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Properties of Three Capillary Fluids in the Critical Region William D. Machin Department of Chemistry, Memorial University of Newfoundland, St. John’s, Newfoundland, Canada A1B 3X7 Received December 18, 1997. In Final Form: November 2, 1998 Nitrogen, xenon, and sulfur hexafluoride when adsorbed on a mesoporous controlled pore glass all exhibit capillary condensation at temperatures less than Tc, the bulk critical temperature, and all exhibit a capillary critical temperature, Tcc, where adsorption-desorption hysteresis vanishes. For all temperatures less than Tcc, the density and surface tension of these capillary liquids are essentially the same as bulk liquid. Within the intercritical region, Tcc < T < Tc, these capillary fluids at, or near, saturation pressure are also liquidlike. We conclude that fluid behavior at the capillary critical point is not analogous to fluid behavior at the bulk critical point. The capillary critical point is identified as the point at which homogeneous nucleation of the vapor phase within capillary liquid replaces Kelvin failure as the mechanism responsible for abrupt capillary evaporation on desorption. We also discuss the phenomenon of critical depletion reported by Findenegg and colleagues.
As the temperature of bulk liquid coexisting with its vapor is increased, the density of the liquid decreases while the density of the vapor increases, both becoming equal at the bulk critical temperature Tc. Both phases are welldefined and isothermal evaporation/condensation is reversible. This is not the case for liquids contained in small pores (capillary liquids). While analogous capillary phase transitions are commonly observed they are usually irreversible, observed for example, as adsorption/desorption hysteresis in physical adsorption isotherms. Nevertheless, in common with bulk phases, the difference between the density of capillary liquid and capillary vapor at the transition point decreases as the temperature T increases eventually becoming zero at the capillary critical temperature Tcc. Isotherms are reversible for all T > Tcc.1-3 A capillary phase diagram, comparable to the bulk phase diagram, is readily constructed, but the capillary critical point is always shifted to lower temperatures, i.e. Tcc < Tc. An example of such a phase diagram, based on this work, is shown in Figure 1, other examples based on experimental or theoretical studies may be found elsewhere.4-7 The low density (capillary vapor) branch always lies at higher average densities than bulk vapor and the high density (capillary liquid) branch usually lies close to that for bulk liquid. It is well-known that the properties of bulk liquids exhibit remarkable behavior at the critical point, but for capillary liquids two properties are especially relevant. Liquid density and surface tension are both required in any application of the Kelvin equation to the analysis of hysteresis effects. In this work we show that the density and surface tension of capillary liquid nitrogen, xenon and sulfur hexafluoride are essentially the same as bulk at all temperatures less than Tcc. We also show that within the (1) Machin, W. D. Langmuir 1994, 10, 1235. (2) Murdey, R. J.; Machin, W. D. Langmuir 1994, 10, 3842. (3) Burgess, C. G. V.; Everett, D. H.; Nuttal, S. Langmuir 1990, 6, 1734. (4) Burgess, C. G. V.; Everett, D. H.; Nuttal, S. Pure Appl. Chem. 1989, 61, 1845. (5) Rocken, P.; Tarazona, P. J. Chem. Phys. 1996, 105, 2034. (6) Page, K. S.; Monson, P. A. Phys. Rev. E 1996, 54, 6557. (7) Brown, A. J.; Burgess, C. G. V.; Everett, D. H.; Nuttall, S. In Characterization of Porous Solids IV; McEnaney, B., Mays, T. J., Rouquerol, J., Rodriguez-Reinoso, F., Sing, K. S. W., Unger, K. K.; Eds.; Royal Society of Chemistry: Cambridge, 1997, p 1.
Figure 1. Phase diagram for capillary liquid xenon. Solid line, coexistence curve for bulk xenon; +, density of capillary liquid xenon at saturation pressure; O, average density of capillary gas at the lower closure point of the hysteresis loop. Isotherms are reversible for T > Tcc.
intercritical region Tcc < T < Tc the dense capillary fluids are similar to the bulk liquids. Experimental Section The adsorbent is controlled-pore glass (CPG-100, Pierce Chemical Co.) having a nominal pore diameter of 100 Å and a narrow range of pore sizes. The pore volume is ca. 0.66 cm3 g-1, and the surface area (BET, N2) is 135 m2 g-1. Nitrogen (Matheson, Research Purity, 99.9995%) and xenon (Matheson, Research Purity, 99.995%) were used as supplied. Sulfur hexafluoride (Matheson, SEC 99.95%) was stored over degassed CaA zeolite and vacuum sublimed prior to use. The volumetric adsorption apparatus is constructed of stainless steel components and fittings able to withstand pressures in excess of 3000 psi (21 MPa). The pressure transducer (Paroscientific, Model 42K-101) reads to 2000 psi (14 MPa) with a stated accuracy of 0.01%. Maximum pressures encountered are less than 1000 psi (7 MPa). Temperatures are determined with a platinum resistance thermom-
10.1021/la971393r CCC: $18.00 © 1999 American Chemical Society Published on Web 12/15/1998
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Machin Table 1. Adsorbate Properties and Capillary Critical Temperatures adsorptive
Figure 2. Example adsorption-desorption runs for xenon on CPG-100. While these plots have the outward appearance of adsorption isotherms, they are actually isosteres (or quasiisosteres) in that the total amount of adsorptive present for a given run is constant. The greatest amount adsorbed corresponds to the lowest temperature for the run and each point is separated by ca. 1 K. As temperature is increased from the initial (low) temperature the amount adsorbed decreases, the equilibrium pressure (P) and the saturation pressure (Po) increase, but the relative pressure (P/Po) may increase or decrease depending on the relative changes in P and Po. Note that the amount adsorbed at saturation pressure (P/Po ) 1) and the corresponding temperature are readily determined. (i) ], b, desorption, adsorption when 11.99 mmol xenon present in the system, lowest temperature 239 K, highest 271 K, not all points are shown. (ii) O, +, 16.63 mmol, temperatures 259274 K. (iii) 4, +, 23.40 mmol, temperatures 278-286 K. Note that the isosteres are completely reversible at sufficiently high temperatures. eter with an accuracy of ca. (0.002 K. At low temperatures where the density of the adsorbed fluid is much greater that of the bulk, the excess amount adsorbed is determined with an accuracy of 0.1% or better, corresponding to the accuracy of the corresponding equation of state for the bulk fluid,8-11 but at temperatures and pressures approaching bulk critical conditions this error may be larger. The estimated error in relative fugacity is ca. (0.001. Adsorptive surface tensions have been calculated using equations published elsewhere.12-14 The operating procedure is similar to that described previously.15 With a given amount of adsorptive in the system, the temperature is increased in ca. 1 K steps from the lowest temperature to the highest. At each step, temperature, pressure, and amount adsorbed are determined, then the process is reversed with temperature decreasing in ca. 1 K steps. If hysteresis is absent, points obtained while decreasing temperature coincide with those obtained with increasing temperature. Three example runs are shown in Figure 2. The pressure and amount adsorbed at integer temperatures are interpolated from these data and complete isotherms are constructed from numerous runs carried out with different amounts of adsorptive. (8) Younglove, B. A. J. Phys. Chem. Ref. Data Suppl. 1982, 11, 1. (9) Biswas, S. N.; Trappeniers, N. J.; Hoogland, J. H. B. Physica 1984, 126A, 384; Physica 1988, 149A, 649. (10) Sifner, O.; Klomfar, J. J. Phys. Chem. Ref. Data 1994, 23, 63. (11) Biswas, S. N.; Ten Seldam, C. A. Fluid Phase Equilib. 1989, 47, 67. (12) Baidakov, V. G.; Khvostov, K. V.; Muratov, G. N. Russ. J. Phys. Chem. 1982, 56, 497. (13) Rathjen, W.; Straub, J. Proc. Symp. Thermophys. Prop. 1977, 7, 839. (14) Jasper, J. J. J. Phys. Chem. Ref. Data 1972, 1, 841. (15) Machin, W. D. J. Chem. Soc., Faraday Trans. 1992, 88, 729.
N2
Xe
SF6
Adsorbate Properties temperature range/K 91-125 239-300 critical temperature/K 126.26 289.73
275-324 318.7
Capillary Critical Temperatures/K Tcc (experiment) 121 277 Tcc (eq 5) 121 277
305 302
Results and Discussion Isotherms have been determined at integer temperatures for the adsorptives listed in Table 1. Also given in this table are bulk critical temperatures and the corresponding capillary critical temperatures. Isotherms are either type 4 (nitrogen) or type 5 (xenon, sulfur hexafluoride) with H1 hysteresis16 for all T < Tcc. Hysteresis is absent within the intercritical region, Tcc < T < Tc, but all isotherms exhibit a well-defined plateau at high relative fugacities. This plateau is absent in supercritical isotherms T > Tc. In appearance, these isotherms are similar to those presented earlier.1,15 The results divide into three groups: (i) the capillary subcritical region T < Tcc, where hysteresis is present, (ii) the intercritical region Tcc < T < Tc where hysteresis is absent, and (iii) the supercritical region T > Tc. We first consider the subcritical region. Two important properties of capillary liquids are surface tension and density or molar volume. With type 4 or 5 isotherms the latter may be calculated from the amount adsorbed at saturation pressure16,17 as follows. The pore volume Vp of a rigid, inert adsorbent should be constant, apart from very small changes arising from thermal expansion effects. When this is true, the total amount of adsorbate contained in the pores nt is related to the experimental adsorption excess ne as
nt ) ne + Vp Fvap
(1)
where Fvap is the density of the bulk vapor. If at saturation pressure the pores are filled with capillary liquid having the same properties as bulk liquid, then
nto ) neo + Vp Fovap ) Vp Foliq
(2)
where Foliq is the density of bulk liquid and the superscripts refer to the various quantities at saturation pressure. It follows from eq 2 that
Vp ) neo/(Foliq - Fovap)
(3)
Therefore when neo is plotted vs Foliq - Fovap a straight line with slope equal to Vp and zero intercept should be obtained. That is, since Foliq - Fovap f O as T f Tc, then neo must also go to zero at Tc. These expectations are confirmed for all subcritical isotherms (see Figure 3), and we conclude that at saturation pressure capillary liquid nitrogen, xenon, and sulfur hexafluoride in CPG-100 have essentially the same density as that of the corresponding bulk liquids. The surface tension of capillary liquid may be estimated from the position of the sharp knee on the desorption branch of the isotherm.7 The knee is associated with the emptying of the largest pores or with a percolation (16) Sing, K. S. W.; Everett, D. H.; Haul, R. A. W.; Moscou, L.; Pierotti, R. A.; Rouquerol, J.; Siemieniewska, T. Pure Appl. Chem. 1985, 57, 603. (17) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity, 2nd ed.; Academic Press: New York, 1982; Chapter 3.
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Figure 3. Adsorption excess at saturation pressure, neo vs ∆F: ], nitrogen; +, xenon; O, sulfur hexafluoride. The lower curve refers to the capillary subcritical region, T < Tcc, a vertical bar marks ∆F at Tcc for each adsorbate. Note that these results are well-described by eq 3. The upper curve refers to the intercritical region Tcc < T < Tc and, for clarity, has been displaced upward by 5.0 mmol g-1. The slope of the dashed line in each case corresponds to the adsorbent pore volume, 0.661 cm3 g-1.
Figure 4. Plot of RT ln(f/f o) vs reduced temperature. At T < Tcc f/f o refers to the relative fugacity at the knee on the desorption branch of the hysteresis loop while at T > Tcc f/f o refers to the relative fugacity at the isotherm shoulder. ], nitrogen; +, xenon; O, sulfur hexafluoride. The dashed lines serve as a guide to the eye.
threshold. For a rigid inert adsorbent the pore radius at this point should be constant and the position of the knee, (f/f o)k, Fliq - Fvap (or ∆F), surface tension γ, and pore radius rp are related through the Kelvin equation18
RT ln(f/f o)k ) -2γ/rk∆F; rk ) rp - t
∆Phn ) [16πγ3/3kT ln(kTzs/h)]1/2
(5)
where k, h are the Boltzmann and Planck constants and z is the number of molecules within the pore cavity. The Laplace pressure difference ∆PL across a curved liquidvapor interface is given by18
(4)
where t is the thickness of the adsorbed multilayer film as estimated from the amount adsorbed at the lower closure point of the hysteresis loop. The temperature at which the surface tension vanishes, i.e., where capillary liquid cannot exist, may be determined from a plot of RT ln(f/f o)k against temperature,7 as shown in Figure 4. For all three adsorptives, RT ln(f/f o)k extrapolates to zero close to the bulk critical temperature Tc. When bulk surface tension values are used to calculate pore radii, we find for all three adsorptives that rp values are relatively constant over a wide range of reduced temperatures and are in satisfactory mutual agreement: nitrogen, 0.72-0.96 T/Tc, rp ) 9.1 ( 0.4 nm; xenon, 0.83-0.96 T/Tc, rp ) 9.0 ( 0.7 nm; sulfur hexafluoride, 0.86-0.96 T/Tc, rp ) 9.9 ( 1.0 nm. The quoted uncertainties correspond to one standard deviation. While these results do not prove that capillary (18) Everett, D. H. Colloid Science 1973, 1, 123.
liquid has the same surface tension as bulk liquid, they are most readily explained with that assumption. In any case, it is clear that the surface tension of capillary liquid does not become zero at the capillary critical temperature (Tcc). Within the intercritical region Tcc < T < Tc, hysteresis is absent but the isotherms still exhibit a plateau extending from saturation pressure to a steep shoulder marking the onset of extensive capillary evaporation. Capillary fluid at saturation pressure shows behavior similar to that of subcritical capillary liquid in that neo varies linearly with ∆F, as required by eq 3, but neo is slightly larger than expected (see Figure 3). Hence the average density of the capillary fluid at saturation pressure is slightly greater than that of the corresponding bulk liquid. Furthermore, RT ln(f/f o)sh plotted against temperature also extrapolates to zero close to the bulk critical temperature (see Figure 4), but calculated pore radii (eq 4) are not constant. Nevertheless, these results suggest that the capillary fluid at saturation pressure is liquidlike. If this is the case then capillary evaporation (desorption) in this region must follow a different path than that followed in the subcritical region. While model pore shapes are usually taken to be cylinders or slits, a more general model regards the pore structure as a series of cavities interconnected by smaller necks or windows.19 Adsorption proceeds by the growth of liquidlike multilayers on pore walls followed by the collapse of the core vapor phase (bubble) within each cavity (capillary condensation). At T < Tcc capillary evaporation (desorption) occurs when the liquid-vapor interface in the pore neck becomes unstable (eq 4) and all of the capillary liquid within the pore abruptly evaporates. Hysteresis occurs since the desorption process differs from adsorption. When capillary evaporation can occur by the growth of a vapor bubble within the pore cavity, i.e. the converse of the process leading to capillary condensation, hysteresis should be absent. Formation of a bubble within capillary liquid at reduced or negative pressure occurs by homogeneous nucleation of the vapor phase. Fisher20 has shown that the pressure ∆Phn at which this occurs is determined by the liquid surface tension
∆PL ) Pliq - Pvap ) (RT∆F)ln(f/f o)
(6)
When |∆Phn| > |∆PL| the menisci in the necks enclosing the pore cavity will become unstable (Kelvin failure, eq 4) before homogenous nucleation can occur in the capillary liquid in the pore cavity, but when |∆Phn| < |∆PL| homogeneous nucleation (eq 5) will occur before Kelvin failure. Consequently, the isotherms will be reversible when ∆Phn/∆PL < 1 (intercritical region) and irreversible (hysteresis) when ∆Phn/∆PL > 1 (capillary subcritical region). The temperature at which ∆Phn/∆PL ) 1 determines the capillary critical temperature. Values of Tcc estimated this way (see Figure 5) are in excellent agreement with experimental values (see Table 1). These results suggest that at all T < Tc dense capillary fluid at, or close to, saturation pressure is liquidlike, having (19) Mason, G. Proc. R. Soc. London 1983, A390, 47. (20) Fisher, J. C. J. Appl. Phys. 1948, 19, 1062.
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Figure 5. Estimation of the capillary critical temperature from eqs 5 and 6. Isotherms are reversible when ∆Phn/∆PL < O and exhibit hysteresis when ∆Phn/∆PL > O. This figure shows the results obtained for xenon. The vertical bar near the center of the figure indicates the capillary critical point. Similar plots are obtained for nitrogen and sulfur hexafluoride.
density and surface tension comparable to the corresponding bulk liquid. Consequently, the capillary critical point originates not from true critical behavior of the capillary fluid at Tcc but from the availability of a reversible desorption path (homogeneous nucleation) for T > Tcc. These conclusions differ from widely accepted models for capillary phase behavior5,6,21,22 which generally predict true critical behavior at the capillary critical point. Recently, however, Bhatia and Sonwane23 have shown that pore criticality may arise from the “instability of the surface layering mechanism”, but the density and surface tension of the capillary condensate remain similar to bulk liquid. With respect to other experimental studies, Brown et al.7 have reported isotherms for xenon and carbon dioxide adsorbed in porous Vycor glass. They find that the density of these adsorbates at the capillary critical point is essentially the same as that for bulk liquid but is significantly larger at lower temperatures. As well, their analysis also suggests that the surface tension of capillary condensate becomes zero at Tcc. At the present time we cannot reconcile the differences between their work and our own, but it may be significant that Vycor porous glass has low porosity (ca. 0.3) and small pore size (rp ∼ 3-4 nm) compared to CPG-100 (porosity ca. 0.6, rp ca. 9 nm). Findenegg and co-workers have made an extensive study of the near critical capillary phase behavior of sulfur hexafluoride in several controlled pore glasses with different pore diameters: CPG-10 (7.7 nm), CPG-240 (24.2 nm), and CPG-350 (31.3 nm). Their experimental results are presented as isotherms or as isosteres24-28 but all show features relevant to this work. These are (i) a diminution in the extent of hysteresis as the temperature is increased, (ii) a capillary critical temperature less than the bulk (21) Evans, R.; Marconi, U. M. B.; Tarazona, P. J. Chem. Soc., Faraday Trans 2 1986, 82, 1763. (22) Thommes, M.; Findenegg, G. H.; Schoen, M. Langmuir 1995, 11, 2137. (23) Bhatia, S. K.; Sonwane, C. G. Langmuir 1998, 14, 1521. (24) Findenegg, G. H.; Thommes, M.; Michalski, T. Special Publication ESA SP 1992, ESA SP-333, In Proceedings of the VIIIth European Symposium on Material and Fluid Sciences in Microgravity; European Space Agency: Geneva, 1992; Vol. 1, 795. (25) Thommes, M.; Findenegg, G. H. Langmuir 1994, 10, 4270. (26) Schoen, M.; Thommes, M. Phys. Rev. E 1995, 52, 6375. (27) Schoen, M.; Thommes, M.; Findenegg, G. H. J. Chem. Phys. 1997, 107, 3262. (28) Thommes, M.; Schoen, M.; Findenegg, G. H. Lect. Notes Phys. 1996, 464 (Materials and Fluids under Low Gravity), 51.
Figure 6. Critical depletion of sulfur hexafluoride at Tc. When sufficient adsorbate is present to ensure a slight excess of bulk liquid at the critical point, a sharp minimum in the adsorption excess is observed; otherwise, no minimum is observed. Total amount of SF6 present; O, 24.43 mmol; 4 23.90 mmol; +, 23.03 mmol.
critical temperature, and (iii) isotherms and isochores exhibit a plateau as the bulk saturation pressure is approached. The adsorption excess at saturation pressure neo increases as (Tc - T) increases as we have observed (see Figure 3). For example, neo estimated from the isosteres, shown in Figure 3 of ref 25 when plotted according to eq 1 (of this work), is linear with respect to ∆F and extrapolates to a value close to zero at Tc. From these isosteres they have mapped out the capillary coexistence curve for sulfur hexafluoride in CPG-240 and CPG-350. They find, as we do for CPG-100, that the density of capillary liquid is only slightly larger than that of bulk liquid and that the capillary coexistence curve is narrower than the bulk curve and displaced to lower temperatures (see Figure 1). While the experimental phenomena they observe are very similar to those described in this work, they conclude that capillary fluid exhibits true critical behavior at the capillary critical point. This implies that pore fluid at saturation pressure within the intercritical region should not be liquidlike. Nevertheless, isotherms and isochores in this region reach a plateau in the amount adsorbed as saturation pressure is approached and the corresponding density of the pore fluid is similar to that of bulk liquid. At subcritical temperatures when excess adsorptive is present at saturation pressure it may be apportioned between three fluid states: (i) vapor external to the adsorbent novap, (ii) bulk liquid external to the adsorbent noliq, and (iii) capillary condensed liquid filling the pores in the adsorbent, not , hence
N ) novap + noliq + not
(7)
where N is the total amount of adsorptive. Part of the capillary condensed liquid is regarded as the adsorption excess noe and depends on not and the density of the bulk vapor phase (see eq 2). Equation 3 may be rearranged to give noe ) VP (Foliq - Fovap), and since at all subcritical temperatures Foliq > Fovap, then noe will be positive, decreasing to zero as T f Tc (see Figure 3).
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On the other hand, at temperatures equal to or greater than Tc the adsorptive is apportioned as (i) a homogeneous fluid external to the adsorbent, next at density Fext, and (ii) a fluid within the pores of the adsorbent, nint at average density Fint, where N ) next + nint and the adsorption excess is given by
ne ) VP(Fint - Fext)
(8)
When Fint > Fext, then ne > O, but if Fint < Fext, then ne will be negative. If sufficient adsorptive is present to ensure a slight excess of external bulk liquid at all subcritical temperatures then the calculated adsorption excess as the temperature of the system is increased, or decreased through the critical point, ne may show a pronounced minimum and may even become negative. This behavior is shown in Figure 6 for sulfur hexafluoride in contact with CPG-100. Negative values for ne at the critical temperature suggest that the average density of the
capillary fluid at this point is less than the density of the external bulk fluid. This phenomenon, first described by Thommes et al.,22 as “critical depletion” should occur with all systems where the foregoing conditions apply. We conclude from these results that nitrogen, xenon, and sulfur hexafluoride, when capillary condensed in CPG100, retain the density and surface tension of bulk liquid. The capillary critical point, where hysteresis vanishes, is not analogous to the bulk critical point, but it occurs when capillary evaporation of pore liquid can take place by homogeneous nucleation of the vapor phase within the pore cavity. Acknowledgment. This work was supported by the Natural Sciences and Engineering Research Council of Canada. Professor P. D. Golding has provided much useful comment and discussion. LA971393R
