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Characterization of Microporous Carbons Using Adsorption at Near Ambient Temperatures Jacek Jagiełło,† Teresa J. Bandosz, and James A. Schwarz* Department of Chemical Engineering and Materials Science, Syracuse University, Syracuse, New York 13244-1190 Received January 29, 1996X The adsorption isotherms of CF4 and SF6 measured at various temperatures near ambient are used to obtain information about micropore structure of two commercial carbons. This information is extracted from the comprehensive analysis of the data on three levels of assumptions. First from the temperature dependence of adsorption the isosteric heats of adsorption are calculated using nonparametric regression. Heats of adsorption allow for qualitative comparison of adsorption energetics between different carbons. The more detailed description of this energetics is given in terms of the adsorption energy distributions. Finally the micropore size distributions are calculated from the adsorption energy distributions assuming that the adsorption energy of molecules confined in micropores is determined by their sizes and geometry. No assumptions are made about the analytical form of the distribution. Effects of model assumptions, such as gas-solid interaction parameters or the pore wall thickness, on the calculated micropore size distributions are discussed. In order to obtain better insight on pore size distribution and geometry of actual carbons the results of this work are compared with the results of an independent chromatographic method based on the exclusion effect.
Introduction The microstructure of activated carbons is a key factor in determining their use in practical applications. Characterization of meso and micropore structure is traditionally based on the analysis of nitrogen or argon adsorption isotherms measured at 77 K. The mesoporosity is evaluated from the isotherms in the pressure range from about 10 to 760 Torr, which is relatively easy to measure; whereas in order to study adsorption in micropores, it is necessary to perform measurements at very low pressures (e10-5 Torr). It is known that measurements at such low pressures require sophisticated equipment and are less accurate than those at higher pressures.1 The objective of the present paper is to extract information about carbon microporosity from adsorption isotherms of carbon tetrafluoride, CF4, and sulfur hexafluoride, SF6, measured at temperatures near ambient over the pressure range 1-760 Torr. Under these conditions, which are supercritical for CF4 and far from condensation for SF6, the adsorption takes place predominantly in micropores due to the enhancement of the adsorption potential in such pores.2 The advantage of such experimental conditions compared to conventional nitrogen isotherms at 77 K is that in order to study adsorption in micropores we avoid measurements at very low pressures. We analyze the adsorption data at three levels with increasing requirements for invoking model assumptions. First we use nonparametric regression to calculate the isosteric heats of adsorption with no assumptions of the adsorption model. Heats of adsorption reflect the magnitude of gas-solid and gas-gas interactions and are related to micropore sizes, geometry,3 and adsorbate properties. Second, we assume that carbons are hetero* Author to whom correspondence should be addressed. † Permanent address: Department of Fuels and Energy, University of Mining and Metallurgy, 30-059 Krako´w, Poland. X Abstract published in Advance ACS Abstracts, May 15, 1996. (1) Maglara, E.; Pullen, A.; Sullivan, D.; Conner, W. C. Langmuir 1994, 10, 4167. (2) Everett, D. H.; Powl, J. C. J. Chem. Soc., Faraday Trans. 1 1976, 72, 619. (3) Grillet, Y.; Llewellyn, P. L.; Reichert, H.; Coulomb, J. P.; Pellenq, N.; Rouquerol, J. In Characterization of Porous Solids III: Rouquerol, J., Rodriguez-Reinoso, F., Sing, K. S. W., Unger, K. K., Eds.; Elsevier: Amsterdam, 1994; p 525.
S0743-7463(96)00093-5 CCC: $12.00
geneous from the point of view of adsorption energy and we calculate the adsorption energy distributions which characterize this heterogeneity. In our final analysis it is assumed that the adsorption energy of molecules confined in micropores is dependent on pore sizes and the micropore size distributions are derived from the calculated adsorption energy distributions. Unlike in earlier works,4,5 no assumptions are made about the analytical form of the distribution. Analysis of Adsorption Data Nonparametric Regression (NPR). The NPR is used to simultaneously fit adsorption isotherms measured at different temperatures. This procedure was recently applied to describe adsorption of alkanes and alkenes on silica samples.6 Here it is used to assess the magnitude of the experimental error in the data and to calculate the isosteric heat of adsorption, Qst, in a model independent manner. Briefly, it is assumed that over a limited range of temperatures the isosteric heat of adsorption, Qst, is temperature invariant
[(
)]
∂ ∂ ln p ∂T ∂(1/T)
)0
(1)
v
where p, v, and T are pressure, the amount adsorbed, and the temperature, respectively. This assumption is commonly applied whenever isosteric heats are calculated from adsorption data. Integrating eq 1 twice, one obtains the general relationship7
ln p ) g(v)/T + f(v)
(2)
where both functions g(v) and f(v) are dependent only on the amount adsorbed. When these functions are represented by polynomials a virial type of equation is (4) Jagiełło, J.; Schwarz, J. A. Langmuir 1993, 9, 2513. (5) Jagiełło, J.; Bandosz, T. J.; Putyera, K.; Schwarz, J. A. J. Chem. Soc., Faraday Trans. 1995, 91, 2929. (6) Jagiełło, J.; Bandosz, T. J.; Putyera, K.; Schwarz, J. A. In Fundamentals of Adsorption V; LeVan, M. D., Ed.; in press. (7) Czepirski, L.; Jagiełło, J. Chem. Eng. Sci. 1989, 44, 797.
© 1996 American Chemical Society
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obtained.7-9 However, due to the numerical instability of higher order polynomials, the virial equation may only be used for certain shapes of isotherms whose accurate fit does not require polynomials of higher order. This approach restricts the class of applicable isotherms and may also introduce artifacts. Here we propose to use more stable and flexible representations for g(v) and f(v) which are obtained by the linear combinations of cubic B-splines10
ln p )
1
m
m
cjφj(v) ∑bjφj(v) + ∑ Tj)1 j)1
(3)
where φj are cubic B-spline functions and bj and cj are the coefficients. To describe the data by eq 3, we apply a procedure analogous to that used for fitting a single curve.11 This procedure requires an estimate of the optimal smoothing parameter which controls the degree of smoothing the data. For this purpose we use the socalled generalized cross-validation method.12 With coefficients established from the fit of eq 3, it is straightforward to evaluate the isosteric heat of adsorption from the expression m
bjφj(v) ∑ j)1
(4)
Qst ) -R
where R is the universal gas constant. Adsorption Energy Distribution. The quantitative description of adsorption energy heterogeneity is usually given by the adsorption energy distribution (AED). The AED characterizes the adsorbate-adsorbent system and may be evaluated from the experimental adsorption isotherm, v. The relationship between v and the unknown AED, χ, is given by the integral equation
v(p,T) )
∫Rβθ(p,T,q) χ(q) dq
K θ ωθ + q θ(q,p,T) ) 1 + exp p 1-θ RT
(
q(x) ) up*(x)
)]
∂q ∂x
Here we apply the same general approach assuming the slitlike model for the shape of carbon micropores.2,16 The potential, up, of interaction between a gas molecule and a pore is given in this case as a sum of potentials, us, of gas-solid interactions with single walls
up(z) ) us(z) + us(x - z)
(9)
where z is the distance of the molecule from the surface atoms nuclei of one of the pore walls which are separated by the distance x. As a pore wall we consider a graphite crystallite consisting of n parallel planes.17 The potential of interaction of a molecule with the wall is given by the sum of contributions of Lennard-Jones (LJ) potentials integrated over a plane n
us(z) ) 2πFsgsσ2gs
(6)
(8) Bandosz, T. J.; Jagiełło, J.; Schwarz, J. A. Langmuir 1993, 9, 2518. (9) Jagiełło, J.; Bandosz, T. J.; Putyera, K.; Schwarz, J. A. J. Chem. Eng. Data 1995, 40, 1288. (10) De Boor, C. A Practical Guide to Splines; Springer-Verlag: New York, 1978. (11) Silverman, B. W. J. R. Statist. Soc. B 1985, 47, 1. (12) Craven, P.; Whaba, G. Numer. Math. 1979, 31, 377. (13) Ross, S.; Olivier, J. P. On Physical Adsorption; Interscience Publishers: New York, 1964. (14) Jagiełło, J. Langmuir 1994, 10, 2778-2785.
(8)
φ(x) ) χ(q)
-1
where p and T are experimental temperature and pressure, K is the pre-exponential factor of Henry’s law constant, and ω is a parameter that represents the average interaction energy between adsorbate molecules. To solve equation 5 with respect to χ(q) we apply the numerical method, SAIEUS, described elsewhere.14 The AED, χ(q), is represented in this method by a linear combination of B-spline functions.10 The problem is solved using regularization combined with non-negativity con-
(7)
of the gas-solid interaction potential, up, is based on the assumed pore model. Using function q(x) and the calculated AED, one can evaluate the micropore size distribution (MPSD), φ(x), from the relationship
(5)
where θ(p,T,q) is the so-called local adsorption isotherm which describes adsorption on surface sites having adsorption energy q. The function θ is the kernel of the integral equation and its mathematical form represents the accepted model of adsorption. The Hill deBoer (HdB) isotherm is based on the mobile adsorption model which is usually assumed for higher temperatures.13 Thus under our experimental conditions we use the HdB adsorption isotherm
[
straints. The choice of the optimal regularization/ smoothing is based on the analysis of a measure of the effective bias introduced by the regularization procedure and a measure of uncertainty of the solution. Such an approach ensures that the maximum available information is extracted from the data while avoiding artifacts. Relationship between Energies of Adsorption and Pore Sizes. It was proposed by Everett and Powl2 that one can evaluate micropore sizes of carbons with uniform pore structure using adsorption data measured in Henry’s law region and adsorption potentials calculated for model micropores. This approach was later extended4,15 to the analysis of adsorption data on heterogeneous carbons measured over a broader range of pressure. The basic assumption in this approach is that the energy of adsorption in micropores is determined by their sizes. The relationship between the pore size, x, and the adsorption energy, q, defined by the well-depth, up*
[(
)
σgs
2
∑ j)1 5 z + (j - 1)d
(
10
σgs
)] 4
(10)
z + (j - 1)d
where Fs and d are the density of graphite and the separation distance of its lattice planes and gs and σgs are the Lennard-Jones (LJ) potential parameters for interactions between a gas molecule and a carbon atom in the graphite lattice. For an infinite number of lattice planes, eq 10 can be approximated by18
us(z) )
[(
2πFsgsσ2gsd
) ( )
2 σgs 5 z
10
σgs z
σ4gs
4
-
]
3d(0.61d + z)3 (11)
(15) Jagiełło, J.; Schwarz, J. A. J. Colloid Interface Sci. 1992, 154, 225. (16) Stoeckli, H. F. Carbon 1990, 28, 1. (17) Mays, T. J.; Seaton, N. A.; McEnaney, B. Extended Abstracts, Carbon ’94 1994, p 244. (18) Steele, W. A. Surf. Sci. 1973, 36, 317.
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Figure 1. Comparison of selected experimental isotherms of CF4 and SF6 on carbons G and M.
Figure 2. Adsorption isotherms of CF4 on carbon G: experiment (open symbols); fit by NPR, eq 3 (dotted lines); fit by SAIEUS, eq 5 (solid lines).
Table 1. Results of Fitting to the Experimental Data of Equation 3 Using Nonparametric Regression, NPR, and Equation 5 Using the SAIEUS Procedure NPR
SAIEUS
sample
σ (mmol/g)
δ (mmol/g)
ln(K0) (Torr)
δ (mmol/g)
G-CF4 G-SF6 M-CF4 M-SF6
0.016 0.019 0.003 0.027
0.014 0.017 0.002 0.024
14.5 15.9 13.4 14.4
0.018 0.020 0.004 0.025
The function q(x) and its derivative used in eq 8 are evaluated numerically from the up potential. Results and Discussion For our study we chose two carbons: Carbosieve G (G) (Suppelco) and Maxsorb (M) (Kansai Coke and Chemicals Inc.). Both carbons are microporous with micropore volumes of about 0.5 and 0.8 cm3/g, respectively. According to the manufacturers Carbosieve G is a molecular sieving carbon and Maxsorb has pores predominantly of radius in the range of 10-20 Å. Adsorption isotherms of CF4, and SF6 measured for these carbons at three or four temperatures near ambient over the pressure range 1-760 Torr were reported in tabular form elsewhere.9 In Figure 1 we compare selected isotherms of both gases for both carbons. In the first step of our data analysis we describe adsorption isotherms by eq 3 using NPR. In the fitting procedure we consider weighted deviations of the curve from the experimental data. The statistical weight of an experimental point is inversely proportional to the standard deviation of the measurement error. On the basis of the method of the measurement, in our case we assume a constant error variance in the measured v values. To account for this error we introduce weights, w (scaling factors), which are calculated as a function of v
(∂ ln∂v p)
-1
w(v) )
(12)
Since the function ln(p) vs v is not known a priori, the fitting is performed iteratively. Assuming all wi ) 1 an initial fit is obtained and then eq 12 is applied. Usually after two or three steps this process converges. The NPR can also provide an estimate of an error variance, σ2, of the data. We estimate σ2 using a method based on the concept of the “equivalent degrees of freedom” proposed by Wabha.19 In Table 1 the standard deviations, σ, and the root mean square (rms) errors of fit, δ, for all systems (19) Wahba, G. J. R. Statist. Soc. B 1983, 45, 133.
Figure 3. Adsorption isotherms of SF6 on carbon M: experiment (open symbols); fit by NPR, eq 3 (dotted lines); fit by SAIEUS, eq 5 (solid lines).
are reported. To illustrate the goodness of the fit using eq 3, the fitted curves are compared with experimental data in Figures 2 and 3 for two adsorption systems. As a result of our NPR fitting of isotherms the isosteric heats of adsorption, Qst, are calculated according to eq 4. This approach enables us also to evaluate the uncertainty of the calculated Qst values using the law of propagation of errors and the estimated σ of the data. In Figure 4 we plot the variations of Qst vs v for all systems indicating uncertainty in the form of error bars which represent standard deviations in the calculated Qst values. It is seen that the Qst values are considerably higher for carbon G than those for carbon M for both adsorbates. Heats of adsorption reflect the magnitude of the combined gassolid and gas-gas interactions. Assuming that the gasgas component is of the same order of magnitude for both carbons, it follows from Figure 4 that the average gassolid adsorption potential is higher for carbon G. The enhancement of adsorption potential is related to the pore sizes as well as to their shapes.2,20 For example, this enhancement is greater in cylindrical pores compared to that in slitlike ones since the curvature of the surface results in an effective increase in the density of interacting centers in the solid.2 In other words the more the adsorbate molecule is surrounded by the solid, the higher is its adsorption potential. If pore size is taken to mean some average measure of the distance between the pore (20) Biba, V.; Spitzer, Z.; Kadlec, O. J. Colloid Interface Sci. 1979, 69, 9.
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Figure 5. Adsorption energy distributions, AED. Figure 4. Isosteric heats of adsorption calculated from the adsorption model using eq (14) (solid lines) and from the NPR using eq 4 (dotted lines), error bars represent standard deviations of the calculated values.
walls, we can infer that a higher adsorption potential corresponds to a smaller pore. This proposal justifies a qualitative comparison between the carbons microporosity; in the case of our two carbons the conclusion is that carbon G has smaller pores than carbon M. To characterize energetics of adsorption in more detail, we calculate AED from eq 4 using the HdB equation as the local isotherm. The HdB equation contains two nonlinear parameters K and ω. The latter parameter can be estimated according to Steele21 as ω ) 2.4gg. The preexponential factor of Henry’s law constant, K, is weakly temperature dependent, and for the mobile adsorption model this dependence is given by22
K ) K0T1/2
(13)
where K0 is a temperature independent constant related to the force constant of vibration perpendicular to the surface. In order to evaluate K from the data, it is necessary to use isotherms obtained at different temperatures because otherwise parameter K and energy q in eq 6 are grouped together and it is impossible to separate them. Knowledge of K is important since this parameter is responsible for the position of the AED on the energy scale. On the other hand the shape of the AED is not dependent on K. This property allows for the following approach which we apply here. First we calculate AED from one isotherm using the SAIEUS procedure with an arbitrary K0. The resulting AED is generally shifted compared to the “true” AED by a certain value on the energy scale. Next we adjust K0 by fitting the initial AED to all available isotherms of the system. Having K0 we apply the SAIEUS procedure again to all isotherms simultaneously. This process may be repeated iteratively; we found no significant changes in the solution after the first cycle. The results of SAIEUS analyses are reported in Table 1 where δ is the rms error for the simultaneous fit of eq 5 to all isotherms of the system. The isotherms calculated from this equation are added to Figures 2 and 3 to compare with the data and with the curves obtained from NPR. The results show that the fit using SAIEUS is almost as good as that obtained by using NPR which indicates that (21) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon: Oxford, 1974. (22) Jaroniec, M.; Sokołowski, S.; Rudzin´ski, W. Z. Phys. Chem. (Leipzig) 1977, 258, 818.
the adsorption model represented by eq 5 is consistent with the data. The agreement is also seen (Figure 4) between the isosteric heats of adsorption obtained from NPR and those calculated from the equation
∫Rβ(∂θ/∂T) χ(q) dq/∫Rβ(∂θ/∂ ln p) χ(q) dq
Qst ) -RT2
(14) based on the assumed adsorption model. Figure 5 presents the AEDs of both gases for both carbons. The average adsorption energy follows the same trend as the average Qst value. The adsorption energies of both gases are significantly higher for carbon G than for carbon M. The description of adsorption energies in terms of AED is more precise than that based on the isosteric heat of adsorption since the adsorption energy, q, is related only to gas-solid interactions. However, the separation of the gas-solid from the gas-gas interactions is achieved here by assuming the adsorption model given by eqs 5 and 6 which makes the AED description model dependent. Before discussing our approach to evaluate the micropore size distribution, we digress to mention a direct method for calculation of pore size distributions (PSD) proposed by Seaton et al.23 In this method the PSD is calculated from the following integral equation
v(p) )
∫xx
max
min
φ(x) F(p,x) dx
(15)
where F(p,x) is the fluid density in a single pore of size x. The density function, F, which is the kernel of this equation, is calculated for assumed pore models using the methods of statistical mechanics. Since the original work23 where the local density functional theory was used, the methods for evaluating F(p,x) have been in continuous refinement. More advanced versions of density functional theory are applied to corroborate and/or improve the results which are verified using computer simulations.24,25 The direct methods based on eq 15 are very promising; however, they are model dependent and require extensive numerical calculations. Our approach is also model dependent but computationally more simple, and although less rigorous it provides clear insight on the relationship between adsorption (23) Seaton, N. A.; Walton, J. P. R. B.; Quirke, N. Carbon 1989, 27, 853. (24) Lastoskie, C.; Gubbins, K. E.; Quirke, N. Langmuir 1993, 9, 2693. (25) Olivier, J. P.; Conklin, W. B.; Szombathely v., M. In Characterization of Porous Solids III; Rouquerol, J., Rodriguez-Reinoso, F., Sing, K. S. W., Unger, K. K., Eds.; Elsevier: Amsterdam, 1994; p 81.
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Figure 6. Variations of the CF4 and SF6 adsorption energy (potential well depth) as a function of the effective pore width calculated for the pores of infinitely thick walls using eq 11. The LJ potential parameters are given in Table 2.
Figure 7. Variations of the CF4 adsorption energy (potential well depth) as a function of the effective pore width calculated for the pores of different wall thickness, n, using eq 10.
Table 2. Parameters of the Gas-Gas and Gas-Solid Lennard-Jones Interaction Potential molecule
gg/k (K)
σgg (Å)
gs/k (K)
σgs (Å)
CF4 SF6(1) SF6(2)
152 201 155
4.70 5.51 5.46
65.2 74.8 65.9
5.05 4.45 4.43
energetics and pore structure. The MPSD is obtained in two steps: first the AED is calculated from integral eq 5, where the theoretical kernel is used, and then the MPSD is evaluated using relationships (7) and (8). This method, however, can be applied only to the adsorption in micropores of molecular dimensions because then the gassolid interaction potential plays the dominant role. Assuming the slitlike pore model we also consider different thicknesses for the pore walls.17 For this model the variations of the adsorption energy versus pore width can be calculated using eqs 10 or 11. The LJ parameters used in these calculations are given in Table 2. The LJ gas-solid parameters are obtained from the parameters of gas-gas26 and C-C in graphite21 according to the Lorentz-Berthelot rules. Reference 26 lists one set of parameters for CF4 but several sets for SF6. To address the problem of an ambiguity in the choice of the appropriate LJ parameters, we consider in the case of SF6 two widely different sets of parameters. The plots of the adsorption energy, q, versus effective pore width, h, presented in Figure 6 are obtained for pores with infinitely thick walls, eq 11. The effective pore width, h, is defined as the distance between the surface atoms nuclei corrected for the diameter of a carbon atom in the graphite lattice
h ) x - 3.4 Å
Figure 8. Micropore size distributions of carbon G calculated from the corresponding AEDs of CF4 and SF6 assuming infinitely thick pore walls.
(16)
The effect of the pore wall thickness on the adsorption energy of CF4 is illustrated in Figure 7. The greatest difference in energies is observed for pores of wall thickness between one and two graphite planes. Adsorption energies in pores with walls thicker than three graphite planes are almost equal to those of infinitely thick walls. Since the function q(h) is used as a link between AED and MPSD, it is important to note that it has the following properties regardless of the parameterization and assumed thickness of pore walls. The maximum value of q is about twice as large as that for a flat surface and it corresponds to the pore width, h, approximately the value of the diameter of the adsorbed molecule. For increasing h, after (26) Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1964.
Figure 9. Micropore size distributions of carbon M calculated from the corresponding AEDs of CF4 and SF6 assuming infinitely thick pore walls.
the maximum, the decline in adsorption energy is initially steep but becomes independent of pore width for pores wider than double the diameter of the adsorbed molecule. It follows that the calculated MPSD will be more affected by model and experimental errors for larger compared to smaller pores. In Figures 8 and 9 we present the MPSDs for both carbons calculated from the corresponding AEDs of CF4 and SF6 assuming infinitely thick pore walls. Comparing the MPSDs for the two carbons, we obtain consistent
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Figure 10. Effect of pore wall thickness on the micropore size distributions of carbon G calculated from the CF4 results.
Figure 11. Effect of pore wall thickness on the micropore size distributions of carbon M calculated from the CF4 results.
information, regardless of the adsorbate used; carbon G is characterized by a very narrow MPSD centered between 6 and 7 Å whereas the MPSD for carbon M is broader and centered in the range of 8-10 Å. Comparing results obtained for different adsorbates, we find better agreement between maxima in MPSDs calculated from CF4 and SF6 when for SF6 the second set of LJ parameters is used. On the other hand, in the case of carbon M the first set of LJ parameters for SF6 gives better agreement in the shapes of the distributions calculated for both adsorbates. It was shown recently using GCEMC simulations17 that the variation in pore wall thickness may have a significant effect on adsorption in micropores and thus should be considered in the analysis of micropore structure. The simulated nitrogen density isotherms in micropores are shifted to lower pressures as wall thickness increases.17 It follows that calculating MPSD from the adsorption isotherm using eq 15, one will observe a shift of the distribution to wider pores when thicker pore walls are assumed. In our approach this effect is a consequence of the increase in adsorption energy, for a given pore width, with the pore wall thickness (Figure 7). To illustrate this effect we present in Figures 10 and 11 the MPSDs of both carbons calculated for different pore wall thicknesses from the CF4 results. The greatest difference in the calculated MPSDs is observed between the models assuming walls with one and two graphite planes. This difference is especially pronounced in the case of carbon M which has lower adsorption energies than carbon G. In general, the calculated MPSD in the range of lower energies, where q(h) (Figure 7) is flatter, is more sensitive to an uncertainty of the model than that in the range of higher energies. It is beyond the scope of this paper to discuss if a unique model for pore structure is appropriate for characterizing microporous carbons. The actual carbons may contain pores which are heterogeneous from the point of view of both their width and wall thickness. Further investigations, using independent methods, are necessary to obtain additional insight on pore size distribution and geometry found in the actual carbons. One such method was applied to the carbons described in the present study. These carbons were studied27 using gas chromatographic measurements of the retention times of branched alkanes of different critical molecular diameter. Thus their size becomes the major factor for discriminating between pore accessibility. It was found on the basis of this exclusion effect that in the case of carbon G about 40% of the surface of all pores larger than 3.8 Å belong to pores smaller than
6 Å whereas the fraction of the surface in the same range of pore sizes is negligible in the case of carbon M. The result for carbon G is in very good agreement with the MPSD calculated for this carbon from the CF4 adsorption data with the assumption that pore walls are thicker than one graphite plane. It is seen in Figures 8 and 10 that about half of the MPSDs are below 6 Å. In addition, this result in comparison with Figure 10 demonstrates that the model of a single graphite plane for a pore wall is less likely in this case because for this model almost 100% of the pores would have a width below 6 Å. For carbon M the chromatographic result is consistent with the MPSDs presented in Figures 9 and 11, which show a negligible portion of the distribution below 6 Å. It is, however, inconclusive with regard to the problem of pore wall thickness. The MPSDs obtained from SF6 adsorption cannot be compared with the chromatographic results because SF6 molecules do not probe pores smaller than 5.5 Å.
(27) Jagiełło, J.; Bandosz, T. J.; Schwarz, J. A. Carbon 1994, 32, 687.
Conclusion We demonstrate that adsorption isotherms of such gases as CF4 and SF6 measured at various temperatures near ambient provide useful information about carbon micropore structure. This information is extracted from comprehensive analysis of adsorption data using three levels of analysis with increasing number of imposed assumptions. First, from the temperature dependence of adsorption the isosteric heats of adsorption are calculated using nonparameteric regression. Heats of adsorption which represent combined effects of gas-solid and gasgas interactions allow for qualitative comparison of adsorption energetics between different carbons. The more detailed description of these energetics is given in terms of the adsorption energy distribution whose calculation requires an assumption of the adsorption model. Finally the micropore size distribution is calculated from the adsorption energy distribution assuming that the adsorption energy of molecules confined in micropores is determined by their sizes and geometry. It is shown how the calculated micropore size distribution depends on several model assumptions such as gassolid interaction parameters and the pore wall thickness. Further investigations using independent methods are necessary to establish how well the assumed structural and thermodynamic models represent the actual carbons. LA960093R
