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Micropore Size Distribution of Activated Carbon Fiber Using the Density Functional Theory and Other Methods Mustapha El-Merraoui, Masayuki Aoshima, and Katsumi Kaneko* Physical Chemistry, Material Science, Graduate School of Natural Science and Technology, Chiba University, 1-33 Yayoi, Inage, Chiba 263-8522, Japan Received September 17, 1999. In Final Form: January 27, 2000 The pore-size distribution of a series of pitch-based activated carbon fiber samples (A5, A10, A15, and A20) has been determined from the nitrogen adsorption isotherms at 77 K, with the nonlocal density functional theory (DFT), the Dubinin-Stoeckli (DS) method, and the subtracting pore effect (SPE) method using the high-resolution RS plot. The pore-size distributions of all the samples from DFT show a distinguishable peak at small pore widths around 0.6 nm and a broad peak at bigger micropore size. The mean pore widths obtained from DFT and DS methods are similar for samples whose pores are narrow and different for wider ones. The use of the SPE method shows that the mean pore widths are slightly smaller than those obtained by DFT and DS methods.
Introduction Activated carbon has a high surface area and an excellent ability for adsorption of diluted molecules and ions in gas and liquid phases. Therefore, it has been widely used in many industrial and domestic processes such as the removal of contaminants,1,2 gas separation,3,4 gas storage,5-7 and catalytic reactions.8-12 Although activated carbon has a large surface area, the amount adsorbed is mainly dependent on the structure of the pores, particularly micropores. This is because the walls of the pores provide a stronger potential than the external surface area. It is then imperative to assess the porous structure of the activated carbons in order to make better use of the adsorbent for the defined purpose. Accordingly, the determination of the pore-size distribution is of great importance and represents a direct correlation with the amount adsorbed. The particular activated carbon fiber (ACF) has uniform micropores of great pore volume, showing better adsorption characteristic than granulated activated carbon. Hence, a more exact description of the micropore structure is requested for ACFs. Recently modeling of adsorption has made a notable development. Computer simulations can show adsorption mechanisms in different porous materials. The knowledge of the width and geometrical characteristics of the pores as well as the interaction rules governing the fluid-fluid and solid-fluid molecules is the basis information, that the simulation calculations need. Many authors make use of this procedure to interpret experimental data or to elucidate particular phenomena that are not easily observed experimentally. However, the long time of calcu(1) Kaneko, K. Stud. Surf. Sci. Catal. 1998, 120, 635. (2) Adachi, A.; Kobayashi, T. Bull. Environ. Contam. Toxicol. 1995, 54, 440. (3) Sircar, S.; Golden, T. C.; Rao, M. B. Carbon 1996, 34, 1. (4) Sun, J.; Rood, M. J.; Lizzio, A. A. Gas Sep. Purif. 1996, 10, 91. (5) Sun, J.; Brady, T. A.; Rood, M. J.; Lehmann, C. M.; RostamAbadi, M.; Lizzio, A. A. Energy Fuels 1997, 11, 316. (6) Quinn, D. F.; McDonald, J. A. Carbon 1992, 30, 1097. (7) Matranga, K. R.; Myers, A. L.; Glandt, E. D. Chem. Eng. Sci. 1992, 47, 1569. (8) Mehandjiev, D.; Bekyarova, E.; Khristova, M. J. Colloid Interface Sci. 1997, 192, 440. (9) Moon, J. S.; Park, K. K.; Seo, G. Appl. Catal., A 1999, 184, 41. (10) Raymundo-Pinero, E.; Cazorla-Amoros, D.; Marallon, E. J. Chem. Educ. 1999, 76, 958. (11) Martin-Martinez, J. M.; Singoredjo, C.; Moulijn, J. A. Carbon 1994, 32, 897. (12) Mikhalovsky, S. V.; Zaitsev, Yu. P. Carbon 1997, 35, 1367.
lations remains the major inconvenience of the computer simulation method despite the high run speed of the latest computers. The progress of statistical mechanics enabled a few years ago the emergence of a new trend in adsorption science, namely, density functional theory (DFT). In addition to computer simulations the DFT is also established by considering the intermolecular interactions on the microscopic scale. The application of DFT to elucidate the adsorption behaviors has been expected. Several works using the DFT method succeeded in obtaining the poresize distribution of activated carbon.13-19 The present work aims first to evaluate the poresize distribution of a series of ACFs and show the comparison with the conventional procedures as the DubininRadushkevitch-Stoeckli (DS) and the subtracting pore effect (SPE) methods. Experimental Section A series of pitch-based activated carbon fibers A5, A10, A15, and A20 were provided by Osaka-Gas Co. Measurements of nitrogen adsorption isotherms were performed gravimetrically at 77 K. Preevacuation conditions of the samples are 383 K and 0.1 mPa for 2 h. The experimental equipment allows measurement of the relative pressure until 10-6.
Theoretical Basis of the DS, SPE and DFT Methods DS Method. This equation is derived from the assumption that adsorption in micropores proceeds by pore filling rather than a layer by layer development of a thin film against the walls of the pores. In contrast to the Dubinin-Radushkevitch theory (DR), which gives good results for structurally homogeneous microporous carbons, the DS theory assumes that the adsorbent contains inhomogeneous micropores. The DS equation is thus considered as a generalization of the DR equation. The (13) Seaton, N. A.; Walton, J. P. R. B.; Quirke, N. Carbon 1989, 27, 853. (14) Peterson, B. K.; Walton, J. P. R. B.; Gubbins, K. E. J. Chem. Soc., Faraday Trans. 2 1986, 82, 1789. (15) Olivier, J. P.; Conklin, W. B.; Szombathely, M. V. In Characterization of Porous Solids III; Rouquerol, J., et al., Eds.; Elsevier: Amsterdam, 1994. (16) Olivier, J. P. J. Porous Mater. 1995, 2, 9. (17) Lastoskie, C. M.; Gubbins, K. E.; Quirke, N. J. Phys. Chem. 1993, 97, 4786. (18) Lastoskie, C. M.; Gubbins, K. E.; Quirke, N. Langmuir 1993, 9, 2693. (19) Neimark, A. V.; Ravikovitch, P. I. Langmuir 1997, 13, 5148.
10.1021/la991242j CCC: $19.00 © 2000 American Chemical Society Published on Web 03/28/2000
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pore-size distribution evaluated from the DS equation20 is widely accepted as a reference test. The distribution is supposed to obey the normal Gaussian function:
[
]
W00 (x - x0)2 dW0 ) exp dx 2δ2 δx2π
(1)
Here W00 is the total micropore volume, x0 is the average half-width of the micropore, and δ is the variance of the Gaussian distribution. The values of these parameters can be obtained by fitting numerically the following analytical isotherm expression (DS equation) to the experimental adsorption isotherm data:
W)
W00 2x1 + 2mδ2A2
[
exp -
mx02A2
]
× 1 + 2mδ2A2 x0 1 + erf δx2x1 + 2mδ2A2
[ ( 1 m) βk
)]
(2)
2W (aT - aext)
(3)
The theoretical basis for the SPE method was given by use of the grand canonical Monte Carlo simulation (GCMC).25 DFT Method. A system is specified by the grand canonical ensemble, in which bulk chemical potential µ, volume V, and temperature T are defined. The model for the nonlocal grand potential functional is 3
Ω[F(r)] ) kBT F(r)[ln(Λ F(r)) - 1] dr +
∫F(r)f
˜ (r)] ex[F
dr + 1/2
∫F(r′)w(|r - r′|; F˜ (r)) dr′
(5)
Here w is the truncated weighting function to the second order of the smoothed density as reported by Tarazona.28 At the equilibrium density profile the grand potential functional Ω reaches its minimum. Since the pore widths of the present samples are localized mainly in the micropore range, we can presume that the pores have a slit shape.30 The interaction potential between two nitrogen molecules that are separated by a distance r is given by the Lennard-Jones potential
UN-N(r) ) 4N-N
Here β ) 0.33 for a nitrogen adsorbate molecule, k is an empirical constant which links the mean half-width of the micropores to the adsorption energy,21 and A is the Polanyi potential.22 SPE Method. This method offers a good assessment of the total specific surface area aT derived from the slope of the linear line that joins the origin and the point of the RS plot23 with the abscissa 0.5. The external surface area aext is obtained from the slope of the linear part of the RS plot for RS beyond 1.5. By means of the subtracting pore effect (SPE) method,24 the mean pore-width w of a microporous carbon material of slit-shaped pores and micropore volume W can be straightforwardly calculated by
∫
F˜ (r) )
(( ) ( ) )
2
( )
w)
per molecule, which depends on the smoothed density characterized by F˜ (r). The estimation of the excess Helmholtz free energy per molecule can be explicitly described by the Carnahan-Starling equation.27 In contrast to the local density functional theory, the nonlocal density functional theory28,29 introduces the smoothed density concept, which is defined as
∫∫F(r)F(r′)u (|r - r′|) dr dr′ ∫F(r)[µ - V (r)] dr (4) att
ext
where F(r) is the density of the fluid at the position r, Vext(r) is the external potential under which component fluid is influenced, Λ is the de Broglie thermal wavelength for the adsorbate molecule, uatt is the long-range fluidfluid attractive potential derived from the WCA approximation,26 and fex is the excess Helmholtz free energy (20) Dubinin, M. M.; Polyakov, N. S.; Kateava, L. I. Carbon 1991, 29, 481. (21) Dubinin, M. M.; Stoeckli, H. F. J. Colloid Interface Sci. 1980, 75, 34. (22) Polanyi, M. Trans. Faraday Soc. 1932, 28, 316. (23) Gregg, S. J.; Sing, K. S. Adsorption, Surface Area and Porosity; Academic Press: London, 1982. (24) Kaneko, K.; Ishii, C. Colloid Surf. 1992, 67, 203. (25) Setoyama, N.; Suzuki, T.; Kaneko, K. Carbon 1998, 36, 1459.
σN-N r
12
-
σN-N r
6
(6)
which is characterized by the depth N-N of the potential at the minimum of UN-N at rm ) 21/6 σN-N and σN-N is the nitrogen molecular diameter. In practice we adopt the WCA approximation where the intermolecular interaction is divided into a hard-sphere potential of diameter dHS31 and an attractive part uatt represented by
uatt(r) )
{
-N-N 0 < r e rm UN-N(r) r > rm
(7)
We perform the computations of the DFT by taking dHS equal to the nitrogen diameter σN-N. If it is assumed that atoms which constitute the surface of the pore wall interact with the confined nitrogen molecules through the Lennard-Jones pairwise potential, the total interaction potential of the surface and a molecule of nitrogen can be written as32
Vext(z) ) 2πFCC-NσC-N2 × ∆
[( ) ( ) 2 σC-N 5 z
10
-
σC-N z
4
-
σC-N4
]
3∆(0.61∆ + z)3
(8)
Here z is the vertical distance of the nitrogen molecule from the wall of the pore, ∆ is the layer spacing of the graphite, FC is the atomic density of the graphite, and C-N and σC-N are the Lennard-Jones energetic and geometric parameters for a nitrogen molecule and graphite carbon atom. For a slit-shaped pore where the surfaces are separated by a distance H (carbon-carbon internuclear distance), the confined nitrogen molecule interacts with the two opposite surfaces. The total external potential VText is then described by
VText(z) ) Vext(z) + Vext(H - z)
(9)
(26) Weeks, J. D.; Chandler, D.; Anderson, H. C. J. Chem. Phys. 1971, 54, 5237. (27) Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635. (28) Tarazona, P. Phys. Rev. A 1985, 31, 2672; Phys. Rev. A 1985, 32, 3148. (29) Tarazona, P.; Marconi, U. M. B.; Evans, R. Mol. Phys. 1987, 60, 573. (30) Stoeckli, H. F. Carbon 1990, 28, 1. (31) Barker, J. A.; Henderson, D. J. Chem. Phys. 1967, 47, 4714. (32) Steele, W. A. Surf. Sci. 1973, 36, 317.
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El-Merraoui et al. Table 1. Henry Constant for Nitrogen Adsorptions on A5, A10, A15, and A20 at 77 K sample
kH (mol‚g-1‚Torr-1)
sample
kH (mol‚g-1‚Torr-1)
A5 A10
0.29 0.19
A15 A20
0.31 0.44
Table 2. Dubinin-Stoeckli Parameters for A5, A10, A15, and A20
Figure 1. Nitrogen adsorption isotherms on A5, A10, A15, and A20 at 77 K. Key: O, A5; 0, A10; ], A15; 4, A20.
The DFT model isotherms have been computed over the range 10-10-1 of the relative pressure p/p0. The average number density of confined nitrogen molecules is calculated from the density profile at defined relative pressure and pore-width H as
〈F(p/p0, H)〉 )
1 H
∫0H F(p/p0, z) dz
(10)
Prior to the computation of the pore-size distribution by using DFT, we first constructed a series of nitrogen adsorption isotherms for pores varying from 0.5 to 4 nm by step of 0.1 nm. The shape of the pore size distribution g is assumed to obey the bimodal log-normal function 2
g(H) )
∑ i)1
Wi exp[-(ln(H) - Hi)2/2δi2] δiHx2π
(11)
The parameters of the distribution function Wi, Hi, and δi are adjusted in order to give the best least-squares fitting of the theoretical model to the experimental results by minimizing the quantity L n-1
L)
(∫H ∑ i)0
Hmax min
g(H)〈F(pi/p0, H)〉 dH - Y(pi/p0))2
(12)
where n is the number of experimental points and Y(pi/p0) is the experimental amount adsorbed at pi/p0. Results and Discussion Pore-Size Distribution from the DFT Method. Figure 1 presents experimental adsorption isotherms of nitrogen on the samples of activated carbon fibers at 77 K. According to the IUPAC classification the shape of the isotherms is of type I, indicating that the samples are predominantly microporous with a very small external surface area.23 The maximum amount of nitrogen adsorption increases from A5 to A20. However, the sharpness of the knee present in the low relative pressures decreases as the maximum adsorbed amount increases. The sharper uptake indicates the presence of the stronger interaction of a nitrogen molecule with the pore wall. Hence the ACF having a great adsorption capacity should have wider micropores. In addition to the isotherms shown in Figure 1, the representation in a logarithmic scale of the relative pressures in Figure 2a-d allows a clear understanding of
sample
W00 (mg‚g-1)
δ (nm)
x0 (nm)
A5 A10 A15 A20
262.1 376.6 456.8 670.1
0.06 0.31 0.21 0.34
0.50 0.62 0.66 0.77
the adsorption occurring at low relative pressures. Figure 2a-d shows also the results of the DFT fitting to the experimental adsorption isotherms. The fitting is good over the whole experimental range of the relative pressures for the samples A5 and A10. However, for the samples A15 and A20 an S-swing shape of the DFT model fitting curve appears at low-pressure range; it can be an indication that the imposed two modal pore-size distribution functions are limited and further functions should give a better result for fitting of the experimental data. The DFT model adsorption isotherms of nitrogen on graphitic carbon for 1, 1.5, and 2 nm pore-widths w with the use of w ) H - σC, are collected in Figure 3. Here the relationship between H and w is not necessarily simple.33 In the present study, the simplest approximation was used. In the range of micropore-widths w between 1 and 2 nm the adsorption starts to increase noticeably in the range 10-5-10-4 of the relative pressure. For w bigger than 1 nm the increase of the adsorption amount passes by two transitions. The first step is ascribed to the monolayer formation that is not clearly apparent at 1 nm pore width but starts to be evident for 1.5 and 2 nm pore widths. The second step is described by a step rise of the adsorptive amount attributed to the cooperative pore filling which occurs at values of the relative pressures more than 10-2. The cooperative pore filling cannot be described by the Kelvin equation, and thereby we do not use the term of capillary condensation. An estimation of the Henry constant (kH) has been carried out in a low-pressure range at which a quasi-no interaction between adsorbed molecules should occur. Table 1 collects the calculated Henry constants for the samples A5, A10, A15, and A20 using the simplest relationship (W ) kHp). Different kH values are observed, but the fitting at the low relative pressure range must be improved before rigorous discussion. Pore-Size Distribution by the DS and SPE Methods. The fitting of the Dubinin-Stoeckli equation to the experimental data was carried out for all the samples. The parameters of fitting are collected in Table 2. The variance is sensitively depending on the starting value of the lowest relative pressures at which the fitting calculations have been done. Indeed, the lower the value of the relative pressures, the smaller the variance. However, the volume and mean width of the micropores remain inalterable. To avoid the effect of the external surface area, the fitting was performed until 0.2 of the relative pressure. Nevertheless, it is worthy to note that the main change in the results emanated from the fitting until 0.2 of the relative pressure and the whole data concerns the variance parameter. To have a comparison of fitting curves derived from the Dubinin-Stoeckli and the DFT methods, (33) Kaneko, K.; Cracknell, R. F.; Nicholson, D. Langmuir 1994, 10, 4606.
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Figure 2. Experimental, DFT, and Dubinin-Stoeckli fitting isotherms of nitrogen at 77 K. Key: open circle, experimental data; solid line, DFT fitting; dashed line, Dubinin-Stoeckli method; (A) A5; (B) A10; (C) A15; (D) A20. Table 3. Mean Pore Widths of A5, A10, A15, and A20 from DS and DFT Methods w (nm)
Figure 3. Computed isotherms from DFT of nitrogen on graphitic uniform porous carbon with different values of w (w ) H - σC) at 77 K. Key: solid line, w ) 0.1 nm; dashed line, w ) 1.5 nm; dash-dotted line, w ) 2.0 nm.
both of the curves are shown in Figure 2a-d. As it is noticed, the quality of fitting by Dubinin-Stoeckli is not as good as the one described by DFT. The discrepancy between the Dubinin-Stoeckli fitting curve and the experimental data is important for pressures approaching the saturation one. At low-pressure range the difference is also observed. In Figure 4a-d, comparison of the pore-size distributions arisen from the DS equation and DFT adsorption model is given for A5, A10, A15, and A20, respectively. As
w (nm)
sample
DS
DFT
sample
DS
DFT
A5 A10
1.01 1.24
1.03 1.09
A15 A20
1.32 1.55
1.12 1.85
calculated from the DS equation, the mean width of the pores contained in the samples extends from 0.9 to 1.5 nm. A5 has a uniform pore width, and the distribution widths increase sensitively with the average pore width for the other samples. The mean widths of the pores in the two types of distributions are comparable (Table 3). The distributions from DFT have a sharp peak around 0.6 nm and a broad peak at larger pore widths. The separation of the two peaks increases for larger pore-size samples. Furthermore, the lowest peak of pore-size distribution from the DFT is absent in the DS distribution. This is obvious, because the DS equation involves a one-mode Gaussian function that should give less accuracy in the determination of the pore-size distribution for a wide range. The RS plots in Figure 5 were obtained by using the standard nitrogen adsorption isotherm of a nonporous carbon black (M no. 32). The RS plots exhibit a clear f-swing in the low RS range as for RS < 0.2. The f-swing marks the filling mechanism of the adsorption, which occurs before the completion of the monolayer adsorption in the micropores.23,24 However, the c-swing enhancement is noticed for A15 and A20 only. This enhancement takes place for RS > 0.5. From the two straight lines in Figure 5 all of the values of the total and external specific areas, micropore volume, and hence the average pore width can be obtained. The total specific surface area is related directly to the
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Figure 4. Pore-size distributions of A5, A10, A15, and A20. Key: solid line represents a bimodal log-normal function using the nonlocal DFT adsorption model; dashed line represents the Dubinin-Stoeckli equation; (A) A5; (B) A10; (C) A15; (D) A20.
the maximum amount of nitrogen adsorption on the samples. These results show a similarity with the previous analytical methods for small micropore samples. However, for samples of large micropores the discrepancy is noticeable. SPE results seem to display a smaller mean pore size compared to the two methods used before. Conclusion
Figure 5. RS plots of nitrogen adsorption isotherms at 77 K on A5, A10, A15, and A20. Key: O, A5; 0, A10; ], A15; 4, A20. Table 4. Micropore Structure of A5, A10, A15, and A20 from Nitrogen Adsorption Isotherms at 77 K Using the SPE Method
A5 A10 A15 A20
micropore vol (mL‚g-1)
tot. surface area (m2‚g-1)
pore width (nm)
0.34 0.48 0.61 0.96
920 1160 1350 1770
0.75 0.83 0.91 1.09
value of the slope derived from the linear line, which passes by both the origin and experimental point at RS ) 0.5. The slope of the second straight line drawn in Figure 5 gives an estimation of the external specific surface area of the sample, and its intercept with the adsorption axis determines the micropore volume. The values of the average pore size involved from the SPE method are listed in Table 4. The mean pore widths increase in accord with
The SPE method provides a precise method when monodisperse micropores are highly present in the porous material. The one-mode Gaussian distribution used in the DS method represents a constraint for the determination of the real pore-size distribution, and the DR equation which is the Kernel function in the generalized adsorption isotherm is not always appropriate for all the available pore groups. The DFT approach provides better information about the pore-size distribution even at wide pore-width range. The modeling by DFT remains, however, conditioned by idealized assumptions such as the smoothness of the adsorbent surface, no connection between pores, exclusion of chemical functional groups on the surface, and the shape of the pores. The modification brought by Olivier16 on the attractive fluid-fluid potential through the mean field theory gives a realistic approach to the pore-size determination. However, more efforts for the development of the DFT method are needed. For example, the improvement of the DFT method for the micropore size distribution determination should be done by comparison with the experimental studies using a systematic series of activated carbon samples whose pore wall chemistry and crystallinity are controlled. Acknowledgment. This work was supported by a Grant in-Aid for Scientific Research B from the Japanese Government. LA991242J
