Physicochemical Foundations for Characterization of Adsorbents by

Review of Kelvin's Equation and Its Modification in Characterization of Mesoporous ... is assumed in the Dubinin–Astakhov adsorption isotherm equati...
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Langmuir 1997, 13, 6589-6596

6589

Physicochemical Foundations for Characterization of Adsorbents by Using High-Resolution Comparative Plots M. Jaroniec* and K. Kaneko Department of Chemistry, Faculty of Science, Chiba University, 1-33 Yayoi, Inage, Chiba 263, Japan Received July 10, 1997. In Final Form: September 22, 1997X

Theoretical foundations for the comparison of adsorption isotherms at the range of low surface coverages are presented in order to facilitate the use of this high-resolution comparative analysis for characterization of porous adsorbents. The well-known models of adsorption on heterogeneous and microporous solids were used to explain the different types of the high-resolution comparative plots observed experimentally. As expected, the presence of micropores contributes mainly to the upward curvature on the high-resolution plot, but this curvature can be enlarged or reduced significantly by the difference in the surface properties of the compared adsorbents. It is shown that a proper selection of the reference solid allows a quantitative characterization of microporosity and surface heterogeneity of porous adsorbents.

Introduction Comparative plots are commonly used in adsorption to characterize various porous adsorbents, i.e., to evaluate their structural properties such as the volume of micropores (pores of widths below 2 nm), the external surface area of mesopores (pores of widths between 2 and 50 nm), and the total surface area.1-4 The main idea behind comparative plots is to utilize the differences which exist between adsorption processes taking place on a nonporous surface and in the micropores for characterization of porous adsorbents. Physical adsorption of gases and vapors on a nonporous surface or on the mesopore surface occurs via a layer-by-layer mechanism, whereas adsorption in micropores resembles the volume filling mechanism. In the case of porous solids containing both micropores and mesopores, e.g., active carbons and active carbon fibers, the volume filling of micropores occurs first at low pressures and is followed by the formation of a multilayer film on the mesopore walls, and finally, the remaining empty space inside mesopores is filled via a capillary condensation process.1 Thus, the dependence of the amount adsorbed on a porous solid plotted (compared) against the amount adsorbed on a reference nonporous solid is linear at higher pressures because the layer-by-layer adsorption occurs on both solid surfaces.1-5 However, at low pressures, the adsorption mechanisms on the solid studied and the reference adsorbent can be different, resulting in nonlinear behavior of the initial segment of the comparative plot. It should be noted that the linear segment of the comparative plot at higher pressures is rather insensitive on the choice of the reference solid because after the total filling of micropores and completing the first adsorbed layer, the surface effects * Permanent address: Department of Chemistry, Kent State University, Kent, OH 44242: phone, 330-672-3790; fax, 330-6723816; e-mail, [email protected] and jaroniec@kentvm. kent.edu. X Abstract published in Advance ACS Abstracts, November 15, 1997. (1) Gregg, S. J.; Sing, K. W. W. Adsorption, Surface Area and Porosity, 2nd ed.; Academic Press: London, 1982. (2) Sing, K. S. W.; Everett, D. H.; Haul, R. A. W.; Moscou, L.; Pierotti, R. A.; Rouquerol, J.; Siemieniewska, T. Pure Appl. Chem. 1985, 57, 603. (3) Rouquerol, J.; Avnir, D.; Fairbridge, C. W.; Everett, D. H.; Haynes, J. H.; Pernicone, N.; Ramsay, J. D. F.; Sing, K. S. W.; Unger, K. K. Pure Appl. Chem. 1994, 66, 1739. (4) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988. (5) Sing, K. S. W. Ber. Bunsen-Ges. Phys. Chem. 1975, 79, 724.

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are negligible and the film formation is mostly controlled by adsorbate-adsorbate interactions. The slope of the linear segment in this range is proportional to the external surface area, whereas its intercept determines the maximum amount adsorbed in micropores, which can be converted to the micropore volume.1 There are several types of comparative plots such as the t-plot,6-11 Rs-plot,1-5,12-16 and θ-plot,17 which differ only in the manner of presenting the standard adsorption isotherm measured on the reference solid. In the case of the θ-plot, the standard isotherm is expressed in terms of the surface coverage θ, which is the ratio of the amount adsorbed to the monolayer capacity.17 The thickness of the surface film on the reference solid, t, which is obtained by multiplication of the surface coverage θ by the monolayer thickness, is used to construct the t-plot.6 In the Rs-method,1 the amount adsorbed on a porous solid is plotted against the reduced standard adsorption Rs, which is defined for a nonporous reference solid as the ratio of the amount adsorbed at a given relative pressure p/p0 to the amount adsorbed at p/p0 ) 0.4. Recent developments in the commercial instrumentation allow accurate gas adsorption measurements at very low pressures, which are essential for comparing the surface properties of various adsorbents.4 The lowpressure adsorption isotherms are often utilized to determine the adsorption energy distribution functions for different solids, and consequently to compare their surface properties.4-18 Although this approach is attractive and popular, it has also some disadvantages.4,18 The evaluation of the adsorption energy distribution from (6) Lippens, B. C.; de Boer, J. H. J. Catal. 1965, 4, 319. (7) Sing. K. S. W. Chem. Ind. (London) 1967, 67, 829. (8) Broekhoff, J. C. P.; Linsen, B. G. Physical and Chemical Aspects of Adsorbents and Catalysts; Academic Press: New York, 1970. (9) Lamond, T. G.; Price, C. R. J. Colloid Interface Sci. 1969, 31, 104. (10) Rand, B.; Marsh, H. J. Colloid Interface Sci. 1972, 40, 478. (11) Ternan, M. J. Colloid Interface Sci. 1973, 45, 270. (12) Sing, K. S. W. In Surface Area Determination; Everett, D. H., Ed.; Butterworths: London, 1970. (13) Parfitt, G. D.; Sing, K. S. W.; Urwin, D. J. Colloid Interface Sci. 1975, 53, 187. (14) Roberts, R. A.; Sing, K. S. W.; Tripathi, V. Langmuir 1987, 3, 331. (15) Carrot, P. J. M.; Roberts, R. A.; Sing, K. S. W. Carbon 1987, 25, 59. (16) Sing, K. S. W. Carbon 1989, 27, 5. (17) Jaroniec, M.; Madey, R.; Choma, J.; McEnaney, B.; Mays, T. J. Carbon 1989, 27, 77. (18) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1991.

© 1997 American Chemical Society

6590 Langmuir, Vol. 13, No. 24, 1997

experimental adsorption data requires assumption of a model for the local adsorption isotherm and inversion of the ill-posed integral equation of adsorption, which is not an easy numerical task.4 Thus, the adsorption energy distributions calculated from the low-pressure adsorption isotherms are model-dependent functions. The aim of the current work is to present physicochemical foundations for an alternative method of evaluating the surface properties of various adsorbents, which utilizes the low surface coverage range of the comparative plot. Note that conventional comparative plots are usually constructed on the basis of adsorption data from the multilayer range, whereas in the current paper the submonolayer part of the comparative plot is mostly considered. In order to distinguish the comparative analysis, which employs adsorption data measured from very low pressures to the monolayer completion, from the conventional one, the term “high resolution” (HR) is used. The idea of this plot was introduced previously19-22 for estimating the size of the micropores present assuming that their distribution is relatively narrow as well as for analyzing both micropores and mesopores in activated carbon aerogels.23 Also, this idea was verified by using computer-simulated adsorption isotherms for micropores of different sizes.24 However, practical applications of the HR comparative plot showed that the choice of the reference adsorbent is essential for interpreting its nonlinear behavior and consequently for estimating the size of micropores.25-27 For instance, the HR comparative plot for the fluorinated carbon showed a nontypical behavior, i.e., deviated downward, which was attributed to lowering the carbon surface energy due to fluorination.27 The later result indicates that the nonlinear behavior of the HR comparative plot can be caused not only by the difference in adsorption mechanisms (e.g., the volume filling of micropores contained in the adsorbent studied vs the mutilayer adsorption on the reference solid surface) but also by the difference in the surface properties of both adsorbents. It will be shown that the source of the nonlinearity of the HR comparative plot is the difference in the adsorption energies for both adsorbents, which can be caused by the existence of micropores in the adsorbent studied and/or by its different surface properties (e.g., surface functionality and/or heterogeneity) in comparison to the reference solid surface. The simple adsorption models such as Langmuir, Langmuir-Freundlich (LF), and Dubinin-Radushkevich (DR) will be used to predict different possible shapes of the HR comparative plot. Some experimental evidence for the theoretically predicted HR plots will be presented for selected carbon adsorbents. Physicochemical Foundations of the HR Comparative Plots General Concept of the Comparative Plot. Comparative plots such as the t-plot and Rs-plot have been (19) Kaneko, K.; Ishii, C. Colloids Surf. 1992, 67, 203. (20) Kaneko, K.; Ishii, C.; Ruike, M.; Kuwabara, H. Carbon 1992, 30, 1075. (21) Kaneko, K. In Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: Amsterdam, 1997; p 679. (22) Kenny, M.; Sing, K. S. W.; Theocharis, Ch. In Fundamentals of Adsorption; Suzuki, M., Ed.; Kodansha: Tokyo, 1993; p 323. (23) Hanzawa, Y.; Kaneko, K.; Pekala, R. W.; Dresselhaus, M. S. Langmuir 1996, 12, 6167. (24) Setoyama, N.; Suzuki, T.; Kaneko, K. Carbon, in press. (25) Choma, J.; Jaroniec, M. Pol. J. Chem. 1997, 71, 380. (26) Kruk, M.; Jaroniec, M.; Choma, J. Carbon, in press. (27) Setoyama, N.; Li, G.; Kaneko, K.; Touhara, H. Adsorption 1996, 2, 293.

Jaroniec and Kaneko

mostly used for assessment of microporosity in various porous solids.5-16 For a porous solid the total amount adsorbed, n, is equal to the sum of the amount adsorbed in the micropores, nmi, and the amount adsorbed on the mesopore surface, nme, i.e.

n ) nmi + nme

(1)

After expressing the quantity nme in terms of the relative surface coverage θme ) nme/n°me, where n°me is the monolayer capacity for the mesopore surface, one can rewrite eq 1 as follows:

n ) nmi + n°meθme

(2)

It is assumed that at higher pressures all micropores are filled, i.e., nmi ) n°mi ) constant, and the layer-by-layer (multilayer) adsorption occurs on both the mesopore surface of the adsorbent studied as well as on the surface of the reference solid, and subsequently is followed by the capillary condensation inside mesopores.1 However, it has not been always realized that the linear equation of the comparative plot can be obtained from eq 2 only by assuming the equality of θme and θr, where θr ) nr/n°r is the ratio of the amount adsorbed, nr, on the reference solid to its monolayer capacity, n°r.17 Taking into account in eq 2 that nmi ) n°mi ) constant (because at higher pressures all micropores are completely filled) and θme ) θr, the well-known linear form of the comparative plot is obtained:

n ) n°mi + n°meθr

(3)

Equation 3 shows that at higher surface coverages, the dependence of the amount adsorbed on the solid studied plotted as a function of the surface coverage θr is linear, with the intercept equal to the micropore capacity and the slope equal to the monolayer capacity of the mesopore surface. Let us consider the physical meaning of the assumption θme ) θr. This equality can be used in the range of high pressures because the multilayer formation is not strongly affected by the surface properties of the compared adsorbents (see eq 3) or in the entire pressure (surface coverage) range when two solids possess identical surface properties, i.e., when their interactions with adsorbate molecules are the same. If the surface properties of both solids are different, which is usually the case, θme is not equal to θr and eq 3 cannot be used to describe the HR comparative plot. In this case, eq 2, which is much general than eq 3, should be used. As can be seen from eq 2, the shape of the HR comparative plot can be influenced by the microporosity (the first term of eq 2) and surface properties of mesopores (the second term of eq 2). In order to discuss this influence, firstly we consider the case of mesoporous adsorbents of different surface properties (i.e., adsorbents without micropores), secondly the effects of microporosity on the HR comparative plot, and, finally, both the effects of microporosity and different surface properties. Surface Effects. In order to illustrate the influence of the difference in the surface properties of both adsorbents, we assume that the solid studied has no micropores, i.e., n°mi ) 0. In this case, eq 2 becomes

n ) n°me θme

(4)

Since the comparative θ-plot requires that the amount adsorbed on the adsorbent studied, n, is plotted as a

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Langmuir, Vol. 13, No. 24, 1997 6591

function of the relative surface coverage on the reference solid, θr, the relative coverage θme in eq 2 should be expressed in terms of θr. First, the simplest adsorption model, i.e., Langmuir adsorption model,28 will be used to demonstrate the effect of the difference in the adsorption energies of both adsorbents on the behavior of the HR θ-plot. According to the Langmuir model adsorption on a mesoporous adsorbent studied, i.e., eq 4, is expressed as follows

n ) n°me

Kmep 1 + Kmep

(5)

where p is the equilibrium pressure and Kme is the Langmuir constant for adsorption on the mesopore surface of the adsorbent studied. Since each point of the comparative plot is found by plotting the values of adsorption on both adsorbents at a give pressure, p, against each other, the pressure p in eq 5 can be expressed in terms of θr using again the Langmuir model for adsorption on the reference solid

p)

called Langmuir-Freundlich (LF) isotherm4

θr

(6)

Kr(1 - θr)

θme )

where Kr is the Langmuir constant for adsorption on the surface of the reference solid. Substitution of p in eq 5 through eq 6 leads to the following expression for the HR plot of θ as a function of θr

θ ) θme )

Figure 1. HR θ-plots predicted by the Langmuir adsorption model (eq 7) for different values of κ.

κ θr

(7)

1 + (κ - 1) θr

The quantity κ is the ratio of the Langmuir constants for both adsorbents, which can be expressed as follows:4

(

)

me - r κ ) Kme/Kr ) exp RT

(8)

where me and r are the adsorption energies for a given adsorbate on the adsorbent studied and the reference solid, respectively, T is the absolute temperature, and R is the universal gas constant. Note that the simple form of eq 8 was obtained by assuming the equality of pre-exponential factors in the K-constants for nitrogen on both solids, which is a good approximation commonly used in the theory of physical adsorption on heterogeneous solids.4 As can be seen from eq 7, the HR θ-plot is linear (i.e., θme ) θr) in the submonolayer region for κ ) 1, which implies that the adsorption energies for both adsorbents are identical, i.e., me ) r. When the interaction of the adsorbate with the adsorbent studied is stronger than that with the reference solid, i.e., me > r (then κ > 1), the HR θ-plot shows an upward deviation in comparison to the straight line θme ) θr (see Figure 1). However, a downward deviation is obtained for the opposite case, i.e., κ < 1, which implies that me < r. An interesting question is to examine the effect of the energetic heterogeneity on the behavior of the HR θ-plot. In order to estimate quantitatively this effect, we assume that the reference solid is energetically homogeneous (i.e., Langmuir eq 6 can be applied to describe adsorption on this solid), and the adsorbent studied is energetically heterogeneous and its heterogeneity can be characterized by a symmetrical quasi-Guassian distribution of the adsorption energy. In the latter case, the submonolayer adsorption can be represented by the so(28) Langmuir, I. J. Am. Chem. Soc. 1918, 40, 1361.

(K*me p)c 1 + (K*me p)c

(9)

where K*me is the Langmuir constant corresponding to the average adsorption energy and 0 < c < 1 is the heterogeneity parameter related to the width of the energy distribution. For c approaching zero, this distribution becomes much more broad, indicating greater energetic heterogeneity of a given adsorbent. Expressing the equilibrium pressure p in eq 9 by eq 6 leads to the following equation for the HR θ-plot

θ ) θme )

(κ*θr)c (1 - θr)c + (κ*θr)c

(10)

where κ* ) K* me/Kr. In order to eliminate the effect of the average adsorption energy and only consider the influence of the energetic heterogeneity on the behavior of the HR θ-plot, we assumed the same average adsorption energies * for both adsorbents. Then Kr ) K* me, i.e., κ ) 1, eq 10 becomes

θ ) θme )

(θr)c (1 - θr)c + (θr)c

(11)

It is easy to show that θme ) 0.5 when θr ) 0.5 and at this point the HR θ-plots calculated according to eq 11 for different values of c intersect (see Figure 2). In the case of a symmetrical energy distribution an increase in the energetic heterogeneity of the adsorbent studied, i.e., a decrease in the value of c, increases nonlinearity of the θ-plot but does not change the position of the intersection point (see Figure 2). However, for two adsorbents showing asymmetrical distribution functions of the adsorption energy, a change in the dispersion of these distributions implies a simultaneous change in their average adsorption energies, which results in changing not only the nonlinearity of the high resolution plot but also its position in relation to the straight line θme ) θr. In this case, both effects illustrated in Figures 1 and 2 can be observed simultaneously and the intersection of the θ-plot with the straight line θme ) θr can appear at the different value of θr than 0.5 or not appear at all if the difference in the average adsorption energies is large. This is illustrated

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Figure 2. HR θ-plots predicted by the Langmuir-Freundlich adsorption model (eq 11) for different values of the heterogeneity parameter c.

in Figure 3, which displays the HR θ-plot curves calculated according to eq 10 by assuming different values of κ* (i.e., different average adsorption energies for both adsorbents) and one value of c ) 0.5 (i.e., fixed energetic heterogeneity of the adsorbent studied, which differs from that for the reference adsorbent; the latter is assumed to be energetically homogeneous, c ) 1). A more general equation for expressing θme as a function of θr can be obtained when gas adsorption isotherms for both the mesopore surface of the adsorbent studied and the reference nonporous solid are described by the LF equation. In this case, the LF isotherm for adsorption on the mesopore surface of the solid studied is given by eq 9 and the isotherm equation for adsorption on the reference solid has also the same form

θr )

(K*r p)cr

(12)

1 + (K*r p)cr

where K*r is the LF constant analogous to K* me and cr is the heterogeneity parameter for the reference solid. It is easy to show that eqs 9 and 12 give the following formula for θme c

θ ) θme )

(κ*) (θr)

Figure 3. HR θ-plots predicted by the Langmuir-Freundlich model (eq 10) for the fixed value of the heterogeneity parameter, c ) 0.5, and different values of κ*. The dotted line represents the plot θ ) θr.

situations. Equation 14 represents the case when both compared adsorption systems obey the LF model and satisfy the condition: K* me ) K* r. However, eq 11 describes the case when gas adsorption on the mesopore surface of the adsorbent studied obeys the LF equation (9) and adsorption on the reference solid obeys the Langmuir equation (6). In other words, eq 14 reduces to eq 11 when cr ) 1, which implies that ν ) c. A purely mathematical similarity of eqs 11 and 14 as well as 10 and 13 makes more difficult a quantitative interpretation of the HR θ-plots for energetically heterogeneous solids. In a general case, one can obtain the ratio of the heterogeneity parameters, ν, and the value of (κ*)c. If the reference solid is energetically homogeneous, then the heterogeneity parameter of the adsorbent studied can be found because in this case ν ) c. Microporosity Effects. In order to consider only the influence of microporosity on the HR θ-plot, we assume that the adsorbent studied does not contain mesopores, i.e., θme ) 0. Taking into account this assumption in eq 2, we have

n ) nmi

(15)

ν

(1 - θr)ν + (κ*)c(θr)ν

(13)

where ν ) c/cr is the ratio of the heterogeneity parameters for both adsorbents and κ* ) K*me/K* r. When ν ) 1 (surface heterogeneity of both adsorbents is identical) eq 13 has a similar form as eq 7, which has been obtained for the Langmuir model, but instead of κ contains (κ*)c, where c is the same heterogeneity parameter for both adsorbents. When this heterogeneity parameter is equal to unity, eq 13 fully reduces to eq 7 derived for the Langmuir model. Of course, when cr ) 1, then ν ) c (energetically homogeneous reference solid) and eq 13 reduces to eq 10. An interesting case generated by eq 13 is for κ* ) 1, which implies the equality of the average adsorption energies for both adsorbents

In contrast to adsorption on energetically heterogeneous solids, where many analytical isotherm equations are available,4,18 in the case of adsorption in micropores, the well-known analytical isotherm is the DR equation29 or its modified forms.30 Thus, the influence of microporosity on the HR θ-plot can be modeled by using isotherms for adsorption in uniform micropores obtained via computer simulations and/or density functional theory calculations (as shown elsewhere24) or by employing the DR analytical isotherm equation. For illustrative purposes, we will substitute into eq 15 the following form of the DR equation to represent adsorption in micropores, nmi.31

[ (wA βλ ) ]

nmi ) n°mi exp -

2

(16)

where

θ ) θme )

(θr)ν (1 - θr)ν + (θr)ν

(14)

Although eq 14 resembles eq 11, they describe different

(29) Dubinin, M. M.; Radushkevich, L. V. Dokl. Akad. Nauk SSSR, Ser. Khim. 1947, 55, 331. (30) Jaroniec, M.; Choma, J. Chem. Phys. Carbon 1989, 22, 197. (31) Jaroniec, M.; Lu, X.; Mady, R.; Choma, J. Mater. Chem. Phys. 1990, 26, 87.

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Langmuir, Vol. 13, No. 24, 1997 6593

A ) RT ln(p0/p)

(17)

Above A is the adsorption potential defined in eq 17, p0 is the saturation vapor pressure, β is the similarity coefficient of the adsorbate, w is the micropore width, and λ is the coefficient of the inverse proportionality between w and the DR characteristic energy. Combination of eqs 15 and 16 gives

[(

θ ) θmi ) exp -

)]

wRT ln(p0/p) βλ

2

(18)

Expressing the equilibrium pressure p in eq 18 by eq 6, we get

[(

θ ) θmi ) exp -

)]

wRT ln[Krp0(1 - θr)/θr] βλ

2

(19)

The HR θ-plots calculated according to eq 19 for w ) 0.50 and 0.75 nm and the parameters βλ ) 3.96 kJ nm/mol, Kr p0 ) 100 (BET constant), and RT ) 0.64 kJ/mol are shown in Figure 4. The observed curvature of the HR θ-plots is as expected. The plot corresponding to the smaller micropore width shows much larger upward deviation from the straight line. Since a decrease in the micropore width causes an increase in the adsorption energy,32 the effect shown in Figure 4 is analogous to that in Figure 1, but much more pronounced. Unfortunately, the DR equation cannot generate the stepwise adsorption isotherms analogous to those obtained for larger micropores via computer simulations and/or density functional theory calculations,33-35 and therefore it can be used to describe the volume filling in small micropores only,36 which leads to the presence of the socalled filling swing on the HR θ-plot.19,20 In order to demonstrate the presence of the cooperative swing on the HR θ-plot, which reflects the filling of adsorbate inside larger micropores, one needs to use the simulated adsorption isotherms as was shown elsewhere.24 Finally, it should be mentioned that a quantitative evaluation of the microporosity effects by using the HR comparative analysis is a complex issue. Even if the adsorbent studied possesses only micropores, their different shapes, sizes, and distribution can cause different curvatures of the HR comparative plots. In addition, the curvature of the HR comparative plot can be perturbed by possible surface heterogeneity of the micropore walls. This effect will be similar to that shown in Figure 2. Another important issue is related to the selection of the reference nonporous surface. It would be desirable that the surface properties of the micropore walls of the adsorbent studied and the reference solid are identical. The complexity of the HR comparative analysis increases when the adsorbent studied possesses mesopores in addition to micropores. This situation is briefly discussed in the section below. Microporosity and Surface Effects. The most complex HR comparative analysis is for adsorbents possessing both micropores and mesopores. This case is described by the general eq 2. In order to illustrate how are the microporosity and surface effects manifested in the HR comparative analysis, we will express eq 2 in (32) Jagiello, J.; Schwarz, J. A. J. Colloid Interface Sci. 1992, 154, 225. (33) Seaton, N. A.; Walton, J. P. R. B.; Quirke, N. Carbon 1989, 27, 853. (34) Cracknell, R. F.; Gubbins, K. E.; Maddox, M.; Nicholson, D. Acc. Chem. Res. 1995, 28, 281. (35) Olivier, J. P. J. Porous Mater. 1996, 2, 9. (36) Kruk, M.; Jaroniec, M.; Choma, J. Adsorption 1997, 3, 209.

Figure 4. HR θ-plots predicted by the Dubinin-Radushkevich adsorption model (eq 19) for different micropore widths w ) 0.50 (curve a) and 0.75 nm (curve b) by using βλ ) 3.96 kJ nm/mol, Krp0 ) 100 and RT ) 0.64 kJ/mol. The dotted line represents the plot θ ) θr.

terms of the relative adsorptions

θ ) f°mi θmi + f°me θme

(20)

where θ ) n/(nmi° + nme°) is the overall relative adsorption on the adsorbent studied, fmi° ) n°mi/(n°mi + n°me) and fme° ) n°me/(n°mi + n°me) denote respectively the fractions of the micropore and mesopore adsorption capacities. Equation 20 is convenient for analyzing the microporosity and surface effects in the high resolution range, i.e., for θ < 1, because in this range analytical isotherm equations can be employed to express adsorption in micropores, θmi, and monolayer adsorption on the mesopore surface, θme. If one needs to analyze the comparative plot outside the θ < 1 range, isotherm equations for multilayer adsorption should be used because θ is greater than unity due to multilayer formation on the mesopore surface. At first, let us consider the simplest situation, when the adsorbent studied possesses micropores and mesopores, but the surface properties of mesopores are identical with those of the reference solid, i.e., θme ) θr. In this case, eq 20 simplifies as follows:

θ ) f°mi θmi + f°me θr

(21)

If the microporosity effects are represented by the DR eq 19, one gets the following formulas for the HR θ-plot:

[(

θ ) f°mi exp -

)]

wRT ln[Kr p0(1 - θr)/θr] βλ

2

+ f°meθr (22)

In this case, the curvature of the HR θ-plot is essentially the same as that illustrated in Figure 4 with the decreasing magnitude of the upward deviation when the fraction of micropores, fmi°, decreases. When the surface properties of mesopore walls and reference solid are different, θme in eq 20 cannot be replaced simply by θr, but its functional dependence on θr is more complex, e.g., Langmuir type (see eq 7) or LangmuirFreundlich type (see eq 10). For instance, when the microporosity contribution to the HR θ-plot is described by the DR model (see eq 19), and the surface effects for the adsorbent studied and the reference solid are represented respectively by the LF model (see eq 10) and the

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Figure 5. HR θ-plots predicted calculated according to eq 23 by using c ) 1.0, βλ ) 3.96 kJ nm/mol, Krp0 ) 100, RT ) 0.64 kJ/mol, and assuming the equality of the f-fractions, i.e., fmi° ) fme° ) 0.5. The solid curves were calculated for w ) 0.5 nm and different values of κ*: 10.0 (curve a), 1.0 (curve b), and 0.1 (curve c). The dashed line was calculated for w ) 1.0 nm and κ* ) 0.5. The dotted line represents the plot θ ) θr.

Langmuir model (see eq 7), then the HR θ-plot is described by the following formulas:

[(

θ ) f°mi exp -

)]

wRT ln[Krp0(1 - θr)/θr] 2 + βλ (κ*θr)c (23) fme (1 - θr)c + (κ*θr)c

It is easy to see that eq 23 has been obtained by substituting eqs 10 and 19 to eq 21. A more general expression for the HR θ-plot can be obtained from eq 21 by assuming that adsorption in micropores of the adsorbent studied obeys the DR model (eq 16), whereas adsorption isotherms for both the mesopore surface and the reference solid can be described by the LF eqs 9 and 12, respectively. In this case, the formulae for the HR θ-plot is

[(

θ ) f°mi exp -

)]

wRT ln[Krp0(1 - θr)/θr] 2 + βλ c (κ*) (θr)ν (24) f°me (1 - θr)ν + (κ*)c(θr)ν

As can be seen from eqs 23 and 24, the overall curvature of the HR θ-plot is determined by the contributions of micropores and mesopores, i.e., fmi° and fme°. When the fractions fmi° and fme° are comparable, both effects arising from microporosity and mesoporosity are significant and the overall curvature o the HR θ-plot depends on the difference in the surface properties of the compared solids. If the adsorption energy for the adsorbent studied is greater than that for the reference solid, i.e., κ* > 1, both the microporosity and the difference in the adsorption energies generate the upward curvature as shown in Figures 1 and 4, and the total deviation is enlarged (see curve a in Figure 5). However, for κ* < 1, which represents the situation when the adsorption energy for the reference solid is greater than that for the adsorbent studied, the microporosity and the difference in the adsorption energies generate opposite curvatures as illustrated in Figures 1 and 4. Sometimes, for larger micropores and κ* < 1 both effects can even compensate (see the dashed line in Figure

5). Usually, the microporosity effects are much stronger than the surface effects, which leads to the S-shape of the HR θ-plot (see curve c in Figure 5). The curve b in Figure 5, which lies between the curves a and c, represents the case of identical surface properties of the mesopore walls and reference solid, i.e., κ* ) 1. In addition, the surface heterogeneity effects can be significant when surface heterogeneities of both adsorbents differ, e.g., when c lies between zero and unity and cr ) 1. In this case, the overall curvature of the HR θ-plot is perturbed by adding an upward deviation at low values of θr and a downward deviation at higher values of θr as shown in Figure 2. This effect is less visible when an enlarged upward deviation is generated by the micropores present and κ* > 1; however, it becomes more significant when a compensation in the curvature of the HR θ-plot occurs due to κ* < 1. However, when the surface heterogeneities of the compared adsorbents are similar, then ν in eq 24 is close to unity and the curvature shown in Figure 2 is small. It disappears completely when ν ) 1. Experimental Illustration It was shown in the theoretical section that simple models of adsorption on heterogeneous and microporous solids allowed an explanation of the different types of the HR θ-plot (see Figures 1 to 5), which were observed experimentally25-27 or constructed on the basis of adsorption isotherms obtained via computer simulations.24 The later study has been done for siltlike micropores of different sizes in order to establish a relationship between the shape of the HR comparative plot and the micropore size.24 Three experimental illustrations are shown below in order to demonstrate that the HR comparative plots reported previously25-27 can be explained in terms of the models discussed in the theoretical section. In the first illustration, the Sterling FT-G carbon black prepared by heating at 2973 K in a neutral atmosphere and supplied by the Laboratory of the Government Chemist (Teddington, U.K.) was used as the reference solid. This carbon of the specific surface area of 11.5 m2/g possesses a very narrow Gaussian-type distribution of the adsorption energy25 and was used by Olivier35 as a model of the homogeneous carbon surface. To our knowledge, the Sterling sample is currently the best example of the carbon surface of high energetic homogeneity. Employing the Sterling carbon as the reference solid, two HR θ-plots were made by using nitrogen adsorption isotherms measured at 77 K on the unmodified and oxidized SAO carbon blacks (see Figure 6). Nitrogen adsorption data for the Sterling and unmodified SAO samples were reported previously.25 The unmodified SAO sample is a furnace soot prepared from antracene oil and heated at 1223 K for 6 h in argon atmosphere.25 The oxidized sample was obtained by treating SAO with concentrated nitric acid at 363 K.37 The BET specific surface areas of the unmodified and oxidized SAO samples are respectively equal to 95 and 205 m2/g. While unmodified SAO sample is a mesoporous carbon black, its treatment with concentrated nitric acid caused a significant increase in the BET specific surface area due to the formation of a small amount of micropores (about 6% of the total pore volume). Thus, a comparison of the unmodified SAO soot with the Sterling carbon demonstrates the influence of the difference in the surface properties of the compared solids on the HR θ-plot. Average adsorption energies for both non-microporous carbon blacks are expected to be similar because both samples were thermally treated in a neutral atmosphere. However, their energetic heterogeneity is different due to (37) Choma, J.; Jaroniec, M. Pol. J. Chem. Submitted.

Characterization of Adsorbents

Figure 6. Experimental HR θ-plots for the unmodified (filled circles) and oxidized (open circles) SAO carbons prepared by using the Sterling carbon as the reference solid. The dotted line represents the plot θ ) θr.

the differences in the raw materials used and preparation conditions, which led to the materials of different degrees of graphitization. As shown in the previous work,25 the SAO carbon is much more heterogeneous in comparison to the Sterling sample, which is considered to have a homogeneous surface. Thus, the expected curvature of the HR θ-plot for the unmodified SAO sample is as that shown in Figure 2. A quick inspection of the curve for the SAO sample (see Figure 6) shows that really its shape resembles the model plots in Figure 2, only the inflection point is below 0.5 indicating that the average adsorption energy for the SAO carbon is slightly lower as that for the Sterling material. In the case of the oxidized SAO sample (see Figure 6), the creation of micropores enhances significantly the upward deviation of the HR θ-plot and the resulting curve resembles those in Figure 4, but its S-shape remains due to the difference in the energetic heterogeneity of the compared adsorbents. In order to illustrate the influence of the reference solid on the HR θ-plot, the unmodified SAO sample was compared against the BP-280 carbon black (40 m2/g), which is produced by the Cabot Corporation by pyrolysis of hydrocarbons. Standard nitrogen adsorption data for this carbon are reported by Kruk et al.38 As can be seen from Figure 7, the HR θ-plot for the unmodified SAO carbon compared against the BP-280 carbon black is different than the plot constructed by using the Sterling carbon as the reference solid (see Figure 6). Instead of an S-curved HR θ-plot shown in Figure 6, a relatively straight line was obtained when the BP-280 carbon was used as the reference solid (see Figure 7). Since both carbons have no micropores, the lack of curvature of the HR θ-plot shown in Figure 7 indicates that the average adsorption energies and surface heterogeneity of these samples are similar. The influence of the difference in the surface properties of the compared solids on the HR θ-plots is further illustrated in Figure 8, which was prepared on the basis of nitrogen adsorption data for unmodified and fluorinated non-microporous carbon blacks. Nitrogen adsorption isotherm for the unmodified NPCII carbon black of the BET specific surface area of 81 m2/g was reported by Kaneko et al.20 The solid line with diamonds in Figure 8 represents the HR θ-plot for the NPCII carbon in comparison to the Sterling carbon, which can be considered (38) Kruk, M.; Jaroniec, M.; Gadkaree, K. P. J. Colloid Interface Sci. 1997, 192, 250 .

Langmuir, Vol. 13, No. 24, 1997 6595

Figure 7. Experimental HR θ-plot for the unmodified SAO carbon (open circles) prepared by using the BP-280 carbon black as the reference solid. The dotted line represents the plot θ ) θr..

Figure 8. Experimental HR θ-plots for the fluorinated NPCII carbon sample compared with respect to the unmodified NPCII carbon (the solid line with filled circles) and the Sterling carbon (the solid line with open circles). The solid line with diamonds represents the HR θ-plot for the unmodified NPCII carbon prepared by using Sterling as the reference solid. The dotted line represents the plot θ ) θr..

as an energetically homogeneous solid. The curvature of this plot indicates that the surface of NPCII is energetically heterogeneous in comparison to the Sterling carbon and its average adsorption energy is much lower than that for Sterling because the θ-plot intersects with the straight line much below 0.5 (see discussion relating to Figure 3). The HR θ-plots for the fluorinated NPCII carbon are represented in Figure 8 by solid lines with circles. These plots were made by comparing nitrogen adsorption isotherm measured for the fluorinated NPCII carbon27 with that for the unmodified NCPII sample (filled circles) as well as that measured on the Sterling carbon (open circles). A comparison of the unmodified and fluorinated NCPII samples shows that fluorination caused a decrease in the average adsorption energy (the intersection point is below 0.5) and increased the energetic heterogeneity of the carbon surface (the S-shaped HR θ-plot). A quantitative estimation of the surface heterogeneity of the fluorinated sample can be done on the basis of the HR θ-plot constructed by using Sterling as the reference solid

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(the solid line with open circles in Figure 8). An inspection of this solid line shows that the surface heterogeneity of the fluorinated sample is significant and its average adsorption energy is smaller than that for the Sterling surface. The observed significant downward curvature of the HR θ-plot for the fluorinated sample compared with Sterling is a combination of the curvatures arising from the surface heterogeneity of the unmodified NPCII carbon (the solid line with diamonds in Figure 8) and the surface heterogeneity induced by fluorination (the solid line with filled circles). Conclusions It was shown that simple models of gas adsorption on heterogeneous and microporous solids can be used to explain the main features of the HR comparative plots. While microporosity alone causes an upward curvature of the HR comparative plot, the difference in the surface properties of the compared adsorbents can lead to either upward and downward deviations of this plot. Therefore,

Jaroniec and Kaneko

in the case when both factors are present, an enlargement of the upward curvature of the HR comparative plot or its reduction can be observed depending on the difference in the average adsorption energies of the compared solids. The current work demonstrates the complexity of the HR comparative analysis when both microporosity and surface heterogeneity are present and the importance of the selection of the reference solid. It was shown that this analysis is especially useful for estimating microporosity when the reference solid of similar surface properties is available as well as for comparing the surface heterogeneity of mesoporous solids. Acknowledgment. M.J. acknowledges the Japan Society for Promotion Science for supporting his 2-month stay at Chiba University. The authors thank Professor J. Choma for providing the SAO samples and fruitful discussion. LA970771P