Langmuir 1987, 3,661-667
661
I
Figure 2. Scheme of the experiment for observing the local Fredericks transition: (1)NLC drops: (2) LB film; (3) a mica substrate. step of the LB staircase was 25 b, for polar and 50 b, for nonpolar films. A NLC (MBBA, 5CB) drop was placed on each step of the LB-staircase' (Figure 2). The orientational transition was observed through disappearance of the optical pattem characteristic of the planar orientation of the drop (Figure 3). The effective range for the van der Waals forces on the order of several hundred angstrom was found. It should be noted that r, is affeded by the spontaneous polarization of the LB films when the latter is allowed for by symmetry reasons (LB layers of X-and X-type*). In conclusion, our estimate for r. 1W500 b, coincides with the values obtained independently with more sophisticated experimental t e c h n i q u e ~ . ~ J ~
a
Figure 3. MBBA drop on a mica cleavage screened with an LB film. The screen thickness 6 = 100 (a), 150 A (h).
So, the application of liquid crystals and LB multilayers is an effective tool for investigations of the anisotropy of the solid surfaces. Taking into consideration the retardation effect could not change the order of r,.
-
(I)Blinov, L. M.;Davydow, N. N.;Sonin, A. A; Yudin, S. G. Kristolbpzphia 1984.29. 531. (8) Blinov, L. M.;Davydova, N.N.;h a r e v , V. V.; Yudin, S.G. Fiz. Tuerd. Tela (Leningrad) 1982,24, 2886.
b
Registry No. MBBA, 97402-82-9; stearic acid, 57-11-4 (9) Rabinovich, Ya I. Kofloidn. Zh. 1977,39,1094. Derjaguin, B. V.; Rabinovich, Ya. I.; Churaev, N. V. Nature (London) 1978, 272, 313. Rabnivich,Ya. L; Derjaguin, B. V.;Churaw, N.V.Ado. Colbid Interfoee Sci. 1982, 16,63. (10) Israelashvili, J. N.;Tabor,D. Roc. R. Soc. London A 1972,331, 19. Horn, R. G.; Jsraelashvili,J. N.;Perez, H . J. Phys. (Le8 Ulis, Fr.) 1981.42, 39.
-
Role of Capillary Condensation in Adsorption at High Relative Pressurest J. C. Melrose P.O.Box 12474,Dallas, Texas 75225 Received October 24,1986. In Final Form: January 28, 1987
A model combining pendular ring and Frenkel-Halsey-Hill adsorption theory is described. The model is useful in interpreting the effect of capillary condensation on adsorption data at high relative pressures. It is particularly suitable for cases in which the adsorbent can be considered as a relatively dense packing of spherical particles. The model is also useful in deriving the corrections which are required if mercury porosimetry data are to be compared with gas adsorption data for adsorbents of this type.
Introduction It has long been known that adsorption at high relative conpressures is complicated by the process of densation. It is not as well recognized, however, that the retentionof wettingliquid in media is complicated by the process of adsorption. In this paper a model is described in which both processes are several features of the model differ significantly from the model used in the pioneering work of Kiselev and co-workers.' The presentmodel is used to predict the amount of pillary-held liquid, relative to the total amountof adsorbate, memodel is also used to compare the high-pressure
wrtion of a mercurv w m i m e t w extrusion curve, in which i h r p t i o n is absent, &h the corresponding curve in which adsorption is included. As in the case of other models of this type, only isothermal states of thermodynamic equilibrium are considered. Also, the model is restricted to states that are completely reversible with respect to adsorption and desorption, as well as to capillary condensation. on the The model is in part based on previous theory of capillary condensation. In this work exact expressions were given for (1) the volume of liquid retained by capillary forces in the region of contact between two
'Presented at the "Kiselev Memorial Symposium",60th Colloid and SurfaceSeience Sympium, Atlanta, GA, June 15-18,1986; K. S. W.Sing and R. A. Pierotti, Chairmen.
ckm.(
(1) Aristov, B. G.; Karnaukbov, A. P.; Kiselev, A. V. Russ. J. Phya. ~ ~pami.) ~ i ~g62,36,115g. . ( 2 ) Melrose, J. C. AIChE J. 1966, 12, 986. (3) Melrase, J. C.; Wallick, G. C. J. Phys. Chem. 1967, 71, 3616.
0743-7463/87/2403-0661~1.~/0 0 1987 American Chemical Society
662 Langmuir, Vol. 3, No. 5, 1987
spherical particles and (2) the corresponding interfacial curvature. This previous model is now extended to include the effect of an adsorbed film on that part of the solid surface which is not in contact with capillary-held liquid. For this purpose a modified form of the Frenkel-HalseyHill adsorption theory4i5is used. As a critical feature of the model, both the chemical potential and the density of the adsorbate component are taken to be the same in the adsorbed film as in the capillary-held liquid. Effects due to vapor phase nonideality and to liquid-phase compressibility are then easily evaluated.2*6However, the much more complicated effects due to interface curvature and three-phase contact line properties are neglected. At the same time, it is recognized that the Gibbs dividing surfaces which define the volumes of adsorbate held as adsorbed film and as capillary condensate must be continuous in the region of the three-phase contact line. A simple approximation which takes this requirement into account is therefore introduced.
Melrose
(4)Halsey, G. D., Jr. J. Chem. Phys. 1948,16,931. Hill, T. L. J.Chem. Phys. 1949, 17, 590, 668. (5) Steele, W. A. J. Colloid Interface Sci. 1980, 75, 13. (6) Melrose, J. C. J. Colloid Interface Sci. 1972, 38, 312. (7) Hough, D. B.; White, L. R. Adu. Colloid Interface Sei. 1980, 14, 3. (8) Bukowiecki, S. T.; Straube, B.; Unger, K. K. In Principles and Applications of Pore Structural Characterization; Haynes, J. M., Rossi-Doria, P., Eds.; J. W. Arrowsmith Bristol, 1985.
aSL+‘LG
FILM
I I
Physical Properties of t h e Model With respect to the adsorbent (solid phase) the usual assumptions apply. That is, the adsorbent is taken to be inert and incompressible, with a rigid (nondeformable) structure. The density, p s , and Hamaker constant, HSs, correspond to those of vitreous (amorphous) silica: p s = 2.25 g ~ m - Hss ~ ; = 6.55 X J. The value of the Hamaker constant is that given by Hough and White,7 while the density agrees with one of the samples studied experimentally by Bukowiecki et a1.6 The structure of a model adsorbent can be defined in terms of particle size and shape, porosity, 4, and coordination number, N . A quasi-regular packing of uniformly sized spheres is therefore assumed, with sphere radii, R , of 50,100, and 200 nm. The corresponding specific surface areas, A,, using the solid density specified above, are 26.67, 13.33, and 6.67 m2 g-l. For each sphere size it is assumed that the porosity (relative void volume) is 0.400 and the coordination number (number of neighboring particles in contact with a given particle) is 8.0. These parameters are in approximate agreement with the sample described by Bukowiecki et a1.8 as displaying a “primitive hexagonal” structure. Since it is assumed that the packing is quasiregular, rather than random, the effect of liquid bridges between neighboring spheres which are not in actual contact may be neglected. In the case of the adsorbate component, the model assumes physical properties which correspond, as in previous work,2to those of liquid argon at its normal boiling point: T = 87.29 K. The density of the liquid phase at saturation vapor pressure, pLo, the surface tension, u, and the gasphase second virial coefficient, B, are thus respectively 3.492 X lo4 mol m-3, 12.54 mN m-l, and -2.354 X lo-* m3 mol-l. An estimated value of Hamaker constant HLLmay be based on the relationship with surface tension proposed by Hough and White.‘ This estimate is HLL= 3.0 X J. Further properties of the liquid adsorbate which are considered are the isothermal compressibility, (&)L, and the changes in density and compressibility which occur
‘SG
I
P,< \
PG0
x
Figure 1. Adsorbed film with pendular ring. when the vapor pressure is reduced. These properties are estimated, as previously,2from the following approximate relationship between compressibility and density:
(PT)L= [PLRT[C(PL/PL’)- DII-’
(1)
Here, R is the gas constant, 8.3144 J mol-l K-l, and the nondimensional constants C and D have the values C = 109.4 and D = 90.8. Use of this relationship involves an extrapolation of measured data into the region of negative liquid-phase pressure.2 As indicated by the work of Bukowieki et a1.,8it appears that actual adsorbents can be prepared which have properties conforming rather closely to the model just described. Also, experiments involving both gas adsorption and mercury porosimetry can be carried out on such adsorbents.8 The recent study of Smith and Stermerg on random packings of amorphous silica particles provides further evidence that mercury porosimetry can be used with such adsorbents. In order to apply the model described above to the case of mercury extrusion, values of the surface tension of mercury and the contact angle of mercury on vitreous silica must be assumed. The usual values of 480 mN mT1and 140’ are close to those used by Buckowieki et a1.8 and by Smith and Stermer.g These values are therefore used in the present work. Application of Pendular Ring Theory to Capillary Condensation Figure 1 is a schematic diagram showing two particles in contact, together with both an adsorbed film and liquid held as capillary condensate. The fractions of the total volume of adsorbate which are to be assigned to the film and to capillary condensate are best defined by a rigorous application of the concept of a Gibbs dividing surface. This problem arises even in a cylindrical capillary tube, as was recognized by Fosterlo and others.l’J2 In the present case the boundary between the two regions may (9) Smith, D. W.; Stermer, D. L. J. Colloid Interface Sci. 1986, 121, 160. (10) Foster, A. G. Discuss. Faraday Soc. 1948, 3, 41. (11) Barrett, E. P.; Joyner, L. G.; Halenda, P. P. J . Am. Chem. Soc. 1951, 73, 373. (12) Brewer, D. F.; Champeney, D. C. Proc. Phys. SOC.1962, 79, 855.
Langmuir, Vol. 3, No. 5, 1987 663
Capillary Condensation i n Adsorption
be defined by the sphere radius which is perpendicular to both the anticlastic (saddle-shaped) liquid/gas dividing surface and the synclastic (convex) adsorbed film/gas dividing surface. This radius forms an angle, P,with the line joining the centers of the two spheres. As in previous work,2 this angle is called the filling angle. Since a regular packing of spherical particles with a coordination number of 8 will include pore openings formed by three particles in triangular array,8 the maximum value of P which is permitted is 30°. In other words, by restricting the application of the model to values of P equal to or smaller than 30°, coalescence of pendular rings during a process of increasing relative vapor pressure (adsorption) will not occur. Thus, the model assumes reversibility with respect to adsorption and wetting liquid retention (desorption) processes, and hysteresis is absent. The minimum value of \k which is permitted is loo,since it is expected that the liquid phase will approach a state of mechanical instability for a filling angle which is still of appreciable magnitude. The profile of the dividing surface for the liquid/gas interface is not a circular arc, since the Gibbs "surface of tension" for this interface must be a surface of constant mean curvature. (We note here that gravitational and other external fields are assumed to be negligible.) Such a surface is called a nodoid or, sometimes, a pendular ring. It will also be recognized in the present model that the liquid/gas dividing surface is displaced a small distance from the surface of constant mean curvature which corresponds to the Gibbs surface of tension. This small distance is taken to be equal to the thickness of the adsorbed film. Thus, the surface of tension is assumed to be tangent to the two solid spheres of radius R. Similarly, the liquid/gas dividing surface is assumed to be continuous with and tangent to the adsorbed filmlgas dividing surface. It is probable that these assumptions involve an approximation in which the magnitude of the curvature of the surface of tension is underestimated. However, for the purposes of the present work, the error so introduced will be neglected. As previously noted, the effect of interface curvature on surface tension is also neglected. Exact equations for the mean curvature, J , the confined volume, V, and the interfacial area, A , of pendular ring (nodoid) surfaces were reported in previous work.2 Values of these geometrical parameters, expressed in nondimensional form, have been tabulated3 for various values of the filling angle, \k. The capillary pressure, Pc, is thus directly proportional to the nondimensional nodoid curvature, JRI2,
It will also be convenient to represent the confined volume as a fraction of the total pore space. This fraction, referred to as the confined volume saturation, S,,, is then proportional to the nondimensional nodoid volume, VIR3,
m)();
3N 1 - 4 SCV = iG(
(3)
The volume of liquid held as capillary condensate is only partly accounted for by S,,, the saturation which is equivalent to the nodoid volume. An additional volume is required because, as noted above, the liquidlgas dividing surface is displaced a small distance from the nodoid surface. It is assumed that this additional volume can be simply expressed by the product of the adsorbed film thickness and the area of the nodoid surface. Thus, it is convenient to express the nondimensional nodoid area,
;
c
I
w LO[
4 $ 0.7
1 3 0J 0.6
1
2
3
4
5
6
7
8
SATURATION OF PENDULAR RING FLUID 1 % )
Figure 2. Fractional surface area for combined wetting film and pendular ring surface.
AIR2, as a fraction, This expression is
FLG,
of the total solid surface area.
(4)
It is noted that the corresponding fraction for the film/gas (or solidlgas) interface is N F S G = 1 - -(1- COS \k) (5) 2 The correction to S, which accounts for the additional volume of capillary-held liquid can now be written as
Here, t is the thickness of the adsorbed film at the solid/gas interface. Similarly, the correction to the total saturation which is due to the adsorbed film itself is given by (7)
The sum of the two corrections is thus proportional to the combined fraction, F L G + F S G . The sum of F L G and F S G is plotted as a function of S,, in Figure 2. It is seen that this sum decreases rather rapidly from an initial value of unity, as the saturation corresponding to the confined volume increases to about 0.2%. Subsequently, as the pendular ring area increases, the decrease in FLG+ F S G is more gradual. The maximum value of S, shown in Figure 2 corresponds to a filling angle of somewhat less than 30'. It should be noted that the relationship shown in Figure 2 is independent of particle size and hence of the specific surface area of the adsorbent. As Wade13J4has shown, the type of curve shown in Figure 2 is experimentally accessible. Pendular ring theory also applies to high-pressure mercury porosimetry? In this case the wetting phase is mercury vapor, while the nonwetting phase is liquid mercury. The contact angle, 19,formed at the three-phase contact line is 40°,as measured through the wetting phase. Pendular ring parameters for this value of 0 have also been tab~lated.~ Since a cylindrical capillary tube is almost always assumed in interpreting mercury porosimetry data,l5-I7a (13) Wade, W. H. J . Phys. Chem. 1964, 68, 1029. (14) Wade, W. H. J . Phys. Chem. 1965,69, 322. (15) Joyner, L. G.; Barrett, E. P.; Skold, R.J . Am. Chem. SOC.1951, 73, 3155.
Melrose
664 Langmuir, Vol. 3, No. 5, 1987
Table I. Relative Vapor Pressures and Densities for Liquid Held as Capillary Condensate" sphere radius, R, nm filling angle, Qj 10 15 20 25 30 '01 1
' 70-5
1b-3
tb-4
50
200
100
a
11
01
D
01
11
0.538 0.773 0.875 0.926 0.955
0.967 0.987 0.993 0.996 0.998
0.734 0.880 0.936 0.962 0.977
0.984 0.993 0.996 0.998 0.999
0.857 0.938 0.967 0.981 0.988
0.992 0.997 0.998 0.999 0.999
= PGIPG';11 = PLIPL'
Figure 4 shows how the magnitude of the CPW correction varies as the measured mercury extrusion saturation increases. Using the tabulated3 confined volume values for 8 = Oo and 40°, the following approximation to the CPW correction can be derived,
1b-2
v JR3
Figure 3. Pendular ring theory: curvature vs. volume.
AS(CPW)
0.49[S(8 = 40°)J'.36
(8)
Here, saturations are expressed as percent of total pore volume. It is seen that the fractional correction, ASIS, can be as large as 0.84.
I
0.04 0
1
2
3
4
SATURATION FROM MERCURY INJECTION (%)
Figure 4. Converging pore wall correction for pendular ring
volume.
comparison between model results for the same values of the ratio Jlcos 8 is of interest. Such a comparsion would show no effect of varying 8 if the cylindrical tube configuration were applicable. In the case of the pendular ring geometry, however, this comparison does show a significant contact angle effect. This effect is shown in Figure 3. Thus, if mercury porosimetry data are to be compared with either adsorption or wetting liquid retention data, the use of the traditional scaling factor, u cos 8, will require a further correction. That is, the wetting phase saturation measured by mercury porosimetry will be smaller than that measured by adsorption or wetting liquid retention at the same value of the ratio, Pc/u cos 8. The reciprocal of this ratio, of course, is often taken as a measure of "effective" pore size. It should be emphasized that this difference in saturation is over and above the saturation difference arising from an adsorbed film of the wetting phase, as given by eq 6 and 7. In the case of mercury porosimetry, of course, no adsorbed film exists. As shown in Figure 3 the difference in saturation reflected in the curves for the two different values of 8 tends to decrease as the saturation itself decreases. This is because the pendular ring geometry approaches that of two parallel flat walls (slit geometry) as the filling angle, \k, decreases to zero. In this limit, as in the case of a cylindrical tube, the traditional factor, u cos 19,is the appropriate scaling factor. However, in the range of filling angles of interest here (loo < \k < 30°)this limit is not approached, and a significant contact angle effect remains. The discrepancy in predicted wetting phase volume will be referred to as the converging pore wall (CPW) correction. (16) Winslow, D. N. J. Colloid Interface Sci. 1978, 67, 42. (17) Conner, W. C.; Cevallos-Candau, J. F.; Weist, E. L. Langmuir 1986, 2, 151.
Chemical Potential of Adsorbate Equilibrium between adsorbate held as capillary condensate and adsorbate in the form of a f i i at the solid/gas interface requires equality of chemical potentials throughout the system, (9) PL = Mfilm = PG Denoting differences in chemical potential as Ap = po p, where po is the chemical potential in the reference state of saturation vapor pressure, eq 9 can be written as
= &klm = A& (10) As in previous work2t6the chemical potential differences for both the gas and liquid phases can be expressed in terms of the relative vapor pressure, a,and the relative liquid phase density, q. These expressions are &L
ApG
= -RT In a
+ S P G o ( l - a)
(11)
Here, eq 1has been used to represent the change in liquid phase compressibility with decreasing relative density. A further relationship which follows from eq 1 involves the change in liquid-phase pressure,
I t is easily seen that eq 10-13 are equivalent to two independent equations relating the relative vapor pressure to the relative liquid-phase density. For specified values of the capillary pressure, Pc, these two equations have been solved simultaneously by means of a simple iterative procedure. Since Pc is directly related to the filling angle, \k, through eq 2, it is convenient to express these results in tabular form for various values of \k. Such a table is presented as Table I. As will be noted, for relative vapor pressures of 0.7 and above, the decrease in liquid-phase density is never more than about 2 % . This small change in density justifies the simple extrapolation procedure represented by eq 1. As pointed out in previous work,2,6eq 10-13, together with eq 2, provide an exact expression relating the interfacial curvature, J, to the relative vapor pressure and relative liquid density. Neglecting terms involving the
Capillary Condensation in Adsorption
Langmuir, Vol. 3, No. 5, 1987 665 100
Table 11. Reduced Chemical Potential and Thickness for Adsorbed Films of Argon on Vitreous Si’lica PoIPoO PLIPLO ArIRT Pc, bar t , nm 0.75 0.80 0.85 0.90 0.95
0.9844 0.9880 0.9913 0.9944 0.9973
0.2796 0.2167 0.1577 0.1021 0.0497
70.1 54.4 39.6 25.7 12.5
0.478 0.519 0.577 0.666 0.846
50
20
second and higher powers of (1 - a), and retaining only the dominant term involving the vapor pressure, the following approximate expression is obtained:
CAPILLARY PRESSURE (BAR) 10
J = -(pLoRT/a)In CY (14) If the reciprocal of J is regarded as a measure of the “effective”pore size, eq 14 is of course the classical Kelvin relationship. Assuming, as is done here, that the surface tension, u, is independent of J , eq 14 underpredicts the effective pore size by 3.6-4.0% as the relative pressure varies from 0.99 to 0.70. It is likely that the curvature dependence of u will produce an even larger effect for relative pressures below about 0.95.
Application of Frenkel-Halsey-Hill Theory Since exact theories for adsorption at high relative pressures are not available,18the approximate relationship known as the Frenkel-Halsey-Hill (FHH) equation4f’has been used in the present study. Two forms of the equation are discussed in the literature. In one ~ o I T I I the ~ J change ~~~ in chemical potential, Apah, is taken to be inversely proportional to the third power of the film thickness, t. The proportionality factor is then the product of an energy of interaction and the molecular volume. An alternative form, discussed by Blake,22replaces Aprh by the product of the liquid-phase density and the chemical potential difference, 4% This product is then called the disjoining pressure.= Justification for this procedure is given by Abrikosov et al.,24 who refer to the term pLApFfimas the “van der Waals part” of the film chemical potential, referred to unit volume of liquid adsorbate. It should be noted that an implicit assumption in this procedure is that the density of the film is the same as that of the capillary-held liquid at the same relative vapor pressure. The proportionality factor in the relationship between pLApLfilmand t-3 now involves only an energy interaction parameter. This parameter is known as the Hamaker constant. Thus, where HsLdenotes the Hamaker constant for the interaction between the solid adsorbent and the adsorbed film, the FHH equation can be written as PLAptilm =
HsL/(~T~~)
(15)
The interaction energy, HSL,can in turn be expressed in terms of the interaction energies (Hamaker constants) for each individual phase. An approximate form of this expression is
HsL = (HssHLL)1’2- HLL (16) For the purposes of the model calculations reported here, the modified form of the FHH equation represented by (18)Rowley, L. A.; Nicholson, D.; Parsonage, N. G. Mol. Phys. 1976, 31,389. (19)Dollimore, D.; Heal, G. R. J. Colloid Interface Sci. 1970,33,508. (20) Nicholson, D. J. Chem. Soc., Faraday Trans. 1 1976, 72, 29. (21)Kanellopoulos, N. K.; Petrou, J. K.; Petropoulos, J. H. J. Colloid Interface Sci. 1983,96,90. (22)Blake, T. D. J. Chem. Soc., Faraday Trans. 1 1975, 71,192. (23)Deryagin, B. V. Colloid J. U.S.S.R.(Engl. Transl.) 1955,17,191. (24)Abrikosov, A. A.;Gorkov, L. P.; Dzyaloshinski, I. E. Methods of Quantum Field Theory in Statistical Physics; Dover Publications: New York, 1975.
R = 200 nm, 1
0.1
0.2
=
1
,
1
I
0.5
1.0
2.0
5.0
30‘
10.0
W E T T I N G FILM T H I C K N E S S Inm)
Figure 5. Relationship between capillary pressure and film
thickness.
eq 15 and 16 is believed to be a sufficiently realistic approximation. Values of the film thickness were calculated by using the values of the Hamaker constants, Hss and HLL, specified previously. Table 11presents some typical results. In connection with this approach to the prediction of film thickness values, the study carried out by Steele5 on the FHH equation is of interest. In this work the interaction energy for thick films of argon on graphite was reported to be 3.08 kT. Combining this result with a molecular diameter for argon of 0.3405 nm yields a value J for the Hamaker constant, HsL. The of 5.81 X corresponding value of Hss is then estimated to be 25.9 X J. This is not inconsistent with a result calculated from excitation energy and refractive index data for graphite. In this calculation the nonretarded form of the Lifshitz f o r m ~ l a was ~ ~ sused. ~ ~ This value of Hss is also in fair agreement with some of the previous estimates of Hss for gra~hite.~’ The work reported by Blakezzwas concerned with the adsorption of n-octane and n-decane on a-alumina. Films of more than 20 nm in thickness were found to be consistent with dispersion force calculations in which retardation is taken into account. Reference was also made to earlier workz8 with n-decane films of less than 5-nm thickness adsorbed on the oxidized surface of aluminum foil. In this case retardation effects are clearly absent, and the inverse third power relationship predicted by the FHH equation appears to hold almost to monolayer coverage. The value of the calculated Hamaker constant, HsL,for the n-decanelalumina system was 4.72 X J. For the system considered in the present work, argon adsorbed on vitreous silica, the calculated film thickness values are shown as a function of capillary pressure in Figure 5. Since now the Hamaker constant, HsL,is only 1.43 X J, the film thickness remains well within the range in which nonretarded dispersion force theory applies. For the three sphere radii considered here, the maximum thickness varies from 0.87 to 1.38 nm. These limits correspond to surface coverages ranging from 2.5 to 4 molecular layers. The minimum thickness, assuming that this is specified by a filling angle of loo, varies from about 1 layer to 1.7 layers. A more realistic minimum thickness is probably the thickness which corresponds to a relative pressure of about 0.8. This thickness is about 1.5 layers. (25)Gregory, J. Adu. Colloid Interface Sci. 1969,2,396. (26)Israelachvili, J. N.; Tabor, D. Prog. Surf. Membr. Sci. 1973,7,1. (27)Visser, J. Adu. Colloid Interface Sci. 1972,3,331.
666 Langmuir, Vol. 3, No. 5, 1987
Melrose 100
2
i 2 010
\
t
CURVE
w L
A ~ m, 2 j 1
50
26.67
100
- 1 c
0.80
A , nm
\
I
0.90
R E L A T I V E VAPOR P R E S S U R E ,
0.95
1.00
Po/P,"
Figure 6. Saturation fraction for pendular ring fluid after correction for wetting film thickness. The dependence of thickness on capillary pressure, as seen in Figure 5, is described approximately by an inverse one-third power relationship. This relationship would be exact if the compressibility and vapor-phase nonideality corrections to the Kelvin equation were in fact negligible. It may be noted that in this approximation the capillary pressure and the chemical potential change per unit volume (disjoining pressure), pLApfilm,are identical in magnitude.
Results and Discussion The film thickness values derived from the FHH equation can now be used to obtain the contributions to the total adsorbate saturation which arise when adsorption is present. These contributions were calculated from eq 6 and 7. Thus, the saturation due to capillary-held liquid is (17) SL = S, + (As),, while the saturation due to the adsorbed film is Sfilm = (As),, (18) The total saturation, S,, is then given by S, = SL+ Sfilm= S,, + (As),, + (AS)s, (19) Using these results it is also possible to evaluate the fraction of the total adsorbate saturation which is due to capillary condensation alone. This fraction is given by
f
= SL/(SL+ Sfid
(20)
Figure 6 shows how the fraction of total adsorbate attributed to capillary condensate increases with increasing relative pressure. As in previous plots the curves terminate at the relative pressures which correspond to a filling angle of 30'. At a given relative pressure, the fraction of capillary-held liquid increases with decreasing sphere size and increasing specific surface area. This is contrary to what might have been expected on intuitive grounds. If the same comparison is made for the limiting values of the relative pressure, the opposite trend is of course observed. In this case the fraction of capillary-held liquid increases from about 74% to 86% over the sphere size range from 50 to 200 nm. It may also be noted that an extrapolation of the curves shown in Figure 6 appears to indicate that the capillaryheld liquid accounts for 100% of the total saturation at relative pressures which are less than 1.0. This extrapolation, however, has no physical significance, since it is not permitted by the model assumptions. Similarly, there is a minimum relative pressure at which the model assumptions again break down. As noted previously, this minimum relative pressure, 0.8, corresponds to a film
1
\
13.33
I 0.85
\
7 0.1
I
I
I
0.5 1.0 2.0 5.0 10.0 20.0 0.2 TOTAL WETTING PHASE SATURATION, S, (%)
50.0
Figure 7. Effect of sphere size on relationship between capillary pressure and total wetting phase saturation (capillary pressure curve).
thickness of about 1.5 molecular layers. This thickness probably corresponds to the minimum thickness for which the FHH equation provides a reasonable model of adsorption. As pointed out above, the results of Blake et aLZ8 do appearz2to conform to the FHH equation down to a surface coverage of this magnitude. The relationship between capillary pressure and total saturation when adsorption is present is shown in Figure 7. It is seen that for each sphere size the magnitude of the slope of the log Pc vs. log S, plot is about the same. This slope is always much larger than that corresponding to the log Pc vs. log S,, plot (Figure 3). This is a consequence of the much different dependence of adsorbed film thickness on capillary pressure (Figure 5). Another characteristic feature of the model results shown in Figure 7 is the maximum saturation corresponding to a filling angle of 30'. For the sphere sizes shown, the maximum saturation varies from 10.6% to 14.2%. At higher saturations, as noted previously, pendular rings coalesce and pore filling occurs. The minimum saturations corresponding to a filling angle of 10' should also be noted. These range from 1.4% at a capillary pressure of about 38 bar to 3.3% at a capillary pressure of about 150 bar. Since these minimum values are probably too low in the case of the two smaller sphere radii, the values corresponding to a relative pressure of 0.8 and a capillary pressure of about 54 bar are also of interest. These saturations range from 1.2% to 5.1%. A final application of the present model which is of interest is the comparison between a mercury porosimetry (extrusion) curve and the corresponding curve obtained from adsorption data. Since for mercury extrusion the phase which is confined as a pendular ring is mercury vapor instead of a liquid, no adsorption contributions to the wetting phase saturation exist. Also, the converging pore wall (CPW) effect discussed above and shown in Figure 4 must be taken into account. Thus, two significant effects contribute to the difference between the two types of capillary pressure vs. saturation relationship. These effects are combined in the results shown in Figure 8 for a sphere radius of 100 nm. Here the mercury extrusion curve has been scaled so as to correspond to the adsorption curve. The scaling factor which was used is the traditional factor, Q cos 0. It is seen from Figure 8 that at equivalent capillary pressures, substantial differences in saturation must occur. A t a capillary pressure of 25 bar, which corresponds ap(28) Blake, T. D.;Cayias, J. L.; Wade, W. H.; Zerdecki, J. A. J. Colloid Interface Sci. 1971, 37, 678.
Langmuir, Vol. 3, No. 5, 1987 667
Capillary Condensation i n Adsorption
In this form of the FHH equation, P is determined from vs. volume of gas adsorbed. a log-log plot of log (PGo/Pc) 50 If gas adsorption data taken at high relative pressures are plotted in this way, it is usually found that P is less than the theoretical value, ranging from 2.1 to 2.8. These values CAPILARY 2 o of P appear to depend on the nature of the adsorbent. PRESSURE Thus, the lowest values of r are observed for low-energy (BAR) 10 solid surfaces such as Teflon and polyethylene, while the MODEL - PREDICTED highest values are found for high-energy surfaces such as 5 (AFTER CORRECTING ocose R = 100 n m rutile and anatase.29 This trend appears, however, to be F R O M 3 6 7 . 7 mNm-' T O 1 2 . 5 4 mNm-') complicated by a surface area effect. Carrott et aL30report 2 2 values of r for relatively low-surface-area samples of alu0.1 0.2 0.5 1.0 2.0 5.0 10.020.0 50.0 mina, rutile, and anatase. These values are similar to those T O T A L W E T T I N G PHASE SATURATION ( % I for various samples of silica with higher surface areas. The Hamaker constant, HSL,for the SiOz/argon interaction is Figure 8. Predicted capillary pressure curves showing correction required when converting mercury porosimetry data to adsorption only about one-third as large as that for the AlZO3/argon data. or TiOz/argon interaction. Thus, higher surface areas appear to compensate for lower values of HsL. proximately to a relative vapor pressure of 0.9, the difSome, but not all, of these empirical findings are ference in saturation is about 3%. This difference is nearly qualitatively predicted by a capillary condensation model. 6% at a relative vapor pressure of 0.97. It follows from First, the existence of capillary condensate requires an these differences that pore size distributions derived from empirical exponent which is less than the theoretical value. mercury intrusion/ extrusion and from vapor desorpSecond, the weaker the solidlfluid interaction, the greater tion/adsorption (aside from effects arising from hysteresis) the effect of capillary condensate will be. This accounts must necessarily differ to some extent. Such differences for the lower values of the exponent which are observed are in fact seen in the early results reported by Joyner et for low-energy surfaces. Third, however, the predicted These differences are also seen in the recent work of effect of increasing surface area does not seem to be conConner et aI.,I7which includes mercury porosimetry data firmed by the empirical evidence. In accordance with the for both intrusion and extrusion. The work reported by results shown in Figure 6 , this predicted effect is to enWinslowlGis more difficult to interpret, since the adsorphance the relative contribution of capillary condensation. tion data were adjusted to agree with mercury intrusion This leads to a lower value of r than would otherwise be data a t an effective pore radius of 25 nm. expected. As pointed out above, the work of Carrott et Models which accommodate both an adsorbed film and al.30suggests an opposite effect of increasing surface area. capillary condensate have been previously p r ~ p o s e d . l J ~ J ~ , ~ ~ In contrast to this previous work, the present model reConclusions stricts the adsorbed film to adsorbate held on that part For adsorbents which can be considered as an ensemble of the total solid surface not in contact with capillary-held of densely packed spherical particles, pendular ring theory liquid. This distinction is in one sense an arbitrary one, can be used to model the contribution of capillary consince it is assumed that both contributions to the total densation to the total volume of adsorbate retained in the volume of adsorbate have the same density, as well as the pore space. The contribution of adsorption per se on that same chemical potential. The distinction adopted in the part of the total surface not in contact with capillary-held present model does, however, emphasize the significance liquid can also be modeled, using the Frenkel-Halsey-Hill of the filling angle in specifying the location of a boundary adsorption theory. Reasonable interaction energies for use between the Gibbs dividing surfaces for the liquidlgas and in this theory can be obtained from estimated values for adsorbed film/gas interfaces. the appropriate Hamaker constants. Also introduced is One of the advantages of a model in which the relevant a simple approximation which takes into account the reGibbs dividing surfaces are clearly defined will become quirement that the Gibbs dividing surfaces for capillaryevident in future work. This work will address several held liquid and for an adsorbed film at the solid/gas inproblems which have not been confronted in previous terface must be continuous. It is found that for a given adsorption models. The first of these problems is that relative vapor pressure the relative contribution of capillary arising from the curvature dependence of surface tension. condensation to the total amount of adsorbate increases The second relates to the line tension and related effects with increasing specific surface area. at the three-phase line of contact. It is believed that the Mercury porosimetry data can also be modeled in the significance of both of these problems can be assessed by high-pressure region in which hysteresis is absent. A the appropriate extensions of the present model. converging pore wall (CPW) effect, which arises when So far as a direct application to commercially available porosimetry data are rescaled to correspond to adsorption adsorbents is concerned, the results provided by the data, is identified. This effect becomes quite significant present model are obviously of limited utility. This is for wetting phase saturations in the range below 5 pore because such adsorbents are characterized by a distribution volume percent. Consequently, both this effect and the of pore sizes and shapes. On the other hand, some of the presence of adsorbed films should be taken into account features exhibited by adsorption data on such systems are in interpreting differences in the pore size distributions predicted, at least qualitatively, by the model results. The derived from porosimetry data and from adsorption data. features in auestion involve the exDonent. r. in the empirical formbf the FHH e q u a t i ~ n . ~ ~ , ~ ~ 100
PREDICTED A D S O R P T I O N CURVE FOR SAME MODEL
,
,
(29) Zettlemoyer, A. C. J . Colloid Interface Sci. 1968, 28, 343.
(30) Carrott, P. J. M.; McLeod, A. I.; Sing, K. S. W. In Adsorption at the Gas-Solid and Liquid-Solid Interface;Rouquerol,J., Sing, K. S. W., Eds.; Elsevier: Amsterdam, 1982.
