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Modeling of Adsorption of Gases on Graphite Surfaces Accounting for the Solid-Fluid Nonadditivity Correction Eugene A. Ustinov* and Julia Kukushkina Ioffe Physical Technical Institute, 26 Polytechnicheskaya, St. Petersburg 194021, Russia
William R. Betz Supelco, Inc., Supelco Park, Bellefonte, Pennsylvania 16823, United States Received September 25, 2010 A series of graphitized carbon blacks have been studied using argon and nitrogen adsorption at their boiling points. Analysis of adsorption isotherms was performed with nonlocal density functional theory (NLDFT) accounting for the Axilrod-Teller equation to describe the effect of nonadditivity of the gas-solid interaction. In our previous study [Ustinov, E. A. J. Chem. Phys. 2010, 132, 194703] we have shown that the nonadditivity effect decreases the attractive component of Ar-Ar interaction in the first molecular layer adjacent to the graphite surface by about 23%. This is a source of a large error (up to 40%) when a standard NLDFT is applied to fitting the low-temperature Ar adsorption isotherm on a graphitized carbon black. A new approach that incorporates the Axilrod-Teller equation into the standard NLDFT diminishes the relative error from 40 to 4%, which suggests that the nonadditivity correction should not be ignored in most adsorption systems including crystalline and amorphous solids. The present study is an extension of our approach to N2 adsorption isotherms at 77.3 K on graphitized carbon blacks. We show that the approach allows to reliably determine the gas-solid molecular parameters, the gas-solid nonadditivity coefficient, the Henry coefficient, and the specific surface area. The surface areas of different carbon blacks determined with the N2 at 77.35 K and Ar at 87.29 K are very close to each other, though in the former case the values proved to be slightly smaller presumably due to nonspherical shape of the nitrogen molecule. A comparison with the Brunauer, Emmett, and Teller method is provided.
1. Introduction Analysis of data on adsorption of gases on the graphitized carbon black surface still retains its significance. The reason is experimental reproducibility of such systems and comparatively small number of free parameters. The latter provides a severe test for theories used for the analysis, which is crucial in more complex cases of application of the theories. An example is the pore size distribution analysis of nanoporous materials. Being an ill-posed task, it requires a very accurate experimental measurements and a well-grounded model including its parameters. Up to now, molecular theories and numerical calculations are known to fail to agree with experimental adsorption isotherms of simple gases on graphitized carbon blacks as far as the additivity of pair interactions is concerned. It is now well established that due to the nonadditivity of pair potentials the internal energy of gases adsorbed in the first molecular layer on the graphitized carbon black surface decreases by about 20%.1-4 A simplified approach based on the nonlocal density functional theory (NLDFT) was first proposed to account for the gas-solid nonadditivity correction.5 An alternative approach has been developed by Do et al. using Monte Carlo simulations and an idea of polarization of adsorbed molecules by the electric field exerted by the graphite surface due to quadropoles of carbon atoms.6,7 A more rigorous *Corresponding author. E-mail:
[email protected]. (1) (2) (3) (4) (5) (6) (7) (8)
Freeman, M. P. J. Phys. Chem. 1958, 62, 729. Sams, J. R.; Constabaris, G.; Halsey, G. D. J. Chem. Phys. 1962, 66, 2154. Sams, J. R. J. Chem. Phys. 1965, 43, 2243. Rauber, S.; Klein, J. R.; Cole, M. W. Phys. Rev. B 1983, 27, 1314. Ustinov, E. A.; Do, D. D. Part. Part. Syst. Charact. 2004, 21, 161. Do, D. D.; Do, H. D.; Kaneko, K. Langmuir 2004, 20, 7623. Do, D. D.; Do, H. D. Colloids Surf., A 2007, 300, 50. Ustinov, E. A. J. Chem. Phys. 2010, 132, 194703.
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approach was published recently,8 which incorporated the AxilrodTeller equation9 into the classical NLDFT. As applied to the system Ar-graphitized carbon black at 87.29 K the nonadditivity correction substantially improves fitting of the adsorption isotherm: the relative error was shown to drop from 38 to 4%.8 The main nonadditivity effect is due to the gas-solid interaction, namely, due to the sum of triplet potentials of one carbon atom and two gas molecules adsorbed in the first molecular layer. In homogeneous systems the role of triplets comprising three molecules is relatively small. Thus, the contribution of such triplets to the decrease of internal energy of the uniform liquid does not exceed 4-5%,1-4,8 contrary to 23% in the first molecular layer of argon at a bulk pressure close to the saturation pressure at its boiling point. It suggests that the contribution of the triplets comprising three gas molecules can be approximately accounted for with the usual effective fluidfluid molecular parameters without rather involved explicit calculations. In the present study we shall consider such a simplification accounting for the only contribution of the nonadditivity correction associated with the gas-solid interaction. An advantage of such simplification is that the approach developed in ref 8 can be applied to any gas adsorbed on the graphitized carbon black surface using the gas-solid nonadditivity coefficient as an adjusting parameter. We have taken this opportunity to compare the gas-solid molecular parameters derived from nitrogen and argon adsorption isotherms on the same carbon black samples. A quite accurate fitting of adsorption isotherms with the developed technique based on the incorporation of the Axilrod-Teller equation into the classical NLDFT provides a unique possibility to derive a set of reliable parameters including the surface area. The latter is especially important because (9) Axilrod, B. M.; Teller, E. J. Chem. Phys. 1943, 11, 299.
Published on Web 11/30/2010
DOI: 10.1021/la103852t
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the gas-solid molecular parameters determined from the adsorption isotherm by the least-squares technique are strongly correlated with the surface area, which means that the surface area must be evaluated in the framework of the same molecular theory without invoking any empirical methods like the Brunauer-Emmett-Teller (BET) approach. In order to demonstrate the new approach, we consider nitrogen at 77.3 K and argon adsorption at 87.3 K on a series of high-quality graphitized carbon blacks produced by Supelco Inc.
2. Theory 2.1. Classical Attractive Mean-Field NLDFT. We rely on the Tarazona smoothed density approximation10 for the hard sphere fluid and the Carnahan-Starling equation11 for the hard sphere Helmholtz free energy (repulsive component). The attractive component of the Helmholtz free energy is accounted for with the perturbation Weeks-Chandler-Andersen (WCA)12 potential. It should be noted that in most cases the parameters of the WCA term usually associated with the Lennard-Jones potential well depth ε0 ff and the collision diameter σ0 ff are temperature dependent, which implies that the perturbation potential represents the attractive component of the Helmholtz free energy rather than the internal energy. Therefore, ε0 ff and σ 0 ff can be considered as effective parameters in a density smoothing procedure (i.e., the WCA scheme) for the attractive term of the Helmholtz free energy. Given the true collision diameter σff, the true pair potential εff can be determined with the Percus test particle method13 in the framework of the standard NLDFT. It is turned out that the true values of σff and εff/k obtained with the NLDFT are very close to those used in numerical simulations, while the group composed of effective parameters σ 0 ff3ε 0 ff/k is 11% less than that composed of true parameters σff3εff/k.8 The gas-solid potential is modeled as a sum of 10-4 potentials exerted by the stack of parallel graphene sheets separated by a distance Δ = 0.335 nm. The density distribution along the normal to the graphitized carbon black surface at each bulk pressure was determined by minimization of the thermodynamic functional Ω. The technique from mathematical viewpoint has been exhaustively presented in the literature on NLDFT. For details we refer the reader to our previous paper.8 2.2. Nonadditivity Correction. The nonadditivity effect is revealed at a high density of a gas as a secondary appearance of dispersion forces. In spite of random character of instantaneous dipoles of gas molecules, a total electric field produced by those dipoles in each moment is spatially correlated. For this reason, induced dipoles of neighboring molecules are congruent, which causes an additional interaction between the molecules depending on theirs mutual locations. Any pair of induced dipoles interacts with the instantaneous dipole and, as a consequence, with each other. The additional potential is attributed to a triplet of molecules and obeys the Axilrod-Teller equation:9 u3BD ði, j, kÞ ¼ ν
1 þ 3 cos θi cos θj cos θk rij 3 rjk 3 rki 3
ð1Þ
where θi, θj, and θk are the angles of the triangle formed by the molecules i, j, and k; rij is the distance between the molecules i and j; ν is the nonadditive coefficient. The Axilrod-Teller equation is also valid in the case of unlike molecules or atoms. Thus, it was shown (ref 8) that the main contribution to the internal energy of (10) Tarazona, P. Phys. Rev. A 1985, 31, 2672. (11) Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635. (12) Weeks, J. D.; Chandler, D.; Andersen, H. C. J. Chem. Phys. 1971, 54, 5237. (13) Percus, J. K. In The Equilibrium Theory of Classical Fluids; Frisch, H. L., Lebowitz, A. L., Eds.; Benjamin: New York, 1964; p 113.
210 DOI: 10.1021/la103852t
the system Ar-graphitized carbon black comes from C-Ar-Ar triplets providing the decrease in absolute term of the potential energy of a molecule situated at the first molecular layer by 23%. The role of nonadditivity of pair potentials in liquids is markedly lower and usually does not exceed 4-5%. This means that the appropriate choice of intermolecular parameters allows one to approximately account for the nonadditivity of similar gas molecules interaction in the framework of a standard molecular theory. In other words, all features of adsorption systems associated with the nonadditivity could be reduced to the nonadditivity of gas-solid interactions. This substantially simplifies the task. 2.3. Nonadditivity of the Gas-Solid Interaction. The three-body potential of a gas molecule near the graphitized carbon black surface can be written as follows: XZ Z Fðz1 Þδðz2 - kΔÞξðz1 - z, z2 - zÞ dz1 dz2 uAT 211 ðzÞ ¼ ν211 Fs Δ k
ð2Þ Here F(z1) is the adsorbed gas density at a distance z1 from the surface; δ(z) is the Dirac delta function; Fs and Δ are the number density of the graphite (114 nm-3) and the interlayer distance (0.335 nm) in the graphite lattice, respectively; and k = 0, 1, ... is the number of a graphite layer. Numbers 1 and 2 in the subscript at the triplet potential and the nonadditivity coefficient ν are for the gas molecule and carbon atom, respectively. The weight function ξ(z1,z2) is obtained by integration of the Axilrod-Teller potential (eq 1) assuming that the adsorbed phase is uniform in the XY plane parallel to the carbon black surface (see ref 8 for details). The total three-body potential of the system gas-solid is given by Z AT Fðz1 ÞuAT ð3Þ U211 ½FðzÞ ¼ 211 ðz1 Þ dz1 This potential is an additional term of the grand thermodynamic potential Ω. Minimization of Ω at a given bulk pressure and temperature leads to the density distribution along the normal to the surface. Minimum Ω corresponds to zero value of the functional derivative of Ω with respect to density F(z). This condition can be written as Z δFðz0 Þ 0 dz þ 2fatt ½FðzÞ kT ln½FðzÞ þ fex ½FðzÞ þ FðzÞf 0 ex ½Fðz0 Þ δFðzÞ þ
AT δU211 ½FðzÞ þ VðzÞ - μ ¼ 0 δFðzÞ
ð4Þ
Here fex[F(z)] and fatt[F(z)] are the repulsive and the attractive component of the Helmholtz free energy per molecule, respectively; V(z) is the external potential exerted by the graphitized carbon black; μ is the chemical potential; and F(z) is the smoothed density defined by the Tarazona prescription.10 It should be noted that we did not use the cutoff distance in calculation of the attractive term of the Helmholtz free energy fatt[F(z)]. Instead, a tail correction has been used analytically for regions where the density distribution is uniform. The solid-fluid potential V(z) is defined as a sum of potentials exerted by the stack of graphenes: " # X 2 σ sf 10 σ sf 4 2 ð5Þ VðzÞ ¼ 2πFs Δεsf σ sf 5 z þ kΔ z þ kΔ k Here εsf and σsf are the gas-solid potential well depth and the gas-solid collision diameter, respectively. This completes formulation of the model. Once the density distribution is determined at Langmuir 2011, 27(1), 209–214
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a specified bulk pressure and temperature by solving eq 4, the amount adsorbed can be calculated by integration of the density with respect to distance z.
3. Results and Discussion In the present study we analyzed four samples of graphitized carbon black labeled Carbopack F, C, Y, and B supplied by Supelco (Bellefonte, PA). The BET surface are of the samples varies from 6 to 100 m2/g. Nitrogen and argon adsorption isotherms have been measured with Micromeritics analyzer ASAP 2020 at 77.35 and 87.29 K, respectively. For comparison, we have analyzed N2 and Ar adsorption isotherms on Carbopack F sample published in the literature.14,15 3.1. Method of Analysis. The temperature of the measuring cell is the boiling point of N2 or Ar at the bulk pressure, which is never constant. For this reason, the temperature can slightly change during the measurement and differ from experiment to experiment up to several degrees. This feature could be a source of errors, especially in case of comparison of surface properties of different samples, i.e., Henry coefficients, heat of adsorption, gas-solid molecular parameters, etc. To avoid this drawback, we normalized the temperature to the true boiling point corresponding to 1 atm. Output data enclose the reduced and absolute pressure for each point. We took this advantage to restore the current temperature using bulk properties modeled by the same density functional theory that was applied for modeling of adsorption isotherms. Once the temperature was determined, we recalculated the reduced pressure p/p0 for the same amount adsorbed and the boiling point using the following expression derived from the Clausius-Clapeyron equation: lnðp=p0 Þ = lnðp0 =p0 0 Þ -
q-λ ð1 - Tb =TÞ RTb
ð6Þ
where p0 /p0 0 is the measured reduced pressure at a temperature T; q and λ are the isosteric heat of adsorption and the heat of evaporation, respectively; R is the universal gas constant; and Tb is the boiling point temperature. The isosteric heat of adsorption was calculated with the NLDFT for each point of the adsorption isotherm. In principle, each adsorption isotherm can be treated separately to determine molecular and structural parameters. However, a more effective way is to analyze all adsorption isotherms together meaning that some adjusting parameters are common for all systems. Thus, the gas-solid collision diameter σsf should be the same for C-N2 and C-Ar pairs regardless of the carbon black sample. The same statement is valid for the nonadditivity coefficient ν211 for triplets C-N2-N2 and C-Ar-Ar. In the present study the carbon black surface area is also considered as an adjusting parameter. The reason is that the mean error of fitting of an adsorption isotherm is highly sensitive to the choice of the surface area. This means that the latter can be reliably evaluated in the framework of the same molecular theory (NLDFT in this work) without invoking any theoretically ungrounded empirical approaches such as the BET method. It seems logical to assume that for each sample of carbon black the surface area must be the same, no matter whether nitrogen or argon is used for its determination. This reduces the total number of adjusting parameters, making them more feasible. However, on the basis of our previous investigations, we revealed that the surface area evaluated with nitrogen adsorption isotherm is always slightly (14) Kruk, M.; Li, Z.; Jaroniec, M.; Betz, W. Langmuir 1999, 15, 1435. (15) Gardner, L.; Kruk, M.; Jaroniec, M. J. Phys. Chem. B 2001, 105, 12516.
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smaller than that determined with the argon isotherm. This interesting feature has an explanation, which will be discussed below. At the present stage we introduced a parameter which is the ratio of surface areas determined with nitrogen and argon adsorption isotherm using NLDFT for the same sample of carbon black. This parameter does not depend on the surface, which also allows us to reduce total number of adjusting parameters. At a high surface area of the samples the capillary condensation occurs at pressures close to the saturation pressure. This is because the size of carbon black particles decreases with the increase of the surface area, resulting in the appearance of a system of interconnected mesopores between the particles. Since in the present study we did not account for the capillary condensation/evaporation in the model, we omitted the region of high pressures depending on the sample. Thus, for Carbopack F and C (S
