Langmuir 1991, 7, 2833-2843
2833
Structure Analysis of Porous Solids from Preadsorbed Films Peter Pfeifer,*p+Gregory P. Johnston,*Ravindra Deshpande,t Douglas M. Smith,*Jand Alan J. Hurdg Department of Physics, University of Missouri, Columbia, Missouri 65211, UNM/NSF Center for Micro-Engineered Ceramics, University of New Mexico, Albuquerque, New Mexico 87131, and Division 1841, Sandia National Laboratories, Albuquerque, New Mexico 87185 Received January 11,1991. In Final Form: July 29, 1991 In this paper we explore the idea that the variation of the surface area of an adsorbed film on a porous solid, as a function of coverage (film volume), should yield structural information about the solid. We describe the formalizationsfor relating such film area measurements to surface roughness and micropore size distribution, showingthat under weak thermodynamicand structural assumptionsthe two are uniquely related to each other. The surface roughness of four Cab-0-Si1fumed silicas, two phase-separated glasses (CPGand Vycor), and two silicaxerogels are obtained from combined water-nitrogen adsorption experiments (nitrogen adsorption on preadsorbed water films) and compared to results from scattering and molecular tiling. Also, the surface roughness of various silica gels is assessed from the change in surface area when the surface is coated with surfactant or is derivatized. The micropore size distributions from film area measurements for Vycor and the xerogels are evaluated and compared to the resulta from micropore analysis, including Kelvin equation analysis, of nitrogen adsorption on the bare solids. The agreement between the film area results and the results from other methods, to the extent that they overlap, is excellent throughout. Specifically, we find that all of the Cab-0-Sils and the CPG glass are essentially smooth at length scales up to 1nm. By contrast, Vycor and the base-catalyzed xerogel show an extended fractal behavior, with surface fractal dimension of 2.3 f 0.1 and 2.5 f 0.1, respectively, over a length range of about 0.2-10 nm (combination of all available data). 1. Introduction The small-scale surface roughness associated with the internal structure of porous solids is important in many applications, catalytic and otherwise, involving molecular processes in confined geometries. Typically, this roughness, which we define to occur on length scales significantly less than some characteristic pore or particle size of the solid, is measured via small-angle X-ray or neutron scattering (SAXS, SANS) or molecular tiling (adsorption of monolayers) measurements. Molecular tiling has the advantage of being very selective and sensitive to smallscale surface roughness: The surface area measured by tiling with molecules of diameter d cannot see pores of diameter less than d and therefore, by a variation of d , selectively reveals the presence or absence of such pores. In this way, molecular tiling has been used to detect surface fractality in the range up to 1nm,1-3occasionally even up to 80 nm by using polymers as tiles.'I2 The method also works with nonspherical molecules as tiles if d values are calibrated by tiling of a smooth ~ u r f a c e .A~ disadvantage is the limited variability of d and that the experiments are time-consuming. Scattering, by contrast, probes the surface topography over length scales of about 1-50 nm in a single experimental run. But as a consequence, it is less sensitive to structural features at a particular length scale because the scattered intensity a t any wave vector magnitude q depends on the structure in a whole interval of lengths around 2 r / q . Moreover, it cannot distinguish open from closed pores, may leave considerable structural
* To whom correspondence should be addressed. + University of Missouri.
* University of New Mexico.
8 Sandia National Laboratories.
(1) Avnir, D.; Farin, D.; Pfeifer, P. Nature 1984,308, 261. (2) Farin, D.; Avnir, D. In The Fractal Approach to Heterogeneous Chemistry; Avnir, D., Ed.;Wiley: Chicheater, 1989; pp 271-293. (3) Pfeifer,P.;Avnir, D.; Farin, D. In Large Scale Molecular SystemsQuantum and Stochastic Aspects; Gans,W., Blumen, A,, Amann, A., Ede.; NATO AS1 Series B; Plenum Press: New York, 1991;pp 215-229. (4) Stermer, D. L.; Smith, D. M.; Hurd, A. J . J. Colloid Interface Sei. 1989,131, 592.
ambiguity even without closed pores,3 and is not readily available in most laboratories. There exists therefore a considerable incentive for developing new adsorption-based methods for surface roughness determination. The method explored in this paper has the combined advantages of molecular tiling (sensitivity, selectivity) and of small-angle scattering (probing of a wide range of length scales, no timeconsuming experiments). The basic idea is simple: We replace the tiling with molecules of diameter d by a preadsorbed film of water (or some other species) of thickness d and measure the surface area of the film by nitrogen adsorption. In this way, d can be varied simply by varying the coverage of the preadsorbed film and all one has to do is to eliminate the unknown film thickness in favor of a directly measurable, equivalent quantity. We will show in detail how this can be done and how structural information, generally in terms of micropore size distribution, is obtained from such data. We then apply the resulting computational procedure to extract structural information for a series of silicas which have been characterized previously by other methods. This will serve as an important test for the new method. The fact that the test solids are all silicas will make the study also significant from the viewpoint of comparability of the different samples: Since the samples are chemically identical, any differences in preadsorbed film behavior will necessarily be due to differences in surface geometry. 2. Background The use of "film surface area" as a structural probe of porous solids has been previously proposed for determining a number of porous solid properties. As mentioned, the principle is to measure the variation in surface area of the material coated with a series of preadsorbed films. The first step is to measure the surface area of the dry solid using conventional nitrogen adsorption (Brunauer-Emmett-Teller, BET) analysis. The solid is subsequently warmed, equilibrated with a vapor, and rapidly cooled to
0743-7463/91/2407-2833$02.50/0 0 1991 American Chemical Society
Pfeiferet al.
2834 Langmuir, Vol. 7, No. 11, 1991 film surfacearea A(z)
I
film volume V(z)
Figure 1. Reduced surfaceareas of preadsorbed films as a means to determine coordination number, pore size, and surface roughness (schematic). 77 K, and then the nitrogen surface area is measured again. From the variation of surface area as a function of vapor adsorbate loading, pore structure parameters may be extracted. Conceptually,the variation of surface area with increasing loading is illustrated in Figure 1 for (a) coordination number determination of powder compacts, (b) pore size distributions (micropores, mesopores), and (c) surface roughness. Karasz and co-workers5 first reported the use of gas adsorption (nitrogen and argon) to characterize the structure of water films on a porous solid surface. Wade,817 who was interested in measuring the coordination number of particle compacts (i.e., the average number of particles that a particle touches), adsorbed water on silica and alumina powder compacts and then measured nitrogen surface areas. Also measurements for water films controlled by desorption instead of adsorption were carried out? T o vary the coordination number, powder compacts were pelleted over a wide range of pressures. Coordination numbers calculated using Wade's model of the water adsorption and condensation process resulted in values that were 30-60% larger than expected. However, a more complete model of the water adsorption processsJOyielded coordination numbers in reasonable agreement with porosity-derived values. In another set of investigations, Ferguson and Wadell and Giacobbe et a1.12used preadsorbed water films to study the pore structure of Vycor glass. Ferguson and Wade varied the film thickness over a range of about 0.2-4 nm (our estimate) and measured film areas by nitrogen adsorption. They compared the dependence of film area on film volume with the theoretical expression for a system of cylindrical pores, eq 12,and found complete discordance. Giacobbe et al. deposited water films of estimated thickness 0.3 and 1 nm and used argon to measure the film areas. In addition, they measured the heats of adsorption and desorption of the argon-on-water films. Rather than inferring the pore structure from the dependence of film area on film volume, Giacobbe et al. performed a Kelvin analysis of the respective argon isotherms to obtain the pore size distributions of the water-coated samples. Comparison with the pore size distribution of the bare solid indicated the presence of an extensive micropore regime, in agreement with the calorimetric results and with the conclusions of Ferguson and Wade. We will confirm these findings in our Vycor study below. (5) K a r m , F. E.: Champion, W. M.; Halsey, G. D. J. Phys. Chem.
1911. - - - -, 60. .., 376. - .-.
(6) Wade, W. H.J. Phys. Chem. 1964,68, 1029. (7) Wade, W. H. J. Phys. Chem. 1966.69,322. (8) Venable, R.; Wade, W. H. J. Phys. Chem. 1965,69,1395. (9) Smith, D. M. J. Phys. Chem. 1986,90, 2723. (10) Smith, D. M.; Olague, N. E. J. Phys. Chem. 1987,91, 4066. (11)Ferguson, C. B.; Wade, W. H. J. Colloid Interface Sci. 1967, 24, A---. M (12) Giacobbe, F.; Aylmore, L.A. G.; Steele, W. A. J. Colloid Interface Sci. 1972, 38, 211. I
Figure 2. Model of a preadsorbed H20 film 'smoothing" the surface of a porous solid. The film volume V is measured gravimetrically, and the film area A is measured by Nz adsorption.
More recent qualitative determinations of surface roughness from film surface data have been made in ref 13. Preliminary results of the present, quantitative method have been presented in ref 14. 3. Theory
On a nonplanar surface, the surface area of an adsorbed film, A, is a function of the film thickness, z, as shown in Figure 2. We assume that every point of the film surface has the same shortest distance z to the solid (equidistant film surface). Studies of the film vapor equilibrium on a variety of surface geometries have shown that this equidistance assumption and the associated continuum description of the film are good approximations.1S21 Alternatively, equidistance may be viewed as a property that may not hold for every point of the film surface, but holds on average ("mean-field" theory's); in that case, z is the average shortest distance of a point of the film surface to the solid, and the working hypothesis is that the film area and volume are the same as that of an equidistant film with distance 2. A related question is whether equidistance is preserved in the frozen film, i.e., when the film is cooled to cryogenic temperatures. The answer is yes, based on the following experimental data. The film areas on our CPG sample (section 5 ) , which is smooth a t scales below 1 nm, do not depend on coverage; hence, the frozen water forms a uniform film rather than a dispersed set of microcrystallites. Similarly, the pore size distributions of water-coatedVycor show a uniform reduction of pore radii for most pores, corresponding to water films of uniform thickness.lZ The equidistance property implies that the film area A ( z ) is related to the film volume V(z) by A ( z ) = dV/dz (1) because an infinitesimal film thickness increase of dz leads to a film volume increase of A ( z ) dz. The problem we wish to solve then is how measurements of A as a function of V can be converted into information about the surface geometry of the solid. (13) Ross, S. B.;Smith, D. M.; Hurd, A. J.; Schaefer, D. W. Langmuir 1988, 4, 977. (14) Johnston, G. P.; Smith, D. M.; Hurd, A. J.; Pfeifer, P. In Characterization of Porous Solids II; Rodriguez-Reinoso,F., Rouquerol, J., Sing, K. S. W., Unger, K. K., Eds.; Elsevier: Amsterdam, 1991: pp 179-188. (15) Andelman, D.; Joanny, J. F.; Robbins, M. 0.Europhys. Lett. 1988, 7, 731. (16) Pfeifer, P.; Obert, M.; Cole, M. W. R o c . R. Soc. London, A 1989, 423, 169. (17) Pfeifer, P.; Wu,Y.J.; Cole, M. W.; Krim, J. Phys. Reu. Lett. 1989, 62, 1997. (18) Pfeifer, P.; Cole, M. W. New J. Chem. 1990, 14, 221. (19) Pfeifer, P.; Cole, M. W.; Krim, J. Phys. Reu. Lett. 1990,65,663. (20) Li, H.; Kardar, M. Phys. Reu. E 1990,42,6546. (21) Robbins, M. 0.;Andelman, D.; Joanny, J. F. Phys. Rev. A 1991, 43, 4344.
Langmuir, Vol. 7, No. 11, 1991 2836
Structure Analysis of Porous Solids Before solving the problem in generality, we first consider the special case where the surface of the solid is fractal, with fractal dimension D. In this case, V(z) is proportional to z3-O and therefore A ( z ) is proportional to 22-D by eq 1. By eliminating z from these two relations, one obtains A
size distribution, or surface roughness distribution, that is free of any model assumptions for the surface geometry. It is related to the film volume V(z) in good approximation by VPre(z) = V(Z) - z dV/dz = S,’A(z’) dz’- zA(z)
oc v(Z-DIl(3-D)
(2) Equation 2 describes how the film area on a fractal surface (D > 2) decreases with increasing film volume by progressivepore filling,and how the film area remains constant on a flat surface (D = 2). Thus, for a fractal surface, a log-log plot of A versus V should be linear and have a slope -m (m> 0),which yields the dimension of the surface from D = 2 + ( m / ( m+ 1)).This illustrates how structural information about the surface may be extracted from the dependence of A on V. It also illustrates the sensitivity of the analysis by mapping the theoretidal interval 2 I D < 3 into the wide experimental interval 0 Im < -. For example, if the slope m has uncertainty Am,the dimension D has an uncertainty of only (Am)/(m+ U2. Thus, film area measurement should give more accurate D values than other methods. Our experimental D values in this paper will not show this improvement because we assign more conservative error bars, substantially larger than the standard deviations, to them. We note that eq 2 does not cover the limiting case D = 3 because V(z) is no longer governed solely by D in that case. If D = 3 and the surface is nonuniformly spacefilling,%hevolume V(z)grows logarithmicallywith z; hence, A(z) a z-l and
A a e-V/”t t 2’) Both eqs 2 and 2’ are two-parameter expressions when used to fit experimental data, the parameters being the prefactor and the exponent. A detailed discussion of the prefactors, of the constant appearing in eq 2’, and of a three-parameter generalization of eq 2 when the film thickness approaches the lower limit of the fractal regime is presented in the Appendix. In experimental applications, one also wants to convert a given interval of A and V, in which eq 2 or 2’ is experimentally observed, into a corresponding interval of lengths to which D pertains. For 2 I D < 3 this is easy: From eq 1and V(z) a z3-0 it follows that A = (3 - D)V/z, which gives the thickness z of a film with area A and volume V as z = (3 - D)V/A (3) Thus, eq 3 can be used to convert the interval of A and V into a corresponding interval of lengths z. There exists no similarly simple expression for z when D = 3 (cf. Appendix); in that case z is most conveniently determined from eq 6. All of the above expressions for a fractal surface hold regardless of whether the surface is self-similar or self-affine if, for the latter, the necessary distinction between the local and global dimension is We now turn to the general situation, not necessarily fractal, and show how the dependence of A on Vuniquely determines the pore size distribution. We define V,,,(z), the cumulative volume of pores with radius iz, as the volume of the space outside the solid that is inaccessible to spheres of radius 2.26 This provides a concept of pore (22) Mandelbrot, B.B. The Fractal Geometry of Nature; Freeman: San Francisco, 1982; pp 357-364. (23) Mandelbrot, B. B.In Fractale in PhySiC8;Pietronero, L., Tosatti, E.,Eds.; North-Holland Amsterdam, 1986, pp 3-28. (24) Pfeifer,P.; Obert,M. In The Fractal Approach to Heterogeneous ChemiStry; Avnir, D.,Ed.; Wiley: Chichester, 1989; pp 11-43. (25) Pfeifer, P. Springer Ser. Surf. Sci. 1988, 10, 283.
(cumulative distribution) (4) dV,,,/dz
= -2 d2V/dz2
dA/dz (differential distribution) (5)
-Z
Indeed, the space inaccessible to spheres of radius z can be obtained by centering a sphere of radius z at every point on the film surface in Figure 2 and deleting all points in the film that are inside such a sphere. This amounts to subtracting a layer of area A(z) and thickness z , measured from the film surface, from the film. Thus, the volume to be subtracted is zA(z) if we neglect small effects due to curvature of the film surface. Together with eq 1, this gives eqs 4 and 5. The curvature effects are a t most of the order 22 dA/dz, which is small compared to zA(z). In practice they are in fact much smaller because the overestimates in subtracted volume at points where the film surface is convex and the underestimates a t points where the film surface is concave cancel each other statistically for a solid whose surface is neither convex nor concave. Equations 4 and 1may be regarded as two equations for the two unknown quantities V, and z, assuming the A vs V dependence as given. To solve them, we rewrite eq 1in the form dz = (1/A) dV and consider A as a function of V. Integration yields =
1 s,vA(I.”)dV’
This gives the function V(z) in terms of the experimentally measured function A(V) and leads to the following procedure for obtaining the pore size distribution: (a) One uses eq 6 to calculate for each film volume V the corresponding film thickness z(V). (b) By assigning to each film area A(V) the film thickness z(V), one obtains the film area as a function of film thickness, A b ) . (c) From A(Z), one obtains dV,r,/dz by eq 5. Alternatively, if we denote the measured functional dependence of film area on film volume by a(u) in order to distinguish it from the functions A(z) and V(z) of the film thickness, the computation of dVPm/dz from a ( ~is) summarized by the two equations
Note that we compute dV,,/dz from a( V(z)) rather than from V(z) because the second derivative needed when working with V(z) may be difficult to obtain accurately in practice due to numerical instabilities. Similarly, one would compute the cumulative distribution from Vpom(z) = V(z) - za(V(z)) rather than from V,,,(z) = V(z) - z dV/dz. Equations 7a and b are the general solution of the problem of converting measured film areas and volumes into structural information about the surface. By using small molecules such as H2O to generate the films, pore size distributions all the way down to pore radii z = 0.2
Pfeifer et al.
2036 Langmuir, Vol. 7, No. 11,1991
nm can be measured in this way. There is no restriction on the shape of pores, on whether pores of different sizes are independent (as in the models of traditional p r o simetry), or whether small pores are subpores of larger pores instead. Pores with entrance diameter less than the diameter of the film molecules will be seen as closed pores by the method, which has the advantage (over X-ray and neutron scattering) that only pores of practical interest are actually counted. Perhaps the most prominent feature of the method is that, unlike most other adsorption-based porosimetry methods, it does not presuppose any specific thermodynamic theory of the film solid and film vapor equilibrium. The only ingredient is the equidistance assumption, which is a very weak assumption as we have explained above. In fact, the equidistance hypothesis can be tested by independent experiments if desired. One test is that the experimental function a(u) should be decreasing with increasing u. If this is not satisfied, then either the film is not equidistant (example, droplet formation due to incomplete wetting) or the film grows on an essentially convex solid (example, equidistant film on a sphere). But for most solids of interest the microstructure is far from convex, so that the latter scenario for a nondecrease of a(u) can be ruled out in most cases. A test based on thermodynamics is as follows. Let N(p) be the adsorption isotherm for the preadsorbed film, Le., the number of adsorbed molecules as a function of vapor pressure p, and suppose that the film is adsorbed by van der Waals forces (physisorption). Then, if the film has an equidistant surface a t distance z ( p ) , the relations
hold.’* Here V ( z )is as before, p is the number density of the film, (Y is the coefficient for the vapor-solid van der Waals interaction, kT is the thermal energy, p o is the saturation pressure, and u is the film vapor surface tension. Equations 8 and 9 may be used to compute the functions z ( p ) and V(z)from the experimental isotherm (the details need not concern us here), which then can be compared with the film thicknesses and the function V(z)obtained from eqs 7. A third test involves small-angle scattering from films as a function of coverage.18 Finally, perhaps the most stringent and interesting test is to check whether the analysis in terms of eqs 2, 2’, and 7 yields the same results as structure investigations by conventional methods. It will be seen below that this is indeed the case. We illustrate the conversion of film area data into pore size distribution, eq 7, for two elementary examples. The first one is a flat surface so that a(u) equals a constant C, the surface area of the solid, for all u. Equations 7 in this case correctly give z = V ( z ) / Cand Vpore(z)= 0 for all z. The second example is a fractal surface, for which we have a(u) = C U ( ~ - ~ ) from / ( ~ - eq ~ ) 2 where C is again a constant. Here eqs 7 yield
in agreement with the earlier power law V ( z )a 23-0 and the result VP&) a z3-0 from eq 4. Thus, eqs 7 recover everything that we have derived for the situation where
a0
I
0
I
0
$4 00
.
I
, 01
06
04
08
IO
P/P,
Figure 3. Surface areas, measured by NZadsorption, of water and heptane films on two grades of Cab-0-Sil. The relative pressure p / p o is the water or heptane vapor pressure at which the films were preadsorbed.
the surface is fractal. The result, eq 11,also agrees with a previously obtained expression25 for the pore size distribution of fractals more general than surfaces.
4. Experimental Section Nitrogen adsorption was measured at 77 K using an automated volumetric adsorption analyzer over relative pressures between 0.05 and 0.20 (fivepoints). Larger pressures sometimes resulted in nonlinear BET plots and BET ‘C” values sensitive to the pressure values employed. Before analysis, samples were outgassed at 383 K under vacuum for 3 h. Water adsorption was measured at 293 K, with the uptake being determined gravimetrically using a Cahn microbalance. For film surface areas, after equilibration with water vapor, the samples were rapidly cooledfrom293to 77 K before nitrogensurfacearea measurement. Similarexperimenta usingn-heptane wereperformedpreviously,’* but the actual heptane uptake was not measured. A series of samples were studied includingdense powders and porous solids. For the surface roughness studies, fumed silica (CabO-Si1gradee L90,MS7,HS6, and EH5) was used which had been previouslycharacterized by means of small-anglescattering and moleculartiling.lg Porous solidsstudied were a Vycor phaseseparated Blase from Corning, a controlled pore glass (CPG-350) from Electronucleonics, and two silica xerogels. The xerogels were prepared using either a two-step acid-catalyzed tetraethylorthosilicate (TEOS) synthesisscheme (designated A2),which results in microporosity, or a two-step base-catalyzed TEOS system (designatedB2),which has a broader pore sizedistribution than A2.% A series of B2 silicagels of varying surface areas were prepared by aging wet gels (before drying) in different water/surfactant solutions. Wet gels were cast as &“-diameter cylinders, aged in the reaction fluid for 24 h, washed in five ethanol baths for 24 h each (to remove unreacted monomer), and placed in water/ Triton X-100surfactant solutions containing between 0 and 20 volume % surfactant. The samples were slowly dried at 303 K and outgassed under vacuum at 298 K before film surface area determination by nitrogen adsorption. The weight fraction of Triton X-100on the dried gels was measured by thermogravimetric analysis in an oxygen atmosphere. The samples were then heated to 693 K in air for 24 h to burn out the surfactant film whereafter the surface area was remeasured. 5. Results and Discussion A. Fhmed Silicas. Previ0usly,1~we have reported surface area measurementaof preadsorbed n-heptane films on a series of Cab-0-Si1powders. Figure 3 includes these heptane results for the smoothest and roughest grades (L90and EH5, roughness determined by scattering) as well as the results for preadsorbed water films. Two (26) Brinker, C. J.; Keefer, K. D.; Schaefer, D. W.; Ashley, C. S. J. Non-Cryst. Solids 1982,4%,41.
Langmuir, Vol. 7, No. 11, 1991 2837
Structure Analysis of Porous Solids
-
Heptane 2 0 -
011
\
N
HZO 0
0
+
LPO 1157 H55
A
EH5
A
I 8
-IO
-08
-06
-04
-02
0 0
log ( P / P, 1
Figure 4. Double logarithmic plot (base 10)of film surface area versus relative vapor pressure for preadsorbed water and heptane on the Cab-0-Sils.
interesting points may be observed in Figure 3. For the smooth L90 sample, the area is essentially independent of the film coverage for both water and heptane. This result is consistent with our earlier finding@ from SAXS and molecular tiling that the L90 sample is smooth. In contrast, the film areas for the EH5 sample decrease significantly with increasing coverage, in agreement with the sample's roughness. There are several reasons for the difference in the magnitude of decrease for heptane and water at a given pressure p / p o on EH5. First, the coverage as a function of p / p o depends on the substrate-adsorbate interaction and therefore is not necessarily the same for heptane and water, neither in terms of film volume nor in number of adsorbed molecules. Second, if one assumes that the same number of molecules are adsorbed a t the same relative pressure, then the heptane film is thicker (due to the larger size of the molecule) and should have a lower surface area than the water film, which is what the data show. Third, since nitrogen is smaller than heptane but larger than water, a heptane monolayer may present to nitrogen a significantly reduced surface roughness compared to the bare solid, whereas a water monolayer may present a roughness not much different from the bare solid. This is consistent with the observation that the water film area drops significantly from the dry surface area only after the water film exceeds a monolayer @ / p o> 0.2) and that at high coverage water and heptane approach the same surface area. We have preadsorbed heptane and water films only in a fairly limited range of p / p o in order to stay in the adsorption regime controlled by short-range and van der Waals forces, as opposed to the capillary forces at higher p / p o . The rationale was that the film surface might be no longer equidistant as one enters the capillary condensation regime, although the theoretical evidence18 is to the contrary. Experimental evidence that the film remains equidistant in the capillary condensation regime will be obtained below (see the discussion of eqs 12 and 13 for Vycor). In this limited range, one may assume that the volumetric uptake of heptane and water is proportional to p / p o . The assumption is reasonable because of the low values, -10, of the BET "C" constant for heptane and water. With this assumption, one may test for possible surface fractality, eq 2, by plotting the data in Figure 3 on a log-log scale. The result is shown in Figure 4. It also includes the heptane results for the two intermediate grade Cab-0-Sils (MS7, HS5) studied earlier.13 All six data sets show a power-law behavior. The nominal D values from the slopes for the heptane curves are 2.05,2.08,2.11,and
2.10 for L90, MS7, HS5, and EH5, respectively. The D values from the water curves are 1.98 for L90 and 2.10 for EH5. We estimate an error bar of fO.10 for all D values. The nominal film thickness, calculated from eq 3 and V a p / p o ,varies by a factor of minimally 2 (heptane on L90) and maximally 8 (water on EH5). This spans a length range of up to about 1 nm. Thus, from the viewpoint of scaling and within the limits of our experimental precision, the Cab-0-Sils are all smooth (D = 2) at scales up to 1nm. In order to have a well-defined scaling law, the rule of thumb is that the power law should extend a t least over a decade of lengths. But the question of how large the interval of observation should be depends also on what the power law is used for. Here we have used it to show that what may look like significant surface roughness in Figure 3 is, in fact, insignificant from the viewpoint of fractal behavior. For this, the limited length range probed by these experiments is perfectly acceptable. In other cases, one may wish to compare one D value, over some limited range, with the D value from some other experiment, over the same range, to see whether the two agree-as a step toward establishing or disproving the presence of some scaling regime. One may wish to see whether two D values, from different experiments and over distinct ranges, agree and might indicate the presence of a more extended scaling regime. Or one may simply wish to use D values, nominal and crossover-induced as they may be, to exhibit qualitative trends or pinpoint differences between different materials. In the following, both for the Cab-0-Sils and the porous silicas, we will compare D values obtained from eq 2 with D values from other experiments. Thus we investigate scaling properties of our systems in the step-by-step approach. Except for L90, the D values from film areas are mostly lower than the ones obtained from SAXS and molecular tiling.13 The values from SAXS are 2.0, 2.1, 2.2, and 2.5 for L90, MS7, HS5, and EH5, respectively, with an uncertainty of f0.10. They refer to length scales above 3 nm ( 2 divided ~ by the largest wave vector magnitude in the scaling regime). They provide an example where the D values in neighboring length ranges do agree for L90 and MS7, but are significantly different for HS5 and EH5. The values from molecular tiling are 2.1,2.3, 2.3, and 2.5 (all fO.10) for L90, MS7, HS5, and EH5, respectively. They refer to scales up to 1 nm as in the film area measurements. We attribute the difference between the film area analysis and the tiling analysis to uncertainties in the latter. In fact, the surface area of the solid measured by tiling with heptane (the number of heptane molecules at monolayer coverage times the cross-sectional area of heptane) is 10-2076 lower than the film area of heptane at maximal loading (amounting to about a monolayer, too). Conceptually the two areas should agree because both refer to the surface traced out by a heptane monolayer. So the difference in area suggests that the cross-sectional area of heptane may have to be assigned a value larger than the one assumed in ref 13, which would shift the D values from tiling closer to 2. The same conclusion has been arrived a t from a determination of cross-sectional areas by tiling of smooth reference surface^.^ A refined picture of the surface structure of the Cab0-Sils then is as follows. The L90 sample is smooth throughout. The MS7, HS5, and EH5 samples are rough at scales below 1 nm, but not rough enough to lead to a dimension D significantly larger than 2. At scales above 3 nm, the MS7 sample continues to have a D value of essentially 2 while the HS5 and EH5 samples show a D value of 2.2 f 0.1 and 2.5 f 0.1, respectively.
Pfeifer et al.
2838 Langmuir, Vol. 7, No. 11, 1991 c V
l
CPG
8 gl..l
I
0 0 0
a
g 0.I
1
0
t
o
0
0
0
o
* 10
0 2
0.0
0.0 00
02
06
04
08
0 4
PIP,
Figure 6. Water adsorption isothermsat 290 K for various porous silica samples.
0 6
0 8
I O
PIP, (HzO)
I O
Figure 6. Surface area, measured by Nz adsorption, of water films on the porous silicas as a function of the relative water vapor pressure at which the films were preadsorbed. I
Table I. Pore Structure Parameters for the Porous Silica Samoles
CPG Vycor B2 A2
surface area,
pore volume,
mean pore
m2/g
cm3/g
radius, nm
60 159 918 763
1.0 0.22 1.41 0.41
33 2.8 3.1 1.1
B. Porous Silicas. In addition to these dense silica powders, surface areas of preadsorbed water films were measured on four porous silica samples. The water adsorption isotherms a t 290 K for all four samples are presented in Figure 5. Differences in the isotherms reflect the basic differences in surface area and pore structure of the samples. The surface area of the bare solid from nitrogen adsorption, the pore volume from nitrogen condensation at relative pressure -0.995, and the mean pore radius (twice the pore volume to surface area) for the samples are presented in Table I. The CPG and Vycor samples exhibit fairly narrow pore size distributions in the mesopore regime, and the B2 sample has a wide distribution extending from macropores all the way to micropores, whereas the A2 sample has only micropores. On the basis of the mean pore radius, it appears that water uptake in Figure 5 is only a result of adsorption in the van der Waals regime (no capillary condensation) for relative pressures less than 0.9for CPG, 0.5for Vycor, and 0.3 for B2 and A2. The surface areas of the preadsorbed water films on these porous silicas are presented in Figure 6. The four samples exhibit quite different behaviors. The microporous A2 sample has a high initial surface area that drops dramatically in the vicinity of relative pressure of 0.3,due to pore blockage. In contrast, the CPG sample has essentially a constant film area until a relative pressure greater than 0.9 is reached. The Vycor and B2 samples exhibit intermediate behavior with a steady decline in area as the coverage increases. We divide the analysis into three parts: a fractal analysis, one in terms of classical pore models, and finally the one in terms of the model-free pore size distribution, eqs 7. a. Fractal Analysis. The data of Figures 5 and 6 may be combined to yield film area versus film volume, assuming that the density of the film is that of bulk water at 77 K. The results are shown in Figure 7 in a log-log representation to test whether a fractal power law, eq 2, is obeyed. No proportionality was assumed here between the water uptake and relative pressure. In order to illustrate the differences between the four samples, the
3 L CDO
-
I
-
0 Vycor--* e a2 0 A2
.
I
.a I
I
Figure 7. Surface area of water films on the porous silicas, divided by the surface area of the bare solid, as a function of the film volume. Table 11. Fractal Dimensions D and Associated Film Thickness Ranges, m. and ~ p ufor , the Power Laws in Figures 7 and 9. D zmh,nm z-,nm film CPG Vycor B2 B2 a
2.06 f 0.10 2.31 f 0.10 2.51 fO.10 2.48f 0.10
0.33 0.17 0.18 1.15
1.07 1.05 0.86 10.6
water (Figure 7) water (Figure 7) water (Figure 7) surfactant (Figure 9)
Error bars for D are estimated uncertainties.
film areas have been normalized by the surface area of the bare solid. The CPG, Vycor, and B2 samples exhibit regions of power law behavior from which values of D were extracted. The A2 sample obeys neither a power law nor an exponential law, eq 2'. The D values and the length scale ranges are given in Table 11. The limits Zmjn and were calculated from z = V / Aat the left end of the power law regime and from eq 3 at the right end, respectively. The reason why we do not use eq 3 to estimate Z m h is because for a monolayer or less the film thickness is independent of D (cf. Appendix). It follows from Table I1 that at scales below 1nm the CPG surface is essentially smooth, as expected from the nearly constant film area in Figure 6, but that Vycor and B2 are fractally rough with D values significantly larger than 2. These results may be compared with previous fractal analyses of the CPG family of controlled pore glasses, Vycor, and B2-type gels. From molecular tiling experiments for CPG, Christensen and T o p s ~ found e ~ ~ D = 2.09 f 0.08, and Stermer et al.4 obtained D = 2.21 f 0.10, both
Langmuir, Vol. 7, No. 11, 1991 2839
Structure Analysis of Porous Solids
-a f
. .
-1 0 OOOI
8
m
.
. ....II 001
%
.. .= ..
-..-I
II
I
1:
0
.
.... IO0
Concentration of surfactant in water
Figure8. Variation in surface area of the B2 xerogelas a function of the composition of the solution from which the gel was dried.
over a similar length range as in Table 11. Thus, the D value in Table I1 agrees with the values from tiling within experimental error bars. SAXS studies of CPG by Hohr et a1.% gave D = 2.20 f 0.05 over a length range of 2-30 nm. While this may not seem to be a significant increase in D compared to the value in Table 11,the rapid drop in film area in Figure 7 at film thicknesses above 1nm does signal that the surface is no longer smooth a t those length scales. We conclude from this that the CPG surface is smooth only up to about 1 nm and shows significant roughness at larger scales. For Vycor, surface fractality has previously been investigated only by SAXS and SANS. The result is D = 2.40 f 0.10 over a length range of 2-10 11111.28 The D value in Table I1 matches well with this result and suggests that the Vycor surface scales with one and the same fractal dimension, D = 2.3-2.4, over the entire range from 0.2 to 10 nm. Similarly, the D value of the B2 sample in Table I1 compares favorably with the fractal dimensions of about 2.4 obtained by Brinkermfrom SAXS measurements of B2 aerogels, i.e., of critical-point dried samples of the same initial wet gel. Thus, given that our analysis relies on physical principles very different from those of the other methods and that the length range probed in our analysis overlaps just marginally with the range of the other methods in most cases, we obtain fractal dimensions remarkably consistent with those from other approaches. Table I1 includes a second D value for the B2 sample which was obtained as follows. In a study concerning the effect of pore fluid surface tension on silica gel pore structure during drying, we dried a series of B2 gels from water/surfactant solutions under a wide range of surfactant concentrations. Expecting the surface area of the resulting dried gels to remain constant or increase with increasingsurfactant concentration (lower surface tension), we were surprised to observe that the surface area decreased with increasing surfactant concentration. Subsequent sample analysis indicated that our outgassing conditions did not remove the adsorbed surfactant. The surfactant was apparently blocking access to pores on the surface and therefore had an effect analogous to that of preadsorbed water films. The reduction in surface area as a function of surfactant concentration is shown in Figure 8. Heating of the samples to 673 K in air resulted in removal of the surfactant and eliminated all differences in surface area among the different samples (remeasurement of the surface area after the heat treatment). The mass of surfactant adsorbed on each sample was measured (27) Christensen, S.V.; Tops@, H. Quoted in ref 2. (28) HBhr, A.; Neumann, H. B.; Schmidt, P. W.; Pfeifer, P.; Avnir, D. Phys. Rev. B. 1988,38, 1462. (29) Brinker, C. J. Private communication, 1990.
8
7
010 0 I
01
10
Surfactant film volume ( c m 3 / g )
Figure 9. Surface area of surfactant films on the B2 xerogel, divided by the surfacearea of the bare solid,as a function of the film volume. The leftmostfour data points amount to a coverage of much less than a monolayer and can no longer be interpreted in terms of a continuous film (we estimate the thickness of a monolayer of the surfactant to be about 2 nm).
by thermogravimetric analysis and was converted into surfactant volume, taking the density of bulk surfactant at 293 K as a substitute for the unknown density at 77 K (the overestimate of film volume and thickness introduced by the presumably underestimated density amounts only to a multiplicative constant and does not affect the powerlaw behavior). The so-obtained dependence of surface area on surfactant volume is shown in Figure 9 and obeys a power law at coverage from about half a layer upward. The resulting D value and length range are given in the last entry of Table 11. The limits Zmin and Zmm were calculated as in the case of water films. The agreement with the D value from water films is excellent, although perhaps somewhat fortuitous in view of the more pronounced scatter in the data in Figure 9 and some uncertainty in the film structure. I t suggests that the fractal dimension of 2.5 for the B2 surface may extend over the remarkably wide range from 0.2 to about 10 nm. A related test of using film surface areas as a surface roughness probe has been suggested by Avnir and For porous solids whose surface has been derivatized with groups of different molecular size, the change in surface area as a function of the volume occupied by the surface groups should yield the same structural information as preadsorbed films. Okubo and Inoue31 derivatized a porous silica gel with a series of alkyl groups and measured both the surface area and pore volume with nitrogen. From the difference in nitrogen pore volumes between the underivatized and derivatized samples, the volume occupied by surface groups can be estimated. The log-log plot of surface area versus this volume yields a D value of 2.15. There is a considerable uncertainty in this value, however, because of the limited number of data points (three), the limited length range probed, and the possibility that the immobilized alkyl groups may form a diffuse interface with respect to nitrogen adsorption.32 Nakamura et alSw have derivatized a silica gel with various amounts of trimethylchlorosilane. Their data, when analyzed in terms of eq 2, gives a D value of 2.1. b. Analysis in Terms of Classical Pore Geometries. Next we turn to the question whether the film area versus (30) Avnir, D.; Farin, D. Private communication, 1990. (31) Okubo, T.; Inoue, H. Am. Imt. Chem. Eng. J. 1988,34,1031. MU, (32) Schmidt, P. W.; Avnir, D.; Levy, D.; HBhr, A.; Steiner, M.; A. J. Chem. Phys. 1991,94,1474. (33) Nakamura, Y.;Shinoda, M.;Danjo, K.; Iida, K.; Otauka, A. Adu. Powder Technol. 1990, 1 , 39.
Pfeifer et al.
2840 Langmuir, Vol. 7, No. 11, 1991 I
8
a
-05
c L
B
2
vycor a2 A2
-I 5
-I 5
-1 0
-0 5
00
loa( I-V/Vp,,,)
Figure 10. Test whether the data in Figure 7 obeys eqs 12. Logarithmsare to the base 10. The values used for the total pore volume are those in Table I. The straight lines show eq 12a for n = 1 1 2 and n = 2/a. film volume data in Figure 7 may, entirely or in part, also be explained by nonfractal surface models. For this, we consider the traditional pore models in which departures from a planar surface are modeled in terms of independent slits, cylinders (open-ended), or spheres. They predict that A varies with V as where
n = 0 for slit pores n = 1/2 for cylindrical pores
(12b)
n = 2/3 for spherical pores subject to V < V,,. Here A, denotes the surface area of the bare solid and V,,, denotes the total pore volume. The condition for eq 12 is that the pores have all the same radius R and do not intersect or branch. The result, eqs 12, then follows from calculating V ( z )and A(%)for the indicated geometries and eliminating z as in eq 2. The radius may be recovered from eqs 12 by [(I - n)A,l (13) For the data to obey eq 12, a log-log plot of A / A , versus 1- V/ V,, should yield a straight line with slope 0, l / 2 , or 2/3. The CPG data satisfy this trivially, with n = 0, over the entire range in Figure 6 where the film area is constant. This is equivalent to the result D = 2 from the fractal analysis. It expresses the fact that the surface of a slit pore, a t lengths smaller than the slit radius, is equivalent to a planar surface. For the other three samples, however, Figure 10shows that none of the data sets satisfies any of eqs 12within acceptable accuracy. All three samples would require an exponent n > 1for eq 12a to fit with the data in the region where most data points lie. It follows that there is no model in terms of simple pore geometries that is consistent with the data. The discrepancybetween the models, eqs 12, and the data says that there cannot be just a single pore size in these samples and that, therefore, there must be a whole range of different pore sizes, with small pores being subpores of larger pores. But this is precisely what the fractal analysis in Table I1 state if we use the D values to calculate the pore size distribution from eq 11. Thus, the fractal analysis inTable I1is highly robust with respect to alternative interpretations, and is even suggested by the alternative analyses themselves.
R = V,,,,/
A similar disagreement between the experimental data and the expressions 12 for open-ended cylindrical pores was observed by Ferguson and Wade11 in their study of Vycor. Their conclusion was that, contrary to what the sharp peak at pore radius z = 3 nm of the Kelvin pore size distribution suggests (the same peak, somewhat shifted, can be seen in our Figure 13), there must be a significant fraction of closed-endedpores with radius less than 2 nm, which agrees very well with the fractal model. A low fractal dimension such as D = 2.3 for Vycor implies that the pores in the fractal regime are mostly closed-ended. Furthermore, Ferguson and Wade noted that the A vs V data at large V indicate the additional presence of a significant fraction of pores with radius larger then 3 nm, roughly at z = 5 nm. We now show that also this second conclusion of Ferguson and Wade is consistent with the current, fractal interpretation of the structure of Vycor. To this end we note that if a pore volume smaller than in Table I were used, the data points for Vycor in Figure 10 would be shifted to the left and the two high-coverage points would fall near the line of the spherical pore model. This amounts to partitioning the pore volume into a part that comes from the fractal structure, not relevant for eqs 12, and a part that nominally comes from spherical pores. It suggests that the fractal surface structure of Vycor is superimposedby more or less spherical pores, with a radius of about 4 nm estimated from eq 13 with n = 2/3. These pores would correspond to the 5-nm pores of Ferguson and Wade. In view of the somewhat different characteristics of our sample (Table I) and the one of Ferguson and Wade, we believe that the two estimates agree reasonably well. Such coexistence of two structures, with no interruption of the fractal behavior a t the length scale where the 4-nm pores occur, is in fact what the SAXS/SANS data yielded for Vycor.28 There, Vycor was modeled as a system of interconnected units (particles), the particles having an average diameter of 35 nm and being the carriers of the fractal surface. From this it was concluded that the larger pores must be of interstitial, approximatelyspherical type with diameter -8 nm. We find it pleasing that the scattering data, which infer these interstices from the structure at large scales, and the preadsorptiondata, which infer them from the opposite end, lead to the same conclusion. Since the two leftmost data points for Vycor in Figure 10 correspond to films in the capillary condensation regime, the agreement between the preadsorption and scattering results indicates that the films in the capillary condensation regime are well described by an equidistant film surface. Some component of quasi-spherical geometry, although of a different type, may be present also in the B2 sample. Figure 7 shows that near a film thickness of 0.2 nm the film area locally increases with increasing film thickness. Since no such behavior was observed for any of the other silicas, the possibility that this increase is due to incomplete wetting can be ruled out. Incomplete wetting is also unfavorable on purely chemical grounds. We therefore think that the B2 silica may be one of those rare cases where the microstructure of the surface is dominated by convex features. The radius of the features is presumably less than 1nm because the film area rapidly decreases for thicknesses beyond 0.3 nm. It is unlikely that the increase in film area can be modeled by a random packing of solid spheres, however: For packings with coordination number larger than 2, the net effect of convexity and sphere contacts amounts to a decrease in film area.gJo c. Pore Size Distributions from Equations 7. In addition to the data analysis in terms of various geometric
Langmuir, Vol. 7, No. 11, 1991 2841
Structure Analysis of Porous Solids
*
- I a
IoFllm SA MP malysls
I
-
-C
Fllm SA
-C
Adsorptlm
m
I
I
IO
z(nm)
10
I
100
z(nm)
Figure 11. Pore size distributionsof the B2 xerogel, calculated from film surfaceareas, micropore analysisaccording to Mikhail et al.," and Kelvin analysis. Film SA
+ MPanalysis
+ Kelvin analysis
I
z(nm)
Figure 12. Pore size distributions of the A2 xerogel, calculated from film surface areas, micropore analysisaccording to Mikhail et aLarand Kelvin analysis, respectively. models, we have calculated the pore size distributions for the Vycor, B2, and A2 samples. This was done by the procedure described in eqs 7, using the film area and film volume data in Figure 7. Equations 7a and b were discretized as zi = zi-l
I
Desciptlon
Gi .t,)
+ -(vi - vi-1) - + 21
(14a)
where Ai and Vi are the area and volume of the ith data point (i = 1,2, ...,s) ordered by increasing volumes, zi is the corresponding film thickness, s is the number of data points for the sample, and A, is the surface area of the bare solid with V, = 0 and z, = 0. Equations 14a and b are the trapezoidal rule for integration and the two-point backward difference for differentiation. The results are presented in Figures 11-13. For comparison, the figures also include the pore size distributions from the Kelvin equation analysis of nitrogen condensation and from micropore analysis of nitrogen adsorption at relative pressures between 0.1 and 0.4." Since we have restricted all water films to coverages substantially below complete pore filling, the pore size distributions from the film areas span a correspondingly limited range of pore radii. For example, in the B2 xerogel the maximum film thickness according to eq 14a is 0.95 nm, so that the (34) Mikhail, R. H.; Brunauer, S.; Bodor, E. E. J. Colloid Interface Sci. 1968, 26, 45.
Figure 13. Pore size distributions of Vycor, calculated from film surface areas and Kelvin analysis (adsorption and desorption branch). largest pore radius probed is 0.95 nm (Figure 11). This maximum film thickness of 0.95 nm agrees well with the maximum film thickness of 0.86 nm for B2 in Table 11, calculated from eq 3. A similar agreement between thicknesses calculated from eqs 14a and 3 is found for Vycor. This provides a strong consistency test of the fractalanalysis in Table I1and the model-independent analysis in Figures 11-13. The test is particularly demanding for the B2 sample because of its local increase in film area. The increase is fully included in the calculation of the zi(s from eq 14a, but is averaged out in the calculation of z from eq 3 by virtue of the power law, eq 2. The increase in film area for the B2 xerogel has yet another consequence. It is responsible for the negative as calculated from eq 14b, a t z = 0.2 value of dV,,/dz, nm in Figure 11. Such a negative value is unphysical, in agreement with the fact that eqs 4,5,7b, and 14b for the pore size distribution are inapplicable when the film area increases. The reason for the inapplicability is that an increase in film area corresponds to a convex surface and that eq 4, as discussed in its derivation, is not valid for a convex surface. But the negative value of dV,,,/dz at z = 0.2 nm does not affect the applicability of eq 14b a t the points where the film area decreases. This follows from the independence of both sides of eqs 5,7b,and 14b on the structure below and above z. Thus, the results for d Vwy/dz a t z > 0.3 nm may well be compared to the pore size distributions obtained by other methods. Figures 11-13 show that the agreement between the pore size distribution from film areas and from the other two methods is excellent for all three samples. For this assessment, it must be borne in mind that the film area method is based on a theoretical framework and experimental data completely independent of the other two methods, and that each method has its own limitations. For example, the sharp peak a t z = 2 nm in the desorption curve in Figure 13 is in all likelihood35an artifact due to the inapplicability of the Kelvin analysis to the particular desorption isotherm in Vycor (see also refs 11and 12 and our discussion of eqs 12 and 13 for Vycor). Limitations of the micropore analysis according to Mikhail et al." have been pointed out in ref 36. In the light of such uncertainties, we believe that the results from the three methods, to the extent that they overlap, agree indeed remarkably well. Another type of assessment that should be a valuable (35)Gregg,S.J.;Sing, K. S. W. Adsorption, Surface Area and Porosity, 2nd ed.; Academic Press: London, 1982; pp 154-164. (36) Carrott, P. J. M.; Sing, K.S. W. In Characterization of Porous Solids;Unger, K. K.,Rouquerol,J.,Sing, K. S.W., Kral, H., Eds.;Elsevier: Amsterdam, 1988; pp 77-87.
2842 Langmuir, Vol. 7, No. 11, 1991 tool in further studies is the determination of film areas from immersion calorimetry (see, e.g., refs 37 and 38. The fractal behavior found for the B2 sample and Vycor, together with eq 11,predicts that the film area curves in Figures 11and 13should obey dVwre/dz a 9 - O with the respective D values and z intervals given in Table 11. The falloff of the curves observed for 0.3 nm < z < 1.0 nm (Figure 11)and 0.2 < z < 1.0 nm (Figure 13)confirms this prediction qualitatively. A quantitative test was not performed because of the increased fluctuations of the curves in Figures 11 and 13 compared to Figure 7. The increased fluctuations are due to the fact that Figures 11-13 amount to taking the derivative of the data in Figure 7
-.[ dao
dV 3= a(u)] (15) dz du u=V(z) which always amplifies fluctuations and noise in the data. Equation 15 is a simple corollary of eqs 7b and 1. An example for this amplification is the negative value of dVpore/dZ at z = 0.2 nm in the B2 sample. Thus, when a sample has a fractal surface, the scaling behavior is obtained more accurately from the function a(u) than from dVpre/dz. Conversely, if the goal is to detect specific departures from scaling or from some other behavior, the function dVpre/dz will be the more sensitive one. Finally, we point out that eq 15 may be used to devise discretization schemes other than eq 14b. For instance, the two-point backward difference of eq 15 gives as an alternative to eq 14b. It remains to be seen which of the two schemes (and obvious variants thereof) provides the most robust numerical procedure for routine determinations of dVwre/dz. This will be addressed in a future paper. 6. Conclusions The structural information about surface roughness and microporosity obtained from film area measurements agrees very well with previous results from SAXS/SANS and molecular tiling, both for nonfractal surfaces and for surfaces that show fractal behavior over a limited length range. In some cases, the film area analysis has filled gaps in the length range probed by previous experiments (lengths below 2 nm for Vycor). In other cases, the analysis has provided structural information, with a high degree of internal consistency, of samples that have not been studied well before (A2 and B2 xerogels). The modelindependent pore size distributions calculated from the measured film areas and film volumes agree with the distributions from traditional methods, within the expected accuracy of the latter and within the range of mutual overlap. This indicates that the basic assumption of the film area method, namely, that the film surfaces are equidistant, is satisfied under a wide variety of circumstances, including films in the capillary condensation regime. For future tests of the method, we have suggested specific isotherm analyses, scattering experiments, and calorimetric measurements.
Acknowledgment. We thank D. Avnir and D. Farin for useful discussions, and W. A. Steele for bringing refs (37) Partyka, S.; Rouquerol, F.; Rouquerol, J. J. Colloid Interface Sci. 1979,68, 21. (38) Rouquerol, J.; Crillet, Y.; Francois, M.; Pokier, J. E.; Cases, J. M.
In Characterization of Porous Solids; Unger, K. K., Rouquerol, J., Sing, K. S. W.,Kral, H., Eds.; Elsevier: Amsterdam, 1988; pp 317-322.
Pfeifer et 41. Table 111. Fitting Functions for the Various A versus V Relations on a Fractal Surface. 25DC3 D=3 general, eqs A.3 and A.3/ A = (C3 + C2V)c~ A = C*e-CiV film 4 monolayer, eqs A.6 and A.6’ ‘4 = (C3 + C Z V ) ~ I A = CS-C~” A = Cg-CiV film >> lower limit of fractal regime, A = C2VC1 eqs A.5, A.5’, 2, and 2’ The fitting parameters C1, Cz, and CBare not the same in all entries. Their form in terms of D, Z n h Amf, and Vmf is obtained by comparison with the referenced equations.
11and 12 to our attention. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this work (P.P.).
Appendix In eqs 2 and 3, we have assumed that the film thickness z is well above the lower limit of the length range within which the surface is D-dimensional. The corresponding - ~ for all mathematical idealization is that V(z) 0: ~ 3 holds z. In practice, however, the film thickness may fall near or even below this lower limit (e.g., the fractal regime might begin only at 1nm), in which case there may be significant departures from a pure power law for V(z) and hence from eqs 2 and 3. We describe here the modifications necessary under such circumstances. As a byproduct, this leads to a natural treatment of the case D = 3. We assume that the surface is D-dimensional at scales above some reference length zref and has an unknown structure at scales below this length. Thus, Zref may coincide, but need not, with the lower limit of the fractal regime. The film area then obeys A(z) = Aref(z/z,ef)2-D 2 I D I 3 (A.1) for 2 Zref, where the constant Aref is the film area at film thickness zref. If Zref is not the lower limit of the fractal regime, then eq A.1 holds also for z values smaller than Zref. The reason why the film area, but not necessarily the film volume, obeys a power law for z 2 2r.f is that the area A(z) depends only on the structure of the solid at scale z, whereas the volume V(z) depends cumulatively on all structure below z including the unknown regime below Zref. When we integrate eq A.l to obtain V(Z) for z 1 Zref, the unknown structure below Zref enters through the film volume V,fat film thickness Zref (undetermined integration constant): 2ID> Lower Limit of t h e Fractal Regime. As explained above, the volume Vref encapsulates all the unknown surface structure at scales below z,,fand therefore is an independent constant. If the fractal behavior of the surface extends to scales much smaller than Zref,however, then the structure below zrefis no longer unknown and the volume V,,fis determined by the constants Aref, %ref,and D, provided that D < 3. In that case eq A.2 continues to hold for 2 > Zref and V >> Vref which makes the constants V,,fand 1drop out. The expressions A.4 and A.5 supply the prefactors omitted in the earlier discussion. For D = 3, the situation where the fractal behavior extends to scales much smaller than zrefis different. In that case eq A.2' continues to hold for 2 > lower limit of the fractal regime. c. The C o n s t a n t Vref When zref = Monolayer Thickness. In this case one has Vref = ArefZref regardless of D because molecules of thickness z,f cannot probe surface features smaller than z,f and form a monolayer whose volume is indistinguishable from that on a flat surface with areaA,f. The expressions for V(z) that result from substituting V,f = Arefz,f into eqs A.2 and A.2' have been obtained previously.16 The corresponding expressions for eqs A.3 and A.3' are
(A.6')
subject to V? AreFmf. They describe the corrections which need to be made in eqs 2 and 2' when the film thickness decreases down to a monolayer. Clearly, no corrections are necessary for D = 2 and D = 3. For 2 < D C 3, there are corrections because the term D - 2 in eq A.6 introduces departures from the power law, eq 2. This term is small for low D, but becomes dominating at monolayer coverage for D > 2.5. d. Number of Fitting Parameters. The number of fitting parameters available in the various expressions for A versus V depends on whether we take some of the constants Zref, Aref, and Vref as given by experiment, or whether we consider them to be adjustable. We take them as adjustable in order to ensure an unbiased data treatment, i.e., in order not to force the fitted curve to pass through any particular data point (Vref, Aref) or to assume knowledge of the film thickness z,f before A,f or V,f has been determined. The independent fitting parameters under this condition are identified in Table 111. Their number is at least two and a t most three. e. Film Thickness i n Terms of A and V. To obtain the generalization of eq 3, one may formally solve eq A.l for Aref and substitute into eqs A.2 and A.2'. But this gives transcendental equations for z in terms of A and V which are not useful and require knowledge of %ref and Vref. Analogous constants are required for the calculation of z from A or V alone, Le., directly from eq A.l, A.2, or A.2'. Thus, if the A versus V data do not obey a pure power law and hence eq 3 does not apply, the simplest and most reliable way to get z is from eq 6. When applying eq 6 to the A versus Vfunction, eq A.3', e.g., to reconstruct the relation A.2', one has to replace the left-hand side of eq 6 by z - zrefand the lower limit of integration by Vref in order to account for the condition V 1 Vref in eq A.3' and that the film thickness at volume Vref equals zrefby eq A.2'. This illustrates that it is not necessary to have data down to zero film volume in order to implement eq 6. One can always start the integration at volume Vref if the corresponding film thickness z,f is known.
