Carbon 44 (2006) 653–663 www.elsevier.com/locate/carbon
Pore size distribution analysis of activated carbons: Application of density functional theory using nongraphitized carbon black as a reference system E.A. Ustinov a, D.D. Do
b,* ,
V.B. Fenelonov
c
a
Scientific and Production Company ‘‘Provita’’, 6, Prospect Kim, St. Petersburg 199155, Russia Department of Chemical Engineering, University of Queensland, St. Lucia, Qld., 4072, Australia Boreskov Institute of Catalysis, Siberian Branch of the Russian Academy of Science, Novosibirsk, Russia b
c
Received 9 August 2005; accepted 23 September 2005 Available online 8 November 2005
Abstract The application of nonlocal density functional theory (NLDFT) to determine pore size distribution (PSD) of activated carbons using a nongraphitized carbon black, instead of graphitized thermal carbon black, as a reference system is explored. We show that in this case nitrogen and argon adsorption isotherms in activated carbons are precisely correlated by the theory, and such an excellent correlation would never be possible if the pore wall surface was assumed to be identical to that of graphitized carbon black. It suggests that pore wall surfaces of activated carbon are closer to that of amorphous solids because of defects of crystalline lattice, finite pore length, and the presence of active centers, etc. Application of the NLDFT adapted to amorphous solids resulted in quantitative description of N2 and Ar adsorption isotherms on nongraphitized carbon black BP280 at their respective boiling points. In the present paper we determined solid–fluid potentials from experimental adsorption isotherms on nongraphitized carbon black and subsequently used those potentials to model adsorption in slit pores and generate a corresponding set of local isotherms, which we used to determine the PSD functions of different activated carbons. 2005 Elsevier Ltd. All rights reserved. Keywords: Activated carbon; Adsorption; Nongraphitic carbon; Carbon black; Modeling
1. Introduction Development of industrial applications of micro- and mesoporous materials requires a reliable characteristic of their porous structure. One of the most popular and relatively simple methods of evaluation of the pore size distribution function (PSD) of different adsorbents is based on analysis of nitrogen and argon adsorption isotherms at their respective boiling points. Over the past few decades there has been an increasing interest in characterization of mesoporous materials, stimulated by the synthesis of highly ordered mesoporous silica such as MCM-41 and
*
Corresponding author. Fax: +61 7 3365 2789. E-mail address:
[email protected] (D.D. Do).
0008-6223/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.carbon.2005.09.023
SBA-15. The advantage is that the pore size of those materials can be independently determined with the Xray diffraction technique, providing a convenient way of verification of different conventional adsorption-based methods of PSD analysis. All theories developed for PSD analysis rely on a reference adsorption isotherm of a nonporous solid, which is assumed to have the same chemical structure as that of the pore wall surface. Some theories such as the Barrett, Joyner and Halenda (BJH) method [1], the modified BJH version recently developed by Kruk, Jaroniec and Sayari (KJS) [2,3], the Broekhoff and de Boer (BdB) theory [4,5] and its modifications [6– 9] use the reference system directly in the form of the experimental t-curve. The Horvath–Kawazoe [10] (HK) method developed for microporous carbonaceous materials uses the 10-4 LJ potential derived for the perfect
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graphite sheets constituting the pore walls. More sophisticated nonlocal density functional theory (NLDFT) [11– 16] also implicitly relies on a reference adsorption isotherm of a nonporous solid because the solid–fluid molecular parameters are chosen to correlate the experimental reference isotherm. Hence the correct choice of a reference system is very important to reliably obtain PSD. In the case of activated carbons it is common to take the graphitized thermal carbon black (GTCB) as the reference solid to determine the PSD by the NLDFT or the HK method, which implies perfect crystalline structure of the pore walls. As a pore model the slit pore of infinite extent is assumed in the vast majority of works devoted to characterization of carbonaceous materials. However, one should bear in mind that topology of activated carbons is very complex and still far from fully characterized [17]. Besides, in spite of some evidence of the existence of a large amount of crystalline phase in activated carbons [18], it is unlikely the pore wall surface to be identical to that of GTCB. Some authors suggest a nongraphitized carbon black (NGCB) as a reference solid for PSD analysis of activated carbons [19–22]. This is indirectly justified because of a rather poor correlation of nitrogen adsorption isotherms in activated carbons with the NLDFT, when GTCB is used as the reference solid. This poor correlation results in the existence of an artificial gap around 1 nm pore width in the PSD function. The convolution task using a regularization method is known to be an ill-posed problem. It means that any systematic deviations between the experimental adsorption isotherm and the correlated function make the PSD function highly unreliable. There could be at least two possible reasons for such deviations. One is the assumption of infinite extent of the slit-shaped model. The other one is the unrealistic assumption of pore walls having perfect graphitic structure. In the present paper we rely on the idea that the pore wall surface is amorphous and the nongraphitized carbon black BP280 [23] is taken as a reference solid for PSD analysis of a series of activated carbons. The nongraphitized carbon black is highly energetically heterogeneous [22–26], which requires development a special method of application of NLDFT to such surfaces. Recently we developed an approximate way of accounting for an amorphous surface in the framework of NLDFT [9,27,28]. This method was shown to be a useful tool for excellent description of nitrogen and argon adsorption at their respective boiling points on nonporous silica, MCM-41 and SBA-15 samples. This paper presents a further development of this method for analysis of nitrogen and argon adsorption on NGCB and in slit pores having the same pore wall chemical structure. We show the applicability of this method to characterization of a series of activated carbon analogous to AX21. Although our procedure still relies on the assumption that pore walls of activated carbon have the same chemical and physical structure as that of NGCB, the resulting excellent description of adsorption isotherms
of numerous activated carbon samples lends great support to this assumption of amorphous surface. 2. Model Thermodynamic equilibrium in the grand canonical ensemble corresponds to the global minimum of the following grand thermodynamic potential: Z Þ þ uext ðrÞ l dr X ¼ q½f ðr; q; q ð1Þ are the local and smoothed fluid densities, where q, q Þ is the molar Helmholtz free energy; respectively; f ðr; q; q uext(r) is external potential exerted by the solid; l is the chemical potential. The task is reduced to finding a fluid density profile in a confined space of the pore corresponding to the minimum of X . The molar Helmholtz free energy is split into the ideal term kT[ln (K3q) 1], excess term fex ð qÞ due to repulsion, and the intermolecular interaction potential u(r) due to attraction: Þ ¼ kT ½lnðK3 qÞ 1 þ fex ð qÞ þ uðrÞ f ðr; q; q
ð2Þ
Here k is the Boltzmann constant and K is the de Broglie is a weighted average wavelength. The smoothed density q Z ðrÞÞ dr0 ðrÞ ¼ qðr0 Þxðjr r0 j; q ð3Þ q ðrÞÞ is the weighting function. In the where xðjr r0j; q Tarazona version of NLDFT [29,30] the smoothed density is approximated as follows: 2 ðrÞ ¼ q 0 ðrÞ þ q 1 ðrÞ 2 ðrÞð q qðrÞ þ q qðrÞÞ
where i ðrÞ ¼ q
Z
qe ðr0 Þxi ðjr r0 jÞ dr0
ð4Þ
ð5Þ
for i = 0, 1, 2. Functions x0(r), x1(r), and x2(r) depend only on the distance r from a given point and defined to reproduce the hard-sphere direct correlation function [30]. Here we replaced the local fluid density q in the integrand by the effective density qe for the sake of generality to account for the contribution of the solid to the smoothed density in the vicinity of the surface. We discuss this subtlety in Section 2.3, but here it should be emphasized that in the original definition of the smoothed density the effective density qe is the local fluid density. 2.1. Repulsive potential for the fluid–fluid and fluid–solid interaction The excess Helmholtz free energy in the Tarazonas NLDFT is given by the Carnahan–Starling equation [31] for a reference hard-sphere fluid: fex ð qÞ ¼ kT
4g 3g2 2
ð1 gÞ
;
p g ¼ d 3HS q 6
ð6Þ
E.A. Ustinov et al. / Carbon 44 (2006) 653–663
where dHS is the equivalent hard-sphere diameter, and g is the dimensionless density. The excess free energy increases from zero at g ¼ 0 (infinitely diluted fluid) to infinity at g ¼ 1 (extremely dense fluid). The increase of the smoothed density reflects the decrease of the smoothed void volume v ¼ 1 g. We use the (local) void volume v ¼ 1 pd 3HS q=6 in the definition of the excess Helmholtz free energy [9,27,28] as with this choice the void volume is taken to be zero whenever the point under consideration is inside the solid region. What this simply means is that fluid molecules do not penetrate into the solid. Formally v is zero if the local density q takes the maximum possible value qm ¼ 6=ðpd 3HS Þ. The latter can be interpreted as a result of replacement of fluid molecules by the solid atoms also contributing to the decrease of the void volume down to zero, with the effective solid density being qm. Thus, the density in the integrands of Eq. (5) is the local fluid density outside the solid and qm inside the solid. The integrals are taken over the distance jr r 0 j within one fluid–fluid collision diameter rff for i = 0 and 2 and two collision diameters rff for i = 1 [29,30]. Consequently, we have the standard Tarazonas scheme for calculating the smoothed density at a distance larger than two fluid–fluid collision diameters from the surface. However, at a distance less than two collision diameters from the surface solid atoms contribute to the increase of the effective smoothed density. It means that in the proximity of the solid the excess Helmholtz free energy additionally increases, which accounts for the contribution of the solid–fluid repulsive potential. Therefore the repulsive potential between fluid molecules from one hand and between fluid molecules and solid atoms from the other hand could be approximately accounted for consistently in the framework of the same Carnahan–Starling Eq. (6). From our viewpoint it makes sense in the case of amorphous solids when the fluid and the solid are both disordered media, which justifies application of the same procedure to fluid–fluid and fluid–solid interactions. 2.2. Attractive potential for the fluid–fluid and fluid–solid interaction The fluid–fluid attractive potential u(r) is defined in the framework of mean field approximation: Z 1 uðrÞ ¼ qðr0 Þ/ff ðjr r0 jÞ dr0 ð7Þ 2 where /ff(r) is the attractive Weeks, Chandler and Andersen (WCA) potential of two molecules separated by a distance r [32]: 8 > < eff ; r < rm 12 6 /ff ðrÞ ¼ 4eff ½ðrff =rÞ ðrff =rÞ ; rm < r < rc ð8Þ > : 0; r > rc Here eff is the potential well depth; rm = 21/6rff is the distance at which the potential is minimum; rc is the cutoff distance.
655
Further development of this approach requires introducing only attractive solid–fluid potential because the repulsive solid–fluid potential has been already accounted for with the excess Helmholtz free energy as discussed in Section 2.1. In our previous papers we relied on the WCA perturbation scheme with the attractive solid–fluid potential well depth esf and the solid–fluid collision diameter rsf determined by the least-squares fitting. This has led us to excellent correlation of nitrogen and argon adsorption isotherms on nonporous amorphous silica [9]. In the case of nongraphitized carbon black the attractive WCA solid–fluid potential (described by two molecular parameters, rsf and esf) did not provide a good correlation, therefore in the present paper we use experimental adsorption isotherms of the reference NGCB directly to determine the pairwise attractive potential by the least-squares fitting. Besides, in order to avoid any irregularities in the solid– fluid potential function we used the Tikhonov regularization method [33]. 2.3. Accounting for the solid–fluid transition zone In the case of amorphous solid the surface atoms are dispersed about the plane located at the first central moment of the solid density. This is implicitly accounted for in the developed model because both repulsive and attractive solid–fluid interactions are considered in the same framework as those for the fluid–fluid interactions, and the fluid by definition is an amorphous medium. Such a dispersion exists even when the void volume v was assumed to be a Heaviside step function. However, to make the model consistent we introduce here a dispersion of the solid mass at the surface in the form of the following error function: Z 1 1 nðzÞ ¼ pffiffiffiffiffiffiffiffi exp z2 =ð2d2 Þ dz ð9Þ 2pd z Here d is the variance, and z is the distance from the surface. The function n(z) is defined as the average solid density related to the density far away from the surface, where n(z) (z < 0) is unity. The first central moment of this function coincides with the position of the surface. For large and positive z, n(z) asymptotically tends to zero. Hence, in the proximity of the surface the effective local density in the integrands of Eq. (5) may be expressed as follows: qe ðzÞ ¼ qðzÞ þ qm nðzÞ
ð10Þ
Having set the effective local density in the proximity of the surface, the smoothed density and, consequently, the excess Helmholtz free energy can be found by Eqs. (4)–(6). 3. Results and discussion 3.1. Nitrogen and argon adsorption on nongraphitized carbon black In this section we consider argon and nitrogen adsorption isotherms at their boiling points on nongraphitized carbon black Cabot BP 280 [22,24].
E.A. Ustinov et al. / Carbon 44 (2006) 653–663
10.0
1.0
0.1 10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
p/p 0
a
100.0
-2
Amount adsorbed (µmol m )
where qs is the graphite density (114 nm3), and D is the graphite layer spacing (0.335 nm). Values of the solid–fluid potential well depth esf/k were determined from the Henry law regions of the adsorption isotherms on graphitized carbon black. The solid–fluid collision diameter rsf was calculated by the Lorentz–Berthelot rule. All molecular parameters used for prediction of nitrogen and argon
100.0
-2
Figs. 1 and 2 show adsorption isotherms of nitrogen at 77 K and argon at 87.3 K on the sample BP 280 in linear and logarithmic scales, respectively. Dashed lines are plotted with the conventional NLDFT in which the fluid–fluid molecular parameters are taken from Neimark et al. [34]. The solid–fluid potential for the dashed lines was calculated by the 10-4-3 Steele equation [35]: " # 2 r10 r4sf r4sf ext 2 sf u ðzÞ ¼ 2pqs esf rsf D 4 ð11Þ 5 z10 z 3Dð0:61D þ zÞ3
Amount adsorbed (µmol m )
656
80
-2
Amount adsorbed (µmol m )
100
60
10-6 20
b
0.0
0.2
0.4
0.6
0.8
1.0
10-5
10-4
10-3
10-2
10-1
100
p/p 0
Fig. 2. Nitrogen (a) and argon (b) adsorption isotherms on nongraphitized carbon black Cabot BP 280 (logarithmic scale). Notations are the same as those for Fig. 1.
p/p0
a 100
80
-2
Amount adsorbed (µmol m )
1.0
0.1
40
0
60
40
20
0 0.0
b
10.0
0.2
0.4
0.6
0.8
1.0
p/p0
Fig. 1. Nitrogen (a) [24] and argon [22] (b) adsorption isotherms on nongraphitized carbon black Cabot BP 280 at 77 and 87.3 K, respectively (linear scale). (Circles) Experimental points, dashed line is plotted with the conventional NLDFT for the solid–fluid molecular parameters adjusted to GTCB. (Solid line) Correlation with NLDFT developed for adsorption on amorphous surface.
adsorption isotherms on NGCB by conventional NLDFT are listed in Table 1. One can see from the figures that the conventional application of NLDFT to nitrogen and argon adsorption on NGCB leads to noticeable deviations from experimental adsorption isotherms. Especially large divergences are observed in the logarithmic scale (Fig. 2). The important feature of predicted isotherms is the presence of the Henry law region at approximately p/p0 < 5 · 105 for nitrogen and p/p0 < 5 · 104 for argon, while experimentally the Henry law is not observed at all. The reason is the energetic heterogeneity of the NGCB. One possible way to account for the energetic heterogeneity is to impose an energy distribution function on the isotherm in the framework of the patchwise model. However, like in the case of N2 and Ar adsorption on nonporous silica [27], a superposition of theoretical isotherms calculated with different values of the potential esf/k does not provide an acceptable correlation of experimental adsorption isotherms on NGCB in the region of middle and high values of the reduced pressure.
E.A. Ustinov et al. / Carbon 44 (2006) 653–663 Table 1 Molecular parameters for Ar and N2 adsorption on carbon black rsf (nm)
118.05 94.45
0.3305 0.3575
0.3380 0.3575
58.01 56.10
0.3353 0.3488
0.8 0
-8 0.4 -16 0.2
-24 -1
0.0 0
1
a
2
3
4
z/σff 1.0
8
0.8 0 0.6 ext
u e /kT
Now we turn to the approach developed in this paper. The task is to determine the dependence of the external solid–fluid potential uext(z), which is required in the grand thermodynamic potential (Eq. (1)). As it was stated above, this potential is purely attractive because the repulsive potential is already accounted for with the excess free energy. It is necessary first to designate the variance d in Eq. (9), which is associated with the surface roughness. At present stage we do not have any independent assessment of the variance for the nongraphitized carbon black surface. Qualitatively we have found that d is not identically zero because otherwise the theoretical isotherm does not match the experimental data precisely. We find that when the variance is greater than about 0.02 nm, excellent correlation against the experimental isotherm data is always achieved. However, we need to stress that using only the adsorption isotherm does not allow us to determine the true variance, which physically describe the surface roughness. This is due to the fact that if the variance is incorrectly chosen the solid–fluid potential is adjusted accordingly to compensate for the incorrect value of variance for the surface roughness. This compensation fails when the variance is close to zero. The solid–fluid potential for nitrogen and argon determined with the new approach is shown in Fig. 3. Shortdashed line shows the change of the relative solid density near the solid surface corresponding to the variance d = 0.02 nm. The surface (i.e., the first central moment of the solid mass) is positioned at the coordinate z = 0. To the left of this boundary the relative solid density tends to unity. Long-dashed line is the attractive term of the solid–fluid potential determined by the least-squares fitting from the experimental adsorption isotherms. This is a monotonic function similar to a resulting attractive potential, which could be obtained by the integration of the WCA solid–fluid pair potential over the solid. However, it is more rigorous and informative to determine the attractive term of the solid–fluid potential directly from the adsorption isotherm, which has been successfully done. The repulsive term of the solid–fluid interaction can be estimated as follows. Given the density profile at a specified st bulk pressure, one can determine the smoothed density q corresponding to the standard Tarazonas prescription (Eqs. (4) and (5)), assuming the solid does not contribute to the excess Helmholtz free energy. In this case the const will be less than the actual ventional smoothed density q in the vicinity of the surface, and, consmoothed density q sequently, the excess free energy fex ð qst Þ is less than fex ð qÞ at a distance z < rff. Therefore, the repulsive term of the solid–fluid potential could be defined as follows:
0.6
ξ(z)
esf/k (K)
1.0
ξ(z)
dHS (nm)
ext
rff (nm)
8
u e /kT
Ar, 87.3 K N2, 77 K
eff/k (K)
657
-8 0.4
-16 0.2
-24
0.0 -1
0
b
1
2
3
4
z/σff
Fig. 3. Solid–fluid potential in the case of N2 (a) [24] and Ar (b) [22] adsorption on NGCG Cabot BP 280. Short-dashed line shows the dispersion of the solid density in the vicinity of the surface. Dash-dotted line shows the potential, which would be exerted by the crystalline graphite surface. Long-dashed line is attractive term determined from experimental adsorption isotherms by least-square fitting. Solid lines denote resulting solid–fluid potentials at p/p0 (from the bottom to the top): 0, 105, 104, 103, 102, 101, and 1. The decrease (in absolute term) of the resulting potential with the increase of the bulk pressure reflects the effect of surface energetic heterogeneity.
qðzÞ fex ½ qst ðzÞ urep ðzÞ ¼ fex ½
ð12Þ
Hence, the effective solid–fluid potential accounting both the attractive and repulsive terms and corresponding to the conventional way of application of the NLDFT to crystalline solids is given by rep ext uext e ðzÞ ¼ u ðzÞ þ u ðzÞ
ð13Þ
Solid lines in Fig. 3 show the dependence of the effective solid–fluid potential on the distance from the surface determined at different values of the bulk pressure. The important feature is that the potential increases (the absolute value decreases) with the bulk pressure due to the increase of the repulsive term urep with loading. This unusual, but not totally unexpected, behavior of the potential may be interpreted as follows. First fluid molecules are adsorbed on the most active surface sites and block them. For this reason further adsorption at higher pressure occurs on less
E.A. Ustinov et al. / Carbon 44 (2006) 653–663 60
50
-2
active sites. So the subsequent increase in pressure involves weaker surface sites, which is reflected by the shifting of the potential curve toward larger potentials. It allows us to come to the conclusion that the energetic heterogeneity inherent to amorphous surfaces is embedded into the developed approach, so it is not necessary to involve any surface distribution function with respect to the solid–fluid potential. The second feature of the effective solid–fluid potential is that the potential well at any bulk pressure is markedly wider than that corresponding to the Steele Eq. (11) (dash-dotted line in Fig. 3). The reason is that in contrast to the crystalline surface centers of surface solid atoms of the amorphous solid are dispersed around the plane identified as the surface. A consequence of such dispersion is that there are concavities and convexities on the surface reflecting the surface roughness in atomic scale. It means that some fluid molecules could reside very close to the geometrical surface between neighboring solid atoms, while another molecules cannot approach close to the surface due to repulsion forces exerted by protruded solid atoms. This dispersion smoothes the average repulsive potential, which makes the potential well wider and closer to the surface compared to any potential derived by integration of the 12-6 LJ pair potential over the slab or 2D plane. The effect of smoothing of the repulsive potential comes mainly from the excess Helmholtz free energy, which is a function of the smoothed density and is enhanced by the distribution of the solid density about the geometrical 2D surface. In other words, the smoothing effect exists even if n(z) is a Heaviside step function. As we mentioned above, at this stage the variance d of the function n(z) is not exactly known for the carbon black BP 280, but the choice of a value for this variance is found to be not very significant. We address this problem to our future communications. The result of application of our approach to nitrogen and argon adsorption on nongraphitized carbon black BP 280 is demonstrated by solid lines in Fig. 1 and 2. As seen from the figures, calculated curves nearly coincide with experimental adsorption isotherms over the entire pressure range. It allows us to extend this method to slit pores of activated carbons. The structure of the fluid adsorbed on the NGCB is shown in Fig. 4 in the case of nitrogen adsorption. Different curves reflect the local density distribution nearby the graphite surface at different specified reduced bulk pressures. As one can see from the figure, there are density oscillations, however they are much less pronounced compared to the case of adsorption on the GTCB surface. One more interesting feature is that the main density maximum of the first molecular layer corresponds to the distance from the surface approximately equal to 0.57rff, while in the case of GTCB the density maximum coincides with the potential minimum at a distance of about 0.97rff. This interesting peculiarity of adsorption on nongraphitized carbon black again is resulted from the dispersion of surface carbon atoms around the geometrical surface, which in
Density (µmol m )
658
40
30
20
10
0.95
0.9
0.1
p/p0=0.99
0 -2
0
2
4
6
8
10
12
z/σff
Fig. 4. Density distributions of nitrogen adsorbed on nongraphitized carbon black BP 280 at 77 K. The reduced bulk pressure is incremented by 0.1 in the range from 0.1 to 0.9. Calculations were made with the developed approach. The dashed line denotes the saturated bulk liquid density.
average allows fluid molecules to approach closer the surface than in the case of basal graphite plane. In other words, strong repulsive forces exerted by graphitized carbon black keep fluid molecules further from the surface than in the case of amorphous surface. 3.2. Modeling of adsorption in slit pores To model adsorption in the slit pore of infinite extent formed by two parallel amorphous graphite surfaces it is necessary to account for the enhancement of the potentials exerted by the opposite pore walls. The attractive potential exerted by the wall uext is already determined, hence the potential inside the slit pore having width H is given by ext ext uext slit ðzÞ ¼ u ðzÞ þ u ðH zÞ
ð14Þ
The pore width is defined as the distance between the first central moments of solid density corresponding to each wall. It means that in this case we do not make any difference between the geometrical and the available (physical) pore width as we do when the pore wall is identified with the basal graphite surface. The repulsive term of the solid–fluid potential is accounted for by the excess free energy like in the case of NGCB. The smoothed density is calculated by Eqs. (4) and (5), with the effective local density in the integrand of Eq. (5) being set as follows: qe ðzÞ ¼ qðzÞ þ qm ½nðzÞ þ nðH zÞ
ð15Þ
As an example, Fig. 5a presents the set of selected local nitrogen isotherms on slit pores having wall surface identical to that of NGCB Cabot BP 280 in the range of pore width from 0.25 to 3 nm. The pore width as a function of the number of isotherms shown in Fig. 5a is presented in Fig. 5b. The dashed line bounds the region (to the right from the line) of the equilibrium phase transition. One
E.A. Ustinov et al. / Carbon 44 (2006) 653–663 40
-3
Density (µmol cm )
30
20
10
0 10-9
10 -8
10-7
10 -6
10 -5
10-4
10 -3
10-2
25
30
10 -1
100
p/p 0
a 3.5
3.0
Pore width (nm)
2.5
2.0
1.5
1.0
0.5
0.0 0
b
5
10
15
20
35
Number of isotherm
Fig. 5. Local nitrogen isotherms at 77 K in slit pores of infinite extent (a) for the pore widths shown as a function of the isotherm number (b). The chemical structure of the pore wall surface is assumed to be the same as that of nongraphitized carbon black Cabot BP 280.
can see from the figure that the set of local isotherms is quite regular compared to that for slit pores constituted by parallel basal graphite surfaces (see for example Fig. 3 in [36]). In the general case the entire set of local isotherms generated for the PSD analysis corresponds to the sequence of widths H = (2N + 1)rff/m, where m is 30 (Dz = rff/m is the integration step). We determine N by the following recurrent formula: N ðkþ1Þ ¼ N ðkÞ þ int½ð10g 1ÞN ðkÞ ð16Þ where k is the isotherm number. The starting value of N is 10, which corresponds to the pore width of 0.25 nm. For the parameter g we took the value 0.02. Such sequence reproduces the linear dependence of the pore width on the isotherm number for small pores and logarithmic one for larger pores. 3.3. Characterization of activated carbons with N2 adsorption isotherms using slit pore model with amorphous walls We provided the PSD analysis of a series of activated carbons of a superactive porous carbon of type Amoco PX- 21 (later known as Anderson AX-21 and Kansai
659
Maxsorb) which obtained via chemical activation of a carbonaceous precursor with potassium hydroxide at 700– 900 C [37–42]. Recently these nanoporous carbonaceous materials attract much attention because of their effectiveness for methane [40] and hydrogen storage [41,43], resulted from their very large pore volume [39] and surface area [44]. At present the search of optimal methodology of producing such materials has been carried out in different laboratories with varying of the precursor and conditions of the chemical activation [37,38,40–42,45,46]. In this paper we investigate samples of such materials manufactured at Kemerovo Institute of Carbon & Carbon Chemistry of Siberian Branch of the Russian Academy of Science (Kemerovo) [46]. The carbons were prepared analogously as AX21 commercial carbon and we designated them as Kemerit. Mixture of urea and 1,2,3 benzotriazol was used as a carbon precursor with molar relation 2:1 (for samples 1.1 and 1.2) and 1:1 (for samples 2.1 and 2.2). For the chemical activation potassium hydroxide was used with mass two times of the mass of the precursor. Chemical activation of samples 1.1 and 2.1 was carried out at 700 C for 30 min. Samples 1.2 and 2.2 were additionally treated for 22 min at 900 C. After washing the samples were dried in vacuum at 150–200 C. Nitrogen adsorption isotherms were measured at 77 K with ASAP-2020 (for the region of low pressures) and ASAP-2400 volumetric adsorption analyzers (Micromeritics). Typical transmission electron micrograph (TEM) recorded with a JEM 2010 electron microscope at resolution of 0.14 nm operating at 200 kV is presented in Fig. 6. It is seen from the figure that the porous structure of the samples is far from the ideal slit pore model, which is a peculiarity of activated carbons obtained with chemical activation. The 3D packs of graphens typical for activated carbons obtained with oxidizing activation are nearly absent, which was confirmed by the XRD technique. In such a structure each graphene is available from both sides, which explains the abnormal high surface area and the micropore volume. The variety of structure of real porous materials with random packing of cross-linking elements forces us to use a simplified geometrical model for numerical calculations, and the slit pore model is a natural choice. At this stage the requirement of an adequate description of the real system can be reduced to quantitative fitting of adsorption isotherms and reasonable correlation of the PSD analysis with independent measurements of total surface area and the pore volume. One can expect that the super-simplified slit pore model of infinite extent may be used at least as a unified method of comparison of different porous materials. The convolution task we solved with the conventional Tikhonov regularization method [33]. Figs. 7 and 8 present N2 adsorption isotherm in activated carbon Kemerit-1.1 and corresponding PSD functions. In these figures we compare three versions of NLDFT. The first one is the original NLDFT based on the model of slit pore having crystalline pore walls identical
660
E.A. Ustinov et al. / Carbon 44 (2006) 653–663
Fig. 6. High-resolution transmission electron micrograph of a Kemerit sample.
60
(a) 50 40 30 20 10 60 50
(b)
40
30
30
20
20
10
10
0
0
-2
40
50
µmol m
Amount adsorbed (mmol g-1)
0 50
(c)
40 30 20 10 0 10-6
Fig. 8. Pore size distribution function of activated carbon Kemerit-1 obtained with different versions of NLDFT: original NLDFT (a), improved NLDFT accounting the nonadditivity factor (b), and the amorphous surface-based approach developed in this work.
10-5
10-4
10-3
10-2
10-1
100
p/p0
Fig. 7. Nitrogen adsorption isotherm in activated carbon Kemerit-1 at 77 K. (Circles) Experimental data; (lines) correlation with the original NLDFT (a), NLDFT accounting for a nonadditivity factor [37] (b), and the new approach (c). Calculations in the first two cases are based on graphitized carbon black taken as a reference system, and the third theoretical isotherm is calculated accounting for amorphous surface of the pore walls. Dashed line is nitrogen adsorption isotherm on the graphitized carbon black at 77 K.
to that of graphitized carbon black. The second version differs from the previous one by accounting for a nonadditive factor for the solid–fluid potential [47], which significantly improves the description of the experimental adsorption isotherm on GTCB. The third version is the approach developed in the present paper and relied on the assumption on amorphous structure of the pore wall surface. A striking result is that in the case (a) and (b) NLDFT does
not provide an appropriate correlation of the experimental adsorption isotherm despite 120 local adsorption isotherms were involved into the PSD analysis. It is important to bear in mind that the convolution task is an ill-posed problem, which means that the reliable PSD function by no means could be obtained in the case of systematic divergence of the experimental and the calculated isotherms. Note, however, that in the second case the correlation is better compared to the original NLDFT. The question is, however, why the improved NLDFT version [47] that properly describes nitrogen adsorption isotherm on GTCB surface, correlates the experimental isotherm in the activated carbon so poorly (which is persistently observed for nearly all activated carbons)? The answer probably lies in the apparent similarity of the calculated isotherm for the activated carbon sample and the experimental nitrogen adsorption isotherm on GTCB in the reduced pressure region from 104 to 103. It is seen from Fig. 7b that the S-shaped shoulder in the calculated isotherm reproduces the part of N2—GTCB adsorption isotherm in the range of the first molecular layer coverage. It suggests that the deviation is caused by wrong choice of the reference solid, i.e., the GTCB, rather than any drawback of NLDFT. In contrast to the previous two cases, the approach relied on
E.A. Ustinov et al. / Carbon 44 (2006) 653–663
the assumption of amorphous pore wall surface leads to excellent correlation of the adsorption isotherm (Fig. 7c). It is also of interest to compare the PSD functions obtained with the three NLDFT versions (Fig. 8). Note that in cases (a) and (b) the pore width is determined as the available width, namely Hin = H D. The upper curve corresponding to the original NLDFT version is the most complex one, which clearly discords with the simple form of the initial adsorption isotherm. The pronounced gap around 1 nm pore width always appears in PSDs obtained with NLDFT versions based on GTCB as a reference solid, pointing to its artificial nature. The new approach leads to the smoothest and therefore more plausible PSD function. Figs. 9 and 10 demonstrate nitrogen adsorption isotherms and PSDs for the series of Kemerit samples. Some results of the PSD analysis are presented in Table 2. In Table 2 Hav and d are the average pore width and the variance, respectively, with the average pore with being defined as follows: Z H max dV =V 0 d log H ð17Þ ðlog H Þ log H av ¼ d log H H min
-1
Amount adsorbed (mmol g )
20
10 X
X XX X
XX XX
XX
XX
XX
X
XX XX
XX
XX
XX
X XX
XX
XX
X
X
X
X
10-5
10-4
10-3
10-2
3
X
10-1
1.5
X XX X X XXX
1.0
X X X X X XX XX X X X X X X XXX XXXXX XX XXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX X
X X X XX X X X X X XX X X X XXXXXXXXXX
0.5
1
10
100
relative pressures 0.05–0.30. The specific surface area of nitrogen molecule x was taken as 0.162 nm2. Interestingly, the ratio S/SBET is very close to 0.8 for all samples, which suggests that the BET method could be used for quick estimation of the specific surface area of activated carbons by taking the nitrogen projection area as 0.130 nm3. It is seen from Table 2 and Fig. 10 that the developed approach allows us to trace the effect of manufacturing conditions of activated carbons on the pore structure. The additional treatment of the samples (1.2 and 2.2) at higher temperature during the chemical activation slightly increases the average pore size and the dispersion of the PSD function. The effect of temperature depends on the composition of the precursor. At higher molar fraction of urea (samples 1.1 and 1.2) the increase of the temperature of activation leads to a very small decrease of the micropore volume and the increase of the mesopore volume. On the contrary, at lower urea—1,2,3 benzotriazol ratio (samples 2.1 and 2.2) the additional treatment at higher temperature significantly decreases the micropore volume, while the mesopore volume remains nearly the same. It is also seen that the pore width, pore wall surface area, microand mesopore volume decrease with the decrease of urea in the precursors composition at a specified temperature. The approach is shown to be a promising tool to reveal the effect of technological parameters and the precursor on properties of activated carbons. Further improvement and perfecting of the model should be relied on generalization of experimental data obtained for a wide series of activated carbons by different techniques.
0 10-6
X X
Fig. 10. Pore size distribution functions of Kemerit samples. Sample number: (—) 1.1; (- - -) 2.1; (– Æ –) 1.2; (– · –) 2.2.
X
30
XX XX XXX X X X X
2.0
Pore width (nm)
X X X X X X X X X X X X XX X X X X X X X X X X X
40
2.5
0.0 0.1
70
50
3.0
-1
Differential pore volume (cm g )
3.5
The micropore (less than 2 nm), mesopore, and total pore volumes are designated as Vmi, Vme, and V0, respectively. The surface area S is determined from the PSD using the slit pore model. For comparison we present in Table 2 the surface area SBET determined with the standard Brunauer, Emmett and Teller method in the region of
60
661
100
p/p 0
Fig. 9. Nitrogen adsorption isotherm in the series of activated carbon Kemerit at 77 K. (s) Kemerit-1.1, (h) Kemerit-2.1, (·) Kemerit-1.2 (D) Kemerit-2.2. (Lines) correlation with the developed approach. Table 2 Structural parameters of Kemerit samples determined with the PSD analysis Sample
SBET (m2 g1)
S (m2 g1)
Hav (nm)
d (nm)
Vmi (cm3 g1)
Vme (cm3 g1)
V0 (cm3 g1)
1.1 2.1 1.2 2.2
2873 2550 2899 2118
2315 2014 2292 1678
1.98 1.78 2.28 1.88
0.314 0.273 0.359 0.302
1.004 0.909 0.993 0.750
0.826 0.595 0.993 0.536
1.830 1.505 1.986 1.287
662
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