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GCMC Study on Relationship between DR Plot and Micropore Width Distribution of Carbon T. Ohba and K. Kaneko* Physical Chemistry, Material Science, Graduate School of Natural Science and Technology, Chiba University, 1-33, Yayoi, Inage, Chiba 263-8522, Japan Received October 18, 2000. In Final Form: February 23, 2001 The morphological change of Dubinin-Radushkevich (DR) plots of the N2 adsorption isotherm for graphite slit pores was examined with relevance to the pore size distribution (PSD) and the temperature dependence using grand canonical Monte Carlo (GCMC) simulation. The simulated DR plots with different half-widths of PSD having Gaussian and Gaussian-like distributions of average pore width ) 1.2 nm were almost the same as each other below [ln(P0/P)]2 ) 60; the linearity depending on the half-width of PSD was observed over the [ln(P0/P)]2 range of 10-60. On the other hand, the DR plot deviated downward from the linear relation seriously above [ln(P0/P)]2 ) 60. This is caused by adsorption in the pore for pore widths less than 1.0 nm. The micropore volume calculated from the DR plot for pore widths more than 1.6 nm is underestimated considerably, but that for pore widths less than 1.4 nm is reasonable. The higher the adsorption temperature, the narrower the linear region of the DR plot. The limitation of the Dubinin-Astakov plot is appointed.
1. Introduction Dubinin-Radushkevich (DR) analysis has been devoted to elucidating the micropore structures of various solids regardless of insufficient understanding of the basis of the DR equation. In particular, activated carbon whose micropore structure cannot be determined by X-ray diffraction has been analyzed by the DR equation. The DR equation is given by eq 1.1
W ) W0 exp[-(A/E)n]
n ) 2 E ) βE0 A ) RT ln(P0/P) (1)
Here, W and W0 are the amount of adsorption at the relative pressure P/P0 and the micropore volume, respectively. A is the adsorption potential. β and E0 are the affinity coefficient and characteristic adsorption energy, respectively. The βE0 value can provide the isosteric heat of adsorption qst,φ)e-1 at the fractional filling φ ) e-1 using the enthalpy of vaporization ∆Hvap.
qst,φ)e-1 ) βE0 + ∆Hvap
(2)
Therefore, the linear plot of ln W versus A2 leads to W0 and qst,φ)e-1, which are quite important parameters. In case of carbon materials, the E0 value gives the average pore width w0 by the so-called Dubinin-Stoeckli (DS) relation. Several forms of the DS relation are known.2 The newest one is described by eq 3.3
w0/nm ) 10.8/(E0/kJ‚mol-1 - 11.4)
(3)
w0 estimated from eq 3 will be designated w0DS in this article. The n value is not necessarily equal to 2 and is variable in a wide range; eq 1 gives the Dubinin-Astakov (DA) equation. The combination of eqs 1 and 3 ensures the Gaussian type pore width distribution, if we can get a linear DR plot. Thus, the DR analysis can provide the * To whom correspondence should be addressed: Department of Chemistry, Faculty of Science, Chiba University, 1-33 Yayoi, Inage, Chiba, 263-8522, Japan. Fax: 81-43-290-2788. E-mail: kaneko@ pchem2.s.chiba-u.ac.jp. (1) Dubinin, M. M. Chem. Rev. 1960, 60, 235. (2) McEnaney, B. Carbon 1987, 25, 69. (3) Hugi-Cleary, D.; Stoeckli, F. Carbon 2000, 38, 1309.
essentially important parameters on the micropore structure of activated carbon such as the micropore volume, average pore width, and the isosteric heat of adsorption at φ Z 0.3. Also, the morphological examination of the DR plot has been believed to give the heterogeneous structure of micropores.4 Then, upward or downward deviation of the DR plot at a low P/P0 or high P/P0 region has been associated with the deviation of the pore width distribution from the Gaussian distribution. Marsh gave a good review on this point.5 Although the DR equation has been helpful to understand the micropore structures of carbon in particular, a fundamental understanding of the DR plot is still insufficient.6,7 The DR equation is a phenomenological relation. There is an argument that the DR equation cannot describe the adsorption behavior of the Henry low at a low P/P0 region. Chen and Yang studied the basis of the DR equation using statistical mechanics, giving evidence that there is no contradiction with the Henry law.8 Stoeckli et al. have developed the applicability of the DR equation.9 Nguyen and Do reported a new approach to the DR equation using the average molecular potential theory.10 However, still the DR equation must be studied from a different angle. Recently, the grand canonical Monte Carlo (GCMC) simulation and density functional theory (DFT) methods have been applied to analyze the adsorption isotherm to elucidate the micropore structures.11-16 The comparative study on micropore structures of activated carbon fibers with DR and DFT methods showed that the DR analysis (4) Dubinin, M. M.; Stoeckli, H. F. J. Colloid Interface Sci. 1980, 75, 34. (5) Marsh, H. Carbon 1987, 25, 49. (6) Dubinin, M. M. Carbon 1989, 27, 457. (7) Kakei, K.; Ozeki, S.; Suzuki, T.; Kaneko, K. J. Chem. Soc., Faraday Trans. 1990, 86, 371. (8) Chen, S. G.; Yang, R. T. Langmuir 1994, 10, 4244. (9) Stoeckli, F. Carbon 1998, 36, 363. (10) Nguyen, C.; Do, D. D. Carbon 2000, 38, 1339. (11) Seaton, N. A.; Walton, J. P. R. B.; Quirke, N. Carbon 1989, 27, 853. (12) Nicholson, D. J. Chem. Soc., Faraday Trans. 1996, 92, 1. (13) Bojan, M. J.; Steele, W. A. Carbon 1998, 36, 1417. (14) Lastoskie, C.; Gubbins, K. E.; Quirke, N. J. Phys. Chem. 1993, 97, 4786. (15) Lastoskie, C.; Gubbins, K. E.; Quirke, N. Langmuir 1993, 9, 2693. (16) Olivier, J. P. J. Porous Mater. 1995, 2, 9.
10.1021/la001463l CCC: $20.00 © 2001 American Chemical Society Published on Web 05/18/2001
DR Plot and Micropore Width Distribution
Langmuir, Vol. 17, No. 12, 2001 3667 Table 1. Micropore Volume and Characteric Adsorption Energy in w0 ) 1.2 nm ∆ W0/mL mL-1a W00/mL mL-1b βE0/kJ mol-1 qst,φ)e-1/kJ mol-1 E0/kJ mol-1 w0DS/nm
0.2
0.4
0.8
1.6
0.97 0.87 5.9 11.5 17.9 1.7
0.86 0.86 6.7 12.3 20.4 1.2
0.76 0.85 7.9 13.5 24.0 0.9
0.68 0.84 8.6 14.1 25.9 0.7
a This is determined from the intercept of the extrapolated linear plot. b W00 is the original pore volume used for GCMC simulation.
Figure 1. DR plots corresponding to different half-widths of pore width distribution for average pore width ) 1.2 nm (a) and the Gaussian and Gaussian-like pore width distribution (b): O, ∆ ) 0.2 nm; 4, ∆ ) 0.4 nm; 0, ∆ ) 0.8 nm; ×, ∆ ) 1.6 nm.
Figure 3. DR plots of different average pore width and 0.4 nm half-width at 77 K: O, 0.8 nm; 4, 1.0 nm; 0, 1.2 nm; ×, 1.4 nm; b, 1.6 nm; 2, 1.8 nm; 9, 2.0 nm. Table 2. Micropore Volume and Characteristic Adsorption Energy for Various w0 w0/nm W0/mL mL-1a W00/mL mL-1b βE0/kJ mol-1 qst,φ)e-1/kJ mol-1 E0/kJ mol-1 w0DS/nm
0.8
1
1.2
1.4
1.6
1.8
2.0
0.79 0.79 17.5 23.1 53.0 0.3
0.87 0.83 9.3 14.8 28.0 0.6
0.87 0.86 6.7 12.3 20.4 1.2
0.67 0.88 6.5 12.1 19.8 1.3
0.51 0.88 6.7 12.3 20.4 1.2
0.44 0.88 6.5 12.1 19.8 1.3
0.39 0.88 6.3 11.8 19.0 1.4
a This is determined from the intercept of the extrapolated linear plot. b W00 is the original pore volume used for GCMC simulation.
Figure 2. The magnified DR plots of Figure 1b.
is still very helpful to understand the micropore structure.17 We need to understand the DR analysis using GCMC or DFT methods. In the preceding GCMC simulation study,18 Ohba et al. showed using GCMC simulation that the linearity of the DR plot is obtained for the wider Gaussian distribution of the micropore width and the DS relation holds for a restricted pore width region. In this article, the morphological change of the DR plot with the variation of the pore width distribution and the adsorption temperature studied by the GCMC method is given.
width range of w ) 0.4-2.5 nm with every 0.10 nm difference. GCMC simulation was carried out using the established procedures.19 We used the 12-6 Lennard-Jones potential φff(r) for the fluid-fluid interaction at an intermolecular distance r using a single center approximation.
[( ) ( ) ]
φff(r) ) 4ff
σff r
12
-
σff r
6
(4)
Here, ff and σff are the N2-N2 potential well depth (ff/kB ) 104.2 K) and the effective diameter (σff ) 0.3632 nm), respectively. Steele’s 10-4-3 potential function was used for the interaction potential of an N2 molecule with a single graphite slab.20
φsf(z) ) A
[(
) ( )
2 σsf 5 z
10
-
σsf z
4
-
σsf4
]
3∆C(z + 0.61∆C)3
(5)
2. Simulation The N2 adsorption isotherms at 77 K were calculated with GCMC simulation in a graphite slit pore of the pore
where A is 2πσsf2sfF∆C and z is the vertical distance of the
(17) El-Merraoui, M.; Aoshima, M.; Kaneko, K. Langmuir 2000, 16, 4300. (18) Ohba, T.; Suzuki, T.; Kaneko, K. Carbon 2000, 38, 1879.
(19) Nicholson, D.; Parsonance, N. G. In Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: London, 1982. (20) Steele, W. A. Surf. Sci. 1973, 36, 317.
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w ) H - (2z0 - σff)
z0 ) 0.856σsf
(6)
where H is defined as the distance between opposite graphite surfaces and z0 is the distance of the closest approach between N2 and the graphite surface. The following established GCMC simulation method calculated the adsorption isotherms of N2. The random movement, creation, and removal of molecules make new configurations. They are accepted when they obey Metropolis’s sampling scheme in proportion to exp(-∆E/kT) where ∆E is the change of total energy in the system. We used the rectangular cell l × l × w for the calculation, and this size is replicated two-dimensionally to form an infinite slit-shaped pore. 3. Results and Discussion
Figure 4. DR plots with different Gaussian-like distributions of average pore width 1.2 nm. The right DR plot is obtained from the left pore width distribution system. The broken lines are the Gaussian distribution (left) and its DR plot (right). Table 3. W0 and E0 Values for the Antisymmetric Gaussian Pore Width Distribution regular distribn Figure 4a Figure 4b Figure 4c Figure 4d broken solid line solid line solid line solid line line W0/mL mL-1a W00/ mL mL-1b βE0/kJ mol-1 qst,φ)e-1/ kJ mol-1 E0/kJ mol-1 w0DS/nm calcd av pore width/nm
0.81 1.09
0.89 0.90
0.98 0.91
0.76 0.82
0.86 0.86
5.9 11.4
7.2 12.8
6.4 12.0
6.6 12.2
6.5 12.1
17.8 1.5 1.3
21.8 1.0 1.1
19.4 1.2 1.1
20.1 1.2 1.3
19.6 1.2 1.2
a This is determined from the intercept of the extrapolated linear plot. b W00 is the original pore volume used for GCMC simulation.
molecule from the graphite surface. F is the carbon atomic number density, and ∆C is the interlayer distance of the graphite. sf and σsf are fitted parameters of the N2-carbon potential depth and effective diameter, respectively, which were obtained using the Lorentz-Berthelot rules. The pore width w which is determined experimentally is associated with the physical width H using eq 6.21
3.1. Dependence of the DR Plot on the Pore Width Distribution. The DR plot was constructed using the GCMC simulated N2 isotherms for graphite slit pores having the Gaussian pore width distributions, as shown in Figure 1a. Here, the average pore width w0 is fixed at 1.2 nm and the half-width ∆ of the pore width distribution is changed from 0.2 to 1.6 nm. The linear DR plot in this scale is observed above P/P0 ) 10-4 for all systems. However, all DR plots deviate downward below P/P0 ) 10-4. This downward deviation below P/P0 ) 10-4 is attributed to the primary filling process below P/P0 ) 10-4; cooperative filling begins at P/P0 ) 10-4 from GCMC simulations.22 The sharper the pore width distribution, the more remarkable the downward deviation. The widest system of ∆ ) 1.6 nm has only a slight downward deviation. The simulated adsorption isotherms for the pore width smaller than 1 nm have a sharp jump below P/P0 ) 10-4, giving the stepwise shape in the DR plot below P/P0 ) 10-4. A small contribution of adsorption in such small pores weakens the stepwise feature below P/P0 ) 10-4. Therefore, the evident downward deviation at the low P/P0 range is ascribed to a sharp pore width distribution. As the GCMC simulated adsorption isotherm has no blocking effect, the downward deviation of the DR plot can occur without any blocking effect near the pore entrance. Figure 2 shows the DR plots magnified from those in Figure 1 in order to check the linearity above P/P0 ) 10-4. The linear range is limited to the narrow pressure range of P/P0 ) 10-4 to P/P0 ) 10-2. A system having a sharper pore width distribution gives a single linear relationship of a greater slope, whereas the DR plot of the system of the widest pore width distribution has the smallest slope. Also, the deviation from the linearity depends on the halfwidth ∆. The linear plot for smaller ∆ deviates downward, whereas the linear plot for a greater ∆ deviates upward. These behaviors are completely different from the established understanding; the upward and downward deviations from the linear DR plot are ascribed to the imperfect Gaussian distribution without greater pores and with additional pores of great widths, respectively. Therefore, the conventional understanding of the relationship between the morphology of the DR plot and the pore width distribution should be re-examined. The W0 from the intercept of the extrapolated linear plot is considerably different from the original pore volume used for the GCMC simulation except for the case of ∆ ) 0.4 nm. In case of a greater ∆, the micropore volume is underestimated, whereas it is overestimated for ∆ ) 0.2 nm. Hence, one (21) Kaneko, K.; Cracknell, R. F.; Nicholson, D. Langmuir 1994, 10, 4606. (22) Ohba, T.; Suzuki, T.; Kaneko, K. Chem. Phys. Lett. 2000, 326, 158.
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Figure 5. The DR plots of various Gaussian distributions having the peak at 1.2 nm pore width with the variation of the cutoff position. The right DR plots are obtained from the left distributions. (a) I, O; II, 0; III, 4. (b) IV, 4; V, 0; VI, O.
must be cautious while determining the W0 from the linear DR plot. As the βE0 and qst,φ)e-1 are intensively influenced by the presence of smaller micropores, the system of a greater ∆ gives larger βE0 and qst,φ)e-1. As to the determination of w0 from the DS relation, the determined w0 values are seriously different from 1.2 nm except for ∆ ) 0.4 nm. Therefore, the DR analysis is effective for micropore systems having the pore width distribution of about 30% of the half-width. Table 1 summarizes the W0, βE0, qst,φ)e-1, and w0 values determined by eqs 1-3. Figure 3 shows DR plots for the micropore width distribution of a regular Gaussian type of ∆ ) 0.4 nm as a function of w0. The morphology of the DR plot depends sensitively on the w0 value. The DR plot is described by a straight line for w0 e 1.2 nm, whereas the linearity is lost for w0 g 1.4 nm. For wider pore systems of w g 1.4 nm, a linear range is limited from [ln(P/P0)]2 ) 15-40. Table 2 shows W0, βE0, qst,φ)e-1, and w0DS for various w0. The βE0 value for w0 g 1.4 nm is almost constant, being close to that of w0 ) 1.2 nm irrespective of different w0 values. This is clearly erroneous. The average pore width value estimated from the DS relation is reasonable only for w0 ) 1.2 and 1.4 nm. Hence, the DR plot provides a completely erroneous conclusion in some cases. 3.2. Effect of Antisymmetric Pore Width Distribution on the DR Plot. Figure 4 shows the morphological change of the DR plot with introduction of the antisymmetry in the Gaussian pore width distribution for w0 )
Figure 6. DR plots of 1.2 nm average pore width and 0.4 nm half-width at various temperatures: O, 77 K; 4, 87 K; 0, 97 K; b, 107 K; 2, 117 K.
1.2 nm. The broken curves show the regular Gaussian pore width distribution and the corresponding DR plot. The deficient pore width distribution in the small pore width range shifts the DR plot downward (Figure 4a). This behavior is different from the established understanding.5 The stepwise structure in the low P/P0 range disappears by lack of small pores. The slope of the linear range below P/P0 ) 2 × 10-4 becomes greater. On the
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Figure 7. The relationship between the relative pressure on the downward deviation linear DR plot and the adsorption temperature.
Ohba and Kaneko
not seriously affect the fundamental shape of the DR plot and the linearity of the DR plot below P/P0 ) 10-4 is almost preserved. In the cutoff of the small pore width range (Figure 5a), the cutoff at a greater pore width gives a downward parallel shift of the DR plot. On the contrary, the cutoff at a smaller pore width leads to a downward parallel shift of the DR plot in the case of the cutoff of the wider pores. The linearity of the DR plot is lost because of the cutoff of the half distribution. The shape of the DR plot must vary because of the intense cutoff of the Gaussian type distribution according to the conventional interpretation of the DR plot.5 However, the morphology of the DR plot is insensitive of the cutoff of the Gaussian pore width distribution. Therefore, the evaluation of the micropore width distribution from the shape of the DR plot must be done cautiously. Table 3 shows the variation of the related parameters with the pore width distribution change. The w0DS value is not so different from the average pore width calculated from the distribution. It is noteworthy that the linearity of the DR plot does not sensitively depend on the pore width distribution. 3.3. Effect of Temperature on DR Plot. The adsorption isotherm depends on the adsorption temperature. In particular, adsorption above the boiling point is sensitively influenced by the adsorption temperature. Molecules must be preferentially adsorbed in smaller pores above the boiling point. Hence, the relationship between the DR plot shape and the adsorption temperature should be examined by the GCMC technique. Figure 6 shows the DR plots for the w0 ) 1.2 nm pore system of ∆ ) 0.4 nm at different temperatures. The complete linearity of the DR plot is observed at 77 K. The increase in the adsorption temperature narrows the linear range, and the linear range disappears at 117 K.
Figure 8. DA plots of 1.2 nm average pore width and 0.4 nm half-width at different temperatures: O, 77 K; 4, 87 K; 0, 97 K; b, 107 K; 2, 117 K.
contrary, the excess pore width distribution in the small pore width range (Figure 4b) shifts the DR plot upward. The stepwise structure below P/P0 ) 10-4 does not disappear. The slope becomes smaller. Therefore, the stepwise structure in the simulated DR plot stems from the adsorption jump in narrow pores. The linear DR plot can give reasonable values of the pore volume and βE0 in these heterogeneous pore systems, as shown in Table 3. In the case of the deficient pore width distribution in the wider pore range (Figure 4c), the slope of the DR plot becomes smaller, agreeing with the prediction by the traditional DR analysis, if we focus the discussion in the linear range. As the stepwise structure is not affected by the cutoff of wider micropores, the nonlinear range below P/P0 ) 10-4 does not change. The excess distribution in the wider pore width range in Figure 4d leads to a reasonable result as in Figure 4c. Therefore, this type of heterogeneous structure can be elucidated from the DR analysis for the allowed pore width range (w ) 1.2 nm in this case). As to the DS relation, types of Figure 4c,d do not affect the w0DS, which must be changed according to the variation of the pore width distribution. Table 2 summarizes these parameters. The effect of the sharp cutoff of the Gaussian pore width distribution on the DR plot is shown in Figure 5. This Gaussian pore width distribution has a peak at 1.2 nm and ∆ ) 0.4 nm. This distribution function is cut off at the different pore width. It is noteworthy that the cutoff does
The linear DR plot at 77 K deviates downward at a low relative pressure. The relative pressure on the downward deviation depends on the temperature, as shown in Figure 7. The higher the adsorption temperature, the greater the relative pressure on the downward deviation. Molecules can be adsorbed in smaller pores at a higher temperature, which is similar to adsorption in the lower P/P0 range at 77 K. Consequently, we can regard that the downward curve of the DR plot at 77 K shifts to the higher pressure range at the higher temperature. Hence, the DR analysis of the curve region must have no useful information on the pore structure. Although the n value in the DA equation can be variable according to the system, n ) 3 has been often used for the micropore structural analysis. Thus, the DA plots with n ) 3 were applied to the adsorption data at different temperatures as shown in Figure 8. The linearity of the DA plot is improved. The adsorption data even at higher temperatures show a considerably good linearity. However, this linear DA plot is constructed by adsorption data in the low fractional filling which is not necessarily useful as mentioned above. Therefore, we cannot derive the information on pore structures from this DA plot, even if the DA plot is linear. Acknowledgment. We acknowledge support from the Grant-in-Aid for scientific research by the Ministry of Education, Culture, and Science, Japanese Government. LA001463L
