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Langmuir 1999, 15, 6004-6015
A New Way to Analyze Adsorption Isotherms† Franc¸ oise Ehrburger-Dolle* Institut de Chimie des Surfaces et Interfaces, CNRS, BP 2488, 15, rue Jean Starcky, F-68057 Mulhouse Cedex, France Received September 30, 1998. In Final Form: April 23, 1999
It is shown how a simple change of variable allows analysis of adsorption isotherms from an angle which is very different from the traditional one and eventually yields very simple equations characterizing the growth mechanism of the adsorbed molecule clusters in terms of fractal dimensions. The variable experimentally measured is the relative gas pressure p ) P/P0. The variable involved in most equations describing the variation of the number of adsorbed molecules N is p or the chemical potential µ, which is proportional to ln(p). In the present approach, the variable is a length δ related to the mean free path (∝ P-1) of molecules in the gas phase, at a constant temperature, by means of the following relation: δ ) λ - λ0. It is shown that the derivative dN/dδ determined from the experimental data obtained for very different samples of silica or carbon materials and several adsorbates consists of one, two, or more power law regimes over the whole domain investigated, corresponding to p values ranging between 10-6 and 1. The exponents can be related to a fractal dimension which characterizes the growth of the adsorbed molecule cluster, which is governed by the molecule-molecule and molecule-solid interactions, the surface heterogeneity, the surface fractal dimension, and the diffusion on the solid surface. It follows that the whole adsorption isotherm can be described by one, two, or more equations having all the same analytical form and describing the particular mechanism involved in each regime. It is shown that this new approach can be used to analyze any type of isotherm of adsorption on solid surfaces. However, in the particular case of adsorption on microporous solids characterized by a type I isotherm, which was previously investigated and which will not be considered in the present paper, the physical meaning of the results may be somewhat different. Examples of adsorption of nitrogen and argon on silica and carbon materials are presented and discussed. In the multilayer coverage domain, the results are compared to those obtained with the fractal Frenkel-Halsey-Hill equation.
1. Introduction 1
Very recently, Cerofolini and Rudzinski nicely summarized the evolution of theories of gas-solid adsorption by distinguishing three different periods: (i) The first period they called the “Pioneering Age”, during which the goal was to find equations, that is, analytical relations between the number N of adsorbed molecules and the relative gas pressure p ) P/P0, that were the experimental data set; this goal can be reached either by using realistic theoretical models,2 yielding simple mathematical relations, or by finding empirical equations verified for series of adsorbate-adsorbent systems; the well-known Brunauer-Emmett-Teller (BET) and Dubinin-Radushkevich (DR) equations,3 which are still the most widely used ones, were established during this period. (ii) Investigations performed during the second period (“Middle Age”) were aimed at understanding the mismatch between theories elaborated for model surfaces and experimental isotherms; the concept of heterogeneous surfaces was born, and most of the methods were based on the introduction of a distribution of adsorption energies.4,5 (iii) The most recent † Presented at the Third International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, held in Poland, August 9-16, 1998. * E-mail:
[email protected].
(1) Cerofolini, G. F.; Rudzinski, W. In Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: Amsterdam, 1998; pp 1-103. (2) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: Oxford, 1975. (3) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic Press: London, 1982. (4) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1998.
research in the field of adsorption resulted from the development of computers (“Modern Age”) and therefrom the possibility to perform simulations. The beauty of simulations lies in the fact that one may change ad infinitum all parameters involved in the physical or chemical process under investigation. The comparison between the data generated by simulation and the experimental ones yields eventually information about the mechanism involved during the real process. Unfortunately, the computer is behaving somehow as a “black box”; in other words, there is no analytical relation between the ingredients and the results. Under such circumstances, the only outcome would be to perform simulations in order to find the best fit to each new series of experimental data and to infer a realistic description of the mechanism involved in each particular system and, among others, adsorption of molecules on solid surfaces. This drawback obviously did not escape Cerofolini and Rudzinski,1 who wrote: “The fascination by the new possibilities created by computer simulations is also accompanied by a growing nostalgia for having simple expressions that could be applied by anyone in research and practice.” The present paper is aimed at showing that such equations actually exist and what is their physical meaning. These equations are obtained very simply when adsorption isotherms are examined from a new angle. In previous papers,6,7 I have shown that the derivative of any adsorption isotherm (or at least all the ones I had analyzed) is a power law of the gas pressure within the (5) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1992. (6) Ehrburger-Dolle, F. Langmuir 1994, 10, 2052. (7) Ehrburger-Dolle, F. Langmuir 1997, 13, 1189.
10.1021/la981349r CCC: $18.00 © 1999 American Chemical Society Published on Web 07/13/1999
A New Way To Analyze Adsorption Isotherms
whole range of pressure in the case of microporous solids and, up to the monolayer coverage, in the case of solids exhibiting an external surface. From these features, it followed that the isotherms are described by equations consisting of a power law term (in other words, a Freundlich equation) plus a constant. In a first step,6 the exponent was expressed in the classical form E0/RT (where E0 is a characteristic adsorption energy), which is involved in both Dubinin-Radushkevich and Freundlich equations.5 Obviously, this approach fails when the exponent is negative, as occurs in the case of low-temperature adsorption of nitrogen or argon on carbon surfaces or in narrow pores for which the width is less than about six times the adsorbed molecule size (nanopores). In the second step,7 it was shown that the derivative toward pressure of adsorption isotherms obtained by simulations8,9 using density functional theory (DFT) for nitrogen adsorbed in pores of sizes ranging between 1.94 and 6 molecular size units also displays one or more power law domains. This feature, which allows a direct comparison between experimental and simulated isotherms, was shown to be very useful for a precise determination of the nanopore width of activated carbons.10 However, in the case of adsorption on external surfaces (type II isotherms) onto which multilayers are building, the derivative toward pressure becomes an increasing function of the pressure which cannot be fitted to a power law. The present paper is aiming to show how a realistic change of variable allows one to fit the derivative, with respect to this new variable, by power laws which extend up to saturation pressure and which yield a more straightforward relation between the exponents and a fractal dimension. 2. Experimental Section 2.1. Samples. The silica sample investigated is Aerosil 200. It is a fume silica produced by Degussa and obtained by pyrohydrolysis of SiCl4. Aerosil 200 is an hydrophilic amorphous silica consisting of aggregates of primary particles (average particle size equal to 12 nm) covered by hydroxyl groups. Similarly to fume silica, furnace carbon blacks also consist of aggregates of primary particles. The samples chosen are a reinforcing black (N330) characterized by small aggregates and a conducting grade one (TB#5500). Both samples are produced by Tokai Carbon. The average primary particle sizes are equal to 26.4 and 23.6 nm, and the dibutyl phthalate absorption (DBPA) amounts to 1.01 and 1.55 cm3 g-1, respectively. To investigate the effect of the crystallite size of the carbon black particles, adsorption on a graphitized N330 sample is also investigated. This sample (Vulcan 3 graphitized produced by Cabot) is used as a reference material. Finally, adsorption on extended graphitic planes (about 30 nm) is investigated by using Grafoil, which is exfoliated compressed graphite produced by Union Carbide. 2.2. Adsorption Measurements. A classical volumetric device was used for the measurement of adsorption of nitrogen at 77 K on Grafoil and on TB#5500. The room temperature is regulated to 293 K. The pressure was measured with three different pressure sensors (Barocel 1, 100 and 1000 Torr) in order to cover the whole range of pressure. For all other samples, isotherms of adsorption of nitrogen and argon (77 K) were performed by using an automatic apparatus (ASAP 2000, Micromeritics). Prior to the adsorption measurements, all (8) Lastoskie, C.; Gubbins, K. E.; Quirke, N. J. Phys. Chem. 1993, 97, 4786. (9) Lastoskie, C.; Quirke, N.; Gubbins, K. E. In Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W.; Steele, W. A., Zgrablich, G., Eds.; Elsevier: Amsterdam, 1998; pp 745-775. (10) Ehrburger-Dolle, F.; Gonzalez, M. T.; Molina-Sabio, M.; Rodriguez-Reinoso, F. In Characterization of Porous Solids IV; McEnaney, B., Mays, T. J., Rouque´rol, J., Rodriguez-Reinoso, F., Sing, K. S. W., Unger, K. K., Eds.; The Royal Society of Chemistry: Cambridge, 1997; pp 237-244.
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Figure 1. Isotherm of adsorption of nitrogen at 77 K on Aerosil 200 plotted in logarithmic coordinates. samples were outgassed during 14 h at 393 K in the case of the silica sample and at 473 K in the case of the carbon ones.
3. Classical Ways to Analyze Adsorption Isotherms By far, most gas-solid adsorption isotherm measurements are performed in order to determine the surface area of an adsorbent. It follows that, among the tens of isotherm equations already published,1,3-5 mainly two equations, the Brunauer-Emmett-Teller (BET) equation and the Dubinin-Radushkevich (DR) one, yielding an estimation of the monolayer coverage, are used. These two equations are as follows:
BET equation: N(p) ) CNm p/(1 - p)/[1 + (C - 1)p] (1) DR equation: ln[N(p)] ) ln(N0) - [RT/(βE0)]2[ln(1/p)]2 (2) in which p ) P/P0 is the relative pressure, C and E0 are increasing functions of the gas-solid interaction, and β is a similarity constant characteristic of the adsorbate (for nitrogen at 77 K, β ) 0.33). Figure 1 shows the isotherm of adsorption of nitrogen at 77 K on Aerosil 200, plotted in logarithmic coordinates. The curves calculated by using the BET (dotted line) and the DR (dashed line) equations are also plotted, and the parameters obtained from the fit are indicated in Table 1. In all cases, the limits of the fit are determined from the linear domains in the classical representation of the BET and DR plots. The narrowness of the pressure range within which the data fit to the BET equation clearly appears in Figure 1. It is amazing to see that the DR equation fits the data over quite a broad range of pressure, in the submonolayer coverage domain. As a matter of fact, in the particular case of adsorption of nitrogen at 77 K on any amorphous silica, the fitting range is significantly more extended than that in the case of microporous carbons,7 for which this equation has been developed. The relative difference between the values of the monolayer coverage obtained by the BET and the DR equations ranges between 7 and 13% for all samples (Table 1). As seen here
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Table 1. Characteristic Parameters Obtained from the Fit of the Experimental Data to Brunauer-Emmet-Teller (BET), Dubinin-Raduskevich (DR), and Frenkel-Halsey-Hill (FHH) Equations in the Case of Adsorption of Nitrogen at 77 K on Aerosil 200 and on Several Carbon Materials Aerosil 200
TB#5500
N330
N330 G
BET range of P/P0 Nm (mmol g-1) SBET (m2 g-1)a C
0.05-0.18 2.23 217 83
0.05-0.2 1.89 184 1087
range of P/P0 N0 (mmol g-1) SDR (m2 g-1)a E0 (kJ mol-1)
4 × 10-4 to 6 × 10-3 2.07 202 13.4
10-3 to 0.045 2.13 208 17.9
range of P/P0 exponent Ds (eq 5a) Ds (eq 5b)
0.2-0.98b -0.40 (1.80) 2.60
0.02-0.95b -0.30 2.10 2.70
DR
Grafoil
0.05-0.2 0.71 69 195
0.05-0.2 0.76 74 575
10-3 to 0.1 0.25 24 2141
2 × 10-3 to 0.02 0.76 74 15.7
2 × 10-3 to 0.04 0.82 80 23.1
10-3 to 0.04 0.28 27 22.0
0.01-0.89 -0.39 (1.83) 2.61
0.38-0.75 -0.36 (1.92) 2.64
0.38-0.77b -0.46 (1.62) 2.54
FHH
a
The nitrogen molecular area used for the determination of the surface area is 0.162 nm2. b Upper experimental value.
Figure 2. Derivatives of the measured adsorbed amount N (continuous line) and of the fitting Dubinin-Radushkevich equation (dashed line) with regard to relative pressure, plotted as a function of relative pressure in logarithmic coordinates, in the case of adsorption of nitrogen on Aerosil 200.
and as generally observed, the BET equation yields larger values than the DR equation does, in the case of amorphous silicas, whereas the opposite is observed for most carbon materials. Figure 2 shows that, as in the case of the precipitated silica investigated previously,7 the experimental derivative dN/dp obtained for Aerosil 200 scales with the relative pressure p as follows:
dN/dp ) k′p
(3)
between p ) 5 × 10-5 (which is the lowest limit of this data set) and a value of p close to 0.05, corresponding to the lower limit of the BET domain. The prefactor k′ and the exponent ν are equal to 0.64 and -0.84, respectively. From eq 3 it follows that, within the same range of pressure, the isotherm equation N(p) is as follows:
N ) k0 + k1pν+1
(4)
The parameters k0 and k1 are obtained from the best fit to eq 4 by means of the curve-fitting procedure of the SigmaPlot software using the Marquart-Levenberg algorithm and a weight equal to 1/y2; the exponent R ) ν + 1 is determined from the derivative, and the value of k1 can be checked against that of k′ ) (ν + 1)k1. In the present case, with R ) 0.16, one obtains from the nonlinear regression k1 ) 3.97, yielding k′ ) 0.635, which compares with the value of the prefactor in eq 3 equal to 0.640. Because the value of the integration constant (k0 ) -0.54) cannot be neglected within the range of coverage involved, eq 4 cannot be approximated here by a Freundlich equation. Because, also, this constant is negative, eq 4 does not apply when p f 0. In the present case and in the case of nongraphitized carbon blacks (paragraph 5.3), the expected Henry law domain is not observed at relative pressures as low as 3 × 10-5 even though it is for graphitized carbon black. The experimental determination of the crossover pc between the two regimes requires adsorption measurements down to very low pressures which will be performed soon. Figure 2 also shows that, within the pressure range where the DR equation fits the data, its derivative also yields a realistic fit to the data, but the upper limit is more than 1 order of magnitude below that of the power law. It is worth mentioning that the method of analysis of the isotherm by using its derivative used here is quite different from the derivative summation (DIS) method proposed by Villieras et al.,11 in which the derivative dN/ d(ln p) yields information about energetic surface heterogeneity but not an isotherm equation. In the last 15 years, a large number of researchers also became interested by the determination of the surface fractal dimension Ds of the solid adsorbent. Although several attempts to deduce the surface fractal dimension from a single adsorption isotherm have been proposed,12,13 the fractal Frenkel-Halsey-Hill (FHH) equation,13 which describes the multilayer coverage, became mostly used for this purpose. Its generic form is (11) Villie´ras, F.; Michot, L. J.; Bardot, F.; Cases, J. M.; Franc¸ ois, M.; Rudzinski, W. Langmuir 1997, 13, 1104. (12) Fripiat, J. J. In The Fractal Approach to Heterogeneous Chemistry; Avnir, D., Ed.; Wiley: Chichester, U.K., 1989; pp 331340. (13) Pfeifer, P.; Liu, K. Y. In Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: Amsterdam, 1998; pp 625-677.
A New Way To Analyze Adsorption Isotherms
N ∝ [-ln(P/P0)]m
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(5)
in which m is expressed as:
m ) -(3 - Ds)/3
(5a)
m ) -(3 - Ds)
(5b)
or as
The power law (eq 5) in which m is given by eq 5a, called the van der Waals (VdW) wetting regime, is verified when the molecule-solid interactions prevail; when eq 5b is verified, the power law (eq 5) is called the capillary wetting regime or the capillary condensation (CC) regime, and it occurs when the molecule-molecule interactions can no longer be neglected. In the case of adsorption on a flat surface, Ds ) 2, so that the classical FHH exponent m ) -1/3 is recovered in eq 5a. The value of the exponent m is obtained by a linear regression of the experimental data N plotted as a function of ln(P0/P) plotted in logarithmic coordinates, for relative pressures above 0.4 (see figures 6a and 12a, next in paragraphs 4.4. and 5.4, respectively). In the case of Aerosil 200 and carbon black N330, the values of m (Table 1) agree with the ones obtained by Ismail and Pfeifer14 on the same type of samples. The fitting equation obtained for Aerosil 200 plotted in Figure 1 will be discussed in detail in the next paragraph. 4. Analysis of Isotherms of Adsorption of N2 and Ar on Aerosil 200 by a New Method It was shown in the previous section that the adsorption isotherm cannot be fitted to a single equation. At first sight, this is not too surprising because one may intuitively accept the idea that mechanisms involved during monolayer coverage are different from the ones involved during the formation of multilayers. What is amazing is that the four fitting equations, plotted in Figure 1, all have a different analytical form and involve either the relative pressure p (BET equation and eq 4) or ln p (DR and FHH equations). These features result from the nature of the underlying model. The DR and FHH equations are based on a thermodynamic approach to adsorption in which the actual variable is a chemical potential which is proportional to ln p, at a constant temperature. In a molecular approach, the pressure P can be related to a flux of molecules hitting or leaving the adsorbent surface, as in the case of the BET equation. As a matter of fact, the gas pressure P can also be related to the mean free path λ of molecules:15
λ)
kBT πx2σ2P
(6)
in which kB ) 1.38 × 10-23 J K-1 is the Bolzmann constant, T is the isothermal temperature (T ) 77 K), σ is the collision diameter (σN2 ) 0.375 nm and σAr ) 0.364 nm), and P is the gas pressure. Introducing the saturation pressure P0 (P0 ) 100 kPa for N2 and P0 ) 28.4 kPa for Ar at 77 K) yields
λ)
λ0 (P/P0)
)
λ0 p
(14) Ismail, I. M. K.; Pfeifer, P. Langmuir 1994, 10, 1532.
(6a)
Figure 3. Derivatives of the measured adsorbed amount N with regard to the mean free path λ in the gas-phase plotted in logarithmic coordinates as a function of λ (open circles, upper scale) and as a function of δ (closed circles, lower scale), in the case of adsorption of nitrogen on Aerosil 200.
in which λ0 ) 17 nm, for nitrogen at 77 K, and λ0 ) 63 nm, for argon at 77 K. The next paragraphs will be devoted to the analysis of the adsorption isotherms using λ or better δ ) λ - λ0 as the new variable. 4.1. Analysis of the Derivatives dN/dλ (or dN/dδ). Figure 3 shows the derivative dN/dλ plotted as a function of the mean free path λ (open circles, upper axis). Obviously, because λ scales as (P/P0)-1, dN/dλ scales as λ-ν-2 within the same range. As far as one is dealing with power laws, using p ) P/P0 or 1/p, that is, as a variable yields a power law within the same domain. Actually, the limiting value of λ in the gas phase is not zero but λ0 at which condensation occurs. For this reason it is realistic to replot the curve and to determine its derivative by using the new variable δ ) λ - λ0, which tends to zero when λ f λ0 or when P f P0. Figure 3 shows that when dN/dδ is plotted as a function of δ ) λ - λ0 (filled circles, lower axis), then a second power law is observed. It follows that the derivative dN/dδ consists of two power law domains describing two regimes: (i) regime 1 characterized by an exponent γ1 ) -1.16 () -ν - 2) extending over more than four decades, up to a value of δ corresponding to a relative pressure equal to 0.2, which is slightly above the upper limit of the BET domain, and (ii) regime 2 characterized by an exponent γ2 equal to -1.48. A similar behavior is observed in the case of adsorption of argon (77 K) on the same sample (Figure 4). However, the values of the exponents are not the same as those in the case of nitrogen: the absolute value of γ1 is slightly larger whereas that of γ2 is smaller than the corresponding values obtained for nitrogen. As a matter of fact, any isotherm of adsorption on amorphous silica shows exactly the same trends. Meanwhile, if one is dealing with a mesoporous silica, a third power law regime beginning at the onset of capillary condensation is observed.16 In other words, the derivative dN/dδ of any isotherm of adsorption (15) Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1954; pp 10-16. (16) Ehrburger-Dolle, F. 5th International Symposium on Aerogels, September 8-10, 1997, Montpellier, France.
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Figure 4. Derivatives of the measured adsorbed amount N with regard to the mean free path λ in the gas-phase plotted in logarithmic coordinates as a function of δ (open circles, upper scale) and as a function of ((closed circles, lower scale), in the case of adsorption of argon on Aerosil 200.
on amorphous silicas consists of power law domains characterized by the following scaling equation:
dN/dδ ∝ δγ
(7)
with δ ) λ - λ0 and λ ) λ0(P/P0) ) λ0p . It follows that the new variable δ is proportional to a function f(p) ) (1 - p)/p: -1
-1
δ ) λ0 f (p)
(7′)
The uncertainty of the value of the exponents is estimated by changing the limits used for the least-squares fit of the data (logarithmic plot) and, whenever its possible, by running two or three independent adsorption measurements. It appears that the experimental error coming from adsorption measurement affects the value of N but has little effect on the exponents (the curves may be slightly shifted). Fluctuations of the values of γ resulting from the change of the limits within a realistic domain amount to (0.02 in the present case. Such a small fluctuation is due to a large number of experimental data. When the number of experimental points is smaller, the uncertainty may become close to (0.10. In almost all cases, the correlation coefficient r2 is larger than 0.99 and never becomes smaller than 0.98. Analysis of the curve of the residuals (not shown here) was performed in many cases. 4.2. Adsorption Isotherm Equations. Integration of eq 7 yields the following isotherm equation:
N ) K0 + K1δ+1
(8)
which is verified over the same range of δ as the derivative. The determination of K0 and K1 is obtained from the best fit of the data to the above equation by using the same method as the one described in paragraph 3, yielding the following equations:
for regime 1: N ) -0.54 + 6.17 δ-0.16 for regime 2: N ) 1.69 + 8.27 δ
-0.48
The corresponding curves are plotted in Figure 5a.
Figure 5. Isotherm of adsorption of nitrogen on Aerosil 200: (a) plotted as a function of δ, in logarithmic coordinates; (b) classical plot and fitting curves corresponding to eqs 9a and 9b for regimes 1 and 2, respectively.
Since the new variable δ is proportional to the function f(p) ) (1 - p)/p, eq 8 can also be written
N ) K0 + K1[λ0f(p)]γ+1
(8′)
Because f(p) is a decreasing function of p, it is more logical to consider 1/f(p) as the new variable when writing the general form of the fitting equation (eq 8) in explicit terms of relative pressure. It follow that
N ) K0 + K′1[1/f(p)]-(γ+1) ) K0 + K′1[(p/1 - p)]-(γ+1) (9) The classical plot of an adsorption isotherm N(p) and the fitting curves are shown in Figure 5b and are written as follows in the case of regimes 1 and 2:
(8a) (8b)
regime 1: N(p) ) -0.54 + 3.92[p/(1 - p)]0.16 (9a) regime 2: N(p) ) 1.69 -2.12[p/(1 - p)]0.48 (9b)
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In the low-pressure range (p , 1), eq 9a becomes, as expected, similar to eq 4, the value of the prefactors differing by less than 1.3%. Comments concerning the same negative constant were already made in paragraph 3. Analysis of the physical meaning of the constant K0 and of the prefactor K1 as well as rescaling is currently in progress. In a previous paper7 it was shown that the isotherm of adsorption into microporous solids can be described analytically by one or more equations (depending on the pore width) which are power laws of p up to saturation pressure (p ) 1). It follows that eq 8 or 9 involving δ, that is, (1 - p)/p, as a variable does not apply for microporous adsorbents. This feature is easy to explain because the size of the micropores ( 0.4, already plotted in Figure 1. However, the value of Ds calculated either by means of eq 5a, assuming a van der Waals wetting regime, or by means of eq 5b, assuming a capillary condensation wetting regime (Table 1), does not agree with the one measured by SAXS. Ismail and Pfeifer14 attributed this fact to the existence of an intermediate regime in which both mechanisms operate simultaneously. However, it could also result from a misleading power law fit. To compare more precisely the value of D deduced from the FHH analysis to the ones obtained by means of eq 12, the experimental derivative dN/dX is analyzed (Figure 6b). It appears that, within the same range of pressures as the one involved for the linear regression in Figure 6a, the data cannot be fitted to a single power law, as would be expected from the FHH equation, but to two different power laws: (i) For 0.4 < p < 0.62, the exponent µ1 equals -1.28; from the value of µ1 + 1 ) -0.28 and a realistic assumption of a van der Waals wetting regime (eq 5a), D1 ) 2.16 is inferred. It is equal to the D value determined in regime 1 over more than four decades by means of eq 12 and, therefrom, also to the fractal dimension of the silica particles measured by SAXS. (ii) For 0.62 < p < 0.98 (upper experimental value), the exponent µ2 equals -1.55, yielding D2 ) 2.45, if a capillary wetting regime (eq 5b) is assumed. This value also compares with the one obtained for regime 2. By using the same fitting procedure as above, the two equations fitting to N(X) are as follows:
for 0.40 < p < 0.62: N(X) ) 4.36X-0.28 - 1.14
(13a)
for 0.62 < p < 0.98: N(X) ) 1.71X-0.55 + 1.70
(13b)
Interestingly enough, the constant in eq 13b is almost the
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Figure 7. Comparison between the derivative curves obtained for adsorption of nitrogen on Aerosil 200 and on a series of carbon materials. For the sake of clarity, the value of the slope corresponding to each domain is rounded to the nearest first decimal, in agreement with the estimated upper value of the uncertainty ∆γ ) (0.1 (see paragraph 4.1).
Figure 6. Isotherm of adsorption of nitrogen on Aerosil 200: (a) classical FHH plot; (b) derivative of the measured adsorbed amount with regard to X ) ln(P0/P), plotted as a function of X, in logarithmic coordinates.
same as the one in eq 8b, but on the contrary to what would have been expected, it is not equal to but smaller than that for the BET monolayer. 5. Analysis of Isotherms of Adsorption of N2 on Carbon Materials Nitrogen adsorbed on basal plane surfaces of graphite provides an interesting system for investigating molecular orientational ordering on surfaces. Elastic neutron scattering was used to investigate the structure of nitrogen films adsorbed on Grafoil and the successive structural changes occurring during increasing coverage.19-21 Kjems et al.20 have shown that the observed transitions from one phase to another correlate well with the changes in the (19) Wang, R.; Wang, S.-K.; Taub, H.; Newton, J. C.; Shechter, H. Phys. Rev. B 1987, 35, 5841. (20) Kjems, J. K.; Passell, L.; Taub, H.; Dash, J. G.; Novaco, A. D. Phys. Rev. B 1976, 13, 1446. (21) Wang, S.-K.; Newton, J. C.; Wang, R.; Taub, H.; Dennison, J. R. Phys. Rev. B 1989, 39, 10331.
slope of the adsorption isotherm. Figure 7 (upper curve) shows the derivative dN/dδ of the isotherm of adsorption of nitrogen at 77 K, which was measured on the same Grafoil (kindly provided by H. Taub). As expected, this curve displays several domains. The location of the crossover between each regime will be compared to the one deduced from neutron scattering. Each domain can be fitted to a power law from which the exponent γ is determined. Because γ is an integer number for all regimes (except at higher covererage), D will be an Euclidean dimension, which is expected for ordered structures. Comparison between the results obtained for Grafoil and those obtained for graphitized and nongraphitized carbon blacks is made in order to show the effect of less and less ordered surfaces on the value of the exponents. Ultimately, Figure 7 displays what changes in the growth process of clusters adsorbed on a perfectly flat crystalline surface (Grafoil), on one hand, and on a fractal amorphous surface (aerosil), on the other hand. 5.1. Adsorption of Nitrogen on Grafoil. Up to p ) 2.6 × 10-4, the exponent γ is equal to -2; it corresponds to the exponent ν ) -2 - γ ) 0, in the pressure derivative curve. The equations fitting the data N(δ) in Figure 8a and N(p) in Figure 8b, within the same range of pressures, are respectively
N(δ) ) 8597 δ-1
(14a)
N(p) ) 505p with p , 1
(14b)
Equation 14b is the Henry law, which indicates also that the adsorbed molecules are isolated (gas phase). Thus, they do not form an assembly, and eq 12, from which D ) 3 would be inferred, does not apply.
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N(p) ) 0.22 - (2.3 × 10-5)p-1
(15b)
The limiting value NC ) 0.22 mmol g-1 agrees fairly well with that for the BET monolayer deduced from the fit of the first linear domain in the BET plot, beginning at p ) 2.6 × 10-4 and yielding Nm0 ) 0.25 mmol g-1 and a high BET constant (CBET ) 2141). The exponent characteristic of the third domain shown in Figure 7 for Grafoil is equal to -1. This regime, called here F and observed up to p ) 0.12 (N ) 0.28 mmol g-1) corresponds to the growth of a dense, two-dimensional phase. The ratio between the number of adsorbed molecules at the end of regime C (0.21 mmol g-1) and that corresponding to the end of this regime (0.28 mmol g-1) is equal to 1.3, which agrees with the coverage (θF ) 1.25) at which the surface film is completely incommensurate.20 These observations are consistent with the growth of a two-dimensional film on a perfectly flat surface for which D ) 2 is expected and actually obtained by means of eq 12. Regime F is characterized by the following equations:
N(δ) ) 0.37 - (1.7 × 10-2) ln(δ)
(16a)
N(p) ) 0.32 + (1.7 × 10-2) ln[p/(1 - p)] (16b)
Figure 8. Isotherm of adsorption of nitrogen on Grafoil: (a) plotted as a function of δ, in logarithmic coordinates along with the curves fitting to the data in regime H (continuous, eq 14a), regime C (dash-dottted, eq 15a), regime F (dash, eq 16a), regime H′ (continuous, eq 17a), and regime M (dash-dot-dot, eq 18a); (b) classical plot and fitting curves corresponding to eqs 15b, 16b, 17b, and 18b for regimes C, F, H′ and M, respectively, using the same line types as in (a). The line corresponding to the low-pressure regime H (eq 14b) is not shown.
For the second domain (Figure 7) the slope is close to zero, up to p ) 1.6 × 10-3, at which the number of adsorbed molecules is equal to 0.21 mmol g-1. This point corresponds to the foot of the substep in the isotherm, which was considered as corresponding to the completion of the commensurate monolayer phase, consisting of an in-plane herringbone structure denoted C.19 Equation 12 yields D ) 1, which is consistent with a linear arrangement of molecules. Within this regime, called here C also, the isotherm equation is
N(δ) ) 0.22 - 1.36 × 10-6δ
(15a)
or in explicit terms of relative pressure p and neglecting p as compared to 1 in the term (1 - p)
Interestingly, near the lower limit of the pressure range (1.6 × 10-3 to 0.12), where eq 16b is verified, N becomes a linear function of ln(p), that is, a linear function of the chemical potential. For this sample, the BET plot presents a second linear domain, beginning at the upper limit of the regime F range (p ) 0.12) and yielding Nm1 ) 0.33 mmol g-1 and a much smaller BET constant (CBET ) 21). It appears that the ratio Nm1/Nm0 ) 1.3 of the two values determined by the BET approach corresponds also to the same coverage θF. At relative pressures above 0.12 (corresponding in Figure 7 to δ ) 272 nm) and up to 0.38, dN/dδ scales as δ-2, as it did in the low-pressure range described by the Henry law. Therefore, I named this regime H′. It is likely that this regime describes the growth of the second layer in which the adsorbed molecules remain isolated (i.e., they do not form an assembly, as was the case in regime H). The kink observed at the end of this regime, which appears more clearly in the case of the graphitized carbon black (because of a larger number of experimental points), could be attributed to the organization of the molecules near completion of the second layer. Regime H′ is described by the following equation:
N(δ) ) 0.23 + 7.78 δ-1
(17a)
or expressed in terms of pressure as
N(p) ) 0.23 + 0.46p/(1 - p)
(17b)
Multilayer coverage is characterized by a more disordered arrangement,21 which agrees with the fact that, above p ≈ 0.42, the exponent becomes noninteger and yields, by means of eq 12, D ) 2.7. Within this regime, named M, the data plotted in Figure 8a can be fitted to the following equation:
N(δ) ) 0.26 + 1.74 δ-0.6
(18a)
and the equation fitting the data plotted as a function of pressure (Figure 8b) is
N(p) ) 0.26 + 0.32[p/(1 - p)]0.6
(18b)
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be attributed to limiting size effects: the size of the graphitic planes is significantly smaller in the case of graphitized carbon black, for which Raman spectroscopy yields 7.2 nm,22 than that for Grafoil (≈30 nm). In the low δ (or high-p) domain, which was not investigated for Grafoil, the absolute value of the exponent becomes larger than 2, yielding by means of eq 12 unrealistic D values larger than 3. Similar results are obtained in the case of capillary condensation into mesopores.16 Obviously, in such a case, the model yielding eq 12 does not apply any more because one is dealing with volume filling and no longer with surface coverage. In the case of the nonporous graphitized carbon black, one is dealing with here, this regime, named I in Figure 9a, begins when the number of adsorbed molecules equals 2.9 mmol g-1 that is, at a coverage θ ) 3.8 (the BET plot exhibits a single linear domain with NBET ) 0.76 mmol g-1 and CBET ) 575). Such a result is consistent with the existence of bulk clusters (or islands) above θ ) 3.7, on graphitic surfaces.21 However, because this domain also appears in the case of the non graphitized carbon black (Figure 7), an alternative explanation would be condensation in necks between primary particles. Such regimes will be investigated in detail elsewhere. The equations for the fitting curves (Figure 9) are as follows: (a) regime H (Henry law domain): p < 1.1 × 10-4
N(δ) ) 33057 δ-1
(19a)
N(p) ) 1944p (with p , 1)
(19b)
After normalization to the BET monolayer, it appears that the Henry constant obtained for the graphitized carbon black (KH ) 2558) is slightly larger than the one obtained for Grafoil (KH ) 2020). (b) regime C′: 1.1 × 10-4 < p < 4.2 × 10-3
N(δ) ) 0.76 - (1.39 × 10-3) δ0.5 -3
-0.5
N(p) ) 0.76 - (5.73 × 10 ) p
(20a)
(for p , 1) (20b)
In these equations, the limiting value is equal to the BET monolayer value (Table 1). (c) regime F: 4.2 × 10-3 < p < 5.7 × 10-2 Figure 9. Isotherm of adsorption of nitrogen on graphitized N330: (a) plotted as a function of δ, in logarithmic coordinates along with the curves fitting to the data in regime H (continuous, eq 19a), regime C′ (dash-dot, eq 20a), regime F (dash, eq 21a), regime H′ (continuous, eq 22a), regime C′′ (dash-dot, eq 23a), regime M (dash-dot-dot, eq 24a) and regime I (dash, eq 25a); (b) classical plot and fitting curves corresponding to eqs 20b, 21b, 22b, 23b, 24b, and 25b for regimes C′, F, H′, C′′, M, and I, respectively, using the same line types as in (a). The line corresponding to the low-pressure regime H (eq 19b) is not shown.
5.2. Adsorption of Nitrogen on a Graphitized Carbon Black (N330 Graphitized). Figure 7 shows that the curve obtained for the graphitized carbon black is similar to the one obtained for Grafoil, except in the domain of pressure ranging from 1.1 × 10-4 to 6.7 × 10-4, in which the exponent is now equal to -0.5, yielding, by means of eq 12, D ) 1.5. This pressure range is broader but overlaps the domain attributed to the growth of the commensurate phase (C) in the case of Grafoil, for which D was equal to 1. The value of D is now no longer integer and larger than it was for Grafoil. Instead of the lines characteristic of the herringbone ordering, the pattern shown by the adsorbed molecules becomes that of branching lines, which could
N(δ) ) 1.03 -(4.3 × 10-2) ln(δ)
(21a)
N(p) ) 0.91 + (4.3 × 10-2) ln[p/(1 - p)] (21b) (d) regime H′: 5.7 × 10-2 < p < 0.38
N(δ) ) 0.72 + 15.3δ-1
(22a)
N(p) ) 0.72 + 0.9p/(1 - p)
(22b)
(e) regime C′′: 0.38 < p < 0.48
N(δ) ) 1.79 - (2.11 × 10-2)δ0.95 N(p) ) 1.79 - 0.31[p/(1 - p)]
-0.95
(23a) (23b)
This regime occurs within the same range of pressure as the kink observed in the case of Grafoil. However, because of too low a number of experimental points, it was not possible to determine the fitting equation in the former case. (22) Gruber, T.; Zerda, T. W.; Gerspacher, M. Carbon 1994, 32, 1377.
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Figure 10. Isotherm of adsorption of nitrogen on nongraphitized carbon black N330: (a) plotted as a function of δ, in logarithmic coordinates along with the curves fit to the data in regime 1 (continuous, eq 26a), regime 1′ (dash-dot, eq 28a), regime 2 (dash, eq 30a), and regime I (dash-dot-dot, eq 32a); (b) classical plot and fitting curves corresponding to eqs 26b, 28b, 30b and 32b, for regimes 1, 1′, 2, and I respectively, using the same line types as in (a).
(f) regime M:
N(δ) ) 1.02 + 3.57δ-0.72
(24a)
N(p) ) 1.02 + 0.46[p/(1 - p)]0.72
(24b)
(g) regime I:
N(δ) ) 1.88 + 3.42δ-1.34
(25a)
N(p) ) 1.88 + (7.7 × 10-2)[p/(1 - p)]1.34 (25b) 5.3. Adsorption of Nitrogen on Carbon Blacks (Nongraphitized). For nongraphitized carbon blacks, the average size of the graphitic layers is less than 3 nm, that is, about 10 time smaller than that for Grafoil. This feature affects the growth of the adsorbed molecule assembly very significantly, as shown in Figure 7.
Figure 11. Isotherm of adsorption of nitrogen on carbon black TB#5500: (a) plotted as a function of δ, in logarithmic coordinates along with the curves fit to the data in regime 1 (continuous, eq 27a), regime 1′ (dash-dot, eq 29a), and regime 2 (dash, eq 31a); (b) classical plot and fitting curves corresponding to eqs 27b, 29b, and 31b for regimes 1, 1′, and 2, respectively, using the same line types as in (a).
(a) For large δ values (or low pressures), the Henry law domain is no longer observed; the absolute value of the exponent becomes close to the one obtained in the case of adsorption of nitrogen on amorphous silica particles characterized by a surface fractal dimension Ds equal to 2.16. From recent SAXS measurements23 performed on a series of carbon blacks, including N330 and TB#5500 investigated here, Ds ) 2.20 ( 0.05 is inferred. The values of D calculated by means of eq 12, which are equal respectively to 2.39 and 2.29, agree fairly well. As in the case of silica, this regime will be named regime 1. The equations fitting to the curves N(δ) and N(p) within this domain are for N330 (Figure 10) and TB#5500 (Figure 11), respectively, as follows: (23) Rieker, T.; Misono, S.; Ehrburger-Dolle, F. Langmuir 1999, 15, 914.
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N330: p < 4 × 10-4 N(δ) ) -0.03 + 19.2δ-0.39 N(p) ) -0.03 + 6.4p0.39 (with 0 < p , 1)
(26a) (26b)
TB#5500: p < 4 × 10-4 N(δ) ) -0.04 + 23.8δ-0.29 N(p) ) -0.04 + 10.5p0.29 (with 0 < p , 1)
(27a) (27b)
As already discussed in the case of Aerosil 200, eqs 26 and 27 are not verified when p f 0. Regimes C′ and F, observed for the graphitized carbon blacks, merge into a single domain (regime 1′), characterized by an exponent close to -0.9 for both samples. The fit to the data yields the following equations:
N330: 4 × 10-4 < p < 4.5 × 10-2 N(δ) ) 1.5 - 0.51δ0.081
(28a)
N(p) ) 1.5 - 0.64[p/(1 - p)]-0.081
(28b)
TB#5500: 4 × 10-4 < p < 4.5 × 10-2 N(δ) ) 3.37 - 0.81δ0.10
(29a)
N(p) ) 3.37 - 1.07[p/(1 - p)]-0.10
(29b)
In both cases, the constant in the above equations is larger than that for the BET monolayer (Table 1). As a matter of fact, the BET monolayer value corresponds to the crossover between this regime and the next one, located at p ) 4.5 × 10-2. The value of D ≈ 1.9 (eq 12), which is smaller than the topological dimension (Dtop ) 2), confirms the growth of a monolayer of adsorbed molecules, without piling-up (two-dimensional confinement). This situation is characterized by a negative exponent for p in the isotherm equation, as is the case for adsorption in narrow micropores when the size, expressed as molecular size units, is smaller than about 3.7 (b) For δ < 360 nm (i.e., p > 4.5 × 10-2), the derivative curve, plotted in Figure 7, exhibits shallow waves, which suggests that regimes H′, C′′, and M, observed in the case of larger graphitic domains, get smeared and give rise, as a first approximation, to a single domain, as in the case of Aerosil 200. This domain is described by the following isotherm equations:
N330: 4.5 × 10-2 < p < 0.89 N(δ) ) 0.50 + 2.71δ-0.47
(30a)
N(p) ) 0.50 + 0.72[p/(1 - p)]0.47
(30b)
TB#5500: p > 4.5 × 10-2 N(δ) ) 1.45 + 4.94δ-0.41
(31a)
N(p) ) 1.45 + 1.55[p/(1 - p)]0.41
(31b)
The fractal dimensions D determined by using eq 12 are equal to 2.47 and 2.41 for N330 and TB#5500, respectively. They are slightly larger than the ones obtained in regime 1 (2.39 and 2.29, respectively) but compare to that for Aerosil 200 in regime 2 (D ) 2.48). Since, for all three
Figure 12. Adsorption of nitrogen on carbon black N330 and TB#5500: (a) classical FHH plot, linear regression and curves fit to the data, deduced from the derivative curve shown in part b, in the van der Waals wetting regime (continuous lines and eqs 33a and 34a for N330 and TB#5500, respectively) and in the capillary condensation wetting regime (dashed lines and eqs 33b and 34b for N330 and TB#5500, respectively; (b) derivative of the adsorbed amount with regard to X ) ln(P0/P) plotted as a function of X and linear regression lines.
samples, a similar surface fractal dimension (Ds close to 2.2) is inferred, one may assume that the larger fractal dimension characterizing the growth of the adsorbed molecule assembly results from a realistic lowering of the molecule-solid interaction in the multilayer range. (c) Near the saturation pressure (p > 0.89), in the case of N330, the same behavior (regime I) as in the case of the graphitized N330 is observed. The equations describing this regime are the following:
N(δ) ) 1.77 + 1.80δ-1.20
(32a)
N(p) ) 1.77 + (4.0 × 10-2)[p/(1 - p)]1.20 (32b) 5.4. Comparison with the Results Obtained by Using the FHH Fractal Equation. The usual FHH plots
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obtained for the nongraphitized carbon blacks N330 and TB#5500 are shown in Figure 12a. The exponents deduced from a linear regression through the data above p ) 0.4 are collected in Table 1. In the case of N330, the comments made in paragraph 4.4 for Aerosil 200 do also apply. In the case of TB#5500, a more realistic value (Ds ) 2.10) is obtained when a van der Waals (VdW) wetting regime is assumed. As in the case of Aerosil 200, the derivative curves dN/dX (Figure 12b) cannot be fitted to a single power law within this pressure range. Fitting the first domain to a power law could be somewhat questionable, particularly in the case of N330. Tentatively, a linear regression is performed within the range of X, as indicated in Figure 12b. In this case, the slope is the same for both samples and yields Ds ) 2.37, with a realistic van der Waals regime assumption (eq 5a). The slopes are also equal within the second domain, in which a capillary condensation (CC) wetting regime is expected. Equation 5b yields almost the same surface fractal dimension Ds as the one deduced from the VdW regime. Equations fit to the data plotted in Figure 12a, within each domain, are as follows:
N330: 0.42 < p < 0.70 N ) 2.01[ln(1/p)]-0.21 - 0.93
(33a)
0.70 < p < 0.89 N ) 0.45 [ln(1/p)]-0.63 + 0.70
(33b)
TB#5500: 0.045 < p < 0.78 N ) 3.81[ln(1/p)]-0.21 - 1.1
(34a)
p > 0.78 N ) 0.67[ln(1/p)]-0.65 + 2.33
(34b)
In eq 33b, for N330, the value of the constant is close to that for the BET monolayer, whereas, in the case of TB#5500 (eq 34b), it is larger. The analyses of the FHH plots made in this paragraph for carbon blacks and in paragraph 4.4 for Aerosil 200 also raise a general comment concerning the fit to the power law by a linear regression of the data plotted in logarithmic coordinates. I have shown here that they may yield misleading exponents just because of the existence of finite limits. The only possibility to determine unmistakably an exponent value is to analyze the derivative (and thus to get a large number of data points for the derivative to be plotted safely). This is true in most cases when one is dealing with experimental objects or phenomena which are, by nature, limited by boundaries. The existence of such boundaries may prevent the asymptotic
regime from being reached but does not preclude the physical meaning of a power law which describes the system within the boundaries. 6. Conclusion The new way to look at adsorption isotherms in the case of solids displaying an external surface (type II isotherms), consisting in analyzing the curves obtained by plotting the adsorbed amount N no longer as a function of the relative pressure or its natural logarithm but as a function of length δ related to the mean free path of molecules in the gas phase, appears as a powerful method to get information about the mechanism of growth of the adsorbed clusters. In other words, the derivative of the adsorbed amount with regard to this length δ, plotted as a function of δ, can be considered as a fingerprint of a particular adsorbent-adsorbate system. It also allows one to describe analytically this growth, by means of power laws. As a consequence, the adsorption isotherm can be described by equations which all have the same analytical form, consisting of the sum of a power law term and a constant. The exponent in the power law is related to a fractal dimension which characterizes the arrangement of molecules during a given stage of the growth and which results from all phenomena involved during one particular stage. Obviously, these equations need now to be renormalized in order to display invariants of the systems. Beside the fact that experimental isotherms can be described by very simple equations, there is a new outcome concerning isotherms resulting from simulations. It is not unrealistic to predict that simulated isotherms will also be described analytically by the type of equations obtained empirically, as is actually the case for adsorption in micropores. It follows that comparison between simulated and experimental isotherms will become very simple. It follows also that the phenomena involved during the growth of real adsorbed molecule clusters will be known from the simulated one, described within a given regime, by the same fractal dimension from which, among others, the relative effects of geometrical and energetic heterogeneities will be inferred. Further work aiming to improve theoretical backgrounds of this new way to analyze adsorption isotherms, based up to now on experimental observations, is in progress. Acknowledgment. The author wishes to thank Olivier Barbieri, Montserat Bellido Gil, Julien Dallamano, and Fabien Ozil (ICSI Mulhouse), for performing adsorption measurements and Nicole Dupont (Universite´ Nancy II) and Haskell Taub (University of MissourisColumbia) for stimulating discussions concerning adsorption of nitrogen on graphite. LA981349R
