Langmuir 1993,9, 2513-2517
2513
Relationship between Energetic and Structura1 Heterogeneity of Microporous Carbons Determined on the Basis of Adsorption Potentials in Model Micropores Jacek JagieUot and James A. Schwarz' Department of Chemical Engineering and Materials Science, Syracuse University, Syracuse, New York 13244 Received August 6,1992. In Final Form: October 1,1992 The relationship between adsorption energy and micropore sizes is based on the dependence of the molecule-pore interaction potential well depth as a function of the pore width. The adsorption integral equation is used to analyze experimental adsorption isotherms of several simple gases measured at supercriticaland near-criticaltemperatureson two microporowcarbons. It is observedthat the adsorption systemsarestronglyheterogeneousfrom the standpointof their adsorptionenergeticswhich are characterized by asymmetrical energy distributions with a broadening toward high energies. It is also shown that these distributions correspond to relatively narrow and symmetrical peaks characterizing the distribution of pore sizes. The strong asymmetrical relationship between adsorption energy and pore width magnifies the energetic heterogeneity even though pore size distributions show only modest dispersion.
Introduction Activated carbons are very complex from a structural point of view. It is widely assumed, however, that in the case of microporous carbons the fundamental structural units are graphite-like aromatic monocrystallites; therefore, micropores may be considered as slit-shaped.' The pore volume and the pore structure have a great influence on the adsorption properties of activated carbons. These properties are determined by measurements of sorption isotherms using different gases. Surface chemical functionalities, derived from activation or treatment procedures, may also influenceadsorption. When only physical adsorbate-adsorbent interactions occur, the study of adsorption phenomena allows a quantitative description of the energetic properties of the solid surface, especially the evaluation of surface energy heterogeneity, which can be connected with the microstructural heterogeneity as well as with the chemical nature of the surface. There are anumber of studies devoted to the adsorption on microporous carbons and several methods have been proposed to characterize the carbon micropore structure. A significant group of these methodsu is based on the empirical relationship between small-angle X-ray scattering (SAXS) results and the values of the so-called characteristic energy parameter of the Dubinin-hdushkevich (DR) e q u a t i ~ n .Some ~ researchers6J applied the original Radushkevich8concept that pore size distribution is related to the distribution of Polanyi's potential. A modification of this concept was presented by Stoeckli et al.9JO who proposed to link the adsorption energy distri-
* To whom correspondence should be addressed.
t Permanent address: Institute of Energochemistry of Coal and Physicochemistry of Sorbents, University of Mining and Metallurgy, 30-069 K r a k 6 ~Poland. . (1) Stoeckli, H. F. Carbon 1990,28,1. (2) Stoeckli, H.F.J. Colloid Interface Sci. 1977,59, 184. (3) Dubmin, M. M.; Stoeckli, H. F. J. Colloid Interface Sci. 1980, 75,
34.
(4) Jaroniec, M.; Piotrowska, J. MOnatSh. Chem. 1986, 177, 7.
(6)Dubinin, M. M.;Zaverina,E.D.;Radushkevich,L. V. 2h.Fi.z.Khim.
1947,21,1361.
(6) Spitzer, Z.; Biba, V.; Kadlec, 0. Carbon 1976,14, 161. (7) Horvath, G.; Kawazoe, K. J. Chem. Eng. Jpn. 1983,16,470. (8) Radushkevich. L. V. Zh. Fiz. Khim. 1949.23.1410. . . (9) Stoeckli, H.F; Carbon 1981,19,326. (10) Stoeckli, H. F.; Lavanchy,A.; Kraehenbuehl, F. In Adsorption at the Gas-Solid and Liquid-Solid Interface; Rouquerol,J., Sing, K. S.W.,
Eds.; Elsevier: Amsterdam, 1982.
bution obtained from the Dubinin-Astakhov (DA) equation'' with the pore size distribution using equations describing the adsorption potential in model mimpores.1~19 However, in the evaluation of the adsorption energy distribution, the condensationapproximationwas applied; thus, the results are equivalent to those obtained in terms of a Polanyi's potential distribution. In recent years the statistical mechanicalapproachcalled mean field density functional theory has been used14J6to study gas adsorption in porous systems. On the basis of the local version of this theory, a method for the determination of pore size distributions was proposed.16 According to the researchers, however, results obtained by this approach are accurate for large pores but only qualitatively correct in the case of very small pores. To improve the microporeanalysis,a more accurate, nonlocal mean field theory was recently applied.17 Adsorption in porous materials has also been studied by molecular simulations.lg2° These two powerful methods are now under development, and their results are very promising; however, their practical application requires extensive computer calculations. In our previous paper21we proposed a simplebut realistic treatment of adsorption in micropores of molecular sizes. We assumed that the adsorption energy of molecules confined in such pores is dependent on pore sizes and can be calculated from equationsderived by Everett and Pow112 for model micropores. Consequently,for a heterogeneous micropore structure the adsorption energy distribution is governed by the pore size distribution. We analyzed DR (11) Dubmin, M.M.; htakhov, V. A. Izu. Akad. Nauk SSSR, Ser. Khim. 1971,6. (12) Everett, D. H.; Powl, J. C. J. Chem. SOC.,Faraday !Trans. 1 1976, 72,619. (13) Stoeckli, H. F. Helu. Chim. Acta 1974,57,2195. (14) Peterson,B. K.;Walton, J. P. R. B.; Gubbina, K. E. J. Chem.SOC., Faraday Trans. 2 1986,82, 1789. (15)Evans, R.; Bettolo Marconi, U. M.; Tarazona, P. J. Chem. Phye. 1986,84,2376. (16) Seaton, N. A.; Walton, J. P. R. B.; Quirke, N. Carbon 1989,27, 863. (17) Lastoekie,C.; Gubbins,K. E.;Quirke,N. International Symposium
on Effects of Surface Heterogeneity in Adsorption and Catalysk on Solids, Kazimien Dolny, Poland, 1992. (18) Tan, Z.; Gubbine, K. E. J. Phys. Chem. 1990,94,6061. (19)Matranga, K. R.: Myere, A. L.; Glandt, E. D. Chem. Em. Sci. 1992, 47, 1669.- .
(20) Bojan,M. J.; Vernov, A.; Steele, W. A. Langmuir 1992,8, 901.
(21) Jagiello,J.; Schwarz, J. A. J. Colloid Interface Sei. 1992,154,226.
0743-7463/93/2409-2513$04.00/00 1993 American Chemical Society
J a g i e h and Schwarz
2514 Langmuir, Vol. 9, No. 10, 1993
and DA isotherms using parameters within the range of the values reported in the literature. We found that typical microporous activated carbons which are strongly heterogeneous from the standpoint of their adsorption energetics show relatively narrow peaks characterizing their micropore pore size distribution. In the present work we analyzeexperimental adsorption isotherms obtained for several simple gases on the same activated carbon in order to study the effect of molecular properties on resulting pore size and energy distributions. In addition we compare these results with those obtained for a different carbon which has a typical molecular sieving property. Energetic and Structural Heterogeneity The model of slit-shaped pores created by two parallel solid lattice planes, discuseed in detail by Everett and Pow1,12appears to be the most widely acceptedl*JBPfor describing micropore geometries in active carbons. The gas-solid potential of a molecule confined in such a pore, up,is given by the appropriate sum of moleculeinteraction potentials with single planes, us: up@)= u,(z) + u&2x - 2 )
(1) where z denotes the distance of the molecule from the surfacenuclei of one of the pore walls which are separated by the distance 8.The interaction potential of a molecule with a solid lattice plane is obtained by appropriate integration of the Lennard-Jones potential (12:6)over two dimensionsof the solid plane and is given by the following expression:
where us* is the depth of the potential minimum and ro is the molecular size parameter of the Lennard-Jones potential. The properties of the potential up(z)were described in ref 12. It was shown that for large distances between the pore walls this potential has two symmetrical minima; for smaller separation distances these minima coalesce into a single minimum. Values of the potential depth, up*, range from us*,the value for the interaction potential with a single wall at an infinite separation distance, to the value of 2us* for x = ro. Variation of the potential depth, up*, as a function of the distance parameter, x , is obtained by numerical calculations from eq 1. In this approach the solid is treated as a continuum, which implies that periodic fluctuations of the potential usas a moleculemoves across a surface are neglected. The adsorption energy, e, which is usually defined by the well depth of the gas-solid interaction potential, is in general dependent on several factors. For the case of activated carbons, it may be dependent in some manner on the content of chemical impurities in the form of heteroatoms in the carbon structure, surface chemical groups, and geometrical structure of the surface. From the fact that the adsorption potential in fine pores is very strongly dependent on the pore sizes, it follows that heterogeneity of the pore sizes will have a dominant effect on energetic heterogeneity. Assuming that the heterogeneous micropore structure of activated carbon can be considered as a system of slitshaped pores and that chemical effects can be neglected, we obtain the following relationship between adsorption energies and pore sizes: (22) Rao, M. B.; Jenkins, R. G.; Steele, W. A. Langmuir 1985, I, 137.
= up*(x) (3) It follows that the distribution function of adsorption energies, ~ ( e ) ,is governed by the distribution of pore sizes, &). Since we will consider adsorption at temperatures above the critical or slightly below the critical point, we can assume the monolayer adsorption model in which the adsorption energy distribution x(c) is related to the experimentally measured adsorption isotherm, V, by the following integral equation: €
V@,T) = aOJemB(e,p,T) €1 x ( 4 de
(4)
where p and T denote pressure and temperature, a0 is the total number of adsorption sites of the investigated system, B(c,p,T)is the so-called local adsorption isotherm describing adsorption on sites with adsorption energy e, and €1 and e, are the minimum and maximum values of the adsorption energy of the adsorption system. Several methods have been proposed to solve this equation with respect to ~ ( c ) . The reviews of the existing methods for solution of eq 4 are given in monographs by Jaroniec and made^^^ and by Rudzinski and E~erett.2~The major difficulty in solving this equation, in general, is related to the fact that the solution may not be stable, especially when the limits of integration are not known and when the analytical form of the function x ( e) cannot be assumed a priori. It is convenient to rewrite eq 4, with the aid of assumption 3, in terms of pore sizes: V@,O = aoJm W x ) , p , T )d x ) dx Xlnln
(5)
where x- is the minimum pore size accessible for the gas molecule. We assume that the value of xmh corresponds to the pore size at which the potential of interactions up*( x ) = 0; for the model accepted here x- = 0.8581-0.The pore size and energy distributions are simply related by the following expression:
In the present study, as in our previous paper?l we assume a normal distribution for the micropore sizes. This proposal is based on the fact that such complex chemical processes like carbonization and activation, during which activated carbons are produced, are random in nature; thus, it is likely that resulting pores obey a normal distribution of sizes. In fact we are using the product of x and the Gaussian distribution, &AX) d x ) a x4&) (7) which ensures the physical sense of this function in that it vanishes for x 0, and on the other hand, for the narrow distributions of pores, it very much resembles the original d&). From the analysis of the potential depth as a function of pore size (eq 3) one finds that the enhancement of adsorption energy in pores characterized by separation distances x/ro > 2 is practically negligible. This implies that the adsorption energy in such larger pores is equal to that of the flat surface. The major influence of these pores on the adsorption isotherm is observed for subcritical temperatures at higher relative pressures when multilayer adsorption or capillary condensation occurs. At super-
-
(23) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids, b v i e r : Amsterdam, 1988. (24) Rudzineki, W.;Everett, D. H. Adsorption of Goees on Heterogeneous Surfaces; Academic Press: London, 1991.
Heterogeneity of Microporow Carbons
Langmuir, Vol. 9, NO.10, 1993 2516
Table I. Parameters of Carbon-Adsorbate Systems Used in Calculations adsorbate d(kJlmo1) w(FG)/(kJ/mol) w(HDB)/(kJ/mol)
>
0.25
1
0.2 I
NZ C)4
cz)4 ClH, coz
9.2 12.2 16.7
2.1 3.2 4.7 5.2 5.1
18.0 16.8
3.5 5.3 7.8 8.6 8.5
0.15
critical temperatures adsorption in larger pores is not sensitiveto pore sizes. However, the energy of adsorption in these pores may not be uniform due to structuraldefects or chemical impurities. For the purpose of the present study, we assume that adsorption in this group of pores is described by an adsorption isotherm for an energetically uniform surface with adsorption energy €0. With this assumption eq 5 takes the following form:
B
O.' 0.05
0 10
0
20
40
30
50
60
[ami
FWS
Figure 1. Adsorption isotherms of nitrogen measured on the G210 carbon samp1e.n
v@,n= awJm e(w,p,ncpw dx + a o 2 ~ ( € o , ~ ,(8) n
0.5
=dn
where a01 and a02represent contributions of the number of adsorption sites in micropores and large pores, respectively. The application of the adsorption integral equation requires the assumptionof the form of the local adsorption isotherm, O(p,T,e).In mathematical terms it is simply the kernel in this equation while from the physical point of view it represents the accepted model of adsorption. In our study we consider two simple local isotherms which represent general classes of adsorption models: the Fowleduggenheim (FG)isotherm,%describing localized adsorption, and the Hill-deBoer (HDB) isotherm,% describing two-dimensional mobile adsorption. In both models the average lateral interactions between adsorbed molecules are taken into account. These isotherms can be written in the following general form:
where K is the Langmuir constant, w is a parameter representing the mean interaction energy between adsorbed molecules, and $ is the function which vanishes for the localized adsorption model
e
$ ==for
HDB
$ = 0 for FG
1 1
0.4
0.3
o.2 0.1
I
0
0
10
30
20
40
F w = S [am1
Figure 2. Adsorption isotherms of COSmeasured on the G210 carbon samp1e.n 0.08
0.07
0.03
i
0.02
0.01
(10)
For $ = 0 and w = 0 eq 9 reduces to the Langmuir isotherm.
Analysis of Experimental Data and Discussion We apply the approach described above to the experimental results obtained previously in our laboratory.2728 These resulta consist of systematicallymeasured isotherms
Y
0
L
100
200
300
400
500
600
700
800
Pressure [Torr]
Figure 3. Adsorption isothermsdC2& measured on the MSC5A molecular sieving carbon sample."
of several simple gases on the same sample (commercially available G210 activated earbon manufactured by North American Carbon of Columbus) at supercritical and nearcritical temperatures in the pressure range up to 60 atm. The details of the measurement procedures and physical properties of the carbon sample are given in ref 28. We have chosen these data to study the effect of different adsorbates on calculated adsorption energy and pore size distributions of an activated carbon. On the other hand, in order to compare results for different carbons, we have
also analyzed adsorption isotherms of ethane measured by Nakahara et al.29on molecular sieving carbon (MSC5A manufactured by Takeda Chemical Industrial Co.) Equation 8which represents our model of gas adsorption on microporous carbons at supercritical and near-critical temperatures contains several parameters. Two of them, K and w , are associated with the assumed local isotherm, 6. The preexponential parameter K in the local isotherm is in principle temperature dependent.m It was shown?l however, that it may be treated as a constant over alimited temperature interval. We treat it as one of the adjustable parameters. The values of the lateral interaction param-
(26) Fowler, R. H.; Guggenheim, E. A. Statistical Thermodynamics; Cambridge University Press: Cambridge, 1949. (26) d e h r , J. H. The Dynamical Character ofAA9orption;Clarendon Presa: Oxford, 1963. (27) Agarwal,R.K.; Ph.D. Thesis, Syracuse University,Syracuse, NY, 1988. (28) Agarwal, R. K.; Schwarz, J. A. Carbon lSS8,26, 873.
(29) Nakahara,T.;Hirata, M.; Omori,T. J. Chem. Erg. Data 1974,lS. 310. (30) Rosa, S.;Olivier, J. P.On Physical Adsorption; Inbdence: London. 1964. (31)Jaioniec, Sokdowski, 5.;R u W k i , W. 2.Phys. Chem. (Leipzig) 1977,258, 818.
2516 Langmuir, Vol. 9, No.10, 1993
JagieHo and Schwarz 0.25 .
3- 0.2 e6 0.15 j
4
2
L
0.1
* eJ 3 0.05
1
0
~
~
0 7
0.6
0.8
1.2
1
1.4
16
1.8
2
X'
Figure 4. Pore size distributions in terms of the reduced size
parameter x* obtained for different molecules for the G210carbon sample. I+ [AI
- .
._.-
0
5
..., 6
10 8
7
9
10
11
12
H14
Figure 5. Comparison of pore size distributions for G210 and
MSC-5A samples in terms of pore width, H = 22, and available pore width, H' = H - 3.4 A.
eter, w , for HDB and FG isotherms are estimated according to Ross and Olivier3O from the parameters of the threedimensional van der Waals equation. The energy of adsorption on the uniform flat surface EO and the potential parameter us* are assumed to be equal and are estimated by the differentialheat of adsorption on graphitizedcarbon black at zero coverage on the basis of values reported by Avgul and K i ~ e l e v . All ~ ~ parameters assumed in our calculations are presented in Table I. The remaining five parameters, K, a01, a02, and two parameters describing the function cp(x), are calculated by the nonlinear fitting of eq 8 to the adsorption data measured at different temperatures. In this procedure the following quantity is minimized N
(11)
where N is the number of experimental points and the goodness of the fit 6 is defined as 6 = (d/m1l2 (12) We have tomention here that before startingthe analysis of experimental results we have removed from the original data basem a few experimental points measured at the highest pressures. These points have a tendency to "bend" isotherms upward. The adsorption amounts at higher pressures are stronglyaffectedby the buoyancycorrections which introduce large uncertainty to their values. We have also disregarded data which were measured in (32) Avgul, N. N.; Kiselev, A. V. In Chemistry and Physics of Carbon; Walker, P. L., Jr., Ed.; Marcel Dekker: New York, 1970; Vol. 6.
15
20
25
30
35
40
Adsorption Energy [kJlmoi]
Figure 7. Comparison of energy distributions of the Cz& adsorption for G210 and MSC-SA samples.
temperature intervals smaller than 10 K. The data base obtained after these manipulations remains fully representative for the considered adsorption systems. In order to illustrate the goodness of the fit, we present in Figures 1-3 experimental isotherms compared with the fitted curves. The results of calculationsare given in Table I1 in which the quantities Z and u describe the calculated mean value and standard deviation of the pore size distribution, cp(x). We also report the values of the calculated initial isosteric heat of adsorption, From Table I1 it is seen that the calculated pore size distribution is not significantly dependent on the chosen adsorption model. It seems, however,that the adsorption model has influence on calculated a0 values, which are usually larger in the case of the HDB local isotherm. Both models are only crude approximations, and it is difficult to say which is more adequate to represent the true adsorption isotherm. The fact that both give almost the same results with regard to the function dx)is probably due to the strong energetic heterogeneity which is a dominating factor in the adsorption process, and therefore the choice of adsorption model has a secondary effect on calculated results. For our further discussion we choose, therefore, numerical results obtained for the mobile adsorption model (HDB), which is usually assumed for higher temperatures. In Figure 4 we show pore size distributions, in terms of the reduced variable x* = x/ro, obtained for carbon G210 using different adsorbates. It is seen that except for CO2 all other adsorbates give practically overlappingfunctions cp(x*). In order to explain this result, we have to consider the size parameter, ro, of adsorbate molecules. This parameter takes values from 3.7 A for the N2 molecule to 4.5 A for the C02 molecule when calculated from the gas-
Langmuir, Val. 9,No. 10, 1993 2517
Heterogeneity of Microporous Carbons
Table 11. Results of the Fitting Procedure of Equation 8 to the Exwrimental Isotherms eystem/modeP GNdFG GNdHDB G-CHJFG G-CHJHDB G-C2HJFG W2HJHDB G-C2HB/FG G-C&/HDB G-COdFG G-COdHDB MSC-C&/FG MSC-C2H$HDB
(Klatm) 8.5 8.4 9.1 9.3 10.5 10.6 10.7 10.6 11.0 10.9 10.3 10.5
Wro 1.20 1.21 1.21 1.21 1.19 1.19 1.20 1.20 1.27 1.28 1.14 1.13
dro -
aOll("OYl3)
0.11 0.10 0.098 0.098 0.099 0.098 0.10 0.097 0.11 0.11 0.12 0.11
4.38 5.28 4.99 7.10 6.57 8.37 6.30 7.76 8.48 10.20 3.00 3.67
aoz/(mmol/g) 4.21 6.21 4.31 6.01 2.61 2.41 1.96 2.4 4.02 5.77 0
0
Qltol(kJ/mol) 15.5 15.0 20.8 20.7 30.9 30.9 33.3 33.1 29.3 28.8 34.3 34.3
2.5 2.7 7.3 7.2 6.0 5.9 7.7 7.0 3.8 2.4
G = activated carbon G210,data from ref 27. MSC = molecular sieving carbon MSC-5A, data from ref 29.
phase second virial coefficient.33 We have to realize, however, that kinetic properties of linear molecules will be different in smallpores than in the gas phase. According to the results of Rao et al.,22 the favorable orientation of linear molecules in narrow pores is parallel to the surface. For such an orientation the effective size parameter ro will be related to the cross section perpendicular to the length of the molecule rather than to the average parameter obtained from its bulk gas virial coefficient. Since the pore size distribution should be independent of the adsorbate molecule, it follows from Figure 4 that the CO2 molecule has to be "thinner" than other molecules considered here in order to obtain the common pore size distribution. Taking mean values of the pore size distribution, a h , expressed in reduced units, from Table I1 we find for the ratio of ro between the C02 molecule and the Nz molecule or CH4 the value of 0.94. This can be compared with the results of calculations of Rao et al.22 of critical pore width for unactivated diffusion in slit pores. Their results for this quantity are 5.42 and 5.72 A for C02 and N2 molecules which gives the ratio of 0.95. On the basis of our results, we find that in the case of adsorption in carbon micropores the effective value of ro is practically the same for N2, CH4, C2H.4, and C2H6 molecules while for the C02 molecule this value is 94% of the former ones. The only spherical molecule considered here is the CHrmoleculefor which the gas-gasroparameter isequalto3.8A.u Takingro = 3.4AforC-Cinthegraphite structure and applyin Lorentz-Bertholet combiningrules, we obtain rg = 3.6 for the methane-carbon system. Taking this value, we obtain the pore size distribution of our G210 sample which is shown in Figure 5 in terms of the pore width H = 2x. The upper scale of the plot represents the so-called available pore width H' which is simply H' = H - 3.4 A. In Figure 5 we also plot the pore size distribution for molecular sieving carbon MSC-5A calculated from C& adsorption results with the assumption that ro(CzH6) = 3.6 A. For our calculationswe did not use the methane data because the reported29isotherms were measured only in their initial parts and only at two
1
(33)Hirschfelder, J. 0.;Curtis, C. F.; Bird, R. B. Molecular Theory of Owes and Liquids; Wiley: New York, 1964.
temperatures. The distribution for this carbon is shifted to lower pore sizes and has its mean value slightly below 5 A on the H' scale which is consistent with the molecular sieving property of this carbon. We also note that for this carbon we obtain a zero value for the a02 parameter which is consistent with the fact that this carbon has no mesoor macropores. In Figure 6 we present distributions of the adsorption energy of different adsorbates on carbon G210 which are related to the pore size distribution by eq 6. A very interesting observation about these two types of distributions is their symmetry. While p(x) is very symmetrical, the x(e) function is widened toward higher energies. This is due to the asymmetry of the 4 x 1 dependence which governs the relationship between these functions. The shape of the energy distribution which can be described as a Gaussian distribution with a tailing toward high energies is characteristic for the adsorption systems which can be described by the DR equation.23 We note that due to the fact that the e ( x * ) function has an extremum at x* = 1 the x(c) function calculated from eq 6 has a singularity at this point which is reflected by almost avertical increase in the x(e) plot when e approaches its maximum value corresponding to x* = 1. In our previous paper21we have assumed x* E (1,m)for the region of pore sizes accessible to the adsorbate molecule;here we consider a wider range, x* E (0.858,=), in which the minimum pore size corresponds to a zero adsorption energy. To addressthe effect of these differentapproaches, in Figure 7 we comparethe adsorption energydistributions for C2H6 adsorption on both carbons calculated using shorter and broader intervals for x*. It is seen that the effect of adsorption in pores of sizes between x* = 0.858 and x* = 1on the calculated x function is not significant. This is an important observation because it shows that the uncertainty in the limit of the pore accessibility,which in practice is not known exactly,does not have a significant effect on the calculation results. Acknowledgment. The work was supported by the New York State Energy Research and Development Authority under Contract 139-ERER-POP-90.
