. The analysis is complicated by hysteresis, which can extend the adsorption (desorption) branches of the isotherm to higher (lower) pressure than that associated with vapor-liquid equilibrium in the pore that is implied by the Kelvin equation. Thus, in the Easton and
10.1021/la801972e CCC: $40.75 2008 American Chemical Society Published on Web 10/04/2008
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Figure 2. Schematic representation of the open slit pore geometry considered in this work. Periodic boundaries are used in the x direction. The density is independent of y, making the problem two-dimensional. We also consider a slit pore that is infinite in the x direction with no contact with the bulk. In this case the density depends only on z and the problem is one-dimensional.
Figure 3. Isotherm of cos θ versus R for T* ) 0.9. The results are symmetric with respect to R ) 0.5 where cos θ ) 0.
Machin experiments we see that pore condensation on adsorption occurs above P0 because of hysteresis even though, based on the Kelvin equation, equilibrium pore condensation should occur for P < P0. It seems worthwhile to seek more general insight into this behavior and here we do this using a molecular model that can describe hysteresis in addition to pore condensation and wetting. In particular, we consider a lattice gas model of a fluid in a slit pore using mean field density functional theory (MFDFT).21-24 MFDFT provides a self-contained approach to analyzing how confinement and wetting behavior influence capillary condensation and hysteresis, as illustrated by the work of Evans and coworkers.25,26 We calculate contact angles for the model using Young’s equation and the interfacial tensions from MFDFT. We also use MFDFT to calculate adsorption/desorption isotherms and molecular density distributions. We place special emphasis on investigating changes in the behavior as we change the strength of the solid-fluid interactions. We study pores which are infinite in length without contact with the bulk gas, as well as pores that are finite in length and have direct contact with the bulk. MFDFT (16) Striolo, A.; Naicker, P. K.; Chialvo, A. A.; Cummings, P. T.; Gubbins, K. E. Adsorpt.sJ. Int. Adsorpt. Soc. 2005, 11, 397–401. (17) Liu, J. C.; Monson, P. A. Langmuir 2005, 21, 10219–10225. (18) Liu, J. C.; Monson, P. A. Adsorpt.sJ. Int. Adsorpt. Soc. 2005, 11(1), 5–13. (19) Liu, J. C.; Monson, P. A. Ind. Eng. Chem. Res. 2006, 45, 5649–5656. (20) Fowkes, F. M.; Harkins, W. D. J. Am. Chem. Soc. 1940, 62, 3377–3386. (21) DeOliveira, M. J.; Griffiths, R. B. Surf. Sci. 1978, 71(3), 687–694. (22) Ebner, C. Phys. ReV. A 1980, 22(6), 2776–2781. (23) Bruno, E.; Marconi, U. B. M.; Evans, R. Physica A 1987, 141(1), 187– 210. (24) Kierlik, E.; Monson, P. A.; Rosinberg, M. L.; Sarkisov, L.; Tarjus, G. Phys. ReV. Lett. 2001, 8705(5), 055701. (25) Evans, R.; Marconi, U. M. B.; Tarazona, P. J. Chem. Soc., Faraday Trans. II 1986, 82, 1763–1787. (26) Evans, R.; Marconi, U. M. B.; Tarazona, P. J. Chem. Phys. 1986, 84(4), 2376–2399.
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has hysteresis built into it associated with metastability of dilute and condensed states of the pore fluid.24,27-30 Recent studies using molecular dynamics and Monte Carlo simulations of adsorption/desorption in model porous materials indicate that the mean field picture of adsorption/desorption hysteresis for the slit pore is qualitatively correct.31,32 For finite length pores, our results show the effect upon hysteresis of contact with the bulk liquid or vapor for partial wetting situations. On the adsorption branch for cases where the contact angle is 90° (the situation encountered in mercury porosimetry33,34), contact with the bulk liquid will lead to pore condensation on the adsorption (intrusion) branch at a state close to the vapor-liquid equilibrium state for the pore. On desorption (extrusion), contact with the bulk vapor leads to evaporation from the pore at or just below the bulk saturation chemical potential or pressure. As we noted above, the situation in the Easton and Machin experiments is a contact angle µ0. On desorption we set the bulk state to be liquid for µ g µ0 and vapor for µ < µ0. We note that contact with the bulk liquid has been used to promote pore condensation in nitrogen adsorption experiments.43,44 We first consider a value of the surface field, R ) 3.0, associated with complete wetting of the free pore surfaces. The isotherms for density and grand free energy are shown in Figure 7 together with the results for the infinite pore that were shown in Figure 5a (the behavior of this system at a slightly higher temperature was also studied recently45). The isotherms are similar in finite and infinite pore systems except that the desorption branches of the hysteresis loops are shortened in the finite pore case. This is because contact with the bulk vapor phase promotes evaporation and the desorption step in the hysteresis loop now lies close to the equilibrium pore condensation transition as indicated by the grand free energy isotherm, a phenomenon first noted for this type of model by Marconi and van Swol.28,29 In this system the desorption step is associated with equilibrium behavior while the adsorption branch corresponds to metastable states, consistent with an interpretation commonly made in the literature for openended independent ideal pores.46,47 The density distributions (43) Aukett, P.; Jessop, C. In Fundamentals of Adsorption; LeVan, M. D., Ed.; Kluwer: Amsterdam, 1996; pp 59-66. (44) Murray, K.; Seaton, N.; Day, M. Langmuir 1999, 15, 6728–6737. (45) Monson, P. A. J. Chem. Phys. 2008, 128, 084701.
Figure 11. Adsorption/desorption isotherms of density and grand free energy versus chemical potential for H* ) 6 and R ) 0.55 (corresponding to a contact angle for the free solid surface of about 85°) at T* ) 0.9. (a) Full line gives the results for a finite length open-ended pore (see Figure 1) with hysteresis removed from the bulk fluid behavior and the dashed line gives the results for an infinite pore. (b) Adsorption/desorption isotherm for the open pore but with only vapor states in the bulk.
associated with selected states on the open pore isotherm in Figure 7 are shown in Figure 8. Figure 9 presents adsorption/desorption and grand free energy isotherms for the three partial wetting situations considered for (46) Ball, P. C.; Evans, R. Langmuir 1989, 5, 714–723. (47) Ball, P. C.; Evans, R. Europhys. Lett. 1987, 4(6), 715–721.
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the infinite pore in Figure 6 with the infinite pore results included for reference. We see in Figure 9 that the hysteresis is now greatly reduced on both the adsorption and desorption branches. In Figure 9a for R ) 0.75 we have equilibrium pore condensation below bulk saturation. We see that pore condensation occurs very close bulk saturation with the evaporation step occurring close to the equilibrium vapor-liquid transition for the pore as in the R ) 3 case. In Figure 9b for R ) 0.25 we have equilibrium pore condensation above bulk saturation and we see that pore condensation occurs very close to this state. Again, contact with the bulk decreases the amount of metastability but the weak solid-fluid interaction allows the pore vapor phase to remain thermodynamically stable even when the bulk phase is liquid. The evaporation step on desorption occurs close to the equilibrium vapor-liquid transition for the bulk. We note again that this behavior is similar to that expected for mercury intrusion/extrusion for idealized pore geometries. Figure 9c shows the results for the case of R ) 0.5 where the equilibrium pore condensation occurs precisely at µ0. We may ask why there is any hysteresis at all for the finite length pore in contact with the bulk in this case. The answer lies in the behavior at the pore entrances, revealed in the density visualizations for states on adsorption and desorption shown in Figure 10. In the neighborhood of µ0 the pore remains empty for a narrow range of chemical potentials on adsorption even though the bulk is liquid. Over this range of chemical potentials a vapor-liquid interface is formed at the pore entrance (Figure 10b) and advances slightly into the pore from the liquid side (Figure 10c) before the liquid fills the pore. Equivalent behavior occurs on the desorption branch where now the vapor-liquid interface appears (Figure 10f) and advances into the pore from the vapor side (Figure 10g) before evaporation occurs. This behavior may to some extent be exaggerated by the lattice model used here as well as the mean field approximation. Let us now consider a case that might reasonably be related to experimental situation in the Easton and Machin work. A value of R ) 0.55 in our model gives a contact angle of about 85°, comparable to that for water on graphite.20 Figure 11a shows adsorption/desorption isotherms in this case for both the finite and infinite pores constructed in the same way as those in Figure 9. We see from the grand free energy curve that the equilibrium pore condensation happens just below the bulk saturation point. As in Figure 9 the hysteresis is dramatically reduced through contact with the bulk for the open pore case. Now let us consider a situation for the model where on adsorption we allow the bulk phase to remain as supersaturated vapor. The results are shown in Figure 11b and we see that the adsorption branch for the finite pore is identical to that for the infinite pore, with condensation occurring well beyond bulk saturation. The qualitative resemblance to the results of Easton and Machin1 is evident, although our use of mean field theory greatly extends the range metastable states of the bulk vapor.
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V. Summary and Conclusions In this paper we have discussed the thermodynamics of pore condensation and hysteresis for systems where the bulk liquid partially wets a free solid surface of the same chemistry as the pore walls. Our calculations apply specifically to a lattice gas model of a fluid adsorbed in a slit pore but are intended to model qualitatively experimental results for water adsorption in graphitized carbon black by Easton and Machin1 as well as other partial wetting systems. It is hoped that the discussion presented in this paper will be helpful to those interpreting measurements and calculations of pore condensation for such systems. A key result is that for solid-fluid interactions consistent with partial wetting of the pore walls and contact angles 0.5 in the present model), metastability may extend the adsorption branch of the isotherm to supersaturated states of the bulk vapor. This can happen even though the equilibrium pore condensation occurs below P0. Pore condensation can occur through reaching the stability limit of the pore vapor or contact with liquid formed in condensation of the bulk vapor, phenomena which can both be described by MFDFT, or through density fluctuations inside the pore, which are not included in MFDFT. Our results show the effect of contact with the bulk on facilitating phase changes in the pore on adsorption and desorption. For finite open pores with R > 0.5, the evaporation transition on desorption occurs at a state close to the equilibrium vapor-liquid transition for the confined fluid28,29,45 unless the bulk is maintained in a metastable liquid state. On adsorption, the location of the pore condensation step depends on whether the bulk is liquid or metastable vapor for P g P0. If the bulk is liquid then this promotes pore condensation at a state close to P0. Otherwise, pore condensation will be delayed until either the bulk vapor or pore vapor condenses by reaching a stability limit or by homogeneous nucleation. This latter case is shown in Figure 1 for water in graphitized carbon black and in Figure 11b for the present model. Our results also illustrate the complementary behavior for for systems with R < 0.5 where equilibrium pore condensation occurs above P0. The present treatment is limited to the slit pore geometry and thus is relevant to systems that can be analyzed in terms of slit pores or to other similar geometries such as cylinders. It will be worthwhile to extend the discussion to the case of more complex pore geometries where pore network effects will further impact the behavior.24,30 Acknowledgment. The author is grateful to M. Thommes and J. Edison for comments on earlier versions of the manuscript and to B. Easton for providing a tabulation of the data plotted in Figure 1. This work was supported by the National Science Foundation (Grant No. CBET-0649552). LA801972E
