17428
J. Phys. Chem. C 2009, 113, 17428–17436
Improved Direct Correlation Function for Density Functional Theory Analysis of Pore Size Distributions Ming Zeng,† Yiping Tang,‡ Jianguo Mi,*,† and Chongli Zhong† Laboratory of Computational Chemistry, Department of Chemical Engineering, Beijing UniVersity of Chemical Technology, Beijing 100029, China, and Honeywell Process Solutions, 300-250 York Street, London, Ontario N6A 6K2, Canada ReceiVed: March 28, 2009; ReVised Manuscript ReceiVed: August 12, 2009
An improved direct correlation function is proposed and applied to construct an accurate free energy functional. The theoretical method is extended to analyze the pore size distributions (PSDs) of both slit- and cylindricalshaped carbonaceous materials. With the same parameters for the fluid and material as in the simulation, the model is much more advantageous than the known density functional methods for calculating the adsorption isotherms in individual pores. The overall PSDs for two model carbons are evaluated for CH4, CF4, and SF6 absorbates, and good results are obtained. The results are reliable because the two model carbons have structures that are exactly known. Encouraged by the success with slit pores, we further extend the model to the internal cylindrical space of a single-walled carbon nanohorn. The PSDs of the carbon nanohorn are evaluated, and satisfactory results are again obtained. 1. Introduction Microporous carbonaceous materials have been found to be highly useful for industrial applications, such as gas storage, separations, and catalysts. A great number of these materials, mostly activated carbons, are characterized by broad pore size distributions (PSDs). Understanding the PSD of a material and its relationship to the material’s performance is important in the design and manufacturing of these materials. The development of reliable methods for characterizing porosity has been the focus of numerous research efforts for several decades.1-4 Because activated carbons are noncrystalline materials, characterization by X-ray diffraction or neutron scattering is difficult to implement. Rather, gas adsorption is regarded as the primary method to determine PSDs for such materials, as adsorption is sensitive to both the geometrical and topological characteristics of the porous structure. One can extract the desired PSD by solving the adsorption integral equation5
Nexp(P) )
∫HH
max
min
f(H) NV(P, H) dH
(1)
where Nexp(P) is the experimental adsorption isotherm, NV(P,H) is the kernel of model isotherms in individual pores, and f(H) is the PSD. The accuracy of the PSD depends on the realism of the model for adsorption in individual pores. For sufficiently large pores (larger than 10 nm), the commonly used BarrettJoiner-Halenda (BJH) method6 provides an adequate description. In contrast, for micropores, the molecular texture of the absorbate becomes important, and the BJH approach has some limitations.7,8 Advances in simulation techniques have made molecular simulation a powerful method for predicting adsorption within porous solids. Recently, Monte Carlo simulation has been used extensively to analyze PSDs.9-11 The simulation method, although it yields reliable results for practical applica* Corresponding author. E-mail:
[email protected]. † Beijing University of Chemical Technology. ‡ Honeywell Process Solutions.
tions, is usually quite time-consuming. Meanwhile, statistical mechanical theories provide a simple and flexible way to handle this problem. Density functional theory (DFT), which is based on the idea that the free energy of an inhomogeneous fluid can be expressed as a functional of the spatial variation of the density, has proven to be a very successful method for analyzing the structural properties of inhomogeneous fluids12 and microporous materials.13,14 The theory, originally proposed by Tarazona and Evans15 and Tarazona,16 was constructed based on a smoothed density approximation to the free energy functional of a hard-sphere fluid, Fhs[F(r)]. Although this approach was closely related to the fine-grained generalized van der Waals theory of Nordholm and co-workers,17 the prescription for the smoothed density of DFT was obtained by recognizing that Fhs[F(r)] is the generating functional for a hierarchy of hardsphere direct correlation functions (DCFs). Therefore, the DCF plays a decisive role in constructing the density functional. Because an accurate DCF is quite difficult to acquire, it has typically been represented by a potential function in a variety of functional models for attractive contribution. This type of substitution inevitably causes some deviations. The first density functional method to calculate the PSD was reported by Seaton and co-workers.18,19 Later, Lastoskie et al.20 and Olivier et al.21 developed an improved method with a more accurate nonlocal DFT (NLDFT). In recent years, Neimark et al.22,23 and Do et al.24,25 have employed NLDFT to determine PSDs of both slit and cylindrical geometries. In these models, a simple mean-field approximation was used to implement the attractive contribution to the Helmholtz free energy functional. The discrepancies between these models and molecular simulations originate from the description of the bulk fluid and are exaggerated in inhomogeneous fluids. For example, with the same molecular parameters, NLDFT fails to reproduce the adsorption isotherms of molecular simulations because it neglects attractive forces in intermolecular correlations. This problem is very obvious at low pressure, especially when capillary condensation is involved. As a result, new parameters
10.1021/jp902803t CCC: $40.75 2009 American Chemical Society Published on Web 09/11/2009
Improved DCF for DFT Analysis of Pore Size Distributions have to be regressed separately for fluid-fluid and fluid-solid interactions with specific experimental data. A number of attempts have been made to improve the density functional expression. For the hard-sphere reference term, the fundamental measure theory (FMT) was developed by Rosenfeld,26 and later modified successfully in several forms,27-29 such as the modified fundamental measure theory (MFMT) of Yu and Wu.28 For the attractive term, Tang et al.30 and Sokołowski and Fischer31 proposed DFT approaches that were beyond a mean-field approximation for the first time by taking pair correlation functions into account. Bryk et al.32 applied the first-order mean spherical approximation (FMSA) to describe adsorption in slit-like pores. Kim and Lee33 used the weighteddensity approximation for the contribution of the attractive interaction, which gives a good description of equilibrium density distributions and adsorption isotherms of Lennard-Jones (LJ) fluids at interfaces. Peng and Yu34 used the modified Benedict-Webb-Rubin equation of state and obtained accurate adsorption isotherms of nitrogen. The equation, however, is a mean-field type and needs a set of specific parameters that have to be regressed from experimental data. Very recently, Neimark et al.35 applied quenched-solid DFT to calculate the PSDs of real microporous carbonaceous materials. By considering the wall as rough, the calculated adsorption isotherms were found to exhibit no artificial stepwise layering transitions when compared with the NLDFT results. This shows that quenchedsolid DFT is a potential way to deal with porous carbonaceous materials. The main purpose of this work was to develop an accurate DCF for the density functional model to analyze PSDs of both slit and cylindrical pores with the same parameters as in simulations of fluids and solids. In the present model, the welladdressed MFMT28 is employed to describe the hard-sphere reference term. For the attractive part, we developed a density functional based on the FMSA.36 As in the general case, the free energy is constructed by combining that of the bulk fluid with the weighted density approximation. Instead of using meanfield theory, we applied the DCF to determine the weight function. The adopted DCF was refined by solving the Ornstein-Zernike (OZ) integral equation, in which an accurate radial distribution function (RDF) was treated as the input. The RDF was originally developed by the FMSA and further improved by the simplified exponential approximation.36,37 Because the relationship between the measured structural parameters and the real structure of material is still unclear, we first evaluated the ability of our model with two model carbons that exhibit different porosities,38 termed as Carbon1 and Carbon2, which were generated by computer-based molecular model to match experimental quantities such as the porosity and average d002 spacing obtained from X-ray scattering.38 These virtual porous carbons capture the essential characteristics of real carbons, but (unlike real materials) they have structures that are exactly known because they are computer-generated. To minimize the calculation errors caused by different absorbates, we followed Cai et al.38 in using three species (CH4, CF4, and SF6) as absorbates and combining the three partial PSDs to produce a more complete PSD. Then, we extended the model to analyze cylindrical pores and compare our predictions with those of other theoretical models. It should be mentioned that the interaction between a pair of real molecules close to the wall might be different from their interaction in the bulk phase because of the effects of the wall39 or three-body interactions. This difference is neglected by almost all computer simulations and DFTs, including those in this work.
J. Phys. Chem. C, Vol. 113, No. 40, 2009 17429 In practical applications, the problem can be resolved by empirically fitting experimental data. 2. Theory In DFT, the density distribution of inhomogeneous fluids can be obtained by minimizing the grand potential and solving the Euler-Lagrange equation
(
F(r) ) Fb exp βµ - β
)
δ(Fhs[F(r)] + Fatt[F(r)]) - βVext(r) δF(r) (2)
where Fb is the bulk density; F(r) denotes the density distribution at the distance r; Fhs[F(r)] stands for the hard-sphere reference system, which can be given by the MFMT;28 Fatt[F(r)] accounts for the attractive interaction; µ represents the chemical potential in the ensemble, and Vext(r) is the external force, which is the cause of the fluid inhomogeneity. For spherical fluid molecules, the interaction is traditionally represented by a Lennard-Jones (LJ) 12-6 potential, which can be mapped into a two-Yukawa potential
uTY(r) )
{
∞
, r d -k1ε + k2ε r r
(3)
where k1, k2, z1, and z2 are defined in the Appendix; ε is the energy parameter; σ is the soft diameter for the LJ potential; and d is the effective hard-sphere diameter, which can be calculated from Barker-Henderson perturbation theory40 as
d)
1 + 0.2977T* σ, 1 + 0.33163T* + 0.00104771T*2
T* )
kBT ε (4)
According to the MFMT,28 the hard-sphere repulsive contribution in eq 2 can be represented by
δβFhs[F(r)] ) δF(r)
∂φ
∑ ∫ ∂nhsR w(R)(r - r′) dr′
(5)
R
with
nR(r) )
∫ F(r′) w(R)(r - r′) dr′
φhs ) -n0ln(1 - n3) +
[
(6)
n1n2 - nV1nV2 + 1 - n3
)
n32 n23 - 3n2nV2nV2 1 n3 ln(1 - n3) + 36π (1 - n3)2 n33
(7)
In eqs 5 and 6, R ) 0, 1, 2, 3, V1, and V2, and w(R)(r) arepresents the weight functions, given by28
w(2)(r) ) πd2w(0)(r) ) 2πdw(1)(r) ) δ(d/2 - r)
(8)
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w(3)(r) ) H(d/2 - r)
(9)
r w(V2)(r) ) 2πdw(V1)(r) ) δ(d/2 - r) r
(10)
where H(r) is the Heaviside step function and δ(r) is the Dirac delta function. In the case of planar symmetry, the expressions for weighted densities are well-known.28 To compute the integrals of a function f over the weight functions in eqs 8-10 in cylindrical geometry, the numerical method of Frink and Salinger41 was used. In cylindrical symmetry, the density profile depends only on the radial coordinate, and the result is a onedimensional (1D) problem that must be solved. The problem geometry is represented by a mesh, and the densities at the nodes of the mesh are to be calculated. The integrals are computed numerically, using a precalculated numerical integration stencil41
∫
(R) Nsten
f(r′) w(R)(r - r′) dr′ )
∑ Ci(R)ω(R)fi i)1
(R) where Nsten is the total number of points in the stencil of the Rth weight function. The weight functions are split into their ) r/(4πRr) for w(V1)] and the fundamental prefactors [e.g., C(V1) i H and δ weight functions, ω. All integration stencils, w(R) i , are calculated numerically by finding the contribution to each node from each element. It should be point out that analytical research42 exists that deals with integrals in cylindrical geometry, which was very attractive because the time spent in numerical calculation was much more demanding. For the attractive part, we introduce the hybrid weighteddensity approximation, which was originally proposed by Leidl and Wanger43 for hard-sphere systems and later applied to the attractive contribution by Kim and Lee.33 In this case, the partial derivative of the free energy functional is given by
δβFatt[F(r)] ) βaatt[F(r)] + β δF(r)
∫ F(r′)
δaatt[F(r′)]
ωatt(|r δF(r′) r′|) dr′ (11)
with
catt(r) ) -βu(r),
∫ F(r′) ωatt(|r - r′|) dr′
(12)
In eq 11, ωatt(r) is the weight function, and aatt[Fj(r)] is the Helmholtz free energy per particle for the attractive interaction, which has been well-predicted by combining the FMSA with the simplified exponential approximation for a bulk LJ fluid.37,44 The details are given in the Appendix. Following Denton and Ashcroft,45 we assume that the weight function ωatt(r) can be determined by the attractive part of the direct correlation function (DCF)
ωatt(r) )
catt(r)
∫ catt(r) dr
(13)
This assumption is supported by the following facts: (a) This function is normalized to unity. (b) It is reduced to mean-field theory when we set
(14)
(c) At the bulk limit F f Fb, the functional derivative of eq 11 gives δ2(βFatt[F(r)]) δF(r)δF(r1)
) -catt(|r - r1 |) ) 2βaatt ′ (Fb) ωatt(|r - r1 |) + βFbaatt ′′ (Fb)
∫ω
att(|r′-r1 |)
ωatt(|r - r′|) dr′
(15)
Integrating over three dimensions in eq 15, we have
Fb
( )
∫ catt(r) dr ) 2βFbaatt′ (Fb) + βFb2aatt′′ (Fb) ) β
∂Patt ∂F (16)
which is identical to the attractive part of compressibility equation. According to Tarazona and Evans15 and Tarazona,16 the DCF is crucial to the density functional method. For a spherically symmetrical fluid, c(r) relates to the total correlation function, h(r), or the radial distribution function (RDF), g(r) ) h(r) + 1, through the Ornstein-Zernike (OZ) equation
h(k) ) c(k) + Fh(k) c(k)
(17)
Thus, to obtain accurate functions c(k) or c(r), we employ the exponent form of the RDF as input, which has been found to well reproduce simulation data37
g(r) ) g0(r) exp[g1(r)]
(18)
where g0(r) and g1(r) are solved analytically and also given in the Appendix. As for the external force, the interaction between the LJ fluid and the walls of the slit pore can be simulated by Steele’s 10-4-3 potential46
[(
Vs(z) ) 2πFvεsfσsf2∆ F(r) )
r>d
2 σsf 5 z
) ( ) 10
-
σsf z
4
-
σsf4
]
3∆(0.61 + z)3 (19)
in which Fv is the volume number density of carbon atoms in the slit wall, with Fv ) 114 nm-3; ∆ is the interlayer space, with ∆ ) 0.335 nm; and εsf and σsf are the energy and size parameters, respectively, for solid-fluid interactions, which are determined using the Lorentz-Berthelot combination rules: εsf ) (εsεf)1/2 and σsf ) (σs + σf)/2. The overall external potential is given by
Vext(z) ) Vs(z) + Vs(H - z)
(20)
where H is the pore width, which is defined as the distance between the carbon nuclei in the surface graphite layers of the opposing pore walls. Adsorption isotherms for pores of different widths H are constructed over a range of pressure values P. The excess adsorption NV(P,H) is then obtained from the difference between
Improved DCF for DFT Analysis of Pore Size Distributions the equilibrium density profile F(z) and the bulk density Fb across the width of the pore
1 NV(P, H) ) H
∫0
H
[F(z) - Fb] dz
(21)
In a structureless single-walled cylindrical pore, the solid-fluid potential is given by47
{ [
)]
63 R - r r -10 1+ × Vext(r, R) ) π Fsεsfσsf 32 σsf R r 2 R-r 9 9 r F - , - ;1; -3 1+ 2 2 R σsf R r 2 3 3 F - , - ;1; 2 2 R 2
2
[ [
( )] ( ) ]}
[
(
(
)]
-4
×
(22)
where r is the radial coordinate of the adsorbate molecule measured from the pore center, R is the pore radius, Fs is the surface number density of carbon atoms, and F[R,β;γ;δ] is the hypergeometric series. The corresponding excess adsorption is calculated with
NV(P, R) )
2 R2
∫0R [F(r) - Fb]r dr
(23)
To determine the PSDs of porous carbons, the set of model isotherms in individual pores needs to be fitted to experimental adsorption data by solving eq 1 numerically. The objective function, which was minimized using the Tikhonov regularization method,48 is nP
Func )
∑ i)1
J. Phys. Chem. C, Vol. 113, No. 40, 2009 17431 TABLE 1: Lennard-Jones Parameters for Fluids and Carbon Material molecule
ε/kB (K)
σ (Å)
ref
CH4 CF4 SF6 N2 C
149.9 152.5 200.9 101.5 28.0
3.7327 4.7000 5.5100 3.6150 3.4000
38 38 38 22 38
3.1. DCF of Bulk Fluids. An important property for bulk fluids is the direct correlation function (DCF). In this work, a new DCF is obtained through the OZ equation, in which the key input, namely, the RDF, is derived from the simplified exponential approximation, whereas in the original FMSA, it is determined through a perturbation expansion method. Figure 1 compares the original and new DCFs with simulation data,50 showing that the new DCF is better than that from the FMSA. Figure 2 shows DCFs with different densities, demonstrating quite good agreement with simulation data.51 This success provides a solid foundation for the theory to advance to inhomogeneous fluids. 3.2. Adsorption Isotherms of Slit Pores. To further test the theoretical model, we follow Cai et al.38 by choosing two model carbons as the criteria. Therefore, the adsorption isotherms for individual pores with different widths obtained by molecular simulation take the role of the standard for evaluation. The results of isotherms of the three investigated gases, namely, CH4, CF4, and SF6, in a series of selected pore sizes are shown in Figure 3. In addition, the results given by an NLDFT calculation using the MFMT functional28 and the mean-field attractive term of the Weeks-Chandler-Andersen scheme52 are also depicted
nH
[Nexp(Pi) -
∑ f(Hj) NV(Pi, Hj) ∆Hj]2 + j)1
nH
R2
∑ f(Hj)2 ∆Hj
(24)
j)1
where nP is the number of points on the isotherm; nH is the number of width subintervals in the numerical integration; and R is a smoothing parameter that can be determined by the socalled L-curve method,49 which provides a balance between the fit to the experimental data and the size of the solution. The second term on the right-hand side of eq 24 is used to stabilize the solution. Similarly to Neimark et al.,35 the PSD is assumed to be an arbitrary shape, no specific function is selected because we have no a prior knowledge that the PSD should subject to a particular functional form.
Figure 1. Comparison of the calculated DCFs from the original FMSA and this work with simulation data50 at T* ) 1.5 and F* ) 0.4.
3. Results and Discussion To test the improved DCF, we first compare the new DCF with those from the FMSA and the simulation. Then, we use the thus-obtained DCF to predict the adsorption isotherms in slit pores with different pore widths. The needed parameters for selected fluids and carbon material were taken from refs 22 and 38 and are listed in Table 1. The predictive capability of our model was tested against experimental and simulation data. With the new model, we calculate the PSDs of two slit model carbons and a cylindrical pore by integration of the adsorption isotherms and compare them with the results of geometric analysis and simulation results.
Figure 2. DCFs calculated with different densities at T* ) 1.5. The open symbols are simulation data.51
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Figure 4. Isotherms for the adsorption of CH4 in slit pores with various widths at two different temperatures. The open symbols are simulation data,38 and the solid lines are the predictive results.
Figure 3. Comparison of isotherms for the adsorption of CH4, CF4, and SF6 in slit pores with various widths at T ) 258 K. The open symbols are simulation data,38 the dashed lines are predicted by NLDFT, and the solid lines are predicted by this work.
in the figure. For comparison, the parameters for fluids and carbon used in our model, the simulation, and NLDFT are the same. Because NLDFT is the most widely used theoretical method, this study could reveal the general capability of current theories for various situations. It is evident that, for CH4, CF4, and SF6, two theoretical methods yield different results. The results of NLDFT deviate systematically from the simulation data by overestimating the isotherms, and the deviation increases with increasing pore size. These deficiencies come from the mean-field approximation of the attractive interaction in NLDFT. In contrast, the new model performs much better, giving nearly the same results as the simulation data for various pore sizes. The new model, in consideration of the direct correlation of fluid molecules and the weighted density of the attractive interaction, is capable of predicting the isotherms correctly and
successfully suppressing the exaggerated deviations of NLDFT. The results strongly suggest that the new model can correctly handle the adsorption behavior and that its application to real materials will be more satisfactory. The adsorption isotherms in a series of selected pores are given in Figure 4 for CH4, Figure 5 for CF4, and Figure 6 for SF6. The simulation data were taken from ref 38. One can see that the new method yields satisfactory results. Similarly to simulation data, these isotherms show a decay in the adsorption with increasing temperature, but the tendency is different with increasing pore width. Figure 4 shows that the adsorption of CH4 decreases monotonically as the pore width increase and that the isotherm follows Henry’s law when the pore width is greater than 9 Å. For CF4, the variation tendency is the same as CH4 at higher temperature (T ) 296 K), whereas at lower temperature (T ) 275 K), the curves at 10 and 12 Å cross. For SF6, this type of crossing becomes more frequent. Figure 6 shows that adsorption decreases from 8 to 11 Å and then increases to 14 Å. As the pore width continues to increase, the isotherms decrease again. This behavior is due to the correlation between the pore size and the molecule size, as well as the fluid-fluid and fluid-wall interactions. 3.3. PSDs of Slit Pores. Similarly to Cai et al.,38 we obtained a PSD using each of the three absorbates and then obtained the overall PSD from the three separate PSDs. To calculate the overall PSD, the reliable pore size range should be considered. According to Lo´pez-Ramo´n et al,,53 the reliable pore size range defines a “window of reliability” that is bounded on the left by the smallest accessible pore size and on the right by the pore size at which adsorption becomes substantially linear, and the window of reliability can also be extended by measuring the isotherm at lower temperatures. The reliability of the PSD is improved when a more strongly adsorbing species is used. As a result, the overall PSD is obtained by picking the reliable pore
Improved DCF for DFT Analysis of Pore Size Distributions
J. Phys. Chem. C, Vol. 113, No. 40, 2009 17433
Figure 7. Overall PSD derived from the new model, simulation,38 and geometric analysis38 of Carbon1.
Figure 5. Isotherms for the adsorption of CF4 in slit pores with various widths at two different temperatures. The open symbols are simulation data,38 and the solid lines are the predictive results.
Figure 8. Overall PSD derived from the new model, simulation,38 and geometric analysis38 of Carbon2.
Figure 9. Comparison of the overall PSD derived from the new theoretical model with that from NLDFT.
Figure 6. Isotherms for the adsorption of SF6 in slit pores with various widths at two different temperatures. The open symbols are simulation data,38 and the solid lines are the predictive results.
size range of the CH4 PSD up to 10 Å and the pore size range of 10-35.5 Å from the “corrected” SF6 PSD. In this case, corrected means that the original SF6 PSD is increased by a factor that describes the fraction of pores accessible to CF4. The thus-obtained overall PSDs are plotted in Figure 7 for Carbon1 and in Figure 8 for Carbon2. The results obtained by simulation and geometric analysis are also included in the two figures for comparison. The geometric analysis38 is a direct determination of the distribution of pore volumes of different sizes using a decomposition algorithm for the analysis of a model carbon and can act as a standard for evaluation. Figure 7 suggests that the theoretical value is slightly better than the simulation value when compared to the result of geometric analysis. Particularly, the simulation yields an artificial
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Figure 10. Comparison of the predicted isotherms with the overall PSD for Carbon1 at T ) 258 K. The open symbols are simulation data.38
Figure 12. Adsorption isotherms of nitrogen at 77 K for structureless single-walled carbon nanotubes as a function of reduced chemical potential. The open symbols are simulation data.10
Figure 11. Comparison of the predicted isotherms with the overall PSD for Carbon2 at T ) 258 K. The open symbols are simulation data.38
Figure 13. Comparison of PSD derived from the new model with those from IDBdB(ASA)10 and IDBdB(DH).10
peak near 18 Å that does not exist in the geometric analysis. In contrast, the theory successfully reproduces the real PSD within this range. It is known that the accuracy of the theory is not better than that of the simulation. The “abnormal” phenomenon might come from the different algorithms used to solve the adsorption integral equation and, more importantly, from the flexibility of statistical mechanics theory. The self-consistent theoretical model provides wide adaptability and good predictability. Meanwhile, the relatively simple theoretical model allows more calculations of adsorption isotherms to be performed. To improve the calculated precision, we use more than 300 curves to fit the real isotherms. The accurate individual isotherms and the consecutive pore size curves ensure the finial result. A similar result was obtained for Carbon2, as shown in Figure 8. Figure 9 shows a comparison between our model and NLDFT. With the same parameters and calculation process, the new model yields better results than NLDFT. The deviation of NLDFT originates from its limitation in describing the thermodynamic properties of bulk fluids. With the obtained PSD, the overall adsorption isotherms of Carbon1 and Carbon2 were predicted. Figures 10 and 11 reveal that there is general agreement between the predictive and “true” isotherms38 of the three adsorbates based on two model carbons. These successes are attributed to the special capability of the new model, which uses an accurate DCF to describe the attractive part of inhomogeneous fluid.
3.4. PSDs of Cylindrical Pores. To further validate the new model, we extended its application to cylindrical pores. Compared to slit pores, cylindrical pores provide a more closed system. Because of the large curvature of the small pores, it takes a fairly long time for the molecules to move to a compact structure. It is necessary for molecular simulations to calculate more additional configurations, whereas the theoretical method does not have this limitation. Figure 12 displays the adsorption isotherms of nitrogen at 77 K for structureless single-walled carbon tubes10 with reduced radii of 2, 2.5, 4, and 8. The figure reveals that the predicted curves can faithfully reproduce the simulation data.10 The result of the PSD analysis for the internal space of a single-walled carbon nanohorn is shown in Figure 13. Here, we present the PSDs from the new DFT model and from IDBdB(ASA) and IDBdB(DH)10 simultaneously for comparison. Our results show that the distribution of the cylindrical pore radii of the material is slightly wider than that from IDBdB(ASA) but quite close to that from IDBdB(DH). It has been confirmed that IDBdB theory gives isotherms and PSDs that are in good agreement with experiments. Therefore, the new theoretical model is still reliable when extended to a cylindrical pores. 4. Conclusions In this work, an improved DCF is proposed for density functional theory analysis of the PSD of porous carbons. In the theoretical model, the modified DCF is used to construct
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J. Phys. Chem. C, Vol. 113, No. 40, 2009 17435
weighting functions for the attractive part. Using the same parameters as used for molecular simulations, the model provides an accurate description of adsorption isotherms of CH4, CF4, and SF6 in individual pores. Because the model carbons have structures that are exactly known, the predicted PSDs were validated by standard geometric analysis. These results illustrate clearly the power of the new model. The predictive capability of the new model for single-walled carbon nanohorns with cylindrical pores was further investigated. Comparisons with molecular simulation data confirmed the high fidelity of the model in the description of adsorption isotherms and PSDs of the real carbon nanotube materials. As a summary, this work provides a reliable theoretical tool for evaluating PSDs in different types of microporous carbonaceous materials. Acknowledgment. The financial support of the National Nature Science Foundation of China (Nos. 20576006 and 20876007) is greatly appreciated.
ar ) 2πFβ
∫σ∞ g0(r)
eg1(r) - 1 - g1(r) - g12(r)/[2g0(r)] LJ 2 u (r)r dr g1(r) (28) Q(t) )
S(t) + 12ηL(t)e-t (1 - η)2t3
(29)
S(t) ) (1 - η)2t3 + 6η(1 - η)t2 + 18η2t - 12η(1 + 2η) (30) L(t) ) (1 + η/2)t + 1 + 2η
(31)
where η ) 1/6Fπd3 is the packing factor, uLJ(r) is the original LJ potential, uTY(r) is the Two-Yukawa form of LJ potential, k1 ) k0ez1(σ-d), k2 ) k0ez2(σ-d), k0 ) 2.1714σ, z1 ) 2.9637/σ, z2 ) 14.0167/σ, g0(r) and g1(r) are given by ∞
Appendix
rg0(r) )
The Helmholtz free energy per particle for attractive interaction is given by
∑ (-12η)nC(1, n + 1, n + 1, r - n - 1)
n)0
(32) ∞
aatt ) a1 + a2 + ar
(1 - η)4 (1 + n)(-12η)n × Q2(z1) n)0 D(6, n, n + 2, z1, r - n - 1) ∞
(1 - η)4 (1 + n)(-12η)n × Q2(z2) n)0 D(6, n, n + 2, z2, r - n - 1)
where ar is the revised part brought by the simplified exponential approximation.
∫σ∞ g0(r) u(r)r2 dr ∞ ) 2πFβ ∫d [g0(r) - 1]uTY(r)r2 dr + ∞ σ 2πFβ ∫d uLJ(r)r2 dr - 2πFβ ∫d g0(r) uLJ(r)r2 dr
a1 ) 2πFβ
{[
]
1 + z1d L(z1d) 12ηβε k1 2 3 2 d z1 (1 - η) Q(z1d) z12 1 + z2d L(z2d) 1 σ + 48ηβε k2 2 2 2 9 d z2 (1 - η) Q(z2d) z2 1 σ6 1 σ6 2 σ3 1 + η/2 1 σ 12 + - 48ηβε 2 9 d 3 d 3 d 9 d (1 - η) )-
[
( )]
[()
]}
()
βεk2
12
)
(26)
a2 ) πFβ
[
k22 k12 6ηβ2ε2 )+ d3 2z1Q4(z1d) 2z2Q4(z2d) 2k1k2 (z1 + z2)Q (z1d)Q (z2d)
[
k1 /d 2
Q (z1d)
-
1 σ () [ 9 Q (z d) ] d k2 /d 2
2
12
∫0∞ rg0(r)e-sr dr ) (1 -L(sd)e η)2Q(sd)s2
(34)
∫0∞ rg1(r)e-sr dr βεk1e-sd (s + z1)Q2(sd) Q2(z1d)
-
βεk2e-sd (s + z2)Q2(sd) Q2(z2d) (35)
In eqs 32 and 33, C(n1,n2,n3,r) and D(n1,n2,n3,z,r) are defined as
{
C(n1, n2, n3, r) )
2
- 24ηβ2ε2
(33)
-sd
G0(s) )
-
∫σ∞ g1(r) u(r)r2 dr ∞ σ ) πFβ ∫d g1(r) uTY(r)r2 dr - πFβ ∫d g1(r)uLJ(r)r2 dr
∑
or given in their Laplace forms as
G1(s) )
[() ( )]
∑
rg1(r) ) βεk1
(25)
-
2
1 σ 3 d
6
]
()+ 2 σ (27) 9( d ) ] 3
n3
2
B(n1, n2, n3, i, R) i-1 tRr r e H(r) , n1 + n2 < 3n3 (i - 1)! i)1
∑∑
R)0
(1 + η/2)n2 δ(r) + (1 - η)2n3 2
n3
B(n1, n2, n3, i, R) i-1 tRr r e H(r) (i - 1)! i)1
∑∑
R)0
, n1 + n2 ) 3n3
(36)
{
17436
J. Phys. Chem. C, Vol. 113, No. 40, 2009
D(n1, n2, n3, z, r) ) n3
2
∑∑
(
-zr
e
∫0r C(n1, n2, n3, y)e-z(r-y) dy H(r) )
(-1)iB(n1, n2, n3, i, R) (tR + z)i
R)0 i)1
×
∑
(1 + η/2) -zr e H(r) + (1 - η)2n3 n2
n3
2
∑∑
(-1)iB(n1, n2, n3, i, R) (tR + z)i
R)0 i)1
(
-zr
e
)
(-1)j(tR + z)jrj -e H(r) j! j)0 i-1
tR r
Zeng et al.
, n1 + n2 < 3n3
, n1 + n2 ) 3n3
)
(-1)j(tR + z)jrj -e H(r) j! j)0 i-1
tR r
∑
(37)
with B(n1, n2, n3, i, R) ) n3-i n3-i-k1
∑ ∑
k1)0
1 × (1 - η)2n3
(-1)n3-i-k1(n3 - 1 + k2)!(2n3 - 1 - i - k1 - k2)!
k2)0
k1!k2!(n3 - 1 - k1 - k2)![(n3 - 1)!]2
×
A(n1, n2, k1, tR) (tR - tβ)n3+k2(tR - tγ)2n3-i-k1-k2
(38) n2
A(n1, n2, k1, tR) )
n2!(i + n1)! × i!(n i)!(i + n1 - k1)! 2 i)max(k1-n1,0)
∑
(1 + η/2)i(1 + 2η)n2-itRn1+i-k1
(39)
In eqs 36-39,H(r) is the Heaviside step function, δ(r) is the Dirac delta function, tR(R ) 0,1,2)are the roots of equation S(t) ) 0, tβ and tγ are the two zeros of S(t) other than tR when R is fixed. References and Notes (1) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic Press: New York, 1982. (2) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1998. (3) Fenelonov, V. B. Porous Carbon; Institute of Catalysis: Novosibirsk, Russia, 1995. (4) McEnaney, B.; Mays, T. J.; Rodrı´guez Reinoso, F., Eds.; Fundamental Aspects of Active Carbons. Carbon 1998, 36 (Special Issue), 6. (5) Sips, R. J. Chem. Phys. 1948, 16, 490. (6) Barret, E. P.; Joyner, L. G.; Halenda, P. P. J. Am. Chem. Soc. 1962, 73, 373. (7) Ravikovitch, P. I.; Haller, G. L.; Neimark, A. V. AdV. Colloid Interface Sci. 1998, 76, 203. (8) Lastoskie, C.; Gubbins, K. E.; Quirke, N. Langmuir 1993, 9, 2693. (9) Herdes, C.; Santos, M. A.; Medina, F.; Vega, L. F. Langmuir 2005, 21, 8733.
(10) Kowalczyk, P.; Hołyst, R.; Tanaka, H.; Kaneko, K. J. Phys. Chem. B 2005, 109, 14659. (11) Kowalczyk, P.; Ciach, A.; Neimark, A. V. Langmuir 2008, 24, 6603. (12) Balbuena, P. B.; Gubbins, K. E. Langmuir 1993, 9, 1801. (13) Quirke, N.; Tennison, S. R. Carbon 1996, 34, 1281. (14) Neimark, A. V.; Ravikovitch, P. I.; Grun, M.; Schuth, F.; Unger, K. K. Langmuir 1995, 11, 4765. (15) Tarazona, P.; Evans, R. Mol. Phys. 1984, 52, 847. (16) Tarazona, P. Phys. ReV. A 1985, 31, 2672. (17) Johnson, M.; Nordholm, S. J. Chem. Phys. 1981, 75, 1953. (18) Seaton, N. A.; Walton, J. P. R. B.; Quirke, N. Carbon 1989, 27, 853. (19) Jessop, C. A.; Riddiford, S. M.; Seaton, N. A.; Walton, J. P. R. B.; Quirke, N. In Characterization of Porous Solids (COPS-II). Proceedings of the IUPAC Symposium; Rodrı´guez Reinoso, F., Rouquerol, J., Sing, K. S. W., Eds.; Elsevier: Amsterdam, 1991; p 123. (20) Lastoskie, C.; Gubbins, K. E.; Quirke, N. J. Phys. Chem. 1993, 97, 4586. (21) Olivier, J. P.; Conklin, W. B.; v. Szombathely, M. In Characterization of Porous Solids (COPS-III). Proceedings of the IUPAC Symposium; Rodrı´guez Reinoso, F., Rouquerol, J., Sing, K. S. W., Unger, K. K., Eds.; Elsevier: Amsterdam, 1994; p 81. (22) Ravikovitch, P. I.; Vishnyakov, A.; Russo, R.; Neimark, A. V. Langmuir 2000, 16, 2311. (23) Ravikovitch, P. I.; Neimark, A. V. Colloids Surf. A 2001, 187188, 11. (24) Ustinov, E. A.; Do, D. D. Langmuir 2004, 20, 3791. (25) Ustinov, E. A.; Do, D. D.; Jaroniec, M. J. Phys. Chem. B 2005, 109, 1947. (26) Rosenfeld, Y. Phys. ReV. Lett. 1989, 63, 980. (27) Roth, R.; Evans, R.; Lang, A.; Kahl, G. J. Phys.: Condens. Matter 2002, 14, 12063. (28) Yu, Y.-X.; Wu, J. Z. J. Chem. Phys. 2002, 117, 10156. (29) Malijevsky, Al. J. Chem. Phys. 2006, 125, 194519. (30) Tang, Z.; Scriven, L. E.; Davis, H. T. J. Chem. Phys. 1991, 95, 2659. (31) Sokołowski, S.; Fischer, J. J. Chem. Phys. 1992, 96, 5441. (32) Bryk, P.; Rz˙ysko, W.; Malijevsky, Al.; Sokołowski, S. J. Colloid Interface Sci. 2007, 313, 41. (33) Kim, S. C.; Lee, S. H. J. Phys.: Condens. Matter 2004, 16, 6365. (34) Peng, B.; Yu, Y.-X. Langmuir 2008, 24, 12431. (35) Neimark, A. V.; Lin, Y.; Ravikovitch, P. I.; Thommes, M. Carbon 2009, 47, 1617. (36) Tang, Y.; Lu, B.C.-Y. Mol. Phys. 1997, 90, 215. (37) Tang, Y.; Lu, B.C.-Y. AIChE J. 1997, 43, 2215. (38) Cai, Q.; Buts, A.; Biggs, M. J.; Seaton, N. A. Langmuir 2007, 23, 8430. (39) Sinanoglu, O.; Pitzer, K. S. J. Phys. Chem. 1960, 32, 1279. (40) Cotterman, R. L.; Schwarz, B. J.; Prausnitz, J. M. AIChE J. 1986, 32, 1787. (41) Frink, L. J. D.; Salinger, A. G. J. Comput. Phys. 2000, 159, 407. (42) Malijevsky, Al. J. Chem. Phys. 2007, 126, 134710. (43) Leidl, R.; Wagner, H. J. Chem. Phys. 1993, 98, 4142. (44) Tang, Y.; Zhang, T.; Lu, B.C.-Y. Fluid Phase Equilib. 1997, 134, 21. (45) Denton, A. R.; Ashcroft, N. W. Phys. ReV. A 1989, 39, 426. (46) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: Oxford, U.K., 1974. (47) Tjatyopoulos, G. J.; Feke, D. L.; Mann, J. A., Jr. J. Phys. Chem. 1988, 92, 4006. (48) Tikhonov, A. N. Dokl. Akad. Nauk SSSR 1943, 39, 195. 1963, 153, 49. (49) Hansen, P. C. SIAM ReV. 1992, 34, 561. (50) Llano-Restrepo, M.; Chapman, W. G. J. Chem. Phys. 1992, 97, 2046. (51) Kunor, T. R.; Taraphder, S. Physica A 2007, 383, 401. (52) Weeks, J. D.; Chandler, D.; Andersen, H. C. J. Chem. Phys. 1971, 54, 5237. (53) Lo´pez-Ramo´n, M. V.; Jagiełło, J.; Bandosz, T. J.; Seaton, N. A. Langmuir 1997, 13, 4435.
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