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Comparison of Methods to Obtain Micropore Size Distributions of Carbonaceous Materials from CO2 Adsorption Based on the Dubinin-Radushkevich Isotherm Moise´s L. Pinto,* Ana S. Mestre, Ana P. Carvalho, and Joa˜o Pires Department of Chemistry and Biochemistry, and CQB, Faculty of Sciences, UniVersity of Lisbon, Ed. C8, Campo Grande, 1749-016 Lisboa, Portugal
Micropore size distributions of various carbonaceous materials were obtained using two methods based on the Dubinin-Radushkevich equation and the relationship between the characteristic parameter of this equation and the pore width. One method was the well-known Dubinin-Radushkevich-Stoeckli (DRS) equation, and the other method used the integral adsorption equation without assuming a mathematical distribution for dw0/dE0. These methods were applied to carbon dioxide adsorption data on four activated carbons and eight chars to obtain micropore size distributions. Significant differences were found between the results obtained by the two methods. The DRS method gives an average of the distribution obtained by the method using the integral adsorption equation, because the latter is not constrained to a particular preassumed shape. The results reveal that the differences are more significant for the analysis of adsorption data on chars than on activated carbons. This was attributed to the fact that a less-homogeneous pore structure is expected to exist on chars than on activated carbons that cannot be described by the DRS method and can be described by the method based on the integral adsorption equation. Introduction The characterization the porous structure of activated carbons is often based on the physical adsorption of suitable gases or vapors.1,2 However, when the porous structures of these materials have very narrow micropores (0.30-0.45 nm), standard low-temperature nitrogen adsorption is not appropriate to correctly characterize them, because of diffusion effects at 77 K. A common alternative is to use CO2 adsorption at 273 K, where these kinetic effects are virtually absent. When preparing activated carbons from different precursors or using different conditions, it is useful to compare them by their pore dimensions. This information is important to correlate the pores dimensions with other observed properties, and to define possible applications for the different materials. Carbon materials that are less activated usually have low pore volume and also narrower pores than the more-activated materials.1,2 Chars that are only carbonized without the presence of an activating agent to promote the porosity development, have still narrower pores than activated carbons. In these cases, the porosity characterization by gas adsorption is only possible if the temperature used in the experiments is sufficiently high to minimize the diffusion constraints, allowing the effective determination of equilibrium isotherms. Thus, as stated previously, the CO2 adsorption at 273 K is then a good choice for characterizing these types of samples. During the last decades, the Dubinin-Radushkevich (DR) equation was applied to describe adsorption results on activated carbons. In this work, we discuss the application of this equation to access the pore dimensions, namely, the pore size distribution of carbon samples, using two different methodologies. One is fitting the Dubinin-Radushkevich-Stoeckli (DRS) and the other is fitting the integral adsorption equation to the adsorption data. These methods should be capable of describing various types of distributions. However, as shown below, the method that is chosen to solve the integral adsorption equation and obtain the pore size distribution may constrain the obtained * To whom correspondence should be addressed. Fax: (351) 217500088. E-mail:
[email protected].
results. The analysis of CO2 adsorption results at 273 K on different activated carbons and chars are discussed as examples, and they show that the methodology used in the analysis strongly influences the conclusions. Recently, more-elaborate methods to obtain pore size distributions based on density functional theory (DFT) have been developed, but only few works in the literature are related with CO2 on carbon materials.3,4 These methods are much more difficult to implement, because of their mathematical complexity, and researchers often prefer to use older methods. In fact, the DRS method is still used very much by researchers, and the comparison with the integral adsorption method presented in this work is relevant in this context. To the best of our knowledge, this is the first time that this comparison has been made. Theory The CO2 adsorption isotherms on activated carbons have been successfully described by the Dubinin-Radushkevich (DR) equation.5,6 The development of the DR equation was based on the Polanyi potential theory and has the form
[ ( )]
w(A) ) w0 exp -
A βE0
2
(1)
where A is the adsorption potential (A ) -RT ln (p/p0)), calculated as the compression work of the adsorptive using, as a reference state, the saturated pressure at the experiment temperature (p0 ) 3411.7 kPa for CO2 at T ) 273 K). Therefore, eq 1 presents a relationship between the adsorbed volume w and adsorption potential A. The other parameters are constants that are related to either the adsorbent material or the adsorptive. The w0 parameter represents the limiting adsorption volume filling the micropores, and the parameter β (affinity coefficient; β ) 0.35 for CO2) depends on the adsorptive. The value of E0 (characteristic energy) depends on the porous solid and is related to the micropore width L in activated carbons by
10.1021/ie100080r 2010 American Chemical Society Published on Web 04/09/2010
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-1
L (nm) )
10.8 nm kJ mol E0 - 11.4 kJ mol-1
(2)
Recent comparison with other experimental techniques has supported this relationship.7,8 The combination of eqs 1 and 2 allows one to obtain an average pore width from the analysis of CO2 adsorption data. Nevertheless, for activated carbons with a heterogeneous micropore structure, eq 1 is often insufficient to describe the experimental adsorption isotherm. One alternative is to use an integral adsorption equation of the form1,5,9 w(A) )
∫
+∞
0
[ ( )]
dw0 A exp dE0 βE0
2
(3)
dE0
where it is assumed that the material is formed not only by one type of pore, but by a continuous series of micropores with different w0 values for each E0 value, with a distribution given by dw0/dE0. In this case, dw0/dE0 can be converted to the micropore volume distribution dw0/dL, using eq 2. The problem now becomes how to calculate dw0/dE0 from experimental w(A), which is a typical inverse Laplace transform problem. Assuming a Gaussian distribution for dw0/dE0, eq 3 leads to a known Laplace transform and the so-called DRS equation is obtained:9,10
( ) {
w ) w00 exp(-B0y) exp
[(
) ]}
B0 ∆ y2∆2 0.5 1 - erf y - 2 2 ∆ √2
(4)
where w00 is the total micropore volume, B0 and ∆ are parameters related to the E0 distribution, and y ) [(T/β) log(p0/p)]2. The w00, B0, and ∆ parameters are usually estimated by fitting eq 4 to the adsorption data and used in the calculation of dw0/dB by
[
w00 (B0 - B)2 dw0 ) exp dB 2∆2 ∆√2π
]
(5)
where B is related to E0 by the relation E0 ) 0.001915(1/B)1/2. The gamma distribution was also considered and used in eq 3,7,8,11 and, similar to the Gaussian distribution, it was chosen because it leads to a known Laplace transform. Still, it gives similar results to those of eq 4 with analogous bell-shaped and one-peak distributions. Other distribution function were also attempted, including linear combinations of simple distribution functions.12 The alternative approach presented here is to calculate dw0/dE0 from the experimental data w(A) without assuming a mathematical distribution, that is, without imposing an “a priori” general shape for the micropore distribution. This may be achieved by converting eq 3 into a discrete form of the integral adsorption equation: m
w(A) )
∑w
0i
i)1
[ ( )]
exp -
A βE0i
2
(6)
in which the distribution simply becomes a summation of Dirac delta functions (δ): dw0 ) dE0
m
∑w
0iδ(E0
- E0i)
(7)
i)1
Thus, if an estimation of a given set of (w0i, E0i) pairs that describes the experimental adsorption data is obtained, a discrete distribution of w0 with L (i.e., dw0/dL) could be constructed using eq 2. The
Figure 1. (a) Set of DR isotherms used to fit the experimental data (dotted lines; only 10 are shown for simplicity); (b) the result of a fit to experimental data (open circles), along with the resulting isotherm (solid line) and the individual DR equations with the fitted w0i values, which constitute eq 6. The adsorbed volume (w) is presented as a function of the relative pressure (p/p0), as in standard adsorption isotherms.
method followed for the fit of eq 6 was to setup a matrix of 20 DR functions (see Figure 1a)), by fixing 20 E0i values (to obtain L values equally distributed between 0.35 nm and 2.00 nm), and estimate the corresponding w0i values (w0i g 0; broken lines in Figure 1b) that minimize the deviations to the experimental data (see the solid line in Figure 1b). This approach is similar to those followed by Linares-Solano et al. for high-pressure adsorption results,13,14 although they did not use a regularization parameter as described below. It must be also stressed that the present work presents the first comparison between results obtained using the DRS method and that obtained using the fitting of the integral adsorption equation, namely, when it is applied to various carbonaceous samples. This integral adsorption method approach was been rarely presented in the literature, probably due to the difficulties associated with the fit of the integral equation to the experimental data. Methods that involve only the fit of one equation, such as the DRS method, are usually preferred. However, the fit of the integral equation can be effectively made in a conventional computer spreadsheet file, as will be shown below. Also note that, although the pores above 0.7 nm are not filled at 0.03p/p0 (a pressure that is equal to the atmospheric pressure), they may effectively contribute to the adsorbed amounts up to this pressure. In fact, the lower six DR functions in Figure 1a correspond to pores with diameters in the range of 0.9-2.0 nm, and thus information on this pore size range may be obtained from the CO2 adsorption results at 273 K up to atmospheric pressure, as the example at the end of this section will show. For a successful fit of eq 6 to experimental data, some important details must be considered. The estimation of the combination of isotherms that describes a particular experimental curve can be regarded as a deconvolution process, which is known to be unstable, with respect to small errors in the input. However, this problem can be greatly reduced if the number of input pressures (n) is more than twice the number of isotherms (m) to be fitted (i.e., the number of w0i at each E0i). In the case of the present work, we used 70 values (wex j ; n ) 70) interpolated
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from the experimental isotherms (with 40-60 experimental points). The fitting was done by varying the w0i to minimize the sum of squares of the deviation between the calculated values (wcalc j , using eq 6) and the experimental points at each pressure. Furthermore, to help in the stabilization of the solution, the fitting was constrained to a smooth distribution by also minimizing the second derivative of the distribution, calculated by ∆w0i/∆Li (this will correspond to d2w0/dL2). In this way, w0i should not be regarded as a set of independent variables to be fitted, but more as a solution vector with mutual dependency in their internal coordinates. The final estimator to be minimized in the fit was then given as n
θ)
∑
m
2 (wcalc - wex j j ) + λ
j)1
∑ i)2
(
w0i - w0i-1 Li - Li-1
)
2
(8)
where λ is a weighting factor to control the smoothing of the fitted distribution, called a smoothing factor. Because the first summation in the preceding equation gives final values in the range of 10-5 for the experimental data used, the smoothing factor used in this work was 10-5 in all cases, to give about the same weight to the fitting of the data and the smoothing of the distribution. This methodology was implemented in a Microsoft Excel spreadsheet to calculate the isotherms matrix and fit the w0i values, using the solver functionality to minimize the fitting estimator. The initial w0i values were set to zero before beginning the fitting routine. In this way, the w0i values were increased to minimize the deviations from the experimental data while still maintaining a smooth distribution. The obtained R2 values of the fits were better than 0.999 (normally ∼0.9995), and the final sum of squares of deviations were between 9 × 10-5 and 1 × 10-5 with a sum of ∆w0i/∆Li (the smoothing estimator) from ∼2 to ∼0.02. The spreadsheet used in this work is given in the Supporting Information. A simple test may be used for comparing the two methods used in this work. We may consider a model activated carbon constituted by micropores of 0.6 and 1.4 nm (E0 values of 29.40 and 19.11 kJ mol-1, according to eq 2) with a microporous volume of 0.08 and 0.20 cm3 g-1, respectively. Then, assuming that adsorption may be described using two DR equations, a CO2 adsorption isotherm can be calculated (Figure 2a)). These data can be treated as experimental data and the methods for estimating pore sizes can be applied to them. The results are presented in Figure 2b for the DRS methodology and the method using the integral adsorption equation directly. The DRS methodology (solid line) predicts only a maximum at 1.2 nm. The method presented above (broken line) correctly predicts two maxima: one at 0.6 nm (sharp) and other at 1.35-1.40 nm. As this simple example shows, the principles underlying the DRS equation do not allow it to be applied to micropore structures of activated carbons when various maxima can be clearly found in the distribution.9 In these cases, Dubinin suggested that two DR equations should be used;1 however, from the given experimental data (Figure 2a), it is not always obvious if a set of two DR equations or only one should be used in the fit. The method for fitting the integral adsorption equation presents reliable pore size distributions for carbonaceous materials with homogeneous or heterogeneous micropore structures, and so, is a more robust methodology. Experimental Section Adsorbents. In this study, carbonaceous materials with different degrees of porosity development were assayed (that
Figure 2. Example of the application of the fit of the integral adsorption equation to obtain pore-size distributions: (a) adsorption isotherm of a model activated carbon constituted by micropores of 0.6 and 1.4 nm with a microporous volume of 0.08 and 0.20 cm3 g-1 respectively, and (b) poresize distributions obtained from the adsorption isotherm using the DRS method (solid line) and the integral adsorption method (broken line), represented as the differential pore volume (dw0/dL) versus pore size (L).
is, activated carbons and chars). Regarding activated carbons, laboratory-made samples (AC1 and AC2) were prepared from cork waste powder, according to the procedures detailed in a previous work,15 and were used along with two commercial samples (identified as AC3 and AC4). Chars were prepared in our laboratory from different precursors (namely, waste from the cork processing industry, rope manufacturing industry, and municipal waste treatment residues) that were submitted to treatments at temperatures of 450-650 °C under a nitrogen flow. CO2 Adsorption. The CO2 adsorption isotherms for the different carbon materials was determined at 0 °C, using a conventional volumetric glass apparatus that was equipped with an MKS-Baratron (Model 310BHS-1000) pressure transducer (0-133 kPa). Before the experiments, the samples (∼50 mg) were outgassed for 2 h at 300 °C, under a vacuum of better than 10-2 Pa. Results In Figure 3, the normalized distributions obtained by eq 4 (solid line) and those obtained by fitting eq 6 (broken line), according to the described procedure, are shown. The samples in Figure 3 are the laboratory-made and the commercial activated carbons. A quick inspection of the plots reveals that the distributions obtained by the two methods are relatively similar in shape for samples AC1 and AC2, but are significantly different in the cases of samples AC3 and AC4. As stated previously, the DRS equation (eq 4) can only give Gaussianshaped distributions, and in the AC1 and AC2 cases, these are similar to distributions obtained by fitting the integral adsorption equation (eq 6). However, for the commercial activated carbons (samples AC3 and AC4), it is clear that, according to the new method, a Gaussian distribution is not a good description of the micropore size distribution. In fact, in these two cases, the DRS equation predicts maxima at pore widths where there is not a considerable micropore volume according to the method based on the integral adsorption equation. This is probably
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Figure 3. Micropore distributions of activated carbons obtained by fitting the DRS equation (solid line) and by fitting eq 6 (broken line), according to the described method. The vertical dashed line represents the weighted average micropore size of the distribution obtained by fitting the integral adsorption equation.
caused by the presence of a considerable fraction of large micropores (1.5-2.0 nm) in samples AC3 and AC4 that shifts the distributions to larger widths. Conversely, the low volumes of large micropores in samples AC1 and AC2 contribute to the agreement between the two methods. The average micropore size from the integral equation, calculated by weighting each size by the corresponding volume, is represented in Figure 3 as a dashed vertical line. As can be seen, the DRS distributions have a tendency to present the maximum near the average micropore size, although in the AC3 and AC4 cases, there is not a significant pore volume at that size, according to the fit of the integral equation. In fact, systematic deviations (not random) were observed in the lower and upper part of the range of A values when fitting the DRS equation to the adsorption data. Thus, the DRS distribution gives a micropore size distribution that is an average Gaussian of the actual distribution, and it is only expected to give good description of the actual distribution in cases where the activated carbons have relatively narrow and symmetric distributions, as in samples AC1 and AC2. Both methods were also applied to CO2 adsorption data on the eight chars obtained in our laboratory. The results, which are presented in Figure 4, show marked differences between the two distributions as in the cases of the commercial activated carbons (samples AC3 and AC4). In several chars, the DRS distribution maxima (solid line) are very different from the average micropore size (vertical line) and, in some cases, is predicted at widths at which the integral equation method (broken line) gives very low pore volumes (CHAR4, CHAR5, and CHAR8). The broken line is different from a Gaussian distribution, for all char samples, indicating that the DRS equation cannot give a good description of the micropore size distributions in these cases. For the majority of the chars in Figure 4, the standard lowtemperature nitrogen adsorption was not easy to perform, because the very slow adsorption kinetics makes it very difficult to obtain a true equilibrium isotherm. This indicates that, most probably, the samples contain a considerable volume of very narrow micropores (0.33-0.45 nm). Looking at the distributions in Figure 4, this conclusion is not expected from the DRS distributions, because only a very small fraction of pore volume is predicted in this range. In contrast, in several cases, the fit of the integral equation predicts a considerable fraction of mi-
Figure 4. Micropore distributions of chars obtained by fitting the DRS equation (solid line) and by fitting eq 6, according to the described method (broken line). The vertical dashed line represents the weighted average micropore size of the distribution obtained by fitting the integral adsorption equation.
cropore volume at values of
