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Langmuir 2006, 22, 6238-6244
Features of Nitrogen Adsorption on Nonporous Carbon and Silica Surfaces in the Framework of Classical Density Functional Theory E. A. Ustinov,§ D. D. Do,*,† and M. Jaroniec‡ Department of Chemical Engineering, UniVersity of Queensland, St. Lucia, Queensland 4072, Australia, and Department of Chemistry, Kent State UniVersity, Kent, Ohio 44242 ReceiVed February 16, 2006. In Final Form: April 19, 2006 Equilibrium adsorption data of nitrogen on a series of nongraphitized carbon blacks and nonporous silica at 77 K were analyzed by means of classical density functional theory to determine the solid-fluid potential. The behavior of this potential profile at large distance is particularly considered. The analysis of nitrogen adsorption isotherms seems to indicate that the adsorption in the first molecular layer is localized and controlled mainly by short-range forces due to the surface roughness, crystalline defects, and functional groups. At distances larger than approximately 1.3-1.5 molecular diameters, the adsorption is nonlocalized and appears as a thickening of the adsorbed film with increasing bulk pressure in a relatively weak adsorption potential field. It has been found that the asymptotic decay of the potential obeys the power law with the exponent being -3 for carbon blacks and -4 for silica surface, which signifies that in the latter case the adsorption potential is mainly exerted by surface oxygen atoms. In all cases, the absolute value of the solid-fluid potential is much smaller than that predicted by the Lennard-Jones pair potential with commonly used solid-fluid molecular parameters. The effect of surface heterogeneity on the heat of adsorption is also discussed.
1. Introduction Equilibrium adsorption data on nonporous surfaces are an important source of information on solid-fluid interactions, and the analysis of such data is often used to model adsorption in micropores and mesopores of different shapes and to characterize the pore structure of different porous materials. In this article, we concentrate on the analysis of nitrogen adsorption isotherms for a series of nongraphitized carbon blacks and nonporous silica as representatives of amorphous solids. The main problem hampering a deeper understanding of the mechanism of equilibrium adsorption on amorphous materials is the lack of agreement between experimental and theoretical approaches. Thus, the use of different simplified equations such as Langmuir, Brunauer-Emmett-Teller (BET), and Fowler-Guggenheim (FG) as the kernel in the Fredholm integral1-3 leads to either poor correlation or the appearance of negative parts on the energy distribution function curve, which assumes a complex oscillating form. The use of the FG equation gives relatively better results but requires one to assume a coordination number that is too low to avoid the 2D phase transition. Thus, it was accepted in refs 1-3 that each molecule adsorbed on a site in the surface layer that has at most two neighbors. The nonlocal density functional theory (NLDFT) was applied to the energetic heterogeneity analysis of carbon blacks in refs 4-6. In this case the isotherm for a graphitized carbon black was used as a kernel in the * Corresponding author. E-mail:
[email protected]. † University of Queensland. ‡ Kent State University. § On leave from Scientific and Production Company “Provita”, 6 Prospect Kim, 199155 St. Petersburg, Russia. (1) Kruk, M.; Jaroniec, M.; Bereznitski, Yu. J. Colloid Interface Sci. 1996, 182, 282. (2) Heuchel, M.; Jaroniec, M.; Gilpin, R. K.; Bra¨uer, P.; Von Szombathely, M. Langmuir 1993, 9, 2537. (3) Choma, J.; Jaroniec, M. Langmuir 1997, 13, 1026. (4) Olivier, J. P. In Surfaces of Nanoparticles and Porous Materials; Schwarz, J. A., Contescu, C. I., Eds.; Surfactant Science Series; Marcel Dekker: New York, 1999; Vol. 78, pp 295-318. (5) Olivier, J. P. In Characterization of Porous Solids; Studies in Surface Science and Catalysis; 2000; Vol. 128, pp 81-87. (6) Olivier, J. P.; Winter, M. J. Power Sources 2001, 97-98, 151.
framework of the so-called patchwise model to determine the distribution of the surface area over the depth of the solid-fluid potential well. It was implied that each patch adsorbs fluid molecules independently of others, which actually could be correct only if those patches are of macroscopic size. An alternative and much less time-consuming NLDFT-based model considers the heterogeneity on the microscopic level.7-9 This approach is based on the same consideration of fluid-fluid and solid-fluid interactions under the assumption that both media are disordered in the case of amorphous solids. It allowed us to extend Tarazona’s version of classical NLDFT10,11 to nitrogen and argon adsorption at their boiling points on nonporous silica7 and carbon black.9 The experimental adsorption isotherms were fitted very accurately over the whole relative pressure range of 5 to 6 orders of magnitude. Nevertheless, one should recognize that any equations of the solid-fluid potential derived from the Lennard-Jones (LJ) or other empirical pair potentials are approximate. This justifies the sensitivity analysis of adsorption isotherms not only to the solid-fluid molecular parameters (the potential well depth and the collision diameter) but also to the shape of the solid-fluid potential profile. The effect of the potential on the adsorption isotherm and hysteresis loop was recently analyzed for the case of cylindrical pores in the framework of NLDFT.12 The potential was represented as a superposition of two square wells. They have found that adsorption and the 2D transition in the first molecular layer are highly sensitive to the potential well close to the pore wall. The spinodal condensation and equilibrium evaporation are almost independent of the latter but are strongly affected by the weak “tail” potential. A more rigorous solidfluid potential could be obtained through quantum DFT.13,14 (For a review, see ref 15.) However, this approach is quite involved compared to the classical NLDFT. For this reason, we adhere (7) Ustinov, E. A.; Do, D. D.; Jaroniec, M. Appl. Surf. Sci. 2005, 252, 548. (8) Ustinov, E. A.; Do, D. D. Adsorption 2005, 11, 455. (9) Ustinov, E. A.; Do, D. D.; Fenelonov, V. B. Carbon 2006, 44, 653. (10) Tarazona, P. Phys. ReV. A 1985, 31, 2672. (11) Tarazona, P.; Marconi, U. M. B.; Evans, R. Mol. Phys. 1987, 60, 573. (12) Zhang, X.; Cao, D.; Wang, W. Chem. Phys. 2003, 119, 12586. (13) Hohenberg, P.; Kohn, W. Phys. ReV. B 1964, 136, 864. (14) Mermin, N. D. Phys. ReV. A 1965, 137, 1133.
10.1021/la0604580 CCC: $33.50 © 2006 American Chemical Society Published on Web 06/07/2006
N2 Adsorption on Nonporous Carbon and Silica Surfaces
to an alternative statement of the problem, which can be reduced to finding the solid-fluid potential directly from experimental adsorption isotherms on the basis of classical NLDFT. Recently, we developed a procedure to determine the solid-fluid potential directly from experimental adsorption isotherms,9,16 which gives us an opportunity to compare different surfaces. However, it turned out that finding the potential outside the first molecular layer, where it becomes relatively small, is unstable. (In the enlarged scale between 0 and -1kT the potential oscillates at low value of the regularization parameter.) Meanwhile, it is important to determine the asymptotic pattern of the potential because it reflects the main features of the interaction between the solid and fluid. In the present article, we have done this by a combination of the adjusted potential within approximately 1.5 molecular layers and the power law for the second part of the potential dependence. In this study, we chose a series of commercial nongraphitized carbon blacks described in ref 1 and nonporous silica LiChrospher Si-1000 17 to analyze the effect of the degree of graphitization on the potential profile and differential heats of adsorption. The comparison of potential curves for silica and carbon blacks is also provided.
2. Model Mathematical details of the approach used are described in our previous papers.7-9 Basically, we used the original smoothed density approximation (SDA) of Tarazona,10,11 the CarnahanStarling (CS) equation for the reference hard-sphere fluid18 to account for repulsive forces, and the Weeks-Chandler-Andersen (WCA) scheme19 for the attractive part of the fluid-fluid potential of intermolecular interactions. This approach is completely identical to Tarazona’s SDA at distances larger than two hardsphere diameters dHS from the solid surface. The difference appears only in the vicinity of the surface. In the approach used, the interaction potential between the amorphous solid and the fluid is considered to be a sum of attractive and repulsive terms in the same manner as done for fluid-fluid interaction. The repulsive term is determined via the CS equation as a function of the smoothed density Fj(r). Close to the surface, the additional solid-fluid repulsive potential appears, and this is due to the fact that the void volume available to insert an extra molecule decreases because of the presence of solid atoms. This is equivalent to the increase of the effective smoothed density. Below we present basic equations of the approach to provide its further application for the analysis of the heat of adsorption. The working equation to determine the density profile is obtained from the condition of the minimum of the grand thermodynamic functional and is given by
kT ln[Λ3F(z)] + Ψ[F(z)] + 2u(z) + uext(z) - µ ) 0 (1) where
Ψ[F(z)] ) fex[Fj(z)] +
∫0z
m
δFj(z′) dz′ (2) δF(z)
F(z′) f′ex[Fj(z′)]
Here, k and T are the Boltzmann constant and the temperature, respectively; Λ is the de Broglie wavelength; F(z) is the local density at a distance z from the surface; u(z) is the potential of a molecule due to the interaction with surrounding fluid molecules; (15) Ghosh, S. K. Int. J. Mol. Sci. 2002, 3, 260. (16) Ustinov, E. A.; Do, D. D. J. Colloid Interface Sci., in press. (17) Jaroniec, M.; Kruk, M.; Olivier, J. P. Langmuir 1999, 15, 5410. (18) Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635. (19) Weeks, J. D.; Chandler, D.; Andersen, H. C. J. Chem. Phys. 1971, 54, 5237.
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uext(z) is the external solid-fluid potential; µ is the chemical potential; and fex[Fj(z)] is the excess Helmholtz free energy. The grand potential is determined in the range of distances from zero to a sufficiently large value zm where the properties of the adsorbed phase become indistinguishable from the bulk-phase properties. Ψ[F(z)] is the functional derivative of the excess free energy functional. The excess Helmholtz free energy is given by the CS equation as follows:18
4η j - 3η j2 (1 - η j )2
fex(Fj) ) kT
π η j ) dHS3Fj 6
(3)
The fluid-fluid potential is determined in the framework of the mean field approximation as
u(z) )
∫
1 F(z′) φff(|z - z′|) dz′ 2
(4)
where φff(z) is the pairwise potential of interaction of two molecules having a separation distance z. In the case of the 1D task application of the WCA scheme, we have
φff(z) )
{
|z| < rm πff[z2 - (9/5)rm2] 2 10 4 2πffσff [(2/5)(σff/z) - (σff/z) ] |z| > rm
(5)
The smoothed density is represented as follows10,11
Fj(z) ) Fj0(z) + Fj1(z) Fj(z) + Fj2(z) (Fj(z))2
(6)
where
Fji(z) )
∫Fe(z′) ωi(|z - z′|) dz′
(7)
for i ) 0, 1, 2. Weight functions ω0(z), ω1(z), and ω2(z) depend only on the distance z from a given point, and in the case of the 1D task under consideration, they can be calculated similarly as in ref 20. The effective density Fe in the integrand is defined as follows:
Fe(z) ) F(z)H(z + Fm[1 - H(z)]
(8)
In the above equation, H(z) is the Heaviside step function, and Fm is the maximum density () 6/(πdHS3)) leading to an infinite excess Helmholtz free energy in the case of a homogeneous fluid. Because all weight functions in eq 7 reduce to zero for |z - z′| > 2dHS, the smoothed density is not affected by the surface at a distance greater than two hard-sphere diameters. However, in the vicinity of the surface the solid atoms contribute to the smoothed density, resulting in the increase of the excess Helmholtz free energy, which reflects the appearance of solid-fluid repulsive forces. (For more details, see refs 7 and 8.) The amount adsorbed expressed in terms of the excess loading is given by
a)
∫0z [F(z) - Fg] dz m
(9)
where Fg is the bulk density at a given pressure and temperature. The solution of eqs 1-8 gives the density profile corresponding to any bulk pressure if the solid-fluid potential uext is known. Once the density profile is determined, the amount adsorbed is calculated with eq 9. The solid-fluid potential could be predicted using the solid-fluid molecular parameters and an equation analogous to the WCA perturbation scheme.7 However, in the (20) Ustinov, E. A.; Do, D. D. Langmuir 2004, 20, 3791.
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UstinoV et al.
general case the potential could be affected by the complex chemical structure of the surface. Thus, a direct determination of the potential exerted by the surface from the adsorption isotherm could in principle give some additional information on the surface chemistry (number and type of functional groups, crystalline defects, etc). For this reason, we employed a procedure that allows us to determine the potential profile using the least-squares technique on the basis of the application of the NLDFT version developed for amorphous solids. 2.1. Heat of Adsorption. There are numerous works on the analysis of the heat of adsorption using the Clapeyron equation. For example, the application of NLDFT coupled with the Clapeyron equation to obtain the differential heat of adsorption is considered in refs 21-23. An alternative way to calculate the heat of adsorption using NLDFT was exploited by Olivier.5 He used a thermodynamic definition of the heat of adsorption via the derivative of the internal energy of the adsorbed phase with respect to the amount adsorbed at constant temperature. The advantage of the approach is that both the amount adsorbed and the heat of adsorption can be calculated directly in the same run of an iteration procedure without the need to vary the temperature at a specified loading. Below we consider this method more specifically. The differential heat of adsorption is defined as the heat released upon transferring one mole of the adsorbed fluid from the bulk phase to the adsorbed phase at constant bulk pressure and temperature. In the case of nonporous solids, the boundary between the two phases zm can be taken arbitrarily but sufficiently far away from the surface, where the external potential is close to zero and the fluid density becomes uniform. The position of this boundary is constant in further consideration. Let δn be the number of molecules transferring through the unit surface area of the boundary. For simplicity, let us consider the spherical molecule having a kinetic energy of 3/2kT. Then the balance of energy can be written as follows:
[
] ()
3 p kT + ug δn + g δn ) δU + δQ 2 F
(10)
The term in the square brackets is the sum of the kinetic and potential energy ug of a molecule in the bulk phase, and the first term on the LHS of eq 10 is the energy that enters into the adsorbed phase with δn molecules. The second term in the LHS is the work needed to be done against the bulk pressure. The transferred energy is consumed for the change in internal energy of the adsorbed phase δU and the heat released to the surrounding δQ. The differential heat of adsorption q is the limit δQ/δn at δn f 0, hence it is given by
p ∂U 3 q ) kT + ug + g 2 ∂n F
(11)
The internal energy can be determined from the Gibbs-Helmholtz equation. If we accept that the dependence of the hard-sphere diameter on temperature is insignificant (which was convincingly shown by Neimark et al.24), then the contribution of the excess Helmholtz free energy to the internal energy will be zero. Then, accounting for the temperature dependence of the de Broglie (21) Balbuena, P. B.; Gubbins, K. E. Langmuir 1993, 9, 1801. (22) Pan, H.; Ritter, J. A.; Balbuena, P. B. Langmuir 1998, 14, 6323. (23) Pan, H.; Ritter, J. A.; Balbuena, P. B. Ind. Eng. Chem. Res. 1998, 37, 1159. (24) Neimark, A. V.; Ravikovitch, P. I.; Gru¨n, M.; Schu¨th, F.; Unger, K. K. J. Colloid Interface Sci. 1998, 207, 159.
wavelength for the internal energy one can write
3 U ) kTn + 2
∫0z F(z)[u(z) + uext(z)] dz m
(12)
After substituting the internal energy U from the above equation into eq 11, we can write the heat of adsorption as follows
∫0z [2u(z) + uext(z)]σ(z) dz ∫0z σ(z) dz m
p q)u + gF g
m
(13)
where σ(z) ) ∂F(z)/∂µ. The set of partial derivatives of the local density with respect to the chemical potential can be determined from eq 1 by taking the functional derivative:
σ(z) + F(z)
kT
∫0z
{
m
}
δΨ[F(z)] + φff(|z - z′|) σ(z′) dz′ ) 1 δF(z′) (14)
For practical use, the above equation can be discretized, resulting in a set of equations that can be solved by an iteration procedure. Once the set of σ(z) is determined, the heat of adsorption can be directly calculated from eq 13. Functional derivatives needed to calculate the amount adsorbed and the heat of adsorption depend on the smoothed density approximation. In the case of the Tarazona SDA used in the present article, the functional derivatives are given in the Appendix.
3. Results As mentioned in Introduction, we developed a special technique to determine the solid-fluid potential. The potential curve is presented as a combination of two sections. The first section spanning over approximately 1.5 monolayers is the subject of the fitting using the Tikhonov regularization procedure.25 The second section is approximated by the power law because one can be certain that the power law correctly describes the asymptotic behavior of the solid-fluid potential. The thermodynamic functional was discretized along the distance from the surface with mesh size dHS/30. Therefore, if the boundary between the two sections corresponds to the distance of 1.5 hard-sphere diameters, then there will be 45 points of the potential curve to be determined by the least-squares fitting. Additionally, we determined one or two (including the power) parameters for the second section of the potential. 3.1. Nitrogen Adsorption on a Carbon Black Surface. In this section, we consider a series of three samples of nongraphitized carbon blacks from Cabot Corp. analyzed in ref 1. All nitrogen adsorption isotherms at 77 K are presented in that paper on linear and logarithmic scales, so we will not reproduce those isotherms. Just to show the pattern, Figure 1 presents an N2 adsorption isotherm on Cabot BP 280. The solid line is the correlated isotherm obtained with NLDFT using the fitted solidfluid potential. The asymptotic part of the potential was defined for distances larger than 1.3 hard-sphere diameters for all samples and was approximated by the power function decaying as the inverse third power of the distance from the surface. As can be seen from Figure 1a, the correlated function underestimates the amount adsorbed at high pressures, especially close to the saturation pressure. Formally, this part of the isotherm can be fitted if the asymptotical section of the solid-fluid potential includes the term that decays inversely with the distance. However, such a weakly dependent term seems to be unfeasible from the (25) Tikhonov, A. N.; Arsenin V. Y. Solutions of Ill-Posed Problems; Wiley: New York, 1977.
N2 Adsorption on Nonporous Carbon and Silica Surfaces
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Figure 2. Solid-fluid (attractive) potential determined with the NLDFT version developed for the case of the adsorption of gases on amorphous solids. The solid, dashed, and dashed-dotted lines are for samples BP 120, BP 280, and BP 460, respectively. In the above numbers of samples, the surface area increases. (See Table 1 for details.) Table 1. Molecular Parameters for N2 Adsorption on Nongraphitized Carbon Blacks ff/k (K) σff (nm) dHS (nm) σss (nm) σsf (nm) Fs (nm-3) N2 77 K Neimark et al.
99.92 94.45
0.3510 0.3575
0.3576 0.3575
0.3400 0.3400
0.3455 0.3487
114 114
Table 2. Characteristics of Carbon Blacks Cabot BP Determined with N2 Adsorption at 77 K
Figure 1. Nitrogen adsorption isotherm on nongraphitized carbon black Cabot BP 280 at 77 K1 (a) on a linear scale and (b) on a logarithmic scale. Experimental data, circles; correlation by the approach using the fitted solid-fluid potential, solid line.
viewpoint of the nitrogen-graphite surface interaction even when accounting for the existence of functional groups on that surface. For this reason, we argue that the deviation observable in the vicinity of the saturation pressure is due to the capillary condensation in relatively large pores and the space between primary particles. To avoid any distorting effect of this deviation on the accuracy of the potential to be determined by the leastsquares fitting, we truncated the isotherm at a reduced pressure of about 0.7-0.9 (depending of the sample). The solid-fluid potential or, more correctly, its attractive part is presented in Figure 2 for the group of three samples of carbon blacks (BP 120, BP 280, and BP 460) having BET surface areas from 30 to 78 m2/g. One can see from the Figure that the solidfluid potentials for the samples are quite similar. Also, it can be seen from Figure 2 that the pattern of the potential-distance dependence sharply changes around the distance of 1.3dHS. At this point, the potential becomes unexpectedly small in its absolute term. Using the factor of the inverse cubic function of the distance determined as a fitting parameter, one can estimate the solidfluid potential well depth sf corresponding to the Lennard-Jones equation. For the asymptotic section, we can drop the shortrange repulsive term. Then accounting for the amorphous structure of the graphite, the potential can be approximately presented as follows:
u≈-
(32)πF σ
6 -3
s sf sf
z
(15)
sample
SBET (m2/g)
sf/k (K)
u*/kT
z*/dHS
BP 120 BP 280 BP 460
30 41 78
20.28 17.66 19.92
-9.72 -9.86 -10.02
1.3 1.3 1.3
Here Fs is the number density of amorphous graphite; sf and σsf are the solid-fluid LJ parameters (i.e., the potential well depth and the collision diameter, respectively). It should be noted that the fluid-fluid molecular parameters were determined at 77.35 K using data for the saturation pressure, liquid-nitrogen density at the saturation pressure, and the surface tension. The fluidfluid collision diameter was adjusted to match the surface tension (0.0888 N/m) in the canonical ensemble. We obtained different values for the molecular parameters compared to those published by Neimark et al.24 All of the molecular parameters that we used in our calculations are presented in Table 1. The solid-fluid collision diameter σsf was determined with the Lorentz-Berthelot mixing rule. Some parameters corresponding to carbon blacks are listed in Table 2. In Table 2, u* is the average potential for the first section of the potential curve. There is a tendency for the potential to increase with increasing surface area. This increase in the potential could be attributed to the greater density of functional groups and crystalline defects in larger surface area samples. The most striking result is that the solid-fluid potential well depth determined from the asymptotic part of the potential curve and presented in the third column is quite small. For the system N2-graphitized carbon black, sf/k is about 57 K. In the present stage, it is hard to support this observation theoretically. Perhaps it is somehow associated with the shielding of the dispersive forces by the electrostatic field exerted by surface functional groups or/and by the electron system of carbon atoms residing on dislocations. Alternatively, the rough surface containing defects and functional groups for some reason might be the source of the highly long-
6242 Langmuir, Vol. 22, No. 14, 2006
UstinoV et al. Table 3. Results of Least-Squares Analysis of the N2-Silica Potential z*/dHS
A
g
u*/kT
1.3 1.5 1.3 1.4 1.5 1.6
-7.3006 -7.6326 -7.6159 -7.6817 -7.7419 -7.8283
3.938 3.983 4 4 4 4
-11.07 -9.864 -11.06 -10.44 -9.863 -9.331
F′ssf/k (K/nm2)
|∆| (%)
1607.3 1621.2 1633.9 1652.1
0.96 0.95 0.98 0.96 0.95 0.93
dependence of the solid-fluid potential was defined as
u ≈ Az-g
(16)
Some results are presented in Table 3. As can be seen from the Table, the exponent g is very close to 4. This suggests that the potential is exerted nearly exclusively by surface atoms, which can be identified as oxygen atoms covering the silica surface. For this reason, we admitted that the exponent is exactly equal to 4 and varied the boundary between the first and second parts of the potential. The product F′ssf/k was determined from the following equation:
u ≈ -2πF′ssfσsf6z-4
Figure 3. N2 adsorption isotherm on LiChrospher Si-1000 at 77 K17 on (a) linear and (b) logarithmic scales. Circles represent experimental data. The solid line represents the correlation by the model.
range potential, which is responsible for the adsorption even at a pressure close to the saturation pressure. In the latter case, the sharp increase in the amount adsorbed at high pressures is not due to capillary condensation; therefore, the vanishing of the solid-fluid potential is seemingly an artifact. We leave this issue for further investigations, which should be provided on the basis of more reliable and detailed experimental information. In any case, there seems to be a sharp change in the adsorption mechanism at a distance of around 1.3 hard-sphere diameters. It could be interpreted as the crossover from the localized to nonlocalized adsorption. If this is the case, the potential-distance dependence is virtually a reflection of sequential adsorption on surface active sites distributed somehow over the adsorption potential. 3.2. Nitrogen Adsorption on Nonporous Silica. It is worthwhile to compare the pattern of the potential-distance dependence determined for nongraphitized carbon blacks and that for amorphous silica. To this end, we analyzed the nitrogen adsorption isotherm on nonporous silica LiChrospher Si-1000 at 77 K published in ref 17. The isotherm and its correlation with the NLDFT model discussed above are shown in Figure 3. One can see that the correlation is excellent over the entire pressure range. The solid-fluid potential was determined analogously as in the case of carbon blacks. However, the difference is that in the first stage we determined not only the multiplier for the power function but also the exponent. We also varied the distance at which the power law was applied. The asymptotic distance
(17)
In the above equation, F′s is the surface density defined as the number of surface atoms per unit surface area (m-2); the solidfluid collision diameter was calculated as the arithmetic mean of the collision diameter of nitrogen and oxygen (0.276 nm26). The values of the group F′ssf/k are listed in the fifth column. Neimark and Ravikovitch26 used the value 2253 K nm-2, which is substantially larger than that determined from the asymptotic part of the isotherm. It does not mean, however, that the obtained potential is underestimated (in absolute term) because the above comparison refers only to the asymptotic part of the potential. Again, as in the case of carbon blacks this discrepancy shows that the LJ equation adjusted to the whole potential curve overestimates the actual potential (in absolute term) a large distance from the surface. In the vicinity of the surface, the potential well found by the least-squares fitting is deeper than that predicted with the LJ equation, which compensates for the diminished absolute value of the potential at larger distances from the surface. The root-mean-square error presented in the last column of Table 3 in all cases is less than 1%. Therefore, it is hard to make a choice of the position of the boundary between the two sections of the potential on the basis of the mean error. We took the distance of 1.5dHS because in this case the potential curve is the most smoothed in the vicinity of the junction point. The potential-distance dependence for the system N2-silica is depicted in Figure 4. This potential may be approximated by the WCA equation defined for the system solid atoms-fluid molecules. This has been done in our previous work.7 However, it is worth noting that such an approximation is appropriate only if the WCA equation is integrated over the whole solid volume. In the case when only surface atoms contribute to the solidfluid potential, the WCA scheme significantly deviates from the actual potential. 3.3. Heats of Adsorption. The calculated heat of N2 adsorption on nongraphitized carbon black Cabot BP 280 is presented in Figure 5. The main feature of this dependence is that the heat of adsorption is relatively high at low loading, which points out the existence of active sites on the surface having high interaction (26) Neimark, A. V.; Ravikovitch, P. I. Microporous Mesoporous Mater. 2001, 44-45, 697.
N2 Adsorption on Nonporous Carbon and Silica Surfaces
Figure 4. N2-nonporous silica potential corresponding to the NLDFT version adjusted to amorphous solids and determined by the least-squares fitting. The first section spans from zero to 1.5 hard-sphere diameters (i.e., 0.537 nm from the surface). The second section from 1.5dHS obeys the inverse fourth power of the distance.
Figure 5. Heat of adsorption for the system N2-nongraphitized carbon black Cabot BP 280 at 77 K. The solid line is calculated with the NLDFT developed for amorphous solids. Experimental points are taken from ref 28 for carbon black Spheron. Squares designate the isosteric heat of adsorption, and all other symbols correspond to data obtained calorimetrically.
potentials with nitrogen molecules. Further increases in coverage lead to a sharp decline in the heat of adsorption, which reflects the energetic heterogeneity of the carbon black surface. The ability to describe this fundamental feature without resorting to the patch-wise model is a substantial advantage of the approach. It shows that the energetic heterogeneity inherent to amorphous solids is embedded in the model, which by no means is possible using standard NLDFT. The small hump at a loading of 7 µmol/ m2 is due to the increase in intermolecular interactions. To show the pattern obtained experimentally, we also plot the data for heat of nitrogen adsorption measured calorimetrically for Spheron carbon black27 and from the Clausius-Clapeyron equation.28 There is apparent similarity, though one could not expect quantitative agreement because experimental data were obtained for different types of nongraphitized carbon black. For comparison, the heat of nitrogen adsorption on graphitized carbon black calculated with the standard NLDFT is presented in Figure 6. The main difference between those curves is that in the latter case the heat of adsorption increases with loading in the region (27) Beebe, R. A.; Biscoe, J.; Smith, W. R.; Wendell, C. B. J. Am. Chem. Soc. 1947, 69, 95.
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Figure 6. Heat of adsorption for the system N2-graphitized carbon black at 77 K. The solid line is calculated with the standard NLDFT. Experimental data for carbon black Graphon (symbols) are taken from papers by Beebe et al.27 and Joyner and Emmett.28
of monolayer coverage, which is due to the increase in intermolecular interactions. Once the first monolayer is completed (which approximately corresponds to 9 µmol/m2), the heat of adsorption sharply drops because the formation of the second molecular layer occurs at a larger distance from the surface where the solid-fluid potential is markedly smaller than that in the potential minimum. Calorimetric27 and isosteric28 experimental data for the heat of N2 adsorption on graphitized carbon black Graphon confirm this explanation. Some deviation is mainly due to different methods employed in the determination of the surface area of graphitized carbon black. In refs 27 and 28, the BET method was used, whereas in the case of the standard NLDFT the surface area is determined in the same framework29 and exceeds the BET surface area by 10-20%s. For this reason, the predicted heat of adsorption is smaller than that measured experimentally using the BET surface area. Besides, the degree of graphitization of the Graphon carbon black (surface area of 80.32 m2/g) is not very high. It is worthwhile to note that the heat of vaporization (dashed line in Figures 5 and 6) is calculated with the NLDFT and the determined value is 5.48 kJ/mol, which is very close to the experimental value of 5.58 kJ/mol. For the system nitrogen-nonporous silica LiChrospher Si1000, the predicted dependence of the heat of adsorption on the amount adsorbed is presented in Figure 7. The dependence is smoother than that in the case of the nongraphitized carbon black sample shown in Figure 6. This points to the high energetic heterogeneity of the silica surface, which is also reflected by the absence of the Henry’s law region even at relative pressures on the order of 10-6-10-5 (Figure 3b).
4. Conclusions The developed NLDFT adapted to amorphous solids allowed us to determine the solid-fluid potential for the system nitrogennongraphitized carbon black and nitrogen-nonporous silica. The pattern of the solid-fluid potential dependence on the distance from the surface suggests that there are two regions where different adsorption mechanisms are operating. Close to the surface, nitrogen adsorption seems to be localized and occurs on active sites distributed over the surface. Beyond the first monolayer (or slightly above), the adsorption becomes nonlocalized and occurs as molecular layering. In the latter case, the potential decays in (28) Joyner, L. G.; Emmett, P. H. J. Am. Chem. Soc. 1948, 70, 2353. (29) Ustinov, E. A.; Do, D. D. Part. Part. Syst. Charact. 2004, 21, 161.
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density and at a distance z from the surface, which can be written as follows:
δFj(z') ) δF(z) ω0(|z′ - z|) + ω1(|z′ - z|)Fj(z′) + ω2(|z′ - z|)[Fj(z′)]2 1 - Fj1(z′) - 2Fj2(z′)Fj(z′)
(A1)
This expression is used for the determination of the density profile at a given pressure (i.e., the chemical potential) with eqs 1 and 2. The functional derivative of Ψ[F(z)] is a bit more complex and takes the following form: Figure 7. Calculated heat of nitrogen adsorption on nonporous silica obtained with NLDFT for the adsorption data presented in ref 17.
accordance with the power law. This study shows that the absolute value of the potential exerted by the solid at a distance larger than 1.3-1.5 molecular diameters is markedly smaller than that calculated from the Lennard-Jones equation. In the case of silica, the analysis reliably shows that the solid-fluid potential decays as the inverse fourth power of the distance, which supports the conventional argument that the potential is exerted nearly exclusively by surface oxygen atoms. It was shown that NLDFT is an effective tool for the analysis of the heat of adsorption, which does not require the Clausius-Clapeyron equation. Predicted dependencies of the heat of adsorption in the case of nongraphitized carbon black and nonporous silica confirm the high energetic heterogeneity of these materials and the possibility of applying the developed version of NLDFT without resorting to the patchwise model. The analysis discussed for nonporous materials can be further extended to porous materials, which can provide a more reliable characterization of those materials and an analysis of the heat of adsorption. Acknowledgment. This work is supported by the Australian Research Council.
δFj(z) δFj(z′) δΨ[F(z)] ) f′ex[Fj(z)] + f′ex[Fj(z′)] + δF(z′) δF(z′) δF(z) δFj(z′′) δFj(z′′) zm F(z′′) f′′ex[Fj(z′′)] + 0 δF(z) δF(z′) δ2Fj(z′′) dz′′ (A2) f′ex[Fj(z′′)] δFj(z) δFj(z′)
}
The second functional derivative of the smoothed density appearing in the integrand is given by
δ2Fj(z′′) δFj(z′′) δFj(z′′) ) χ(z′′, z) + χ(z′′, z′) + δF(z) δF(z′) δF(z′) δF(z) δFj(z′′) δFj(z′′) (A3) τ(z′′) δF(z′) δF(z) where
χ(z′′, z) )
ω1(|z′′ - z|) + 2ω2(|z′′ - z|)Fj(z′′)
τ(z′′) )
Appendix On the basis of the Tarazona formulation for a smoothed density approximation,10,11 it is easy to obtain the functional derivative of the smoothed density at a distance z’ with respect to the local
{
∫
LA0604580
1 - Fj1(z′′) - 2Fj2(z′′)Fj(z′′) 2Fj2(z′′) 1 - Fj1(z′′) - 2Fj2(z′′) Fj(z′′)
(A4)
