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Langmuir 2008, 24, 6668-6675
Nitrogen Adsorption on Silica Surfaces of Nonporous and Mesoporous Materials Eugene A. Ustinov* Ioffe Physical Technical Institute, 26 Polytechnicheskaya, St. Petersburg 194021, Russia ReceiVed December 22, 2007. ReVised Manuscript ReceiVed March 10, 2008 We present an accurate comparative analysis of N2 adsorption at 77 K on nonporous silica and the pore wall surface of MCM-41 materials. The analysis shows that in the low-pressure region of N2 adsorption obeys a peculiar mechanism governed by short-ranged forces, which makes the surface curvature effect on the N2 adsorption in mesopores nearly negligible. We used this observation to define more exactly compared to the BET technique the specific surface area of the reference adsorption isotherm on nonporous silica basing on XRD data and linear sections of t-plots. Calculation of the capillary evaporation and condensation pressures seems to confirm our previous finding that the capillary condensation pressure corresponds to the equilibrium transition rather than spinodal condensation at least for pore sizes less than 7 nm. It allowed us to provide more reliable pore size distribution (PSD) analysis of mesoporous silica materials. For example, the PSDs of MCM-41 samples do not show artificial peaks in the micropore range that we obtained in our earlier publications.
1. Introduction Nitrogen adsorption at 77 K is widely used for determination of surface area, pore volume, and pore size distribution (PSD) of micro- and mesoporous materials. The PSD analysis is highly sensitive to the choice of model, the surface area of a reference system and other parameters, and the degree of accuracy of description of experimental adsorption isotherms. The latter often fails even in the case of application of contemporary approaches like nonlocal density functional theory (NLDFT) mainly due to high energetic heterogeneity of the pore wall surface, which was not accounted for in earlier NLDFT versions. Recently, density functional theory was adjusted to account for the heterogeneity of nongraphitized carbon black and silica in the framework of Tarazona smoothed density approximation1–5 and Rosenfeld fundamental measure theory.6,7 In the former case, we named the modified version NLDFT-AS (nonlocal density functional theory for amorphous solids). The Rosenfeld theory8 is known to be extended to mixtures. For this reason, Ravikovitch and Neimark6,7 recently applied the Rosenfeld theory to a kind of binary mixture, which is the solid (quenched component) and the adsorbed gas (annealed component). The authors coined their approach quenched solid density functional theory (QSDFT). In both cases the solid and the fluid are considered as a kind of binary mixture of disordered media. Such an approach allowed us to improve the descriptive ability of NLDFT even in terms of heat of adsorption.5 In its turn, the higher level of description of experimental data revealed some new and, to some extent, unexpected patterns. Thus, we found that the potential inside pores of MCM-41 silica samples nearly does not change with the pore diameter, being very close to the potential exerted by * Corresponding author. E-mail:
[email protected]. (1) Ustinov, E. A.; Do, D. D.; Jaroniec, M. Appl. Surf. Sci. 2005, 252, 548– 561. (2) Ustinov, E. A.; Do, D. D.; Fenelonov, V. B. Carbon 2006, 44, 653–663. (3) Ustinov, E. A.; Do, D. D.; Jaroniec, M. Langmuir 2006, 22, 6238–6244. (4) Ustinov, E. A.; Do, D. D.; Fenelonov, V. B. Appl. Surf. Sci. 2007, 253, 5610–5615. (5) Ustinov, E. A. Adsorption (Accepted). (6) Ravikovitch, P. I.; Neimark, A. V. Studies Surf. Sci. Catal. 2006, 9–16. (7) Ravikovitch, P. I.; Neimark, A. V. Langmuir 2006, 22, 11171–11179. (8) Rosenfeld, Y. Phys. ReV. Lett. 1989, 63, 980–983.
the nonporous silica.9 One more finding was that the equilibrium transition pressure determined with NLDFT excellently fits the dependence of experimental condensation pressure on the pore diameter at an appropriately chosen specific surface area of the reference nonporous silica.10,11 These observations are not consistent with commonly accepted representations, though there is some proof coming from analysis of the temperature effect on the chemical potential below and above the hysteresis critical temperature.12,13 Such a contradiction definitely requires further investigations. This work is dedicated to a comparative analysis of N2 adsorption on nonporous silica and a series of MCM-41 materials aiming at the definition of the true reference N2 adsorption isotherm. The latter is needed to predict the condensation/evaporation pressure dependence on the pore diameter and generate the correct set of local adsorption isotherms (kernel) for the pore size distribution analysis. We also intend to refine and simplify analytical description of nitrogen adsorption isotherm on nonporous reference silica. In our first attempt1 to apply NLDFT developed by Tarazona14,15 to the amorphous silica surface, we proceeded from the assumption that the adsorption potential asymptotically decays as an inverse third power of the distance from the surface. The fitting of the isotherm was not quite well, which forced us to introduce the cutoff distance for the solid-fluid potential as an additional fitting parameter. Hence, we determined the potential exerted by the surface as tabular data consisting of about 100 points.3–5 It has an advantage that we obtain undistorted potential function without adjusting to any equation. However, it makes difficulties in routine calculations. On the other hand, we have found that the potential exerted by silica surface asymptotically decays as the inverse forth-power distance,3–5 which gives rise to application a relevant equation or a simple combination of equations to fit the potential. In our opinion, highly short-range forces contribute to the (9) Ustinov, E. A.; Do, D. D. J. Colloid Interface Sci. 2006, 297, 480–488. (10) Ustinov, E. A.; Do, D. D.; Jaroniec, M J. Phys. Chem. B 2005, 109, 1947–1958. (11) Ustinov, E. A.; Do, D. D. Colloids Surf., A 2006, 272, 68–81. (12) Morishige, K.; Ito, M. J. Chem. Phys. 2002, 117, 8036–8041. (13) Morishige, K.; Nakamura, Y. Langmuir 2004, 20, 5403–4506. (14) Tarazona, P. Phys. ReV. A 1985, 31, 2672–2679. (15) Tarazona, P.; Marconi, U. M. B.; Evans, R. Mol. Phys. 1987, 60, 573– 595.
10.1021/la704011z CCC: $40.75 2008 American Chemical Society Published on Web 06/06/2008
Nitrogen Adsorption on Silica Surfaces
Langmuir, Vol. 24, No. 13, 2008 6669
Table 1. Molecular Parameters for N2-Silica System at 77.35 K εff/k (K)
σff (nm)
dHS (nm)
FSεsf(1)/k (K nm-2)
Rsf(1) (nm)
FSεsf(2)/k (K nm-2)
σsf(2) (nm)
zS (nm)
(s) dHS (nm)
δ (nm)
|∆| (%)
99.92
0.3510
0.3576
1451 1476 1546 1659 1797 1940
0.273 0.273 0.273 0.273 0.274 0.276
28358 27269 23820 18709 12111 5453
0.0425 0.0426 0.0426 0.0421 0.0403 0.0297
0.1049 0.0976 0.0942 0.0896 0.0843 0.0778
0.277 0.278 0.278 0.279 0.281 0.282
0.00 0.02 0.04 0.06 0.08 0.10
0.632 0.617 0.596 0.625 0.764 1.029
adsorption potential close to an amorphous surface like in the case of localized adsorption. Such a representation of the mechanism of adsorption at low coverage explains the absence of the surface curvature effect on the potential in cylindrical pores of MCM-41 samples described in our previous paper.9 We also present results of our new attempt to compare experimental data on condensation and evaporation pressure dependence on the pore diameter with the prediction based on a renewed NLDFT modification.
2. Model In the case of open systems, all versions of nonlocal density functional theory are based on minimization of the grand thermodynamic potential
Ω[F(r)] ) Fid[F(r)] + Fex[F(r)] +
∫ F(r′){u[F(r′)] +
V(r′) - µ}dr′ (1)
Here F(r) is the local density; Fid[F(r)] and Fex[F(r)] are the ideal and excess Helmholtz free energy, respectively; u[F(r′)] is the intermolecular potential; V(r) is the external potential; and µ is the chemical potential. The ideal term of the free energy is
Fid[F(r)] )
∫ F(r′)kT{ln [Λ3F(r′)] - 1}dr′
(2)
where k is the Boltzmann constant and Λ is the de Broglie wavelength. The intermolecular potential expressed in the mean field approximation is
u[F(r′)] )
1 2
∫ F(r′)φff(|r - r′|) dr′
(3)
where φff(r) is the attractive Weeks-Chandler-Andersen (WCA) potential:16
φff(r) )
{
r < rm
–εff
(4)
rm < r
4εff[(φff ⁄ r) - (φff ⁄ r) ], 12
6
Here εff and σff are the potential well depth and the fluid-fluid collision diameter, respectively; rm ) 21/6σff. In the case of Tarazona SDA,14,15 the excess Helmholtz free energy is defined as follows
Fex[F(r)] )
∫ Fe(r′)fex[Fje(r′)] dr′
(5)
where fex[Fje(r)] is the molecular excess Helmholtz free energy defined by the Carnahan-Starling equation17 for the hard sphere fluid
4η j - 3η j2 fex(Fje) ) kT , (1 - η j )2
π 3 η j ) dHS Fje 6
(6)
Here dHS is the hard sphere diameter. The excess Helmholtz free energy fex[Fje(r)]is a function of the Tarazona smoothed density Fje(r):
Fje(r) ) Fj0(r) + Fj1(r)Fje(r) + Fj2(r)(Fje(r))
2
Fji(r) )
∫ Fe(r′)ωi(|r - r′|) dr′,
i ) 0, 1, 2
}
(7)
In the original Tarazona prescription14 Fe(r) in the integrand is just the local fluid density F(r), and functions ωi(r) (i ) 0,1,2) are singlevalued functions of distance and defined to reproduce the hardsphere direct correlation function.15 In order to account for the
contribution of solid atoms to the increase of the smoothed density near the rough surface of the amorphous solid (more exactly, to the decrease of the excluded volume), we defined the effective density (in the 1D case, for simplicity) as follows2–5,11 (s) Fe(z) ) F(z) + F(s)(z)(dHS ⁄ dHS)3
(8)
(s) is the distance from the surface; dHS is the average hard (s) diameter for the solid atom, and F (z) is the solid atoms
Here z sphere density in the solid-fluid contact zone approximated by the error function:
F(s)(z) ) F(s) 0 erfc[z ⁄ (√2δ)]
(9)
F(s) 0
where is the density inside the solid (taken as 66.15 nm-3 accounting for both silicon and oxygen atoms) and δ is the standard deviation, which can be considered as a measure of surface roughness. At a large distance from the surface, F(s)(z) vanishes and the effective density reduces simply to the local fluid density F(z). However, in the vicinity of the surface, the smoothed density and, therefore, the excess Helmholtz free energy additionally increase due to the solid atoms, which indicates the appearance of the solid-fluid repulsive potential Vrep[z,Fe(z)]. Since the repulsive potential is already accounted for, the variable V(z) in the grand thermodynamic potential (eq 1) can be interpreted as the attractive term of the solid-fluid potential Vatt(z). A distinguishing feature of the present approach compared to our previous works devoted to amorphous surfaces is the substitution of the local density F(z) by the effective density Fe(z) in the integrand of eq 5 at the molecular excess Helmholtz free energy. The reason is as follows. Let solid atoms and fluid molecules be indistinguishable (hypothetically), with F(s)(z) ) FL[1 - H(z)], where FL is the liquid density at the saturation pressure p0, and H(z) is the Heaviside step function. Then, minimization of the thermodynamic potential Ω at p0 with respect to F(z) over the range 0 < z < ∞ will give F(z) ) F(s)(z) ) Fe(z) ) FL, which is expected result because no changes should occur across the contact zone of two identical media. In the general case, this means that the solid is not thermodynamically inert anymore. The condition of minimum Ω in the 1D case is
kT ln[Λ3F(z)] + Ψ[F(z), F(s)(z)] + 2u[F(z)] + V(z) - µ ) 0 (10) where
Ψ[F(z), F(s)(z)] )
δFex[F(z), F(s)(z)] δF(z)
(11)
The above functional derivative can be rewritten in an expanded form as Ψ[F(z), F(s)(z)] ) fex[Fe(z)] +
δFe(z′)
∫ F(z′)f ′ [F (z′)] δF(z) dz′ + δF (z′) ) ∫ F (z′)f ′ [F (z′)] δF(z) dz′ (11a)
(s) ⁄ dHS 3 (dHS
ex
e
e
(s)
ex
e
The last term in the right-hand side of the above equation does not (16) Weeks, J. D.; Chandler, D.; Andersen, H. C. J. Chem. Phys. 1971, 54, 5237–5247. (17) Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635–636.
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present in the original Tarazona NLDFT and accounts for the response of the solid on the change in the fluid density. This term becomes negligible far away from the surface because the solid atom density F(s)(z) tends to zero, while in the vicinity of the surface, this term is positive and tends to a constant value inside the solid. The latter decreases the fluid density near the surface preventing penetration the fluid molecules into the solid, which reflects the effect of the solid-fluid repulsive term of the potential. For the sake of convenience, the total (or fictitious) solid-fluid potential defined in the approach can be expressed in the framework of the original NLDFT as follows
Veff(z) ) Vrep(z) + Vatt(z)
(12)
2.1. Repulsive Solid-Fluid Potential in the NLDFT Extended to Amorphous Solids. Let Veff(z) be the potential corresponding to the standard Tarazona14 formulation that ensures exactly the same density distribution calculated with the new approach for a given chemical potential. Then the condition of minimum Ω is
kT ln[Λ3F(z)] + fex[Fj(z)] +
j (z′) dz′ + ∫ F(z′)f′ex[Fj(z′)] δF δF(z)
2u[F(z)] + Veff(z) - µ ) 0 (13)
Since the density distribution and the chemical potential µ are the same, the comparison of the above equation with eq 10 leads to the following expression for the repulsive potential:
Vrep[z, Fe(z)] ) fex[Fje(z)] - fex[Fje(z)] + δFje(z′) δFj(z′) Fe(z′)f′ex[Fje(z′)] - F(z′)fex[Fj(z′)] dz′ (14) δF(z′) δF(z′)
∫
{
}
The important feature of the repulsive term of the potential is that it depends on both the distance and the amount adsorbed, which allows describing the nonlinear section of the isotherm at low coverage in the case of energetic heterogeneity inherent to amorphous surfaces. It should be emphasized that the repulsive term expressed by eq 14 is just an effective value resulting from the comparison of the two NLDFT approaches at a specified bulk pressure. We present this term to illustrate that in the case of amorphous solid the gas-solid interaction involves the attractive component Vatt(z) and a contribution via the excess Helmholtz free energy as well. Note that the effective repulsive potential Vrep[z,Fe(z)] is mainly positive and becomes very large near the surface. However, it could take negative values around the minimum of the potential well because the functional derivative δ\overline F e(z′)/δF(z) in the integrand of eq 14 can be either positive or negative. Some illustration concerning the N2 - nonporous silica potential is presented in the Section 3.2. 2.2. Attractive Component of the Solid-Fluid Potential. As we mentioned above, the potential exerted by nonporous silica asymptotically decays as inverse forth-power distance from the surface, which means that the source of the potential is located on the surface and associates with the surface atoms (presumably oxygen atoms). Because the repulsive component of the gas-solid potential is already accounted for, the attractive component V(z) can be expressed using the WCA-like potential:16
φsf(1)(r) )
{
(1) r < rm
-εsf(1), 4εsf(1)[(σsf1 ⁄ r)12 - (σsf1 ⁄ r)6], ( )
( )
(1) rm rm
(16b)
Here FS is the surface density of oxygen atoms; zS is the position of surface oxygen layer exerted the potential relative to the geometrical surface. In the general case, zS is small but not zero. 2.2.1. Adsorption Potential in the Cylindrical Pore. The potential inside the cylindrical pore was calculated numerically by integration of the sum of pair potentials φsf(1)(r) + φsf(1)(r) (see eqs 15a, 15b) over the cylindrical surface for the solid-fluid molecular parameters determined from the analysis of the reference system. 2.3. Molecular Parameters for the System N2 - Silica in the Framework of NLDFT-AS. The set of molecular parameters (ff, σff, and dHS) is determined with the reference data on the saturation pressure, coexisting liquid density, and the surface tension at the given temperature. On the first step, we determined the hard-sphere diameter dHS and the group εffσff3 by matching the equation of state corresponding to the DFT version against the experimental data on the saturation pressure and the liquid density. Then, we split the group into ff and σff by matching calculated surface tension against its experimental value. Those parameters for nitrogen at 77.35 K are listed in Table 1. The solid-fluid molecular parameters were determined by the least-squares procedure at a specified value of the standard deviation δ (see eq 9). These parameters are also presented in Table 1 at different values of δ to show their sensitivity to the (s) standard deviation. The solid atom hard sphere diameter dHS was -3 determined using the solid atom density of 66.15 nm averaged over silicon and oxygen atoms. We will describe details of determination of the solid-fluid parameters in the next section. The mean error presented in the last column shows that the best fit corresponds to the standard deviation δ of 0.04 nm, at which the mean error is only 0.6%. Hereafter, we use the parameters calculated at δ of 0.04 nm. Nevertheless, the mean error minimum is not pronounced, which suggests that one should invoke other techniques to determine the parameter δ reliably. It should be noted that in the framework of the developed approach the surface atoms centers of an amorphous solid never lie exactly on the 2D plane. Therefore, even if the standard deviation δ is zero, the surface is not as smooth as that of a crystalline solid meaning that to some extent the surface roughness still exists. One can see from Table 1 that at δ ) 0.04 nm the solid-fluid collision diameter for the second component of the potential is 6.4 times smaller than that for the first component. It shows that forces acting near the surface are extremely short-range, which suggests that the adsorption mechanism at low coverage is similar to localized adsorption. There is a small distance of 0.094 nm between the silica surface and the surface exerting the potential. This can be caused by the fact that oxygen atoms slightly run out under the silica surface.
3. Results and Discussion 3.1. Low-Pressure N2 Adsorption on Nonporous Silica vs MCM-41 Materials. In this section, we consider nitrogen
Nitrogen Adsorption on Silica Surfaces
Langmuir, Vol. 24, No. 13, 2008 6671 Table 2. Characteristics of MCM-41 Materials Evaluated with N2 Adsorption Isotherms MCM-41 d100 (nm) D (nm) Sk (m2 g-1) (3.1) (3.9) (4.2) (4.6) (5.1) (5.5) (6.0)
Figure 1. Average density of nitrogen adsorbed in cylindrical pore of MCM-41 silica at the saturation pressure and 77.35 K calculated with the NLDFT-AS. The dashed line denotes the liquid density (28.84 mmol/ cm3).
adsorption on nonporous silica Lichrosper Si-100018 at 77 K in comparison with N2 adsorption isotherms on a series of MCM41 silica samples.19 The main problem is a reliable determination of the surface area of the reference nonporous silica because the surface area affects the solid-fluid molecular parameters and, hence, has significant impact on the predicted condensation/ evaporation pressure in the cylindrical pore at a specified pore diameter. The Brunauer-Emmett-Teller (BET) method gives only an approximate estimation of the specific surface area and should not be used for a delicate analysis. Thus, the BET surface area of the LiChrospher Si-1000 sample determined with nitrogen lies within the range from 18.1 to 26.2 m2/g depending on the pressure interval taken for linearization.18 For this reason, in the present work we rely on an alternative method, which can be expressed as follows. First, we have a relation between the pore diameter D, pore volume Vp, and the d100 interplanar spacing:20
(
D ) Cd100
F(s) 0 Vp 1 + F(s) 0 Vp
)
1⁄2
(17)
3 Here F(s) 0 is the density inside the solid (2.2 g/cm ); C ) (8/ 1/2 1/2 (3 π)) . It is assumed that the pore diameter D corresponds
to the equivalent cylindrical pore having the same volume as the actual pore with the hexagonal cross section. The difference in the surface area of the equivalent cylindrical and hexagonal pores of the same volume is of 5%. Then the actual pore wall surface of the sample is
Sp )
[(
)]
1 2√3 + Vp V d100 p F(s) 0
1⁄2
3.44 4.17 4.51 5.04 5.37 5.52 5.89
3.10 3.95 4.30 4.60 5.13 5.53 5.96
760.0 758.0 721.0 543.4 611.2 740.4 736.0
bk
SN*/bk (m2 g-1)
1.0068 1.0001 1.0381 1.3863 1.2546 1.0109 1.0000
748.7 753.7 726.1 543.7 600.8 745.6 753.7
time defining more precisely the reference surface area and the solid-fluid molecular parameters, finally condensed in Table 1. The dashed line in the figure denotes the liquid nitrogen density (28.84 mmol/cm3). One can see from the figure that the limiting value of average nitrogen density markedly exceeds the liquid density even for pores larger than 10 nm, and has the maximum of 31.54 mmol/cm3 at the pore diameter of 1.7 nm. We did not introduce into our calculations the so-called external surface area Sext. The reason is that the increase of the amount adsorbed in the completely filled pore is mainly due to the increase of the adsorbed phase density with the chemical potential. Therefore, the Sext determined with the t-plot analysis could be substantially overestimated or completely artificial. On the other hand, we found it convenient to use t-plots for determination of limiting amounts adsorbed am by their extending to p/p0 ) 1. Such a technique allows us to eliminate the effect of capillary condensation between particles close to the saturation pressure. The diameters and pore wall surfaces for the series of MCM-41 samples determined with the above-described procedure are listed in Table 2 along with the d100 interplanar spacing. It is noteworthy that the pore diameters determined with the refined procedure are quite close to those determined with the simplified method.20 The reason is that resorting to the external surface compensates the neglect of the compression of nitrogen in pores. Figure 2 presents some selected correlations from the series of MCM-41 (X.Y) materials, with the MCM-41 (6.0) being chosen as the reference sample. The number in parentheses is the pore diameter determined by Kruk et al. using geometrical considerations.19 Each point designates the amount adsorbed in a given sample corresponding to the amount adsorbed in the reference sample at the same pressure. All plots are straight lines passing though the origin up to nearly condensation pressure, which allows us to suggest
(18)
In order to determine the pore diameter D and the pore wall surface Sp, one needs the pore volume Vp. In the simplest case, the pore volume can be estimated as the ratio am/FL, where am is the amount adsorbed at the saturation pressure and FL is the liquid nitrogen density. However, the true average adsorbed phase density is markedly larger than the liquid phase density and substantially changes with the pore diameter, which is shown in Figure 1. We have determined this dependence iteratively, each (18) Jaroniec, M.; Kruk, M.; Olivier, J. P. Langmuir 1999, 15, 5410–5413. (19) Kruk, M.; Jaroniec, M.; Sayari, A. Langmuir 1997, 13, 6267–6273. (20) Kruk, M.; Jaroniec, M.; Sayari, A. J. Phys. Chem. B 1997, 101, 583–589.
Figure 2. Correlation of N2 amount adsorbed in MCM-41 (X.Y) samples with that for the MCM-41 (6.0). (X.Y): (O) 3.1, (Ω) 4.6, (×) 5.1. (b) Nonporous LiChrospher Si-1000.18
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Figure 3. Correlation of the pore wall surface area determined with the low-pressure adsorption isotherms and that determined with XRD data and limiting values of amount adsorbed. The pore diameter (nm): 1, 3.10; 2, 3.95; 3, 4.30; 4, 4.60; 5, 5.13; 6, 5.53; 7, 5.96.
Figure 4. Reconstructed adsorption isotherm of N2 on the reference silica at 77.3 K (see explanations in the text).
that at low loading nitrogen adsorbs on pore walls identically regardless the pore diameter. We already pointed out this peculiarity obtained by comparison of adsorption potentials determined directly from N2 adsorption isotherms on different MCM-41 samples.9 Below we present an additional proof. Let us assume that the above statement is true. In this case, the slope of each straight line shown in Figure 2 is the ratio bk ) SN/Sk, where k is the number of a given sample, and N is the number of the sample taken as a reference. Assume that k ) 1 for (3.1), 2 for (3.9), . . . , N ()7) for (6.0). The values of bk are shown in fifth column of Table 2. Then, Skbk gives the same surface area of the reference sample, i.e., MCM-41 (6.0). A reliable estimation of that area can be defined as the average:
SN/ ) N-1
N
∑ bkSk
(18a)
k)1
Finally, one can determine the surface areas again as SN*/bk (the last column of Table 2). It is essential that the pore wall surface is determined by two different ways. In the first case, we used XRD technique coupled with adsorption data at the saturation pressure (fifth column). In the second case, in order to determine the surface area we used adsorption data in the low-pressure region (the last column). The degree of mutual consistency of these sets of values of surface area is shown in Figure 3. The straight line presented in the figure has the slope of unity and passes through the origin, which again confirms the assumption that there is no any enhancement of the potential due to the surface curvature near the pore wall surface. Otherwise, the pore wall surface determined with the low-pressure region would be underestimated, especially for samples having smaller pores. The correlation plotted for the pair MCM-41 (6.0)-LiChrospher Si-1000 (Figure 2, filled circles) is also the straight line over the wide pressure range. The specific surface area was assumed to be of 19.55 m2/g.18 However, this correlation is not as perfect as previous ones. In particular, the straight line passes close to, but not exactly through, the origin. That is because there is some difference in the surface structure of MCM-41 materials and that of the nonporous silica. The question then is whether one could improve the reference adsorption isotherm to make prediction of the capillary condensation and evaporation pressure in cylindrical pores more reliable. In our opinion, it can be done as follows. In the relative pressure range from 10-4 to 0.1,
Figure 5. Attractive solid-fluid potential (s) at the silica surface. The effective potential at p/p0: (- - -) 0, (- · -) 1. The filled area denotes the solid.
the amount adsorbed per unit pore wall surface a′ref is a linear function of the reference amount adsorbed aref on the LiChrospher Si-1000 sample:
a′ref ) A + Baref
(19)
where A ) 0.2570 and B ) 0.8761. Then, one can combine the section obtained for MCM-41 (6.0) below p/p0 of 0.1 and the section calculated with eq 19 above p/p0 ) 0.1. The parameter B can be interpreted as the ratio of the LiChrospher Si-1000 surface area of 19.55 m2/g used in the paper of Jaroniec et al.18 to the true specific surface area of that sample of 22.31 m2/g. 3.2. Modeling of N2 Adsorption on the Reference Nonporous Silica. The corrected reference N2 adsorption isotherm is depicted in Figure 4. The solid line is calculated with the NLDFT-AS for the solid-fluid molecular parameters presented in Table 1 and determined using the least-squares fitting. The dependence of the potential on the distance from the surface is presented in Figure 5. As we mentioned above, analysis of experimental N2 adsorption isotherm shows that the source of the potential is mainly associated with the layer of oxygen atoms covering the silica surface. The position of that layer seems to be corresponded to minimum of the potential, which is located at a distance of about 0.1 nm from the surface. One can see from the figure that the potential sharply decreases inside a narrow
Nitrogen Adsorption on Silica Surfaces
Figure 6. N2 density distribution near the reference silica surface at 77.3 K and p/p0 ) 0.99. The dashed line denotes the density of liquid nitrogen (28.84 mmol/cm3).
zone near the potential minimum. From our viewpoint, this zone can be associated with a heterogeneous surface layer where adsorption occurs as localized adsorption governed by highly short-range forces. The effective potential depends on the amount adsorbed. The dashed line in the Figure shows the potential at zero coverage. The increase of the loading leads to blocking the most active surface centers, which makes the solid-fluid potential weaker. The upper limit of the potential at the saturation pressure is depicted by the dash-dotted line. Having found the solid-fluid potential, we calculated the density distribution of nitrogen near the surface at different relative pressures. As an example, Figure 6 shows the density distribution at p/p0 ) 0.99. The high narrow density peak can be attributed to the adsorption in the peculiar surface layer and nearly exactly coincides with the position of the potential minimum. Interestingly, at a larger distance up to 1.5 nm the density of nitrogen only slightly deviates from the liquid density. Note that classical density functional theory as applied to a smooth crystalline surface always produces a number of large density peaks near the surface, which manifests itself as a pronounced layering on theoretical adsorption isotherms. For this reason, the conventional NLDFT-based approach does not allow us to describe low-temperature N2 and Ar adsorption on amorphous surfaces quantitatively. 3.3. Adsorption Potential in Cylindrical Pores. The enhancement of the potential in narrow cylindrical pores could significantly affect the amount adsorbed per unit surface area at the same relative pressure and the pressure of condensation/ evaporation. The enhancement substantially depends on whether all solid atoms or only those located on the surface contribute to the potential. In the former case, the potential increases with the decrease of the pore diameter (in absolute terms), especially close to the pore surface.9 In the latter case, the effect of potential enhancement is much weaker, especially near the pore wall. Figure 7 shows the potential calculated inside pores of different diameters for the parameters listed in Table 1. As seen from Figure 7, the potential enhancement is significant for the pore having a diameter of 1 nm. However, for all mesopores (D > 2 nm) the effect of surface curvature is very small. It is especially interesting to note here that the potential remains nearly the same around its minimum regardless of the pore diameter. This is because the contribution of short-range component to the potential is dominant near the wall, so the potential enhancement appears only in micropores. This confirms the idea that nitrogen
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Figure 7. Potential inside cylindrical pores having diameter (from the bottom to the top): 1, 2, 3, and 6 nm. The dashed line corresponds to the flat reference surface.
Figure 8. Relations for the capillary condensation pressure (O) and evaporation pressure (b) versus the pore diameter for nitrogen adsorption at 77 K in a series of MCM-41 silica materials.21 The solid line is the prediction for the equilibrium transition pressure. The dashed line is for the spinodal condensation pressure.
adsorption occurs identically on mesopore walls of siliceous materials and on the nonporous reference silica. 3.4. Modeling of N2 Adsorption in Cylindrical Pores. Having determined the potential in the cylindrical pore, one can calculate the adsorption and desorption isotherms and the pressure of condensation and evaporation. Experimental data21 on the condensation/evaporation pressure dependence on the pore diameter are shown in Figure 8. As seen from Figure 8, the theoretical equilibrium transition pressure excellently fits the experimental data on the capillary condensation of nitrogen in cylindrical pores. We already obtained this result earlier,10,11 but that time we used the specific surface area of the reference silica LiChrospher Si-1000 as an adjusting parameter in the framework of NLDFT-AS. We found that there was the only possibility to match the theory with experimental data by taking the reference surface area of 24 m2/g. In this special case, the predicted capillary evaporation pressure coincided with experimental capillary condensation pressure for all diameters up to 7 nm. It allowed us to conclude that the (21) Kruk, M.; Jaroniec, M. Chem. Mater. 2001, 13, 3169–3183.
6674 Langmuir, Vol. 24, No. 13, 2008
UstinoV
Figure 9. Nitrogen adsorption isotherms on MCM-41 silica samples at 77.3K in the linear (a) and logarithmic (b) scale. The pore diameter (nm): (O) 4.6, (0) 6.0.
adsorption branch of the hysteresis loop is equilibrium. In the present work, we have come again to the same conclusion, but now without any adjustment. We reliably determined the surface area of the reference silica by comparison N2 adsorption isotherm with those measured in MCM-41 silica materials. In doing so, we have found that the reference surface area is 22.31 m2/g. This value is less than 24 m2/g obtained in our previous work, but there is a reason for getting the same pattern presented in Figure 8. In our earlier work10,11 we proceeded from the assumption that all solid atoms contribute to the solid-fluid potential. Having determined the potential exerted by the nonporous reference silica, we transformed this potential to that in cylindrical pore using a general formula.10,11 As a result, the effect of surface curvature on the potential was relatively high, which has led to underestimation of the condensation and evaporation pressure. A small increase of the reference surface area eliminated that deviation. In our present work, we assume that predominantly surface atoms contribute to the solid-fluid potential, which makes the potential enhancement due to the surface curvature much weaker compared to the previous case. For this reason, the condensation and evaporation pressures remain the same at a smaller value of the reference surface area of 22.31 m2/g. An additional argument corroborates the statement of equilibrium transition at the capillary condensation pressure. As seen from Figure 8, the pressure of equilibrium transition determined theoretically excellently fits experimental data in the region of adsorption/desorption reversibility, which is certainly equilibrium. Such a coincidence could hardly be accidental, which confirms the correctness of the above statement. The quantitative description of the adsorption isotherms is possible only accounting for the pore size distribution. However, it is of interest to compare the experimental adsorption isotherm with the single local isotherm calculated for a given pore diameter to ascertain whether there is a similarity between those isotherms. If this is the case, one can be quite confident in the correctness of the model. It is known that a number of the Heaviside step functions like in the Horvath-Kawazoe theory22 can fit any adsorption isotherm, but it does not mean that the theory adequately reproduces the physical mechanism. Figure 9 presents a comparison of N2 adsorption isotherms calculated with the NLDFT-AS with experimental data for two samples of MCM-41 materials. In our case, local isotherms calculated with NLDFT-AS match experimental data fairly well. There is some discrepancy around (22) Horvath, G.; Kawazoe, K. J. Chem. Eng. Jpn. 1983, 16, 470–475.
the condensation pressure, but this is just a sign that the pore size distribution should be imposed, which always reduces the condensation step. There are two points to be emphasized. First, both experimental and predicted amounts adsorbed similarly increase with pressure in the completely filled cylindrical pore, despite our not accounting for the external surface in our approach. This means that the external surface is an artifact caused by the neglect of the adsorbed phase compression with the increase of bulk pressure (chemical potential). Second, the theory excellently fits adsorption isotherms in the low-pressure region (Figure 9b), which again confirms that there is no potential enhancement in the adsorbed layer adjacent to the pore wall. 3.5. Pore Size Distribution Analysis of MCM-41 Materials. To provide the PSD analysis we have generated 176 local adsorption isotherms for the pore diameters from 0.26 to 16 nm. We used a regularization procedure23 based on Tikhonov regularization method.24 The approximating function for the amount adsorbed is given by
y(p ⁄ p0) )
∫ f(D)F(p ⁄ p0, D) dD
(20)
Here f(D) is the distribution function, F(p/p0,D) is the average nitrogen density adsorbed in the pore of diameter D at the relative pressure p/p0. The functional to be minimized is
R)
∑ (yi ⁄ ai - 1)2 + Rφ
(21)
i
where ai is Ith value of experimental amount adsorbed, R is the regularization parameter, and φ is a regularization function taken as23
φ)
∫ [β(df ⁄ dD)2 - (-f ln f) - λf] dD
(22)
The right-hand side of the above equation can be considered as an analogue of grand thermodynamic potential. Usually only the first term in the integrand is used in the stabilizer φ. The second term is the information entropy. The third term allows us to suppress any artificial peaks, with λ being an analogue of the chemical potential. The presence of the entropy term ensures positive values of f, which is quite convenient feature. The regularization procedure makes the PSD function smoother, which (23) Ustinov, E. A.; Do, D. D.; Fenelonov, V. B. Appl. Surf. Sci. 2007, 253, 5610–5615. (24) Tikhonov, A. N.; Arsenin, V. Y. Solutions of ill-posed problems. Wiley; New York, 1977.
Nitrogen Adsorption on Silica Surfaces
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Jaroniec-Sayari (KJS) method.19 The only condition is the quantitative approximation of the capillary condensation pressure-diameter dependence (as far as we use the adsorption branch of isotherm for characterization). However, the approach presented here seems to be the first one that did not require any adjustment.
4. Conclusion
Figure 10. Pore size distribution of MCM-41 materials determined with N2 adsorption at 77.3 K. The regularization procedure was fulfilled at the following parameters: R ) 0.001, β ) 0.01, and λ ) -10.
is equivalent to the decrease of ‘information’ of the pore structure, i.e., to the increase of the information entropy. For the above reasons, eq 22 is the most general form of the regularization function. More details can be found elsewhere.23 The PSD functions for MCM-41 materials are presented in Figure 10. The PSD functions shown in Figure 10 are quite similar to those presented in our previous work.11 The only distinction is that the PSDs nearly do not reveal an artificial population of micropores, which we obtained previously. The reason is that the present approach allows quantitative fitting of the experimental adsorption isotherm in the low-pressure region even by the single local isotherm. As was mentioned above, in our earlier work10,11 we used an overestimated solid-fluid potential that was compensated by the overestimated reference surface area used as an adjustment parameter. Our observation is that the PSD is not sensitive to the model. One can obtain nearly the same results using different versions of NLDFT, an improved Broekhoff-de Boer theory,10 a generalized thermodynamic approach,11 and the Kruk-
We have accomplished an accurate comparative analysis of nitrogen adsorption isotherms on nonporous reference silica and MCM-41 silica materials, having the pore diameter from 3 to 6 nm. It allowed us to determine the specific surface area of the reference silica and the pore wall surfaces of MCM-41 samples with high degree of accuracy. We took this opportunity to determine the potential exerted by silica surface without any adjustment. The analysis has shown that highly short-range solid-fluid interactions contribute to the potential near the surface, which results in negligible effect of the pore diameter on the adsorption on pore walls. It suggests that the adsorption on the rough surface of an amorphous solid obeys a special mechanism similar to localized adsorption. The advantage of the approach is that the potential is defined analytically providing excellent fitting of the reference N2 adsorption isotherm (with mean error less than 1%). We obtained convincing evidence that the capillary condensation step corresponds to the equilibrium transition rather than to the spinodal point. Thus, the capillary condensation pressurediameter dependence coincides with the predicted dependence for the equilibrium transition pressure over the pore diameter range from 2 to at least 7 nm, with no adjusting parameters being used in the analysis. This observation is known to contradict to the classical representation of adsorption in open-ended cylindrical pores. This means that further efforts are needed to clarify the mechanism of adsorption in mesopores. Meanwhile, one can use the NLDFT-AS version for a reliable pore size distribution analysis of mesoporous silica materials. Acknowledgment. Support from the Russian Foundation for Basic Research is gratefully acknowledged (Project No. 06-03-32268-a). LA704011Z
