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Probing the Pore Wall Structure of Nanoporous Carbons Using Adsorption Thanh X. Nguyen and Suresh K. Bhatia* Division of Chemical Engineering, The University of Queensland, Brisbane, Queensland 4072, Australia Received November 30, 2003 Hitherto, adsorption has been traditionally used to study only the porous structure in disordered materials, while the structure of the solid phase skeleton has been probed by crystallographic methods such as X-ray diffraction. Here we show that for carbons density functional theory, suitably adapted to consider heterogeneity of the pore walls, can be reliably used to probe features of the solid structure hitherto accessibly only approximately even by crystallographic methods. We investigate a range of carbons and determine pore wall thickness distributions using argon adsorption, with results corroborated by X-ray diffraction.
Introduction The use of nonlocal density functional theory (NLDFT) for modeling adsorption isotherms of Lennard-Jones (LJ) fluids in porous materials is now well-established,1-5 and is central to modern characterization of nanoporous carbons as well as a variety of other adsorbent materials.1-4 The principal concept here is that in confined spaces the potential energy is related to the size of the pore,6 thereby permitting a pore size distribution (PSD) to be extracted by fitting adsorption isotherm data. For carbons the slit pore model is now well established and known to be applicable to a variety of nanoporous carbon forms, where the underlying microstructure comprises a disordered aggregate of crystallites. Such slit width distributions are then useful in predicting the equilibrium1-5 and transport behavior7,8 of other fluids in the same carbon. Numerous investigations of the effect of pore wall thickness, carbon density, Fa, interlayer spacing, ∆, and pore-pore correlation have shown5,9-12 that the pore wall thickness and carbon density have the most significant impact on adsorption behavior such as pore filling pressure and adsorbed density profile. Despite these findings it is still common to arbitrarily assume infinite pore wall thickness in using the slit pore model, and the associated pore size dependent Steele 10-4-3 potential6 is then employed for estimating the potential energy profile in a pore of any size. The inappropriateness of this assumption has recently been demonstrated in our laboratory, where it has been shown5 that in typical nanoporous carbons having surface area in the range important for practical * To whom correspondence may be addressed. E-mail: sureshb@ cheque.uq.edu.au. (1) Seaton, N. A.; Walton, J. P. R. B.; Quirke, N. Carbon 1989, 27, 853. (2) Lastoskie C.; Gubbins, K. E.; Quirke, N. Langmuir 1993, 9, 2693. (3) Vishnyakov, A.; Ravikovitch P. I.; Neimark A. V. Langmuir 1999, 15, 8736. (4) Bhatia, S. K. Langmuir 1998, 14, 6231. (5) Bhatia, S. K. Langmuir 2002, 18, 6845. (6) Steele, W. A. Surf. Sci. 1973, 36, 317. (7) Bhatia, S. K.; Nicholson D. Phys. Rev. Lett. 2003, 90, 016105. (8) Jepps, O. G.; Bhatia, S. K.; Searles, D. J. Phys. Rev. Lett. 2003, 91, 0126102. (9) Yin, Y. F.; McEnaney, B.; Mays, T. J. Carbon 1998, 36, 1425. (10) Suzuki, T.; Kaneko K.; Setoyama N.; Maddox, M.; Gubbins, K. E. Carbon 1996, 34, 909. (11) McEnaney, B. Carbon 1988, 26, 267. (12) Mays, T. J. Fundamentals of Adsorption 5; Kluwer Academic: Boston, 1996.
application (>800 m2/g) the pore walls must actually be rather thin and comprised of only a very small number (two to three) of graphene layers. For such small wall thicknesses the adsorption potential is much weaker than that obtained for the infinitely thick wall, and the adsorbed amounts can be lower by factors of 2 or more, particularly at low pressures where fluid-solid interactions dominate.5 Here we present a novel technique whereby both pore size and pore wall thickness distributions can be simultaneously determined. We illustrate the method for a variety of nanoporous carbons, showing good correspondence of the results with X-ray diffraction. Theory In our technique, the Tarazona13 NLDFT is applied to calculate the local density profile at the single pore level, with the confining potential taken as the sum of the potentials,6 φwf(z), from opposite walls of the pore
{ ∑ [ ( ) ( ) ]} n-1
φwf(z) ) 2πFscf σcf2
2
10
σcf
5 z + i∆
i)0
-
σcf
4
z + i∆
(1)
z>0 Here n is the number of graphene layers in the pore wall, ∆ is the interlayer spacing, and Fs is the number of carbon atoms per unit area in a single sheet. Following Steele6 we use the values ∆ ) 0.335 nm and Fs ) 38.17 atoms‚nm-2 for carbon. Considering the adsorption of an LJ fluid in a carbon having slit pore width distribution f(H), and random pore wall thickness characterized by the probability distribution p(n), we obtain the overall isotherm ∞
Fˆ (P) )
∑ m)1
∞
p(m)
p(l)∫0 ∑ l)1
∞
Flm(P,H)f(H) dH
(2)
where Flm(P,H) is the local isotherm in a pore of slit width H, with the left wall having l graphene layers and the right wall having m layers, obtained from the NLDFT. Here we assume that the thicknesses of the two opposing walls of a pore are uncorrelated and that the interaction (13) Tarazona, P. Phys. Rev. A 1985, 31, 2672; 1985, 32, 3148. Tarazona, P.; Marini Bettolo Marconi, U.; Evans, R. Mol. Phys. 1987, 60, 573.
10.1021/la036244p CCC: $27.50 © 2004 American Chemical Society Published on Web 03/27/2004
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potential between adsorbed molecules in neighboring pores is negligible in comparison to the fluid-solid potential energy for small molecules used in characterization, justified earlier for nitrogen.5 In the calculations we used the potential parameters σcf ) 0.3362 nm, cf/k ) 58.02 K for argon, and σc ) 0.3349 nm, c/k ) 30.5474 K. Earlier attempts at characterization considering finite pore walls have been largely ad hoc, utilizing arbitrary assumptions regarding the thicknesses of the confining walls,11,14 for example that one wall is infinitely thick while the other has exactly one graphene layer.14 Such assumptions have resulted in unrealistic surface areas and have therefore not led to a viable solution. Our use of the probability distribution p(n) eliminates such arbitrary assumptions. Nevertheless, given the ill-posed nature of the problem15 the determination of p(n) is fraught with much uncertainty, with a multitude of possible solutions. The problem is, however, made tractable by the relation between the mean number of graphene layers in a pore wall, γ, and the surface area
γ)
2No(1 - wa) FsMcS
(3)
which is a modification of that derived recently5 based on mass balance over a pore wall, accounting also for the mass fraction of inorganic impurities (ash), wa, in the carbon. Here No is the Avogadro number, Mc is atomic weight of carbon, and S is the specific surface area of the solid. Equation 3 provides an important constraint to be satisfied by the probability distribution p(n), since γ ) ∞ ∑n)1 np(n), and the pore wall thickness and pore size distributions are then correlated, which serves to restrict the solution space. In practice for n g 5 the potential energy is insensitive to n, and essentially matches that for an infinite wall. Consequently we lump the combined probability for all pore walls having five or more layers into p(4+), leading to 4+
∑ p(n) ) 1,
n)1
4
∑ np(n) + 5p(4+) e γ
(4)
n)1
yielding raw constraints to be satisfied by p(n), n ) 1, 2, 3, 4, 4+. The solution of the distributions f(H) and p(n) is now accomplished by fitting the predicted isotherm Fˆ (P) in eq 2 to its experimental counterpart using Tikhonov regularization,15 combined with a genetic algorithm16 to search the solution space. The L-curve technique17 is used for determining the regularization parameter. In solving for Fˆ (P) the summations in eq 2 are taken over the domain [1,4+], since walls having five or more layers are essentially infinitely thick. Experimental Section The above approach has been applied to argon adsorption data for a variety of commercial carbons in our laboratory, and features of the pore wall thicknesses distribution correlated with X-ray diffraction measurements. The various carbons used were BPL and F-400 activated carbons manufactured by Calgon Corp.; RB2, ROW-0.8, and ROX-0.8 activated carbons produced by Norit, all of which are bituminous coal based; and a synthetic polymer(14) Ravikovitch, P. I.; Jagiello J.; Tolles, D.; Neimark A. V. Carbon’01, International Conference on Carbon (Lexington, 2001). (15) Tikhonov, A. N. Nauk, Dokl. Akaf. SSSR 1963, 49, 153. (16) Zbigniew, M. Genetic Algorithms + Data Structures ) Evolution Programs; Springer-Verlag: Berlin, 1996. (17) Hansen, P. C. BIT 1990, 30, 658.
Figure 1. Pore wall thickness distribution of ACF-15 activated carbon fiber determined by adsorption. Left inset depicts a comparison of the pore size distribution using the present finite wall thickness method and that using the conventional infinitely thick wall assumption. Right inset depicts fitted and experimental argon adsorption isotherms, illustrating the S-shaped deviation using the infinitely thick wall assumption, in the P/Po range lying between 10-4 and 10-3. based activated carbon fiber, ACF-15, supplied by American Kynol Inc. All the carbons were degassed at 300 °C overnight prior to argon adsorption. A Micromeritics ASAP 2010 volumetric adsorption analyzer was used to obtain adsorption data of argon at 87 K in the carbons. The ash content of each carbon was determined gravimetrically following direct combustion in air at 873 K in a Lindberg box furnace (BF51600). X-ray diffraction patterns of the carbons were also obtained using Cu KR radiation (λ ) 0.15406 nm).
Results and Discussion Figures 1 and 2 illustrate the pore wall thickness distribution obtained for the activated carbon fiber ACF15 and activated carbon BPL, with insets depicting the PSD and argon isotherm in each case. Similar results were also obtained for the other carbons, as summarized in Table 1. From this table it is seen that surface area, S, calculated from the proposed model is slightly greater than that obtained if infinite wall thickness is assumed. This is due to overestimation of the gas-wall interaction potential in the latter case, which leads to a shift of fitted pore size distribution to slightly larger pore size, as seen in the insets in Figures 1 and 2. A shift in the pore size distribution to narrower pores has also been noted in the earlier study using arbitrarily sized pore walls,14 with some reduction in the S-shaped deviation between fitted and experimental isotherms, prominent in isotherm fits based on the infinitely thick wall assumption, when the opposing walls differ in thickness. Thus, the infinitely thick pore wall model underestimates surface area, especially for carbonaceous materials with very high surface area and thin pore walls such as activated carbon fiber, ACF-15. Further, as seen in the insets in Figures 1 and 2 the S-shaped deviation in the isotherms is completely eliminated by the current approach. In the case of pore volume, the infinitely thick pore wall model underestimates the value by only a small amount, as seen in Table 1. This is due to the slightly lower density of the condensed argon in the pores with finite thickness walls.
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Figure 2. Pore wall thickness distribution of BPL activated carbon determined by adsorption. Left inset depicts a comparison of the pore size distribution using the present finite wall thickness method and that using the conventional infinitely thick wall assumption. Right inset depicts fitted and experimental argon adsorption isotherms, illustrating the S-shaped deviation using the infinitely thick wall assumption, in the P/Po range lying between 10-4 and 10-3.
Figure 3. Correlation between X-ray diffraction based parallelism indicator, R, and its counterpart, [γ - p(1)]/γ determined by adsorption, for various carbons.
Table 1 also lists the mean number of graphene sheets, γ, in a pore wall for each carbon, determined from the pore size distribution based on eq 3 and the mineral matter content. The latter was estimated from the residue obtained upon burning the carbon to complete conversion. The mean value of γ is found to be very low in general, and in the range of about 1.5-2, underscoring the inappropriateness of the assumption of infinitely thick walls. Besides highlighting the small wall thicknesses the pore size distributions also suggest that the carbon nanopores occur at discrete sizes or widths. This was evident from the sharp peaks in the distributions at discrete sizes from the application of the present approach, as opposed to the broad distribution peaks obtained from the infinite wall thickness approach. Furthermore, an average distance between the first three major peaks in pore size distribution of BPL and ACF-15 carbons from present approach is 3.29 Å, while the corresponding values from the infinite wall thickness model are invariant at 3.06 Å. The former agrees well with the interlayer spacing of activated carbons.18
The left insets in Figures 1 and 2 illustrate this feature for the ACF-15 and the BPL carbons, which was observed for the other carbons as well. The broader distribution peaks from the infinite wall thickness model may be recognized as being due to the lumping of all heterogeneities in the potential energy landscape into a single heterogeneity (the pore size distribution). By differentiation between the pore wall thickness and pore size heterogeneities through the two distinct distributions (but correlated through eq 3), the finite pore wall approach leads to the sharper PSD peaks. These sharp discrete peaks may well indicate that the small nanopores are essentially comprised of deformities related to portions of one or more neighboring graphene sheets missing in the crystallites comprising the structure. While yielding the PSD and mean number of graphene sheets in the pore walls, the approach also provides the pore wall thickness distribution p(n), as shown in Figures 1 and 2 for the ACF-15 and BPL carbons, respectively. In these figures p(5) represents the lumped probability for all the walls having five or more layers, given above as p(4+). The large proportion of single layer walls (4560%) is evident from these figures and was found to be the case for all the carbons examined. While one may expect details of the pore wall thickness distribution to depend on the manufacturing process, the predominance of single sheet walls appears to be a common feature for moderate and high surface area carbons. Further it was noted that the pore wall thickness obtained from the isotherm fit could be well approximated by a generalized Poisson distribution19 for most of the carbons except for ROX0.8, as seen in Figures 1 and 2. In particular for the carbon fiber ACF-15 it corresponds well to the limiting case of the Poisson distribution, corresponding to maximum randomness. Thus, the activated carbon fiber is expected to have a highly disordered structure. Also as shown in Table 1 the ratio of the standard deviation of the fitted distribution to the mean number of additional layers beyond the surface (γ - 1) is essentially constant and in the range of 1.3-1.5 for the coal-based carbons, suggesting a common form of the distribution for these carbons. These
(18) Radovic, L. R. Chemistry and Physics of Carbons; Marcel Dekker: NewYork, 2003.
(19) Consul, P. C. Generalized Poisson Distributions: Properties and Applications; Marcel Dekker: NewYork, 1989.
Table 1. Comparison between Results Obtained Based on the Present Technique and Those Obtained Assuming Infinitely Thick Pore Walls infinitely thick pore wall
finite wall thickness
carbon
S (m2/g)
Vp (cm3/g)
S (m2/g)
Vp (cm3/g)
γ
σ/(γ - 1)
ACF-15 BPL F-400 ROW 0.8 ROX 0.8 RB-2
1446 1023 1012 1034 1161 1125
0.52 0.51 0.55 0.59 0.65 0.47
1745 1173 1158 1198 1348 1294
0.54 0.53 0.57 0.61 0.66 0.49
1.51 2.17 2.14 2.06 1.95 1.94
1.36 1.47 1.51 1.49 a 1.31
a Pore wall thickness distribution of ROX 0.8 does not conform to generalized Poisson distribution.
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features will be explored in further detail in further work. As an independent characterization of the solid phase, the parallelism indicator, R, which is a measure of the fraction of graphene sheets that have at least one parallel neighbor, was determined from the (002) peak of the X-ray diffractograms following Dahn et al.20 The (002) peak comes from constructive inference between X-rays scattered from parallel stacked graphene sheets. As the proportion of graphene layers with parallel neighbors increases, so will R. In terms of the probability distribution p(n) the fraction of graphene sheets in walls having more than one sheet is [γ - p(1)]/γ, which may be expected to have a direct correspondence with the X-ray parallelism indicator R. Figure 3 depicts this relationship for the various carbons studied in our laboratory, illustrated in terms of a plot of R versus [γ - p(1)]/γ. In this figure the open triangles represent the data for the various coalbased activated carbons in Table 1, indicating a strong and nearly linear relationship between the two quantities. Given the somewhat diffuse nature of the (002) peak, the parallelism indicator is not expected to be quantitatively precise. The good correlation between the adsorption and X-ray diffraction (XRD) derived quantities, despite the approximate nature of R, is therefore representative of a (20) Dahn, J. R.; Xing, W.; Gao, Y. Carbon 1997, 35, 830.
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strong correspondence between the two quantities for the carbons derived from a common precursor. The open circle in Figure 3 represents ACF-15, which is seen to deviate from the correlation for the other carbons, probably due to the different precursor and manufacturing process for ACF-15. Further studies with other carbon fibers are being conducted in our laboratory and will be reported in due course. Summary In summary, the above results demonstrate that adsorption can be reliably used to probe the physical structure of the solid phase in carbons, conventionally studied by X-ray diffraction. The good correspondence with XRD, combined with the sharp PSD peaks, suggests that the narrow micropores are defects within the same crystallite, most likely arising from missing portions of neighboring graphene sheets. Nevertheless, some influence of intercrystalline pore space, particularly in the larger pores beyond about 1 nm may be expected. Further, while we have demonstrated the above method with carbons, one may expect the general principles to be valid for other disordered materials. LA036244P
