Application of Density Functional Theory to Analysis of Energetic

Bryan J. Schindler , Lucas A. Mitchell , Clare McCabe , Peter T. Cummings , and M. Douglas LeVan .... Duong D. Do , Eugene A. Ustinov , Ha D. Do. 2008...
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Langmuir 2004, 20, 3791-3797

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Application of Density Functional Theory to Analysis of Energetic Heterogeneity and Pore Size Distribution of Activated Carbons E. A. Ustinov† and D. D. Do*,‡ St. Petersburg State Technological Institute, 26 Moskovsky Pr., St. Petersburg, Russia, and Department of Chemical Engineering, University of Queensland, Qld 4072, Australia Received October 16, 2003. In Final Form: January 14, 2004 A new approach based on the nonlocal density functional theory to determine pore size distribution (PSD) of activated carbons and energetic heterogeneity of the pore wall is proposed. The energetic heterogeneity is modeled with an energy distribution function (EDF), describing the distribution of solidfluid potential well depth (this distribution is a Dirac delta function for an energetic homogeneous surface). The approach allows simultaneous determining of the PSD (assuming slit shape) and EDF from nitrogen or argon isotherms at their respective boiling points by using a set of local isotherms calculated for a range of pore widths and solid-fluid potential well depths. It is found that the structure of the pore wall surface significantly differs from that of graphitized carbon black. This could be attributed to defects in the crystalline structure of the surface, active oxide centers, finite size of the pore walls (in either wall thickness or pore length), and so forth. Those factors depend on the precursor and the process of carbonization and activation and hence provide a fingerprint for each adsorbent. The approach allows very accurate correlation of the experimental adsorption isotherm and leads to PSDs that are simpler and more realistic than those obtained with the original nonlocal density functional theory.

1. Introduction The pore size distribution (PSD) of activated carbon is an important characteristic of its geometrical structure. There are some classical approaches developed to evaluate PSD in the microporous and mesoporous ranges. In the case of micropores, the most popular methods are the Horvath and Kavazoe (HK) approach1 and its modifications.2-5 Recently a method based on the enhanced potential and enhanced pore pressure was introduced by Do and co-workers6-11 to describe adsorption in pores ranging from micropores to mesopores within the same framework of layering and filling. Mesoporous structure of adsorbents may be determined with the Broekhoff and de Boer (BdB),12,13 Barrett, Joyner, and Halenda (BJH),14 and Kruk, Jaroniec, and Sayari (KJS)15-17 methods. More rigorous methods are based on molecular approaches such as grand canonical Monte Carlo (GCMC) simulations18-22 † ‡

St. Petersburg State Technological Institute. University of Queensland.

(1) Horvath, G.; Kawazoe, K. J. Chem. Eng. Jpn. 1983, 16, 470. (2) Saito, A.; Foley, H. C. AIChE J. 1991, 37, 429. (3) Cheng, L. S.; Yang, R. T. Chem. Eng. Sci. 1994, 49, 2599. (4) Dombrowski, R. J.; Lastoskie, C. M.; Hyduke, D. R. Colloids Surf., A 2001, 187, 23. (5) Ustinov, E. A.; Do, D. D. Langmuir 2002, 18, 4637. (6) Do, D. D. A new method for the characterisation of Micromesoporous materials. Presented at the International Symposium on New Trends in Colloid and Interface Science, September 24-26, 1998, Chiba, Japan. (7) Nguyen, C.; Do, D. D. Langmuir 1999, 15, 3608. (8) Nguyen, C.; Do, D. D. Langmuir 2000, 16, 7218. (9) Do, D. D.; Nguyen, C.; Do, H. D. Colloids Surf. 2001, 187, 51. (10) Do, D. D.; Do, H. D. Langmuir 2002, 18, 93. (11) Do, D. D.; Do, H. D. Appl. Surf. Sci. 2002, 78, 1. (12) Broekhoff, J. C. P.; de Boer, J. H. J. Catal. 1967, 9, 8, 15. (13) Broekhoff, J. C. P.; de Boer, J. H. J. Catal. 1968, 10, 368, 377, 391. (14) Barret, E. P.; Joyner, L. G.; Halenda, P. P. J. Am. Chem. Soc. 1962, 73, 373. (15) Kruk, M.; Jaroniec, M.; Sayari, A. Langmuir 1997, 13, 6267. (16) Kruk, M.; Antochshuk, V.; Jaroniec, M.; Sayari, A. J. Phys. Chem. B 1999, 103, 10670. (17) Kruk, M.; Jaroniec, M. Chem. Mater. 2000, 12, 222.

and nonlocal density functional theory (NLDFT).18-33 They are becoming widely used in the PSD analysis of activated carbon adsorbents in recent decades. The derived PSDs from these sophisticated tools show a peculiar feature in that there is an artificial gap in the vicinity of 1 nm pore width.29-31 This feature is persistent for all activated carbon samples considered in the literature. Furthermore, the correlation of data is not very good, especially in the region of monolayer coverage formation. The poor correlation of the isotherm and the artificial gap in the PSD are closely connected to each other, and that points to the deficiency of the conventional NLDFT analysis where the surface is treated as an energetic homogeneous surface. This suggests that the pore wall surface of activated carbon is clearly heterogeneous (in both topology and surface chemistry). This problem may be overcome by introducing the energetic heterogeneity of the pore wall surface or (18) Panagiotopoulos, A. Z. Mol. Phys. 1987, 62, 701. (19) Maddox, M. W.; Olivier, J. P.; Gubbins, K. E. Langmuir 1997, 13, 1737. (20) Samios, S.; Stubos, A. K.; Kanellopoulos, N. K.; Cracknell, R. F.; Papadopoulos, G. K.; Nicholson, D. Langmuir 1997, 13, 2795. (21) Gusev, Yu. V.; O’Brien, J. A.; Seaton, N. A. Langmuir 1997, 13, 2815. (22) Neimark, A. V.; Vishnyakov, A. Phys. Rev. E 2000, 62, 4611. (23) Tarazona, P. Phys. Rev. A 1985, 31, 2672. (24) Evans, R.; Marconi, U. M. B.; Tarazona, P. J. Chem. Soc., Faraday Trans. 2 1986, 82, 1763. (25) Tarazona, P.; Marconi, U. M. B.; Evans, R. Mol. Phys. 1987, 60, 573. (26) Seaton, N. A.; Walton, J. P. R. B.; Quirke, N. Carbon 1989, 27, 853. (27) Lastoskie, C.; Gubbins, K. E.; Quirke, N. J. Phys. Chem. 1993, 97, 4786. (28) Lastoskie, C.; Gubbins, K. E.; Quirke, N. Langmuir 1993, 9, 2693. (29) Olivier, J. P. Carbon 1998, 36, 1469. (30) Neimark, A. V.; Ravikovitch, P. I. Langmuir 1997, 13, 5148. (31) Ravikovitch, P. I.; Vishnyakov, A.; Russo, R.; Neimark, A. V. Langmuir 2000, 16, 2311. (32) Ravikovitch, P. I.; Neimark, A. V. Colloids Surf., A 2001, 187188, 11. (33) Ravikovitch, P. I.; Neimark, A. V. J. Phys. Chem. B 2001, 105, 6817.

10.1021/la035936a CCC: $27.50 © 2004 American Chemical Society Published on Web 03/27/2004

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random distribution of the pore wall thickness.34 The importance of the energetic heterogeneity of the pore surface was long recognized in the literature.35-40 Some authors model surface heterogeneity by introducing a potential with a periodic spatial variation,35-37 while others use the patchwise model.38,39 In this paper, we account for the energetic heterogeneity of adsorbents in the framework of the patchwise model of the surface. The energetic heterogeneity is described by a distribution function with respect to the solid-fluid potential well depth. Our aim is to determine simultaneously the PSD and the energy distribution function (EDF) from low-temperature nitrogen or argon adsorption isotherms. Here we shall assume that the PSD and EDF are not correlated.

and excess molecular Helmholtz free energy, respectively. The latter is associated with fluid-fluid repulsive forces. The third term, uint(z), represents the attractive component of the intermolecular interactions, and the term uext(z) accounts for the solid-fluid interaction. The last term is a correction accounting for multibody interaction, and R is a small parameter.41 Brief comments will be given below in the section devoted to adsorption on carbon black. The excess molecular Helmholtz free energy is usually calculated from the Carnahan-Starling equation42 for the reference hard sphere fluid:

4η j - 3η j2 (1 - η j )2

fex(Fj) ) kBT

2. Model 2.1. Nonlocal Density Functional Theory. The nonlocal density functional theory is well established and widely presented in the literature. In this section, we briefly describe some basic points of the NLDFT and then a procedure to determine the PSD and EDF by treatment of the adsorption isotherm. It is also justified that we exemplified the formalism of NLDFT in the case of a commonly used one-dimensional task, which could be helpful for readers who are unfamiliar with the NLDFT. Besides, we incorporate to the original NLDFT the factor of nonadditivity of solid-fluid and fluid-fluid interactions,41 which substantially improves the correlation of experimental data on carbon black. We also consider the asymptotic solution of the NLDFT for relatively wide mesopores in more detail, as it significantly reduces the computation time needed to generate a comprehensive set of local isotherms. We rely on the slit pore model and Tarazona’s version of the NLDFT. The distribution of density in a confined pore can be obtained by considering an open system in which a pore is allowed to exchange mass with the surroundings. From the thermodynamic principle, the density distribution is obtained from the minimization of the following grand potential written below for the one-dimensional case:

Ω)

∫F(z)[f(z) - µ] dz

(1)

Here F(z) is the local density of the adsorbed fluid at a distance z from one of the walls of the pore; f(z) is the molecular Helmholtz free energy of the adsorbed phase; µ is the chemical potential. The Helmholtz free energy f(z) may be divided into some contributions:

f(z) ) kBT[ln(Λ3F(z)) - 1] + fex[Fj(z)] + uint(z) + uext(z) + Ruintuext/kBT (2) where Λ is the thermal de Broglie wavelength and kB is Boltzmann’s constant. The first and second terms in the right-hand side (RHS) of this equation are the ideal (34) Bhatia, S. K. Langmuir 2002, 18, 6845. (35) Gac, W.; Patrykiejew, A.; Sokolowski, S. Thin Solid Films 1997, 298, 22. (36) Ro¨cken, P.; Somoza, A.; Tarazona, P.; Findenedd, G. J. Chem. Phys. 1998, 108, 8689. (37) Reszko-Zygmunt, J.; Pizio, O.; Rzysko, W.; Sokolowski, S.; Sokolowska, Z. J. Colloid Interface Sci. 2001, 241, 169. (38) Olivier, J. P. In Proceedings of the Fifth International Conference on Fundamentals of Adsorption; LeVan, M. D., Ed.; Kluwer Academic Press: Boston, 1996; p 699. (39) Olivier, J. P. In Surfaces of Nanoparticales and Porous Materials; Schwartz, J. A., Contescu, C. L., Eds.; Marcel Dekker: New York, 1999; p 295. (40) Ustinov, E. A.; Do, D. D. Part. Part. Syst. Charact., in press. (41) Kruk, M.; Jaroniec, M.; Choma, J. Carbon 1998, 36, 1447.

π η j ) dHS3Fj 6

(3)

where dHS is the equivalent hard sphere diameter. Fj(h) is the smoothed density defined as

∫F(r′)ω(|r - r′|,Fj(r′)) dr′

Fj(r) )

(4)

The weighting function ω(|r - r′|,Fj(r′)) may be calculated using the Tarazona prescription.43 For the one-dimensional case under consideration, this prescription allows us to express the smoothed density as follows:

Fj(z) ) S0(z) + S1(z)Fj(z) + S2(z)[Fj(z)]2

(5)

with the coefficients Si being a function of z and given by

Si(z) )

2σ ∫-2σ

ff ff

F(z + z′)ωi(z′) dz′ i ) 0, 1, 2

(6)

where σff is the collision diameter of the fluid molecule and ω0, ω1, and ω2 are simple functions of z/σff defined in such a manner that in the case of homogeneous fluid Fj(z) ) S0(z) ) F(z) and Si(z) ) 0 for i ) 1,2. The explicit form of ω0, ω1, and ω2 in the general case is given in the paper of Tarazona.43 In the one-dimensional case, the reduced form of these functions is presented in the Appendix. The attractive potential of two molecules is modeled using the Weeks-Chandler-Andersen (WCA) scheme:44

{

-ff φ(r) ) 4ff[(σff/r)12 - (σff/r)6] 0

r < rm rm < r < rc r > rc

(7)

Here r is the distance between the molecules; ff is the potential well depth; rm ) 21/6σff is the distance at which the potential is minimum. Note that the potential associated with one molecule is 1/2 of φ(r). In this formula and further, we drop the cutoff distance rc just to avoid superfluous complexity. It is not difficult to account for the cutoff distance in numerical calculations. In all our calculations, we used the value 5σff for rc. In the mean field approximation, it yields the following expression for the attractive part of the Helmholtz free energy in the case of one-dimensional density distribution:

uint(z) )

∫ζ(z′)F(z + z′) dz′

(8)

The function ζ(z) is obtained by integrating the WCA (42) Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635. (43) Tarazona, P.; Marini Bettolo Marconi, U.; Evans, R. Mol. Phys. 1987, 60, 573. (44) Weeks, J. D.; Chandler, D.; Andersen, H. C. J. Chem. Phys. 1971, 54, 5237.

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potential along the xy plane parallel to the pore walls (note the density is a constant in that xy plane):

ζ(z′) )

{

|z′| < rm (9) |z′| > rm

-(1/2)πff[(9/5)rm2 - z′2] πffσff2[(2/5)(σff/z′)10 - (σff/z′)4]

The condition of minimum of the grand potential leads to the following equation:

µ ) kBT ln(Λ3F(z)) + fex[Fj(z)] +

∫F(z′)f ′ex[Fj(z′)]φ(z,z′) dz′ + 2u

int

(z) +

∫ζ(z - z′)uext(z′)F(z′) dz′

(10)

φ(z,z′) ) ω0(|z - z′|) + Fj(z′)ω1(|z - z′|) + [Fj(z′)]2ω2(|z - z′|) 1 - S1(z′) - 2Fj(z′)S2(z′) (11) The solid-fluid potential uext(z) is usually modeled by the Steele equation:45

uext ) 2πFssfσsf2∆[u˜ (z) + u˜ (H - z)] 10 σsf4 σsf4 2 σsf 5 z10 z4 3∆(0.61∆ + z)3

(12)

In this equation, Fs is the number density of carbon atoms per unit volume, ∆ is the distance between adjacent graphite lattice layers, sf is the potential well depth of the solid-fluid potential, and σsf is the solid-fluid collision diameter usually determined by the Lorentz-Berthelot mixing rule:

σsf ) (σss + σff)/2

(14)

where

πdHS3 F 6

C)

16πx2 ffσff3 9

(15)

In these equations, the density is expressed in molecules per m3. The chemical potential corresponding to the equation of state 14 is

8 - 9η j + 3η j2 j - 2CFη j (16) µ ) kBT ln(Λ3F) + kBTη (1 - η j )3

where

u˜ (z) )

j3 1+η j+η j2 - η - CF2 (1 - η j )3

p ) kBTF

η j)

uext(z) + (R/kBT)uint(z)uext(z) + (R/kBT)

In the homogeneous fluid, the equation of state obtained by a combination of the Carnahan-Starling eq 3 and the WCA scheme 7 has the following form:

(13)

For a given value of the chemical potential µ, the RHS of eq 10 is constant, irrespective of any value of z. It corresponds to the definite density distribution in the pore, which makes this equation valid at any distance z from the pore wall. In confined slit pores, such distribution is an oscillation function and may be obtained by an iteration procedure. 2.2. Asymptotical Solution for Wide Slit Pores. Activated carbon usually has a bimodal pore size distribution. One is in the micropore region, while the other is in the mesopore region. To derive the pore size distribution, one has to determine the local isotherms for pores in those two regions. To obtain local isotherms for pores having width greater than several nanometers, a significant amount of computation time is required. To avoid this excessive computation, we propose in this section a procedure to obtain density distribution and the amount adsorbed in sufficiently wide pores. For those pores, we observe that the oscillation of local density is diminished and the solid-fluid potential is practically zero. This strongly suggests that at sufficiently large distance from the pore walls the adsorbed phase behaves very much like the bulk phase at the same chemical potential. (45) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: Oxford, 1974.

Given the pressure in the bulk phase, one can calculate the bulk phase density as a root of eq 14 and then the chemical potential by eq 16. In the central part of a wide slit pore, the adsorbed phase density on the adsorption branch of the isotherm is equal to that in the bulk phase, as the solid-fluid potential is close to zero. However, it should be taken into account that at a sufficiently high pressure eq 16 has two roots for the density at a specified value of the chemical potential. This situation corresponds to the desorption branch of the isotherm. Hence, at the same bulk phase pressure the density in the center of a wide pore is equal to that in the bulk phase along the adsorption branch and significantly higher after the capillary condensation has occurred. Clearly, it is not necessary to determine the density profile in the center of the pore using the iterating procedure in the framework of NLDFT for a nonhomogeneous fluid, since the density distribution is uniform and could be easily determined beforehand with eqs 14 and 16. This means that the NLDFT iteration algorithm could be applied only in the vicinity of the pore walls, where the adsorbed fluid is indeed nonhomogeneous. The solution for the central part of the pore can be used as a boundary condition. The question is, what root of eq 16 should be taken at a specified bulk phase pressure and, consequently, the chemical potential? The answer gives the comparison of the grand potential Ω for these different roots. In this paper, we do not consider a metastable state and rely on equilibrium conditions. So, the true root of eq 16 should correspond to the case when the grand potential is minimal. Such an algorithm saves a lot of computation time. Figure 1 illustrates the agreement of the exact algorithm and the asymptotical algorithm described above for the phase transition pressure (filling pressure) versus pore width H defined as the distance between the carbon atoms centers in the outermost graphite layers of the pore walls. The distance from each pore wall where we determined the exact solution was 3 nm in the region of the pore width from 6 to 10 nm and 4 nm if the pore width was greater than 10 nm. As is seen from the figure, the asymptotical scheme works very reliably, with the calculation time per one local isotherm being independent of the pore width. 3. Results 3.1. Adsorption of Ar and N2 on Graphitized Carbon Black. The conventional NLDFT is known to overpredict the amount adsorbed of nitrogen and argon on graphitized carbon black in the region of the reduced

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Figure 1. Dependence of the filling pressure (the pressure of the equilibrium phase transition) on the slit pore width for nitrogen adsorption at 77 K: (squares) exact solution with NLDFT accounting for the multibody interaction; (filled circles) asymptotical solution. Table 1. Molecular Parameters for Ar and N2 and the Surface Area of Carbon Black

Ar, 87.3 K N2, 77 K

ff/kB (K)

σff (nm)

dHS (nm)

sf/kB (K)

σsf (nm)

R

118.05 94.45

0.3305 0.3575

0.338 0.3575

58.01 56.10

0.3353 0.3488

0.0183 0.0242

pressure above approximately 0.1.27,28,46 The same drawback is also observed in GCMC simulations. One could think that at lower pressure (below 0.1) the correlation of experimental data is much better. Such an illusion is created by the fact that the amount adsorbed in corresponding figures is always presented in linear scale. Yet, using the logarithmic scale, it is easy to check that the description of experimental data in the low-pressure region is only fair, as well. In particular, the molecular approaches show a part of the isotherm that is too steep in the vicinity of the inflection point below the monolayer coverage (nearly 2D condensation), while experimentally the isotherm is quite smooth. On the other hand, one can suggest that if the correlation ability of the advanced molecular approaches is quite poor in the case of nitrogen and argon adsorption on graphitized carbon black, the reliability of the PSD analysis of porous materials with the same methods would be questionable. That is the reason we introduced the term accounting for the nonadditivity of multibody interactions in the simplified quadratic form (see eq 2). The problem of nonadditivity is not new and has been discussed in the literature (see, for example, refs 47-49). The molecular parameters for argon and nitrogen used in analysis of adsorption isotherms on carbon black are listed in Table 1. The potential well depth, the collision diameter for fluid-fluid interactions, and the hard sphere diameter are taken from the paper of Neimark et al.50 Experimental data were taken from Gardner et al. for argon adsorption on graphitized thermal carbon black at 87.3 K51 and from Kruk et al.52 for nitrogen at 77 K on the (46) Olivier, J. P. J. Porous Mater. 1995, 2, 9. (47) Barker, J. A.; Fisher, R. A.; Watts, R. O. Mol. Phys. 1971, 21, 657. (48) de Pablo, J. J.; Bonnin, M.; Prausnitz, J. M. Fluid Phase Equilib. 1992, 73, 187. (49) Miyano, Y. Fluid Phase Equilib. 1994, 95, 31.

Figure 2. Argon adsorption isotherm on carbon black at 87.3 K in logarithmic scale (a) and in linear scale (b): (O) data of L. Gardner et al. (ref 51); (solid line) developed approach accounting for the multibody interaction; (dashed line) the conventional NLDFT.

same material. The argon adsorption isotherm in logarithmic and linear scales is presented in Figure 2. The potential well depth sf/kB, the parameter R for the nonadditivity term, and the surface area S were determined by the least-squares fitting. The surface area found with the NLDFT is 7.94 m2/g, which is higher than the reported Brunauer-Emmett-Teller (BET) surface area of 6.2 m2/g. This was also observed by Olivier.38 The nitrogen adsorption isotherm on the graphitized carbon black at 77 K is shown in Figure 3. As is seen from the figures, the developed approach accounting for the nonadditivity of multibody interactions correlates experimental data very accurately, while the conventional NLDFT fails to describe N2 and Ar isotherms on graphitized carbon black at their boiling points. 3.2. Pore Size Distribution and Energy Distribution Analysis of Activated Carbons. With the exception of the solid-fluid potential well depth, the molecular parameters of Table 1 are used in the analysis of the pore structure of activated carbon. The overall isotherm for an activated carbon having a pore size distribution and an energy distribution is given by

a(p/p0) )

∫∫f(Hin)ξ(sf/kB)[Fa(p/p0,Hin,sf/kB) F] d(sf/kB) d log Hin (17)

Here a is the excess adsorption, Hin ) H - ∆ is the “physical” pore width, and Fa is the volumetric average density, which is a function of Hin, the solid-fluid potential (50) Neimark, A. V.; Ravikovitch, P. I.; Gru¨n, M.; Schu¨th, F.; Unger, K. K. J. Colloid Interface Sci. 1998, 207, 159. (51) Gardner, L.; Kruk, M.; Jaroniec, M. J. Phys. Chem. 2001, 105, 12516. (52) Kruk, M.; Li, Z.; Jaroniec, M. Langmuir 1999, 15, 1435.

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Figure 4. Isotherms in activated carbons: (O) argon in ROX (ref 56) at 87.3 K; (0) nitrogen in AX21 at 77 K (ref 57); (4) nitrogen in Ajax at 77 K; (solid lines) correlation with the developed model; (dashed lines) result of application of the conventional NLDFT.

Figure 3. Nitrogen adsorption isotherm on carbon black at 77.3 K in logarithmic scale (a) and in linear scale (b): (O) data of M. Kruk et al. (ref 52); (solid line) developed approach accounting for the multibody interaction; (dashed line) the conventional NLDFT.

sf/kB, and the reduced pressure p/p0. The pore size distribution function f is assumed to be the one-valued function of the pore width and defined as the derivative of the pore volume with respect to the decimal logarithm of the width. The energy distribution function, ξ(sf/kB), is given by

ξ(sf/kB) )

kB dS S0 dsf

(18)

where S is the area of the pore wall surface having a solidfluid potential well depth less than sf, and S0 is the total pore wall surface. Note that the energy distribution defined above is normalized. To solve the inversion problem of determining the pore size distribution and the energy distribution, we have generated the set of local isotherms at 70 values of sf/kB (from 20 to 90 K), 250 values of the pore width (up to 100 nm), and 500 values of the reduced pressure p/p0 in the region from 1 × 10-10 to 1. We used the Tikhonov regularization method53,54 with some modifications described in detail elsewhere.55 The two regularization functions were applied to the pore size and energy distributions. Figure 4 shows three isotherms of argon and nitrogen adsorption in activated carbons: Norit ROX (Ar at 87.3 K56), AX21 (N2 at 77 K57), and Ajax (N2 at 77 K). Solid lines are the correlation obtained with the present approach. The dashed lines, which are obtained with the (53) Tikhonov, A. N. Dokl. Akad. Nauk SSSR 1943, 39, 195. (54) Tikhonov, A. N. Dokl. Akad. Nauk SSSR 1963, 153, 49. (55) Ustinov, E. A.; Do, D. D. Langmuir 2003, 19, 8349. (56) Ismagji, S.; Bhatia, S. K. Langmuir 2000, 16, 9303. (57) Aukett, P. N.; Quirke, N.; Riddiford, S.; Tennison, S. R. Carbon 1992, 30, 913.

conventional NLDFT, correlate experimental data quite poorly. The same applies to many other cases in the literature. This points to the unreliability of the PSD in view of the ill-posed deconvolution problem. Incorporation of the energy distribution function to the model substantially improves the situation and allows us to describe the experimental isotherm very accurately (solid lines). The pore size distribution functions obtained from the developed approach for the aforementioned samples (solid lines) are depicted in Figure 5. For comparison, the PSDs determined with the conventional NLDFT are plotted in the same figure as the dashed line. The new approach leads to a much simpler PSD function than the original NLDFT gives, especially in the micropore region. Thus, in the case of activated carbon ROX instead of three peaks in the range of physical pore width from 0.4 to 2 nm the developed approach leads to one nearly symmetrical peak. The artificial gap around 1 nm, which is always seen in PSDs obtained with the original NLDFT, has disappeared in our case. The energy distribution function for this sample is presented in Figure 6a. The vertical dashed line corresponds to the value sf/kB ) 58.01 K found for the homogeneous graphitized carbon black. The dispersion of the energy around this value is quite large, with three peaks being detected. The central (major) peak is located close to 58 K, and the dispersion may probably be attributed to imperfectness of the graphite lattice on the pore walls. The high-energy peak around 75 K could be due to active centers and unsaturated bonds of carbon atoms on the pore edges, while the lower energy peak could be associated with the decrease of the solid-fluid potential in finite pores compared to idealized pores with infinitely extended walls. The nitrogen adsorption isotherm at 77 K in activated carbon AX2157 has an unusual slant S-shaped form, and the total pore volume of this sample is very large (1.34 cm3/g). Our approach describes this isotherm extremely well, while the NLDFT based on the assumption that pore walls are energetically homogeneous (dashed line) shows large deviations from experimental data, especially in the neighborhood of monolayer coverage formation. The pore size distribution and the energy distribution function are shown in Figures 5b and 6b, respectively. Interestingly,

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Figure 5. Pore size distribution of activated carbons: (a) ROX, (b) AX21, and (c) Ajax; (solid lines) the new approach; (dashed lines) the conventional NLDFT.

the EDF is quite narrow and is positioned very close to the value of 56.1 (vertical dashed line) determined for carbon black (suggesting that AX21 is less heterogeneous than Norit ROX), yet the PSDs corresponding to the original NLDFT and our methods are rather different, which deserves some discussion. The possible reason for this difference is that the original NLDFT does not account for the effect of multibody interactions in the calculation of interaction energy. Despite unusually large total pore volume, the PSD does not show the presence of mesopores wider than 4 nm. The PSD function of this activated carbon is more complex than that of Norit ROX, but the dispersion of micropores is smaller. In particular, there are no micropores having width less than 0.75 nm. It is fair to say that this activated carbon is more homogeneous than the Norit AC. The nitrogen adsorption isotherm of another commercial activated carbon, Ajax, has quite smoothed form (see Figure 4), yet it cannot be adequately described with the conventional NLDFT (as well as with GCMC simulations58). On the other hand, the approach presented in this paper once again leads to excellent results. The PSD evaluated with our model substantially differs from that obtained with the assumption of an energetically homogeneous structure (Figure 5c). All traditional methods including GCMC and the Do method58 show bimodal micropore structure of this activated carbon. The new approach simplifies the PSD function. The gap within a range of 1 nm and the peak centered at 0.6 nm, observed with conventional NLDFT, vanished and instead the main peak around 1 nm has appeared. However, the energy distribution function presented in Figure 6c is quite (58) Do, D. D.; Do, H. D. Langmuir 2003, 19, 8302.

Ustinov and Do

Figure 6. Energy distribution function of activated carbons: (a) ROX, (b) AX21, and (c) Ajax. Vertical dashed lines denote the energy of graphitized carbon black.

involved. It consists of three peaks and resembles the EDF of Norit ROX. Since Norit ROX and Ajax have the same precursor, coconut shell, the EDF may be regarded as a “fingerprint” indicator to quantify the energetic surface of activated carbon. 4. Conclusion A new approach based on nonlocal density functional theory accounting for the energetic heterogeneity of pore walls is developed in this paper. The nonadditivity of solid-fluid and fluid-fluid interactions is incorporated to the model. The approach allows excellent correlation of subcritical nitrogen and argon adsorption isotherms on graphitized carbon black, which is used as a reference system. It is shown that the conventional nonlocal density functional theory fails to properly describe low-temperature N2 and Ar adsorption isotherms in activated carbons because of the negligence of the energetic heterogeneity of real adsorbents. The developed approach allows one to decompose the pore size distribution and the energy distribution function, which is defined in the framework of the patchwise model. The application of this method to various activated carbons has led to highly accurate correlation of experimental data and shown that extracted PSD functions do not involve the artificial gap around 1 nm pore width and have quite simple form. The advantage of this method is that it allows determination of the energy distribution function of adsorbents, which discloses their individuality as a fingerprint, which is the result of the precursor and conditions of carbonization and activation used. Acknowledgment. Support from the Australian Research Council is gratefully acknowledged.

DFT Analysis of Activated Carbons

Langmuir, Vol. 20, No. 9, 2004 3797

Appendix

{

The weight functions ω0, ω1, and ω2 in the general case are written as follows:43

{

3 jr < 1 ω0(rj) ) 4πσff3 0 rj > 1

(A1a)

3(1 - zj2)/(4σff) 0

zj < 1 zj > 1

3

0.475 - 0.648rj + 0.113rj 0.288/rj - 0.924 + 0.764rj - 0.187rj2 0

5πσff3 2 ω2(rj) ) 144 (6 - 12rj + 5rj ) 0

rj < 1 1 < rj < 2 rj > 2 (A1b)

rj < 1

(A2a) zj < 1

4

2.592zj - 0.339zj )/6 ω1(zj) ) πσff2(0.208 - 3.456zj + 5.544zj2

1 < zj < 2

- 3.056zj3 + 0.561zj4)/6

0

2

{

{

πσff2(0.398 - 2.85zj2 +

ω1(rj) )

{

ω0(zj) )

zj > 2

(A2b)

ω2(zj) )

{

5π2σff5(1 - 12zj2 + 16zj3 - 5zj4)/288 0

zj < 1 (A2c) zj > 1

Here zj ) |z/σff|. The coefficients in eqs A2 are chosen to ensure that the following equalities are satisfied:

(A1c)

rj > 1

where rj ) r/σff. To reduce these expressions to the one-dimensional case, it is necessary to integrate ω0, ω1, and ω2 over the xy plane (parallel to the pore walls) at a specified coordinate z. This leads to the following equations:

∫ω0(z) dz ) 1 ∫ωi(z) dz ) 0

i ) 1,2

This provides the equality of the local and smoothed densities in homogeneous media, where the density gradient is zero. LA035936A