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Langmuir 1999, 15, 5068-5072
Why an Apparent Surface Dimension of Silica Gels May Be Abnormally High A. G. Okunev* and Yu. I. Aristov Federal Scientific Centre, Boreskov Institute of Catalysis, Novosibirsk 630090, Russia Received April 27, 1998. In Final Form: January 11, 1999 In the present paper, some problems of the “surface resolution analysis” to characterize silica gels are addressed. We measured the adsorption of organic adsorbates (fatty acids, aliphatic alcohols, and ketones) from CCl4 solution on mesoporous silicas of various porosity. A simple linear relationship is found between the monolayer thickness and the monolayer volume, which is valid even for the adsorbates of different homology series. A simple model to describe adsorption experiments quantitatively has been developed. The model shows that the curvature of silica mesopores may influence the packing of test molecules in the monolayer and leads to a decrease in the adsorbate monolayer capacity for large molecules, so that measured surface area is underestimated. The effect increases when the ratio of adsorbed layer thickness h to pore radius R increases and may reach some tens of percent. The model explains why the apparent values of silica surface dimension may be abnormally high (Ds > 3).
I. Introduction During the past decade the so-called “surface resolution analysis” has become a standard method for characterizing porous solid surfaces on molecular scale.1,2 The method is based on simple scaling relationship1
nσ ∝ σ-R
(1)
where the constant R is called “scaling exponent”. Equation 1 relates measured specific surface Ssp of an adsorbent and cross-sectional area σ of a test molecule. Actually, the value of Ssp is usually calculated from the number nσ of physically adsorbed molecules of the cross-sectional area σ needed for the surface monolayer coverage: Ssp ) nσσ.3 The exponent R appears to be negative, and commonly lies between -1/2 and 0 for many particular solids of different nature (silica, alumina, magnesia, carbon blacks, charcoals, minerals, etc.). Widespread physical interpretation of R has been done by the fractal geometry which manifests that R can be expressed in terms of the surface fractal dimension Ds1,4
R)
d ln(Ssp) d ln(σ)
)1-
Ds 2
(2)
The fractal dimension Ds characterizes the roughness of a self-similar surface and varies from 2 for smooth surface to 3 for completely rough and irregular surface (for both cases topological surface dimension is Dt ) 2). Under this approach, the Ds value may be determined experimentally from eqs 1 and 2. To do this, one measures the monolayer adsorption of different adsorbates, usually belonging to some common class of compounds (for example, aliphatic alcohols or fatty acids), and then calculates Ds from the slope of the straight line ln(Ssp)-ln(σ). It is worth noting that the surface resolution analysis encounters some mathematical problems, since actual (1) Avnir, D.; Pfeifer, P. Nouv. J. Chim. 1983, 7, 71. (2) Avnir, D.; Farin, D.; Pfeifer, P. J. Chem. Phys. 1983, 79, 3566. (3) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface area and Porosity, Academic Press: New York, 1982. (4) Mandelbrot, B. B. The Fractal Geometry of Nature; Freeman: San Francisco, 1982.
physical adsorption measurements differ from the Hausdorff conception of Ds value determining in several respects.5,6 Another problem appears when we consider the physical basis of this approach, because the tested surface is assumed to be self-similar and rough on atomic scale and, hence, it possesses a number of atomic size defects. Even if the surface entropy is increased, such a strong irregularity seems doubtful regarding thermodynamics of solids with a high interatomic energy such as silica gels, aluminas, etc. (see Appendix). Moreover, abnormally low scaling exponents R < -1/2 (which, within fractal algorithm, readily give abnormally high values of Ds > 3) were obtained for porous silica gels with aliphatic alcohols as the test molecules.7,8 As it was pointed out recently,9 the limited range of experimental data is the inherent problem of the resolution analysis. In our case the available scale is determined by the lowest and the highest cross-sectional areas of test molecules used that varies typically from 20 to 50 nm2 that gives available with adsorption resolution analysis size range between 4.5 and 7 nm. Since upper and lower limits do not differ even by a factor of 2, a solely adsorption technique gives scarce basis for manifesting of the fractality of the studied silicas and other reasons for power law behavior should be considered. If real oxide surfaces have few defects of atomic size, one has to explain (1) why relation 1 with noninteger R is a common case for mesoporous solids, and (2) what is the reason for R to be lower than -1/2. To answer these questions, we have studied the adsorption of fatty acids, aliphatic alcohols, and acetone on serial mesoporous silica gels and used a simple geometrical model for treating results. The model takes into account the influence of confined pore geometry and the adsorbed layer thickness h on monolayer packing in narrow pores and gives additional evidence that steric difficulties of the monolayer formation may lead to the scaling eq 1 with R < -1/2, thus (5) Van Damme, H.; Levitz, P.; Bergaya, F.; Alcover, J. F.; Gatineau, L.; Fripiat, J. J. J. Chem. Phys. 1986, 85, 616. (6) Fripiat, J. J.; Van Damme, H. Bull. Soc. Chim. Belg. 1985, 94, 825. (7) Drake, J. M.; Levitz, P.; Klafter, J. Isr. J. Chem. 1991, 31, 135. (8) Kutsovskii, Ya. E.; Paukshtis, E. A.; Aristov, Yu. I. React. Kinet.Catal. Lett. 1992, 46, 57. (9) Avnir, D.; Biham, O.; Lidar, D.; Malcai, O. Science 1998, 279, 39.
10.1021/la980482g CCC: $18.00 © 1999 American Chemical Society Published on Web 06/10/1999
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Table 1. Some Geometric Characteristics of the Silica Gels Studied adsorbent
Ssp,a m2/g
av pore radiusa 4V/Ssp, nm
K, nm-1
R h , nm
Davisil 150 KCK-2 Davisil 60 KCC-3
272 380 494 525
8.8 6.8 3.3 3.3
0.025 ( 0.02 0.06 ( 0.025 0.06 ( 0.015 0.095 ( 0.15
9(7 3.5 ( 1.5 3(1 2 ( 0.5
a
N2 BET value.
Figure 2. Langmuir plots of the excess adsorption isotherms from Figure 1. Solid lines are the least-squares fits to the data.
Γa )
Figure 1. Excess adsorption isotherms of n-propanol (1), n-hexanol (2), and pentanoic acid (3) on KCC-3. Solid lines are fits of eq 4 to the experimental data.
giving nonfractal explanation of the origin of apparent values Ds > 3. A similar idea was first presented in ref 10, where steric effects were shown to be responsible, within fractal algorithm, for apparent values Ds > 2 in narrow pores with flat walls, for which Ds ) 2 was reasonably expected. Besides, the ratio (monolayer thickness)/(average pore radius) was found to determine the effect scale. II. Experimental Section Adsorption experiments were carried out with commercial silica gels KCC-3 and KCK-2 (both Reachim, the former USSR), and Davisil 60 and Davisil 150 (both Aldrich). Their specific areas and average pore diameters, measured by nitrogen adsorption at 77 K, are given in Table 1. Before adsorption experiments all samples were carefully dried at 180 °C for 5 h in vacuo. n-Aliphatic alcohols (C3H7OH, C6H13OH, C10H21OH), pentanoic acid, and acetone were adsorbed from their solutions in CCl4 at 18 °C. Carbon tetrachloride and hydrocarbons of a spectroscopic purity grade were used without additional purification. The adsorbed amount was determined from the decrease of the adsorbate concentration in solution by measuring the intensity of C-H vibration line in the IR absorption spectra (ν ) 2975, 2920, and 3005 cm-1 for alcohols, pentanoic acid, and acetone, respectively). The IR spectra were recorded at room temperature using a IR-20 and a Specord 75IR spectrometers.
III. Results and Discussion Typical excess adsorption isotherms Γa(xa) of aliphatic alcohols, pentanoic acid, and acetone at 18 °C are presented in Figure 1 for different silica gels. The excess adsorption Γa is expressed by the equation
Γa )
n ∆x m l
(3)
where n is the total number of moles of both adsorbate and solvent, m is the adsorbent mass, and ∆xl is the mole fraction change in the bulk solution. Equation 4 (10) Gavrilov, K. B.; Okunev, A. G.; Aristov, Yu. I. React. Kinet. Catal. Lett. 1996, 58, 39.
nσ Kexl(1 - xl) m 1 + Kexl
(4)
where Ke is adsorption equilibrium constant, relates the total number of moles nσ of both adsorbate and solvent on the interface with the surface excess Γa.11 All isotherms have an ordinary Langmuir shape and reach the plateaux at concentrations lower than 0.02 M. As adsorbate concentration in solution increases further, excess adsorption Γa (according to eq 3) monotonically reduced. At low concentrations the isotherm slopes are usually markedly above the value one may expect regarding the adsorption equilibrium constant calculated for the whole isotherm. This effect most likely results from a strong adsorption mode, discussed elsewhere.12,13 Note that the value nσ and adsorbate monolayer capacity are not the same, but when a strong specific adsorption takes place, these quantities differ only slightly. So, one can find the adsorbate monolayer capacity directly from the slope of the straight line approximating the isotherm in the Langmuir coordinates: xl(1 - xl)/Γa xl (Figure 2). Monolayer volume V may be calculated from the monolayer capacity nσ using the formula
V)
nσµ F
(5)
where F is the adsorbate liquid density and µ is the molecular mass. Strong adsorption mode influence may be neglected, because the corresponding points are very close to the origin and only slightly change the fitting line slope. Monolayer capacities, monolayer volumes, and specific surface areas of the samples, calculated from our experiments, are listed in Table 2. Adsorption equilibrium constant Ke for all pairs “adsorbent/adsorbate” exceeds 1000 thus showing a strong adsorption of adsorbates under study. Figure 3 depicts the measured specific surface area Ssp of the silica gels as a function of cross-sectional area σ of the test molecules. There is a certain tendency for Ssp to decrease as the size of the test molecules increases. However, there is no simple monotonic correlation between these two values. Figure 4 presents a similar dependence plotted in the double logarithmic coordinates ln(Ssp) -ln(σ), as is usually done under “surface resolution analysis”. This presentation allows to obtain an apparent surface dimension for the KCK-2, KCC-3, Davisil 150, (11) Everett, D. H. Trans. Faraday Soc. 1965, 61, 2478. (12) Gavrilov, K.; Kutsovskii, Ya.; Paukshtis, E.; Okunev, A.; Aristov, Yu. Mol. Cryst. Liq. Cryst. 1994, 248, 159. (13) Okunev, A. G.; Paukshtis, E. A.; Aristov, Y. I. React. Kinet. Catal. Lett. 1998, 65, 161.
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Okunev and Aristov
Table 2. Results of the Adsorption Isotherm Measurementsa
Table 3. Cross-Sectional Areas of the Adsorbates and Adsorbed Layer Thickness (h)
adsorbent
adsorbate
nσ, mmol/g
Sapp, m2/g
V, µL/g
Davisil 150
acetone pentanoic acid propanol hexanol decanol acetone pentanoic acid decanol acetone pentanoic acid decanol acetone pentanoic acid propanol hexanol decanol
1.18 0.9 1.85 1.37 1.1 1.36 1.12 1.27 1.98 1.35 1.63 2.07 1.37 2.8 1.84 1.41
247 260 290 251 240 285 324 277 415 390 355 434 396 438 337 307
87 98 138 171 210 100 122 242 146 147 311 152 149 209 230 269
KCK-2 Davisil 60 KCC-3
cross-sectional area adsorbate
σ, nma
σ,c nm2
h, nm
acetone pentanoic acid propanol hexanol decanol
0.348a 0.48a 0.26b 0.304b 0.362b
0.38 0.50 0.24 0.33 0.41
0.36 0.38 0.5 0.68 0.89
a From empirical equation σ ) 6.16 + 0.596σ , where σ l.d. l.d. calculated from liquid density (ref 16). b From empirical equation σ ) 1.46(number of C atoms) + 21.6 (ref 14). c Calculated from adsorption on Davisil 150 under assumption of the flat surface.
a n ) monolayer capacity. S σ app ) specific surface. V ) monolayer volume.
Figure 5. Relation of the volume V adsorbed on the tested silica samples to the volume Vfl adsorbed on a flat surface as a function of the adsorbed layer thickness: (9) Davisil 150, (2) KCK-2, (1) Davisil 60, (b) KCC-3. Solid and dashed lines are linear fits to the Davisil 150 and KCC-3 data, respectively.
Figure 3. KCC-3 specific surface as a function of the adsorbate cross-sectional areas.
determined by the adsorbate cross-sectional area. Among other factors which may influence the monolayer capacity, the most important are chemical adsorbate-adsorbent interaction7,8 and restrictions caused by the confined pore geometry.5,10 As silica gels studied differ mainly by their porous structure, we have focused our attention on the steric peculiarities of the monolayer adsorption. The ratio (monolayer thickness)/(average pore radius) has been previously specified as a factor which may strongly influence the monolayer formation.10 For further analysis of this effect, we have studied adsorbed volume V as a function of the monolayer thickness. The latter is calculated for each adsorbate using a simple formula
h ) vmol/Naσ Figure 4. Determination of the surface fractal dimension of KCC-3 according to the fractal algorithm: adsorption data for all the adsorbates are fitted with solid line, for aliphatic alcohols solely with dotted line.
and Davisil 60 samples that equal 2.4 ( 0.3, 2.7 ( 0.4, 2.2 ( 0.2, and 2.5 ( 0.2, respectively. It is easy to notice that the data for adsorbates of the different homology series cannot be fitted by the same straight line. When all the adsorbates (including N2) are taken into account, the correlation coefficient η for the fitting line is very low and equals 0.61 for KCC-3 and 0.52 for Davisil 150, while when considering the same homology series, the linear trend helds well. Indeed, the correlation becomes much better (η ≈ 0.94) for both samples if solely data for alcohols are approximated. However, the apparent Ds values calculated from the alcohols adsorption equal 4.1 ( 0.6 (KCC-3) and 3.1 ( 0.4 (Davisil 150) that encounters an interpretation problem (see also refs 7 and 8). Thus, our results (see Table 2 and Figure 3) clearly show that silica gel’s monolayer capacity is not uniquely
(6)
where vmol is the molar volume of adsorbate and Na is the Avogadro constant. The adsorbed layer thicknesses calculated for different adsorbates are given in Table 3. In order to compare the adsorbates with different porosity, the monolayer volumes V in Figure 5 are referred to the N2 BET specific surface of the sample and then to the monolayer volume Vfl adsorbed on a unit flat surface. Obtained ratio V/Vfl vs h decreases at large h for all the samples, which may be understood from a simple geometry consideration. Consider two surfaces of the same area, a flat one and one turned into a spherical pore of radius R. At the same layer thickness h, the ratio of the volumes adsorbed on the spherical Vsp and flat Vfl surfaces is given by
Vsp h h2 )1- + Vfl R 3R2
(7)
This expression shows that the volume adsorbed on a concave surface (positive curvature) is always less than that for a flat surface of the same area. Neglecting the
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Langmuir, Vol. 15, No. 15, 1999 5071
term proportional to h2, the ratio for a small enough arbitrary curved piece of surface may be written as
(
) (
)
V S K Vfl S Kh ) 1) 1Vfl Sfl 2 Sfl Sfl 2
(8)
where V is the volume adsorbed on the curved surface, S and Sfl are the areas of the curved and flat surfaces, respectively, and K is the average curvature of the curved surface. Equation 8 demonstrates that the volume gets smaller as the adsorbed layer thickness goes up, which agrees with our results. For mesoporous solids, the scale of the decrease may reach some tens of percent. Indeed, the size of mesopores varies from 2 to 50 nm, whereas aliphatic alcohols like decanol are about 1 nm long, yielding the ratio h/R ≈ 1/50 to 1/2 and, hence, V/Vfl ≈ 0.5 to 0.98. It is easy to notice that the value of specific surface Sapp of the curved surface, measured from adsorption experiments, also decreases with increase of h
Sapp ) nσσ )
V Kh )S1h 2
(
)
(9)
If relation 8 is valid, the slopes of straight lines on Figure 5 allow us to find for each sample the average surface curvature K and the average curvature radius R h ) 2/K, which appear to correlate with the average pore diameter taken from BET measurements (see Table 1). KCC-3 silica has the lowest V/Vfl ratio of about 0.6 for adsorption of n-decanol, and thus the steric difficulties of adsorbate packing for this sample results in a significant decrease of the measured specific surface and too high value of the apparent surface dimension Ds ) 4.1. This means that the model developed allows to consider abnormally high surface dimensions which were earlier observed for adsorption of n-aliphatic alcohols on silica gels (this work and refs 7 and 8). The effect may be explained by a strongly elongated elliptical (rodlike) adsorption conformation of these alcohols.14,15 In the adsorbed state they are linked to the surface by hydrogen bonds and oriented perpendicularly to the surface. Hence, the adsorbed molecule occupies a large volume in the near-wall region, creating steric difficulties for the adsorption of other molecules. The effect is especially pronounced for the adsorption of higher alcohols in narrow pores. Thus, the measured surface area quickly decreases according to (8) with the increase of the alcohol molecule size, providing the sharp slope of a linear fitting to the adsorption data in the ln(N)-ln(σ) coordinates, and yielding abnormally high values Ds > 3. One can use eq 9 to estimate the values of the apparent surface dimension which may be obtained when formally applying the fractal algorithm. Rewriting eq 2 in the equivalent form
(
Ds ) 2 1 -
)
d ln(Sapp) d ln(σ)
and introducing S(σ) from eq 9, we readily obtain that the surface dimension Ds depends on the derivative dh/dσ as well as on the test molecule cross section σ: (14) Farin, D.; Volpert, A.; Avnir, D. J. Am. Chem. Soc. 1985, 107, 3368. (15) Kipling, J. J. Adsorption from Solution of Non-Electrolytes; Academic Press: New York, 1965. (16) McClellan, A. L.; Harnsberger, H. F. J. Colloid Interface Sci. 1967, 23, 577.
(
Ds ) 2 1 +
dh 1 σ j R h -h h dσ
)
(10)
where h h and σ j are replaced by the average values, since they vary in a relatively narrow range. The latter formula shows that the apparent surface dimension calculated according to the surface resolution analysis depends strongly on the relation between h and σ, i.e., on the shape of test molecules. This explains both different Ds values measured with the adsorbates of different homology series and too high surface dimensions Ds > 3. Indeed, for KCC-3 silica and n-alcohols with perpendicular to surface orientation in the adsorbed state, eq 10 gives the apparent surface dimension Ds as high as 3.7. For this estimation we use the empirical equation σ (Å2) )1.46(number of C-atoms) + 21.6 taken from ref 14 and eq 6 to estimate the derivative dh/dσ. The value obtained is beyond the range of a surface dimension and agrees with experimentally measured value 4.1 ( 0.6. Despite the abnormally high values of fractal dimension, it should be noted that when considering the same homology series adsorbates the power law (1) successfully represents all steric and chemical peculiarities of monolayer formation and may be effictively used for the calculation of the unknown cross-sectional areas of test molecules.14 IV. Conclusions The present paper addresses some problems of the surface resolution analysis used for silica gels characterization. Adsorption of organic adsorbates (fatty acids, aliphatic alcohols, and ketones) from CCl4 solution has been measured for several mesoporous silicas of various porosity. A linear relationship is found between the monolayer thickness and the monolayer volume, which is valid even for the adsorbates of different homologous series. A simple model to describe our adsorption experiments quantitatively has been developed. The model takes into account that an adsorbed molecule not only occupies a place on the surface but also a space nearby this place, which may create in narrow pores steric difficulties for the adsorption of other molecules. As a result, the curvature of silica mesopores influences packing of the test molecules in the monolayer and leads to a decrease in the adsorbate monolayer capacity for large molecules, so that the measured surface area is underestimated. The effect increases when the ratio of the adsorbed layer thickness h to the pore radius R increases and may reach some tens of percent. The model explains why the apparent values of a silica surface dimension may be larger than 3. Appendix Consider the surface with one sort of atoms and assume these atoms to have each five nearest neighbors at the normal state, whereas the defect surface atoms have four neighbors. As a result, the defect atoms have an internal energy which is higher by the value E with respect to that for the normal ones. The increase in the internal energy is surely an obstacle for defects formation, but since the defects also increase the disorder in the system we have to take into consideration the entropy change and calculate the free energy of the system. Let the total number of surface atoms be N ) N1 + N2, where N1 ) XN are defect (4-fold coordinated) atoms and
5072 Langmuir, Vol. 15, No. 15, 1999
Okunev and Aristov
{1 -X X} ) 0
N2 ) (1 - X)N are normal (5-fold coordinated) ones. The free energy of this system may be written as
EN + kTN ln
G ) U + EN1 - ST
X)
where U is an internal energy of the system with N normal surface atoms, EN1 is an increase of the system energy due to N1 defects of the 4-fold coordination, and S is a system configuration entropy. The value of S can be calculated through the number W of different ways to realize this state of the system S ) k ln W, when N particles can be replaced between N1 centers of the first type and N2 of the second type (N ) N1 + N2). The total number of the ways to place N particles between N centers equals to N!. Since N1!N2! variations among them are equivalent, the total number of physically distinguishable permutations is W ) N!/(N1!N2!). As for large N the Stirling formula is valid N! ∼ N ln N, hence
G ) U + EXN - kT ln
{
N! (X‚N)!‚((1 - X)‚N)!
}
) U + EXN - kTN{X ln X + (1 - X) ln(1 - X)} The equilibrium concentration of the defects can be found from the relation dG/dX ) 0, which gives
1 1 + exp(E/kT)
It means that for a solid with two types of surface atomssnormal and defectsthe thermodynamically equilibrium concentration of defects is mainly determined by the energy of a defect formation. For solids with strong bonds (E ∼ 0.5 eV), the defects equilibrium concentration at T ) 27 °C is 3 × 10-9. The respective high concentration of about 10% (X ) 0.1) is equilibrium at a temperature as high as 3000 K. Thus, for solids with strong bonds the high equilibrium concentrations of the surface defects can hardly be reached at reasonable temperatures. Evidently, for defects with lower coordination number the concentration is even lower because of the more high value of E. For molecular crystals the bond energy may be lower by a factor of 10-100 and a 10% concentration may be reached even at room temperature. So, this estimation shows that the surface roughness of the molecular size is unlikely to be the equilibrium case for solids with strong bonds like inorganic oxides, hydroxides, and salts. LA980482G
