Langmuir 1989,5, 714-723
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Temperature Dependence of Gas Adsorption on a Mesoporous Solid: Capillary Criticality and Hysteresis P. C. Ball and R. Evans* H . H . Wills Physics Laboratory, University of Bristol, Bristol BS8 1 TL, United Kingdom Received November 18, 1988
Adsorption isotherms have been calculated for two different models of a mesoporous adsorbent. The first (a) models the adsorbent as a system of independent cylindrical pores each with free access to the reservoir of gas, while the second (b) treats the pore system as an interconnected network of the type investigated by Mason. In both cases, capillary condensation of the fluid in a single pore is treated within a simple density-functional theory. Model a attributes hysteresis to the existence of metastable “gas”states in a single pore whereas model b associates hysteresis with pore blocking; i.e., evaporation of the capillary condensate in a pore, which should occur during desorption, is obstructed by “liquid” condensed in constrictions or ”necks”. By comparing the shapes of the calculated isotherms and scanning curves with experimental results for Xe on Vycor, we conclude that hysteresis of adsorption in such adsorbents is better accounted for by the pore-blocking mechanism of the network model b. Regardless of which mechanism prevails, our results predict that hysteresis loops should shrink with increasing temperature and should disappear, eventually, at some capillary “critical”temperature that depends on the average pore size but lies substantially below the critical temperature of the bulk fluid. Such behavior reflects the criticality of the capillary condensation transition in a single pore. The temperature dependence of the experimental data of Nuttall and Everett for Xe on Vycor and several related systems is similar to the temperature dependence we calculate, and their results support the notion of a capillary critical point. 1. Introduction
The adsorption of gases on porous substrates has been the subject of experimental investigation since the end of the last century.’ Explanations for the observed behavior of the measured isotherms have not been lacking,2but it is only in recent years that a satisfactory understanding has emerged of the fundamental physical processes involved. In particular, advances made during the last decade in theories of highly inhomogeneous fluids3” have heralded a realistic description of the microscopic structure and thermodynamic behavior of fluids in confined geometries. Nevertheless, the connection between theoreticians’ models of simple idealized systems on the one hand and the vast body of experimental data from real systems on the other remains tenuous. In many cases the complex, often poorly characterized, structure of real porous solids renders theoretical interpretation ambiguous. The measured adsorption isotherms show a great diversity of behavior, for which several systems of classification have been p r ~ p o s e d .In ~~ general, ~ ~ ~ low-temperature isotherms are distinguished by steep increases in adsorption at pressures p below the saturated vapor pressure, past, of the bulk gas, and this is attributed to capillary condensation of the gas in the pores to a dense, liquidlike state. The mechanism of capillary condensation was recognized by Zsigmondp in 1911. Confining the fluid in a pore with strongly adsorbing walls has the effect of shifting the bulk gas-liquid coexistence curve to a lower pressure. In the vast majority of cases, hysteresis loops are observed; the rapidly rising portion of the isotherm occurs (1)van Bemmelen, J. M. 2.Anorg. Allg. Chem. 1897,13, 233. (2)Gregg, S.J.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic: New York, 1982. (3)Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity, Oxford University Press: New York, 1982. (4)Evans, R. Ado. Phys. 1979,28,143. (5) Nicholson, D.;Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic: New York, 1982. (6)de Boer, J. H. In The Structure and Properties of Porous Materials; Everett, D. H., Stone, F. s.,EMS.; Butterworths: London, 1958. (7)Everett, D. H.In The Solid-Gas Interface; Flood, E. A., Ed.; Marcel Dekker: New York, 1967;Vol. 2,p 1055. (8) Zsigmondy, R. 2.Anorg. Allg. Chem. 1911,71, 356.
0743-7463/89/2405-0714$01.50/0
at a different (higher) pressure on adsorption from that on desorption. The mechanism for hysteresis has prompted much discussion in the literature, the essence of which identifies two possibilities: (i) hysteresis is an intrinsic property of the phase transition in a single idealized pore; (ii) hysteresis is a consequence of the interconnectivity of a real pore network. The former is the basis of Foster’s “delayed meniscus theory”: placed on a more quantitative basis by Cohan.lo At the heart of this mechanism lies the supposition that nucleation of a capillary phase may be postponed beyond the equilibrium transition between “gas” and “liquid” configurationsof the fluid; in other words, the capillary may support metastable states in a manner analogous to superheating or supercooling of a bulk fluid. In most cases, crude assumptions are made regarding the nature of metastability, and the problem of nucleation of pore phases is not addressed directly. The important feature of a network model was recognized7 by Kraemerll and McBain.12 If a pore has access to the external gas phase only via narrow constrictions or necks, then pore blocking may occur, in that evaporation, during desorption, of the capillary condensate in the pore will be obstructed by liquid remaining condensed in the necks. The relative pressure at which a pore empties now depends on the size of the necks, the connectivity of the network, and the state of neighboring pores. Everett’ gives an excellent review of early work and points out the limitations of the various mechanisms of hysteresis that have been proposed. One feature of hysteresis, which has received relatively little experimental attention, concerns the temperature dependence of the hysteresis loops. Microscopic statistical mechanical theories of capillary condensation13J4indicate (9) Foster, A. G. Trans. Faraday SOC.1932,28,646.
(10)Cohan, L. H.J. Am. Chem. SOC.1938,60,433;1944,66,98. (11)Kraemer, E. 0.In A Treatise on Physical Chemistry; Taylor, H. S., Ed.; van Nostrand New York, 1931;p 1661. (12)McBain, J. W. J. Am. Chem. SOC.1935,57,699. (13)Evans, R.; Marini Bettolo Marconi, U.; Tarazona, P. J. Chem. Phys. 1986,84,2376. (14)Evans, R.; Marini Bettolo Marconi, U.; Tarazona, P. J. Chem. Soc., Faraday Trans. 2 1986,82,1763.
0 1989 American Chemical Society
Langmuir, Vol. 5, No. 3, 1989 715
Capillary Criticality and Hysteresis 4
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to be quite common. It was observed by Nuttall and Everett16*in a systematic investigation of the adsorption of xenon on Vycor glass. Isotherms for Xe were measured at a series of temperatures between 183.45 and 273.15 K, corresponding to reduced temperatures T/ T, of 0.63-0.94 (here T,, the bulk critical temperature for Xe is 289.7 K). Some of these results are shown in Figure 1. There is a progressive shrinking of the loops as T increases, which is accompanied by a shift in loop position to higher relative pressures, PIPsat. At the highest temperature reported, T = O.94Tc, the loop vanishes, and adsorption is then reversible. The result for T = O.87Tc is somewhat uncertain; this isotherm may also be reversible.’“ Results16b for C 0 2 on Vycor are similar for TIT, k 0.70 (above the triple point), with the loop vanishing near T = 258 K, Le., TIT, 0.85. Isotherms for Xe adsorption on an activated carbon powder’“ exhibit similar features. The desorption branches are slightly steeper, a loop persists at T 0.89Tc, but the isotherms are reversible for T k O.94Tc. It is tempting to associate the shrinking of the hysteresis loops observed in such experiments with the onset of the capillary criticality predicted to occur in a single pore. We explore this possibility further in the present paper. By carrying out calculations of adsorption isotherms for different models of the porous adsorbent and making comparison with the experimental results,16we also reexamine the problem of the origin of hysteresis. The paper is arranged as follows: in section 2 we describe a mean-field density functional theory of the fluid in a pore; this is then employed, in section 3, in calculations of adsorption for an idealized, independent pore model of the adsorbent. In section 4 we investigate how the isotherms would be altered when the pore space forms an interconnected network. Numerical results for both models, including scanning curves, are presented in section 5. We conclude in section 6 with a discussion of how our results relate to experiment and make further remarks concerning the notion of capillary critical points.
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Figure 1. Adsorption isotherms of Xe on Vycor measured by Nuttall.16 Solid circles denote adsorption; open circles denote desorption. The reduced temperatures are indicated in each case. At the highest temperature, no hysteresis is observed.
that capillary condensation in a single, infinitely long pore can occur only below a certain capillary critical temperature TccaP,which is dependent mainly on pore geometry and size. The bulk critical temperature T, is shifted to a lower value by confinement; the smaller the pore radius or width the lower is TcW. For T 2 TcaP there is no abrupt first-order transition between gas and liquid configurations-adsorption in the pore is a smooth, monotonically increasing function of pressure or chemical potential p. Regardless of which mechanism for hysteresis prevails, one should expect to observe the disappearance of hysteresis loops at some temperature that is high, but still below bulk T,. The experimental situation is rather complicated. In a detailed study of a number of adsorption systems, Amberg et al.lSfound varied hysteretic behavior. Often the loops shrink in size, and eventually vanish, as temperature increases, and this is also observed by other workers.16J7 Sometimes the loops may decrease in extent as the temperature is lowered toward the bulk triple point.ls Loop shapes may differ at different temperatures.ls Nevertheless, the mode of behavior in which loops shrink and vanish with increasing temperature does appear (15) Amberg, C. H.; Everett, D. H.; Ruiter, L.; Smith, F. W. In Surface Activity; Schulman, J. H. Ed.; Butterworths: London, 1957; Vol. 2. (16) (a) Nuttall, S.Ph.D. Thesis, University of Bristol, 1974 (unpublished). (b) Burgess, C. G. V. Ph.D. Thesis, University of Bristol, 1971 (unpublished). Some of the remarkable data of Nuttall and of Burgess is presented by Everett, D. H. In Proc. Fifth Hungarian Colloid Conf., 1988 (to appear). Everett also attributes the disappearance of hysteresis loops to the onset of capillary criticality. (17) Dubinin, M. M.; Bering, B. P.; Serpinsky, V. V.; Vasilev, B. N. In Surface Phenomena in Chemistry and Biology; Daniel, J. F., Pankhurst, K. G. A., Riddiford, A. C., Eds.; Pergamon: Oxford, 1958. (18) Miles, A. J. Ph.D. Thesis, University of Bristol, 1964 (unpublished). Bailey, A. Ph.D. Thesis, University of Bristol, 1965 (unpublished).
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2. Density Functional Theory for Adsorption in
Pores The behavior of simple fluids in confiied geometries has attracted much theoretical investigation in recent years. Alongside computer ~imulation’”~~ and lattice-gas models,25*26 density functional theorSp has proved an invaluable technique for determining the systematics of fluid phase equilibria in idealized pores.13J4 Most of the previous work with this technique utilizes a grand potential functional of the form QvbI = l d r f h ( P ( d ) + 7 2 sl d r d r ’ ~ ( rdr’)4Jatt(Ir ) -
r’l) - j d r
(P -
V W ) p(r) (1)
Here p is the chemical potential of the fluid, fixed by the reservoir. V(r) is the external potential exerted by the solid walls of the pore on the molecules in the fluid; this is chosen to model a particular idealized pore geometry, (19) van Megen, W.; Snook, I. K. Mol. Phys. 1986, 54, 741. (20) Peterson, B. K.;Gubbins, K. E. Mol. Phys. 1987, 62, 215. (21) Heffelfinger,G. S.;van Swol, F.; Gubbins, K. E. Mol. Phys. 1987, 61,1381. (22) Panagiotopoulos, A. Z.Molec. Phys. 1987, 62, 701. (23) Peterson, B. K.;Gubbins, K. E.; Heffelfinger, G. S.; Marini Bettolo Marconi, U.; van Swol, F. J. Chem. Phys. 1988,88,6487 and references therein. (24) Walton, J. P. R. B.; Quirke, N. Chem. Phys. Lett. 1986,129,382. (25) Nakanishi, H.; Fisher, M. E. J. Chem. Phys. 1983, 78, 3279. (26) Bruno, E.; Marini Bettolo Marconi, U.; Evans, R. Physica A (Amsterdam) 1987, 141, 187.
716 Langmuir, Vol. 5, No. 3, 1989
usually a slitlike pore (two parallel walls, each of infinite area but separated by a finite distance H) or a cylinder (of infinite length and of finite interior radius R,). fh(p) is the Helmholtz free energy density of a uniform hard-sphere 3eference" fluid of density p. Attractive forces between molecules in the fluid are treated in mean-field approximation: is the attractive part of the fluid-fluid painvise potential. Functional differentiation of eq 1with respect to p(r) yields an integral (Euler-Lagrange) equation which can be solved for p(r), the equilibrium density profile of the fluid. Thermodynamic properties follow by recognizing that the resulting minimum of the functional Q v [ p ] is the variational estimate for the equilibrium grand potential of the fluid.4 It is important to recognize that eq 1constitutes a crude approximation for the microscopic structure of the fluid. Treating the repulsive forces in a local density approximation (LDA) causes the short-range correlations, which often produce oscillatory density profiles for fluids near walls, to be excluded. More sophisticated theories treat the repulsive forces in a nonlocal fashion, and these have also been applied successfully to confined f l u i d ~ . 2 ~Because ~ ~ ~ - of ~ the complexity involved in modeling a pore network, we have deliberately chosen to use the simplest LDA functional rather than one of the more sophisticated versions. We are concerned with integrated quantities (the adsorption) and with thermodynamic functions (the grand potential) rather than details of the microscopic structure; the former are often rather insensitive to the latterSz7 The properties of solutions resulting from minimizing eq 1have been described at length elsewhere13J4for fluids in both slits and cylinders. For T < TcaP and p < pat, the chemical potential at bulk liquid-gas coexistence, the fluid can exist with two possible density profiles corresponding to dense liquidlike and dilute gaslike configurations in the pore. The equilibrium profile is that with the lower grand potential. At some value of p ( R,
- uw
(5)
Here R is the radial distance from the pore center, e, is a strength parameter proportional to A and to the wall density, and R , is the "true" pore radius. Hereafter, R, refers to the "effective" radius, i.e., the radial distance from the pore center to the p_ointwhere V = a. Clearly R, = R, - uw. The function Q is defined by
O b ) = (1 -
X ) ~ [Sx cos*4~ ~(1 -J f
0
x 2 sin2 4)1/2]-3(6)
The ninth-order polynomial fit for Q ( x ) calculated by Nicholson31was used in our calculations. The only free parameters are ew (the strength of the wall-fluid attraction) and the shape of the pore size distribution g(R,). Reliable data are not available for either of these. Saam and Cole33have modeled the adsorption of helium on Vycor, which is 99% silica,with the same type of wall potential for which they estimated a strength paWe have made the rameter equivalent to ewHe = 4.7kBTCHe. crude assumption that the wall-fluid interaction strength simply scales with the bulk critical temperature of the fluid for a given wall; i.e., e,/kBT, is a constant, characteristic of the substrate. Thus we use the value ewxe = 4.7kBTCXe in our calculations for Xe on Vycor. For this value of e, the temperature of the wetting transition in our LDA theory for the equivalent single planar wall, where the external potential V ( z ) -2~,4,~2-~ as z m, is T, = O.62Tc. The reduced temperatures corresponding to the measured adsorption isothermdsa lie above our theoretical value of Tw/T,, and we except the wall-gas interface in theory and experiment to be wet completely by liquid; i.e., the contact angle should be zero for all the temperatures of interest. Following Saam and Cole, we have assumed a Gaussian pore size distribution:
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where R A is the average pore radius and A is the standard deviation. There are no widely accepted values for these parameters, and, to an extent, they will be inevitably sample-dependent. Saam and Cole33used values of R A = 29.4 A and A = 0.2RA and found that their calculated isotherms for He on Vycor were in reasonable agreement with experiment. I t is generally accepted that Vycor is mesoporous and an estimate which gives an average pore radius of 10-20 molecular diameters would seem sensible. . of We choose RA = 10u (=36 A) and A = 0 . 2 R ~ Because the uncertainty of these values, we also investigated the effects of changing the pore size distribution. In the independent pore model, we associate hysteresis with metastability in a single pore;' i.e., we assume that the fluid may remain in metastable states beyond the equilibrium capillary condensation transition. Although this is the essence of Foster's delayed meniscus t h e ~ r y , ~ CasselP appears to have been the first to associate hysteresis explicitly with metastability. In principle, metastability may occur on both liquid and gas branches-see Figure 3. It has become conventional, however, to assume that hysteresis is due solely to metastability of the gas (33) S a m , W. F.; Cole, M. W. Phys. Rev. B . 1975,11, 1086. (34) Cassell, H.M. J . Phys. Chem. 1944, 48,195.
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iscus, so the amount adsorbed is rapidly reduced as the pressure is lowered below that for capillary coexistence. To summarize, hysteresis in the independent pore model is assumed to take the form of a jump on adsorption at a pressure p a = ps, the pressure at the gas spinodal, whereas the jump on desorption occurs at a pressure Pd equal to that for coexistence, pco. Results of numerical calculations for this model are presented in Section 5. 4. Network Model
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Figure 3. Mechanism of hysteresis arising from metastability in a single pore. The equilibrium transition from gas to liquid occurs at pco. However, the fluid can remain in metastablestates up to the spinodals (ends of the dashed lines). Capillary condensation occurs at a pressure paand evaporation at pd. In the calculations,it is assumed pa = p B ,the gas spinodal, and = pm.
configuration; Le., on adsorption the gas persists beyond the equilibrium transition, whereas on desorption the jump from the liquid to the gas occurs at coexistence of the two phases. This is implied in Cohan’slO original treatment of the problem and remains implicit in Saam and Cole’s33 theory of liquid film instabilities. In neither case, nor elsewhere, is this assumption justified fully. Nevertheless, the resulta of recent density functional calculations96seem to indicate that one can obtain good agreement between theory and experimental observations of hysteresis when such an ansatz is made. More precisely, we assume that on adsorption the gas branch persists to the limit of metastability (the capillary gas spinodal-see Figure 3) before jumping to the liquid branch, whereas the jump from liquid to gas on desorption occurs at coexistence. Proper justification for this scenario must await a better understanding of the nature of nucleation of the pore phases. However, the observations can be rationalized by considering the formation of menisci in pores of finite length. Everett’ argues that the jump on desorption should occur at the relative pressure PIPrnt, which results from the Kelvin equation applied to the hemispherical meniscus that develops in a cylindrical pore filled with liquid. For very large radii,14this Kelvin estimate becomes exact for the equilibrium transition, i.e., for pco/psllt.Everett associates the jump on adsorption with the value of PIPrnt resulting from the Kelvin equation applied to a cylindrical liquid meniscus, assumed to develop between the adsorbed cylindrical liquid film and the gas. The film becomes unstable with respect to unduloid formation. Since the mean radius of curvature of the cylindrical meniscus is equal to 2Rc, whereas that of the hemispherical meniscus is R,, condensation is predicted to occur at substantially larger relative pressures than evaporation. A simple macroscopic treatment of thick film growth in cylinders14 confirms that such films can remain metastable until the relative pressure reaches the value obtained from the Kelvin equation applied to the cylindrical meniscus. These macroscopic arguments are supported by the results of recent lattice gas calculations by Marini Bettolo Marconi and van Swo1,96who have shown that the adsorption gas branch in finite pores is almost the same as for infinitely long pores, but desorption proceeds via a receding men(36)Ball,P. C.; Evans, R. Europhys. Lett. 1987, 4,715. (36) Marini Bettolo Marconi, U.; van Swol, F., to be published. See also: Heffelfinger, G. S.; van Swol, F.; Gubbins, K. E. J. Chem. Phys. 1988,89, 5202.
Although it has been long r e c o g n i ~ e dthat ~ ~ ~the ~ l in~~ terconnected nature of a pore network may play a significant role in adsorption experiments on porous media, this feature is frequently neglected. The important property of a network, which is not present for a single uniform pore, is the possibility of pore blocking, which we described earlier. Following earlier work by Everett and M a s 0 n ~ has ~ 1 ~developed a model that incorporates blocking effects; he shows that such effects can result in hysteresis even when adsorption in a single pore is reversible. The pore space in Mason’s model consists of an assembly of individual pores connected via windows or necks. In order to establish a quantitative connection either with experiment or with microscopic theories of the pore fluid, we must be able to relate the pore filling pressure (condensation) to a characteristic pore dimension, and it is then necessary to assume some particular geometry. Mason is concerned largely with pores formed by the voids between packed rotund particles, but his method is valid for essentially any geometry. Three parameters or functions characterize the pore space: the connectivity C and the pore and window size distributions g ( r ) and f(r). C is defined as the average number of windows per pore, while r is the characteristic dimension that relates to the filling pressure. (In Mason’s treatment, r is obtained from the macroscopic Kelvin equation for capillary condensation.) g(r) ( f ( r ) )is the normalized fraction of pores (windows) with radius r and thus measures the probability of a single, randomly chosen, pore (window) having a filling radius between r and r + dr. The quantity
Q = l r Q d rg(r) 0
represents the fraction of pores which would be filled at a pressure corresponding to radius rQif all the pores behave independently; equivalently, it is the probability of a single random pore being filled at this pressure. Similarly
P = Jrpdr f ( r ) is the probability of a window having a radius less than rp, i.e., the probability of a meniscus not passing through an individual isolated window at the pressure corresponding to r p The variables P and Q take values between 0 and 1;adsorption corresponds to the increase in Q from 0 to 1,and desorption is the decrease of P from 1to 0. By defining these variables, we can describe adsorption in the network in a universal manner; the details of the size distributions and geometry are all incorporated into the normalized functions P and Q. (37) Everett, D. H. In The Structure and Propertie.9 of Porous Materials; Everett, D. H., Stone, F. s., Eda.; Proc. Tenth Symp. of Colston Research Society; Butterworthe: London, 1958; p 95. Barker, J. A Zbid; p 125. (38)Quinn, H. W.; McIntosh, R. In Surface Actiuity; Schulman, J. H., Ed.; Butterworth: London, 1957; Vol. 2, p 122. (39) Mason, G. J. Colloid Interface Sci. 1982,88, 36. (40) Mason, G.Proc. R. SOC.London A 1983,390,47.
Langmuir, Vol. 5, No. 3,1989 719
Capillary Criticality and Hysteresis
In general, the functions g ( r ) and f ( r ) are related only
to the extent that a given pore can have no windows with a radius greater than the pore radius. Mason assumes a more restrictive relationship in order to develop a tractable theory: out of the C windows that open onto each pore, the largest will always have a radius equal to that of the pore itself. This implies that there is no pore blocking in a single isolated pore with access via windows to the bulk gas phase. Mason justifies this assumption for the case of a pore space arising from the packing of equal spheres.41 Whether the same is true for a randomly intersecting network of cylindrical pores is less clear, but it seems unlikely to be grossly wrong-the essential physical processes are unchanged. Given this assumption, it may be shown that g ( r ) and f ( r ) are related; equivalently, this implies a relationship
Q=PC (10) Adsorption and desorption processes in the network are not equivalent. On adsorption the network behaves as the corresponding assembly of independent pores-there is no hindrance to pore filling. Pore blocking arises on desorption, however, and is controlled by the pore space connectivity C and the window radius distribution f ( r ) . For a pore to empty at the equilibrium pressure, it must have direct access to the bulk gas phase. Initially (for p close to pset),most of the pore space is blocked, and virtually no desorption occurs. At some critical pressure, when menisci may pass through a significant fraction of windows, a chain reaction is established that causes abrupt emptying of the system. The desorption branch therefore has a steplike shape, reminiscent of Type E loops in de Boer's classification.s The state of the network at any instant can be characterized in terms of the number of filled and empty pores. How these interconvert depends on the environment of each pore, i.e., on whether the C neighbors are filled or empty. One can express the filling and emptying of the network as a whole in terms of the interconversion of cells, where a cell consists of a pore and its neighbors. This approach yields differential equations which can be solved for the fraction of pore space filled as a function of P or Q. For a given value of C, these normalized isotherms are universal; one need not specify at this point any details about individual pores. An example for C = 4 is shown in Figure 4. The sharp onset of desorption occurs at Pcrit= 2/3 for this value of C. Scanning curves, which are obtained by reversing the direction of adsorption or desorption within the loop, are obtained by solving the differential equations for different initial ~ o n d i t i o n sand , ~ ~these ~ ~ ~are also shown in Figure 4. Desorption in the network model is essentially a percolation processt2Pa and the percolation threshold is sharp. In a real porous solid, surface effects and heterogeneities of the pore space will serve to round the discontinuity at the percolation threshold. We wish to contrast hysteresis due solely to metastability in a single pore with that due solely to network effects. Thus, in the network model we assume that the jump in the amount adsorbed takes place a t capillary coexistence for both adsorption and desorption in an isolated pore; i.e., there is no metastability and hence no hysteresis in isolated pores. This is the same assumption as that of Mason. However, rather than using the macroscopic Kelvin equation to relate the filling pressure to ~~
~
(41) Mason, G. J. Colloid Interface Sci. 1971, 35, 279. (42) Wall, G.C.;Brown, R.J. C . J.Colloid Interface Sci. 1981,82,141. (43) Mason, G. J . Colloid Interface Sci. 1983, 95, 277.
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Figure 4. Adsorption isotherm in the network model for connectivity C = 4. The probability P may be related to p / p m t in a way that depends on the pore size distribution-see text. (a)
Desorption scanning curves. Desorption starts from different places on the adsorption boundary curve. (b)Adsorption scanning curves. Now adsorption starts from different places on the desorption boundary curve. These curves are redrawn from Mason."o
radius, we utilize density functional calculations, with the same parameters as previously, to determine the adsorption r(p) in single pores. We take the same pore size distribution, g ( r ) , and take the connectivity C to be 4: Mason argues that such a value should be typical.3w1 5. Results of Calculations Adsorption isotherms, calculated for the same reduced temperatures as experiment,16are shown in Figure 5 for the independent pore model. Here I' is in arbitrary units but is directly proportional to the measured adsorption. Although there is some qualitative agreement with the experimental results of Figure 1 in terms of temperature dependence-the loops shrink as T increases and vanish between TIT,= 0.87 and 0.94-the agreement is poorer in other respects. The loop shapes observed in experiment are not well reproduced: those from our calculations are rather symmetrical, resembling de Boer's Type A. In experiment, the desorption branch has a more pronounced knee, and r on the adsorption branch is relatively larger at lower pressures. Moreover the positions of the loops, in p/p,,, are reproduced poorly. Altenng the pore size distribution g(RJ has a significant effect only when such changes are rather extreme;32we anticipate that our values of RA and A are reasonable estimates, and introducing extreme deviatons seems unwarranted. The results from the network model are shown in Figure 6. As expected, the adsorption branch is similar to that
720 Langmuir, Vol. 5, No. 3, 1989
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(d) (el (fl Figure 5. Adsorption isotherms calculated for the independent pore model. I’ is in arbitrary units. p is the density of a uniform fluid at the given chemical potential. The ratio p/psat = p/plat. (a) T = 0.63Tc;(b) T = 0.7OTc;(c) T = 0.76Tc;(d) T = O.83Tc;(e) T = 0.87Tc;(f) T = 0.94Tc.
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(el Figure 6. Adsorption isotherms calculated for the network model. (a) T = 0.63Tc;(b) T = 0.70Tc; (c) T = 0.83Tc;(d) T = 0.87Tc; (e) T = 0.94Tc.
of the independent pore model but is shifted to lower pressures. The desorption branch is flat until the pressure corresponding to the critical probability, Pc,it,is reached, at which point the derivative of I’changes discontinuously. This last feature is an artifact of the simple model. There are a few subtleties involved in using Mason’s model in conjunction with a microscopic theory for the phase transition in a single pore. Mason assumes that capillary condensation f i i a pore with liquid identical with the liquid at bulk coexistence. In a microscopic treatment the liquid configuration is highly inhomogeneous, and adsorption on the liquid branch for an individual pore increases slowly with increasing pressure. If we were to assume that this slow variation of r(p)is not susceptible to pore blocking effects, then the desorption branch would not be flat above P,, although the fraction filled remains unity. The horizontal portions of the curves in Figure 6 are therefore an artifact of the model. Allowing r to take on the equilibrium values of the liquid branch in all pores
results in the more realistic loop shapes shown in Figure 7. These still exhibit the artificial sharp kink at PCit; surface effects and other factors would smooth out this kink in a more realstic treatment. There is no provision in Mason’s model for capillary criticality. In order to relate P and Q to PIPsat,it is necessary to relate the pressure to a radius for which pores fill by capillary condensation. When pores are smaller than the capillary critical radius Rcmfor a given temperature, there is no such relationship. We can therefore calculate the isotherm only down to a pressure corresponding to the capillary critical point. The problems associated with pores close to or below their critical points are rather complicated, e.g., do pores with radii much smaller than the critical radius cause pore blocking of larger pores? Here the network model is clearly oversimplified. For T = 0.94Tc, nearly all the pores that contribute significantly to g(r) are supercritical (R, d R,”), so that adsorption in each, and presumably the total adsorption, is reversible. Thus the isotherm at this tem-
Langmuir, Vol. 5, No. 3, 1989 721
Capillary Criticality and Hysteresis
30
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Figure 7. Adsorption isotherms calculated for the network model
when the amount adsorbed on the liquid branch of each pore may vary on desorption even for P > P* 'The loops now elm- at plp;, < 1. (a) T = 0.70Tc;(b) T = 0.83Tc.
perature is taken to be identical with that from the independent pore model. The results for both models show considerable discrepancies with experiment as regards loop sizes and positions. The loops observed experimentallyare usually smaller than in theory and are centred at significantly larger values of p/p,-especially at higher temperatures. There are several possible reasons for this: 1. Poor Choice of Size Distributions g ( r ) . Whilst g(r) is undoubtedly sample dependent, and our choice of parameters is somewhat arbitrary, drastic changes would be required to improve the agreement with experiment. 2. Poor Choice of Wall-Fluid Strength Parameter E,. Again our choice can be queried. However, the value we chose is effectively a lower bound on t, since a smaller value would increase the wetting transition temperature T,, raising this above the (lower) temperatures of the experiments. Increasing ,€ by reasonable amounts does not have a significant effect on the results.32 3. Poor Modeling of Fluid-Fluid and Wall-Fluid Interactions. There are indications from calculations based on somewhat more realistic potential functions32that improving the potentials does not lead to any significant effects. 4. Grossly Inaccurate Density Functional Approximation. The LDA may give a poor description of the phase transition of the fluid in a pore. Comparison32 of results of the LDA with those of a more sophisticated nonlocal theory for the same potentials shows that whilst the former gives a poor account of the structure of the fluid, its predictions for capillary coexistence and for the location of capillary critical points are rather good, provided the comparisons are made well above (or well below) T,. Whether this particular nonlocal theory provides an accurate description of phase boundaries remains to be seen. It certainly gives a good description of the structure
0'3
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Figure 8. Adsorption of Xe on Vycor at 151 K (T = 0.52TJ measured by brown:'^^ (a) desorption scanning curves; (b) adsorption scanning curves. The open circles denote adsorption and the closed circles desorption. The weight of sample is 61.6 mg.
of confined Given the crudeness of our model, for what is a highly complex physical system, perhaps we should not be surprised if only qualitative agreement between theory and experiment is forthcoming. It is clear that the network and independent pore models predict different loop shapes. The steplike desorption branches of the network model, especially in Figure 7, are more akin to what is observed, and the more general implication is that de Boer's Type E isotherms are a consequence of network efforts. While the temperature dependence of the hysteresis loops is controlled in both cases by the nature of the phase equilibria in a single pore, it manifests itself in different ways. In the independent pore model, shrinking of the loops is connected explicitly with shrinking of the van der Waals loop in r ( p ) as T is increased; the extent of the metastable gas branch becomes smaller as T Tpp. In the network model no account is taken of metastable regions, and the widths of the hysteresis loops depend on the widths of the distribution functions g(r) and f(r). Consequently, the width of the loop does not change significantly throughout the temperature range up to 0.94Tc (contrast Figure 5e with Figure 6d). However, the vertical span of the loops does decrease with increasing temperature, reflecting the decrease in the jump in r at capillary condensation as criticality is approached, so there is still an overall trend of loop shrinking. Any proposal for the mechanism of hysteresis should account for the scanning behavior observed experimentally. Scanning curves of Xe on Vycor have been measured by and these are reproduced in Figure 8. The
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(44) Brown, A. J. Ph.D. Thesis, University of Bristol, 1963.
722 Langmuir, Vol. 5, No. 3, 1989
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02
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Ball and Evans
08
IO
P’Psat
lbi
Figure 9. Scanning curves calculated for the network model for T = O.63Tc: (a) desorption scanning curves; (b) adsorption
scanning curves.
shape of these curves is well described by the results of the network model-see Figure 9. On the other hand, as pointed out by Everett,’ his independent domain theory, which is equivalent to our independent pore model, does not describe properly the observed slopes of desorption scanning curves. The independent pore model predicts that for a given PIPsat, r versus p should be steeper for scanning curves starting at higher pressures on the loop, whereas the opposite trend is observed experimentally (Figure 8a) and in the network model (Figure 9a). In keeping with Everett7 and Mason,@we consider this to be important evidence in favor of the network model. 6. Discussion We have argued that considerations of loop shapes and scanning behavior suggest that a network mechanism should be the dominant cause of hysteresis in this system. However, we should be cautious about making generalizations. The structures of different porous materials are sufficiently varied that we should not expect the analysis of a particular system to provide an altembracing description of hysteretic adsorption phenomena. Indeed, we do not wish to imply that single-pore hysteresis does not occur but rather that, if present in this system, it is probably masked by pore-blocking effects in the network. In this respect, we should contrast our present conclusion with that drawn in our earlier study36 of hysteresis in the porous glasses prepared by a sol-gel t e c h n i q ~ e .There ~~ we concluded that if the pore size distributions were as narrow as claimed in the experiment4 (A = 0.025R~in the best samples), pore blocking could not be the dominant
mechanism, and single-pore hysteresis must be invoked to account for the observed hysteresis. Comparison of our present results with experimenP strongly supports the conjecture that the vanishing of hysteresis at temperatures somewhat below bulk T,and the shrinking of the loops that precedes this are a consequence of single-pore behaviour, i.e., of the approach to capillary criticality in individual pores. In passing, we comment on the implications of our results for the feasibility of determining pore size distributions by adsorption methods. There has been much discussion in the literature as to which branch of the hysteresis loop should be used for this purpose.2 Since pore radius is generally extracted from the relative pressure via the Kelvin equation or variants thereof, one should be seeking to identify the equilibrium transition. The Kelvin equation gives the condition for coexistence of liquid and gas phases in a capillary; it becomes an exact condition for such coexistence in the limit where pore radius or width becomes infinite.13J4 In the independent pore model, desorption is associated with capillary coexistence, whereas in the network model the adsorption branch is associated with coexistence, because this branch is determined by independent pore behavior, which is now assumed reversible. Of course it is very likely that in practice both hysteretic mechanisms operate so that neither branch provides an unambiguous identification of the equilibrium transition. Moreover, it is important to recall that the Kelvin equation provides only an asymptotic estimate of the condensation pressure in a single pore; it can fail drastically for small pores, where it severely overestithe value of PIPsatat the transition. Whilst pore characteristics obtained from adsorption measurements may provide a convenient means of cataloguing porous materials, it is unwise to attach too much significance to the physical parameters that are involved. Other, more direct, techniques such as image analysis are likely to be better indicators of the true pore sizes and structures. Before concluding, we should add some remarks about the precise nature of capillary criticality. As emphasized in section 2, our description of the phase transition in pores has been mean-field-like. Consequently, the theory predicts first-order transitions and criticality of these, for fluids in an infinitely long cylinder. However, since a cylindrical pore constitutes a pseudo-one-dimensional system, there can be no true phase transition for T > 0. Thus, one would expect the first-order transition predicted by mean-field theory to be rounded by finite size effects& so that whilst a very rapid rise in I’ may still persist, the equilibrium isotherms would not be vertical. The rounding, in chemical potential or pressure, should be proportional to exp[-(R,/~r)~], where R, is the radius of the pore and u is the molecular diameter.47 The pseudotransition is therefore very sharp for all but the smallest pores. There can be no true critical behavior, but one would still expect to find a pseudocritical isotherm, similar in shape to that of Figure 2b, the mean-field prediction. Metastable branches and sharp first-order transitions have been reported in several computer simulations of fluids in cylindrical presumably these reflect the limited run times of the simulations. Capillary condensation in a slitlike pore is somewhat different. The slit is a pseudo-two-dimensional system, finite in only one direction. True first-order transitions are therefore viable, and criticality will occur4 in a slit of ~~~
(45) Awschalom, D. D.; Warnock, J.; Shafer, M. W. Phys. Reu. Lett. 1986,57, 1607.
~
~
(46) Privman, V.; Fisher, M. E. J. Stat. Phys. 1983, 33, 385. (47) Evans, R. In Les Houches School on Liquids at Interfaces, 1988 (to be published).
Langmuir 1989,5, 723-727
width H at a temperature TcmPsatisfying the criteria
Since the correlation length can diverge only in the direction parallel to the walls, criticality should correspond to the two-dimensional Ising universality class. Thus, on the critical isotherm T = TcmP,we expect the adsorption to behave as
-
ir - rCi
IP -
with critical exponent 6 = 15. This should be contrasted with the mean-field result, 6 = 3; the critical isotherm has a much steeper slope than the mean-field prediction. The would vanish as jump in adsorption A r = I'lisuid Ar
- (TcmP-
T)B
with order parameter exponent /3 = l / ~ rather , than the mean-field value B = 1/2. Although it should be possible to observe such behavior in computer simulation, it is difficult to envisage a real adsorption experiment that would probe the details of a true capillary critical point. A distribution of pore sizes will act to smear out any true criticality. (48) Evans, R.; Marini Bettolo Marconi, U. J. Chem. Phys. 1987,86, 7138.
723
Nevertheless, we believe that our observations do have implications for experimental studies. We would encourage careful and systematic measurements of the temperature dependence of adsorption and, in particular, the disappearance of the hysteresis loops, for well-characterized mesoporous solids. Theory predicts that the phenomenon of (pseudo) capillary criticality should be quite general, and the crudest estimate, based on a slab approximation for the density profile,13J4gives
(Tc- T c m p ) / T5c (a/%) for cylinders. The sol-gel glasses of Awschalom et al.45 mentioned earlier would appear to be attractive candidates for further study. Since glasses with a series of different average radii can be prepared-RA varies from 22 to 187 A in ref 45-these should provide an opportunity to investigate capillary criticality as a function of pore size. Furthermore, if these materials do possess the very narrow pore size distributions that the authors of ref 45 claim, they should exhibit behavior that is closer to that predicted for individual pores, and one would then expect to see rather steep isotherms near the (pseudo) critical point. Acknowledgment. We have benefited from discussions with M. H. Chan, U. Marini Bettolo Marconi, and P. Tarazona. The stimulus for this work arose out of conversations with D. H. Everett, and we are grateful to him for making the data of ref 16 available to us and for instructive comments on our results. This research was supported by S.E.R.C. (U.K.).
Comparison of Self-Assembled Monolayers on Gold: Coadsorption of Thiols and Disulfides1 Colin D. Bain,2 Hans A. Biebuyck, and George M. Whitesides* Department of Chemistry, Harvard University, Cambridge, Massachusetts 02138 Received October 4, 1988. In Final Form: January 4, 1989
Ordered, organic monolayers were formed on gold slides by adsorption from ethanol of HS(CH2)1oCH20H, HS(CH2)loCH3,[S(CH2)loCH20H]2, [S(CH2)loCH3]2, and binary mixtures of these molecules in which one component was terminated by a hydrophobic methyl group and one by a hydrophilic alcohol group. The compositions of the monolayers were determined by X-ray photoelectron spectroscopy (XPS). Wettability was used as a probe of the chemical composition and structure of the surface of the monolayer. When monolayers were formed in solutions containing mixtures of a thiol and a disulfide, adsorption of the thiol was strongly preferred (-75:l). The advancing contact angles of water and hexadecane on monolayers formed from solutions containing mixtures of two thiols, a thiol and a disulfide, or two disulfides were dependent on the proportion of hydroxyl-terminated chains in the monolayer and largely independent of the nature of the precursor species. This observation suggests that both thiols and disulfides give rise to the same chemical species (probablya thiolate) on the surface. This model is supported by the observation by XPS of indistinguishable S(2p) signals from monolayers derived from thiols and disulfides. Introduction Both long-chain alkanethiols and dialkyl disulfides adsorb from solution onto gold and form ordered, oriented monolayer^.^" These self-assembled monolayers have (1) Supported in part by the Office of Naval Research and the Defense Advanced Research Projects Agency. The XPS spectrometer was purchased through a DARPA/URI and is housed in the Harvard University Materials Research Laboratory, an NSF-funded facility. (2) IBM Pre-Doctoral Fellow in Chemistry, 1985-86. (3) Nuzzo, R. G.; Allara, D. L. J. Am. Chem. SOC.1983,105,4481-4483. (4) Nuzzo, R. G.; Fusco, F. A.; Allara, D. L. J . Am. Chem. SOC. 1987, 109, 2358-2368.
been used in examining electrochemical processes,' in promoting adhesion,8 in studying ~ e t t a b i l i t y ,and ~ in (5) Porter, M. D.; Bright, T. B.; m a , D. L.; Chidsey, C. E. D. J. Am. Chem. SOC.1987,109, 3559-3568. (6) Bain, C. D.; Troughton, E. B.; Tao, Y.-T.;Evall, J.; Whitesides, G. M.; Nuzzo, R. G. J. Am. Chem. SOC.1989, 111, 321-335. (7) Li, T. T.-T.; Weaver, M. J. J. Am. Chem. SOC.1984, 106, 6107-6108. Finklea, H. 0.;Avery, S.; Lynch, M.; Furtsch, T. Langmuir 1987,3,409-413. Rubenstein, I.; Steinberg, S.;Tor, Y.; Shanzer, A,; Sagiv, J. Nature 1988, 332, 426-429. Sabatani, E.; Rubenstein, I.; Maoz, R.; Sagiv, J. J. Electroanal. Chem. 1987,219, 365-371. Taniguchi, I.; Toyosawa, K.;Yamaguchi, H.; Yasukouchi, K. J. Chem. Soc., Chem. Commun. 1982, 1032-1033.
0743-7463/~9/2405-0723$01.50/0 0 1989 American Chemical Society
