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Computer Simulation of Argon Adsorption on Graphite Surface from Subcritical to Supercritical Conditions: The Behavior of Differential and Integral Molar Enthalpies of Adsorption Chunyan Fan,†,‡ D. D. Do,*,† Zili Li,‡ and D. Nicholson†,§ †

School of Chemical Engineering, University of Queensland, St. Lucia, Qld 4072, Australia, ‡Department of Storage and Transportation Engineering, China University of Petroleum, Qingdao, China, and §Theory and Simulation Group, Department of Chemistry, Imperial College, London SW72AY, U.K. Received April 18, 2010. Revised Manuscript Received August 31, 2010

We investigate in detail the computer simulation of argon adsorption on a graphite surface over a very wide range of temperature, from below the triple point to well above the critical point. Adsorption over such a wide temperature range has not been reported previously in the form of adsorption isotherms and enthalpy change during adsorption. The adsorption isotherms can be classified broadly into four categories: below the triple point, the isotherms show stepwise character (a strict layering mechanism) with 2D condensation; type II (according to the IUPAC classification) is followed by isotherms at temperatures above the triple point and below the critical point and a sharp spike is seen for isotherms in the neighborhood of the critical point; and finally the typical behavior of a maximum is observed for isotherms above the critical point. For the isosteric heat, the heat curve (plotted against loading) remains finite for subcritical conditions but is infinite (singularity) at the maximum in excess loading for supercritical adsorption. For the latter case, a better representation of the energy change is the use of the integral molecular enthalpy because this does not exhibit a singularity as in the case of isosteric heat. We compare the differential and integral molecular enthalpies for the subcritical and supercritical adsorptions.

1. Introduction The mechanism of the adsorption of gases on homogeneous surfaces such as graphite is best understood with computer simulation because detailed configurations of the adsorbed phase can be analyzed. The aim of this article is to use GCMC simulation to survey the argon-graphite adsorption system over a wide range of temperatures, showing how isotherms and adsorption heats vary from sub- to supercritical domains. Molecular models require a potential energy model for the interaction between adsorbate molecules and an adsorbate molecule and the atoms in the adsorbent surface. A description of the interaction between adsorbate molecules that gives the correct experimental coexistence curve can be achieved (at least for simple molecules) with an effective potential such as the Lennard-Jones model. Although there are discussions in the literature,1-4 no exact theory exists for the interaction between an adsorbate and graphitic adsorbents. The simple effective potential models are not adequate to account for many-body effects, such as repulsions due to induced polarizations caused by surface quadrupoles and three-body dispersion *Corresponding author. Phone: þ61-7-3365-4154. Fax: þ61-7-3365-2789. E-mail: [email protected]. (1) Rauber, S.; Klein, J. R.; Cole, M. W. Substrate screening of the interaction between adsorbed atoms and molecules - Ne, Ar, Kr, Xe, and CH4 on graphite. Phys. Rev. B 1983, 27, 1314–1320. (2) Nicholson, D. First order dispersion energy in the interaction of small molecules with graphite. Surf. Sci. 1984, 146, 480–500. (3) Nicholson, D.; Roger, F.; Cracknell, R. F.; Parsonage, N. G. Evaluation of a model potential function for Ar graphite interaction using computer simulation. Mol. Simul. 1990, 5, 307–314. (4) Nicholson, D. Fundamentals of Equilibria in Adsorption. In Third International Conference on Fundamentals of Adsorption; Mersmann, A. B., Scholl, S. E., Eds. United Engineering Trustees: New York, 1991; pp 3-22. (5) Sinanoglu, O.; Pitzer, K. S. Interactions between molecules adsorbed on a surface. J. Chem. Phys. 1960, 32, 1279–1288. (6) McLachlan, A. D. Van der Waals forces between an atom and a surface. Mol. Phys. 1964, 7, 381–388.

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interactions, mediated through the adsorbent,5-11 that contribute to the solid-fluid (SF) interactions and are particularly manifest at low temperatures when the kinetic energy is a relatively small part of the total energy. Here we model this with an empirical equation10 to study argon adsorption on graphite over a wide range of temperature.

2. Theory To simulate the adsorption isotherms, we use the GCMC simulation;12-14 here we summarize the essential points used in this work. To model the graphite surface, we use a slit pore with a width large enough that the potential energies of interaction exerted by the two walls are not affected by each other. If we use a simulation box with the graphite surface at one boundary and a hard wall at the opposing boundary, then there will be a spurious effect in the concentration profile near the hard wall boundary, especially when we deal with supercritical adsorption. We model the argon-argon interaction with the Lennard-Jones 12-6 equation15 and the (7) Ross, S.; Olivier, J. P. On Physical Adsorption; Interscience: New York, 1964. (8) Everett, D. H. Interactions between adsorbed molecules. Discuss. Faraday Soc. 1965, 40, 177–187. (9) De Boer, J. H. The Dynamical Character of Adsorption. Clarendon: Oxford, U. K., 1968. (10) Do, D. D.; Do, H. D.; Kaneko, K. Effect of surface-perturbed intermolecular interaction on adsorption of simple gases on a graphitized carbon surface. Langmuir 2004, 20, 7623–7629. (11) Do, D. D.; Do, H. D. Effects of potential models on the adsorption of carbon dioxide on graphitized thermal carbon black: GCMC computer simulations. Colloids Surf., A 2006, 277, 239–248. (12) Frenkel, D.; Smit, B. Understanding Molecular Simulation: From Algorithms to Applications; Academic Press: San Diego, 1996; pp xviii, 443. (13) Adams, D. J. Grand canonical ensemble Monte-carlo for a Lennard-Jones fluid. Mol. Phys. 1975, 29, 307–311. (14) Norman, G. E.; Filinov, V. S. Investigation of phase transitions by a Monte Carlo method. High Temp. 1969, 7, 216–222. (15) Do, D. D.; Do, H. D. Effects of potential models in the vapor-liquid equilibria and adsorption of simple gases on graphitized thermal carbon black. Fluid Phase Equilib. 2005, 236, 169–177.

Published on Web 09/22/2010

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Figure 1. Adsorption of argon on graphitized thermal carbon black at 87.3 K. The solid line is the experimental data of Gardner et al.,20 the circles are from the GCMC simulation with no surface mediation, and the squares are from GCMC with surface mediation.

argon-graphite surface interaction with the Steele 10-4-3 equation.16,17 The molecular parameters of the model are the same as those used in our earlier paper.10 The interaction between two molecules close to the surface is mediated by the adsorbent substrate as a third body. We model this with an empirical equation that includes a surface damping factor to describe the reduction in the intermolecular potential energy of interaction of noble gases on graphite.10 A damping constant of 0.005 was found to describe argon adsorption on graphite well. Figure 1 shows examples of experimental and simulated results for the adsorption of Ar on graphite at 87.3 K in the absence and presence of surface mediation. It is clear that the inclusion of surface mediation in the model is essential in the correct description of the experimental data.10

3. Analysis of GCMC Simulation Data 3.1. Isotherm Representation. Given the ensemble average of the number of particles in the system resulting from the grand canonical Monte Carlo (GCMC) simulation, we compute the surface excess concentration as ÆNex æ ÆNæ - NG ¼ ð1Þ A A The excess concentration is defined as the excess above a reference amount, NG, which is taken to be the number of particles that occupy the accessible volume at the same concentration as that of the bulk gas phase (i.e. NG = VGFG). Here, the parameter VG is the accessible volume of the system (i.e., the volume with nonpositive solid-fluid potential energy). Γex ¼

4. Results and Discussions 4.1. Simulation Results. We divide the temperature range into four regions: (region 1) T is below the triple point; (region 2) T is above the triple point and below the critical point; (region 3) T is around the critical point; and (region 4) T is above the critical point. In region 1, we expect a layering mechanism where layers are formed one by one. At this very low temperature, we see 2D condensation in the first and possibly higher layers. Figure 2 shows the isotherm of argon adsorption on graphite at 55 K and (16) Steele, W. A. The physical interaction of gases with crystalline solids: I. Gas-solid energies and properties of isolated adsorbed atoms. Surf. Sci. 1973, 36, 317–352. (17) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: New York, 1974; Vol. 3.

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its isosteric heat versus loading. The 2D condensation of the first two layers can be seen in the isotherm and corresponding regions of constant heat in the isosteric heat plot. The simulated isotherm at 55 K (Figure 2a) agrees qualitatively with the experimental data of Millot,18 shown as the filled circles in the same figure, supporting the potential of the model for describing low-temperature data. We have not found any literature on the isosteric heat of argon at this low temperature to justify the constant heat across the 2D condensation; however, the constant heat of krypton at 77 K (below the triple point of krypton of 115.8 K) has been observed experimentally when krypton is going through a 2D condensation.19 This lends support to the constant heat of argon at 55 K as predicted by the GCMC simulation (Figure 2b) across the 2D transition of argon. Figure 3 shows the GCMC simulation results for temperatures ranging from below the triple point to temperatures close to the critical point (region 2). The isotherm at 77 K, below the triple point of argon, shows indications of clear layering. As the temperature is increased, the plateau at the first and higher layers becomes less clear because of the increasing thermal motion of molecules allowing them to migrate between layers with relative ease. The adsorption behavior at 77 and 87.3 K agrees very well with experimental data in the literature.20-23 Limited experimental data of argon adsorption at 130 and 140 K24 in the submonolayer coverage are also shown in this graph, and we see that the simulation results are consistent with the experimental data. The simulated adsorption isotherms for temperatures above the critical point are shown in Figure 4. Specovius and Findenegg25 (18) Millot, F. Adsorption of the first layer of argon on graphite. J. Phys., Lett. 1979, 40, 9–10. (19) Thomy, A.; Duval, X.; Regnier, J. Two-dimensional phase transitions as displayed by adsorption isotherms on graphite and other lamellar solids. Surf. Sci. Rep. 1981, 1, 1–38. (20) Gardner, L.; Kruk, M.; Jaroniec, M. Reference data for argon adsorption on graphitized and nongraphitized carbon blacks. J. Phys. Chem. B 2001, 105, 12516–12523. (21) Avgul, N. N.; Kiselev, A. V. Physical adsorption of gases and vapors on graphitized carbon blacks. Chem. Phys. Carbon 1970, 6, 1–124. (22) Ross, S.; Winkler, W. On physical adsorption: VIII. Monolayer adsorption of argon and nitrogen on graphitized carbon. J. Colloid Sci. 1955, 10, 319–329. (23) Sams, J. R.; Constabaris, G.; Halsey, G. D. Second virial coefficients of neon, argon, krypton and xenon with a graphitized carbon black. J. Phys. Chem. 1960, 64, 1689–1696. (24) Avgul, N. N.; Kiselev, A. V. Physical adsorption of gases and vapous on graphitized carbon blacks. Chem. Phys. Carbon 1970, 6, 1–124. (25) Specovius, J.; Findenegg, G. H. Physical adsorption of gases at highpressures - argon and methane onto graphitized carbon-black. Ber. Bunsen-Ges. Phys. Chem. Chem. Phys. 1978, 82, 174–180.

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Figure 2. (a) Isotherm of argon adsorption on graphite at 55 K; the filled circles are the experimental data. (b) Isosteric heat versus loading.

Figure 3. Adsorption isotherms of argon on graphite at temperatures below the critical point. Symbols are simulation results, and solid lines are the experimental data.

measured argon adsorption on Graphon at 253 and 298 K, and their results are shown in Figure 4 as dashed lines. The agreement is qualitative because the simulation results are slightly greater than the data, and a possible reason for the small difference is that the surface strength of Graphon is probably not exactly the same as that of graphite. The behavior of supercritical adsorption is different from that shown in Figure 3 in that they generally exhibit a maximum in the isotherm. Other features from these isotherms are the following: (1) The isotherms show a very sharp maximum for T close to the critical point. (2) The pressure at which the maximum occurs (Pm) is greater than the critical pressure (4.9 MPa). (3) The maximum disappears at very high T (T >800 K). (4) The isotherms at different T values cross each other, giving the impression that over a certain range of pressure the excess amount is greater at higher temperature. (5) We do not observe a negative excess at any temperature.26 (6) The maximum excess concentration under supercritical conditions is less than twice the monolayer coverage 15854 DOI: 10.1021/la1024857

concentration of 12 μmol/m2. For example, at 260 K, the maximum excess concentration is less than the monolayer concentration, and at 160 K (which is very close to the critical point), the maximum excess that can be adsorbed is only 2.5-fold as much as the monolayer concentration. This means that in supercritical adsorption the net excess amount that can be adsorbed on a surface is only 2 times the monolayer concentration at best. Even when the net excess amount is less than the monolayer coverage at 260 K, the number of layers that can be seen in the single-particle density can be as high as six at high pressures. The local density distribution oscillates around the bulk phase concentration (Figure 6b) such that the net excess amount is actually less than 1 monolayer concentration. The reason for the sharp maximum in the isotherms at temperatures close to the critical point (region 3) is the large change in the bulk gas density with a very small change in pressure. This suggests that the isotherms under supercritical conditions should be plotted as the excess amount versus the bulk gas density. This is Langmuir 2010, 26(20), 15852–15864

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Figure 4. Adsorption of argon on a carbon black surface at temperatures close to and above the critical point. Some of these simulation data have been presented in ref 26 but are included here to compare against the simulation data obtained in this work. The top dashed line is the experimental data at 253 K, and the bottom dashed line is the experimental data at 298 K.

shown in Figure 5. The maximum is still observed; however, the significant difference between the plot versus gas density and that versus pressure (Figure 4) is the correct behavior of the excess amount with respect to temperature (i.e., the greater the temperature, the smaller the excess for a given bulk gas density), in accord with the physical expectation. Also plotted in this graph are the data obtained by Specovius and Findenegg27 for argon adsorp(26) Do, D. D.; Do, H. D.; Fan, C.; Nicholson, D. On the existence of negative excess isotherms for argon adsorption on graphite surfaces and in graphitic pores under supercritical conditions at pressures up to 10,000 atm. Langmuir 2010, 26, 4796–4806. (27) Specovius, J.; Findenegg, G. H. Physical adsorption of gases at highpressure: argon and methane onto graphitized carbon black. Ber. Bunsen-Ges. Phys. Chem. Chem. Phys. 1978, 82, 174–180.

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tion on Graphon at 253 and 298 K, which are shown by the dashed line. As mentioned earlier, the agreement is only qualitative and this is probably due to the different surface strength of Graphon compared to that of graphite. Next we study the microscopic behavior of the adsorbed phase and highlight the differences and similarities between adsorption under sub- and supercritical conditions. Under subcritical conditions, especially at temperatures well below the critical point, we observe a type II isotherm according to the IUPAC classification.28 The surface excess concentration (e.g., that shown in Figure 3) (28) IUPAC. Reporting physisorption data for gas/solid systems with special reference to the determination of surface area and porosity. Pure Appl. Chem. 1985, 57, 603-619.

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Figure 5. Plots of the adsorption isotherms in terms of bulk gas density under supercritical conditions. Experimental data of argon adsorption on Graphon (BET area of 81 m2/g)25 are shown as dashed lines. The top dashed line is the data at 253 K, and the bottom dashed line is the data at 298 K.

reflects what is really happening in the adsorbed phase because the gas density is very low compared to the density of the adsorbed phase. For example, at 87.3 K the adsorption isotherm shows a layering mechanism, with the clear formation of the first layer followed by the formation of higher layers. This is seen in the plot of the local density of argon as a function of distance from the surface (Figure 6a). The layering mechanism is reflected in the distinct peaks that are demarcated by interfacing regions of zero local density (particularly between the first and second layers). The positions of these peaks do not change with pressure and are highly localized. Note that these peaks are substantially higher than the gas density, which we will show in the log plot in Figure 7b. Let us now turn to the study of the local density versus distance at 260 K (supercritical adsorption). This is shown in Figure 6b. Comparing these distributions with those at 87.3 K (subcritical conditions), we make the following observations: (1) More than (29) Zhou, L.; Sun, Y.; Yang, Z.; Zhou, Y. Hydrogen and methane sorption in dry and water-loaded multiwall carbon nanotubes. J. Colloid Interface Sci. 2005, 289, 347–351.

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one layer is possible with supercritical adsorption. This is in contrast to some suggestions that only one layer is possible with supercritical adsorption.29 However, it is important to stress here that some of these peaks do not signify a gain in the excess amount because the local density oscillates around the gas-phase density in the region close to the bulk gas phase. We quantify this later. (2) The peaks are shallower for supercritical conditions because of the greater thermal motion of the argon particles, allowing greater ease of mobility between layers. (3) The peak height is lower for supercritical adsorption and the peaks are more widely spread than in subcritical adsorption because the adsorbate molecules are more delocalized. The locations of the peaks in the case of supercritical adsorption move closer to those for subcritical conditions as the pressure increases. This is tantamount to saying that the adsorbed phase is compressed. However, as the pressure is increased to 500 MPa (this is a very high pressure), we see the peaks of the first two layers for the supercritical temperature of 260 K actually move closer to the graphite surface, compared to those for the subcritical temperature of 87.3 K. This illustrates the high degree of compression at extremely high pressures. Langmuir 2010, 26(20), 15852–15864

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Figure 6. (a) Local density vs distance for Ar adsorption on graphite at 87.3 K. The local density is presented in Lennard-Jones units (i.e., FLJ = Fσ3, where σ is the collision diameter of argon). (b) Local density vs distance for Ar adsorption on graphite at 260 K.

If the microscopic behavior of the adsorbed phase in supercritical and subcritical adsorption is similar (i.e., layer structure formation with peaks of local density), why do not we see similarity in the adsorption isotherm? In fact, this statement is not entirely correct because if we plot the local density in a logarithmic plot we note that the local density in the so-called adsorbed phase can be less than the gas-phase density for supercritical adsorption. (See Figure 7a for 298 K.) At 1 MPa, we have a single peak in the local density plot, and the local density everywhere is greater than the gas density. As the pressure is increased to 10 MPa, there are two local peaks and they are both higher than the gas density. However, when we increase the pressure to 100 MPa we see three local peaks, but some local densities are actually lower than the gas density. What this really means is that in supercritical adsorption a peak in the local density plot does not correspond to a gain in excess concentration. Although we also see this phenomenon in subcritical adsorption, the domination of peaks is so great that this effect is insignificant. (See the curve for 70 kPa in Figure 7b.) We now concentrate on the local density plots at 298 K (Figure 7a) and particularly note the one at 20 MPa. At this pressure, we observe two peaks and the minimum local density Langmuir 2010, 26(20), 15852–15864

between the first two peaks is the same as the bulk gas density. Interestingly, this pressure is the one at which the surface excess concentration is maximum. We asked whether this observation is repeated for other temperatures and found that this is indeed the case, as shown by the local density distributions for 180 and 200 K in Figure 8, which are a further illustration of this somewhat interesting observation. Thus, we can conclude that when the minimum local density between the first two peaks is the same as the bulk gas density, the surface excess concentration is at its maximum because as the pressure is increased beyond this point the local density profile will oscillate around the bulk gas density and the increase in the depth of the troughs is greater than the increase in the height of the peaks, resulting in a decrease in the surface excess concentration. Because of the oscillation of the local density versus distance, a positive deviation of the peaks from the bulk gas density could be compensated for by the negative deviation of the troughs. To quantify this, we evaluate the following quantity Z IðzÞ ¼

z z0

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ð2Þ 15857

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Figure 7. Local density plot of argon adsorption at (a) 298 and (b) 87.3 K.

where z is the distance from the interface and z0 is the position at which the solid-fluid potential energy is zero. As z becomes very large (far away from the surface), this quantity is equal to the surface excess concentration. We show in Figure 9 plots of I(z) versus z at various pressures for argon adsorption at 260 and 180 K. We make the following observations: (1) At the pressure at which the surface excess concentration is at its maximum, Pm, the plot of I(z) shows an inflection point at the distance marking the end of the first layer. (2) For pressures less than Pm, we observe a continuous increase in I(z) versus z, approaching the overall surface excess concentration at large z. (3) For pressures greater than Pm, we see a damped oscillation of I(z) and it also approaches the overall surface excess concentration as z becomes large. We have presented comprehensive computer simulation results of adsorption isotherms of argon on graphite over a wide range of temperature and now turn to the discussion of the enthalpy change of the system when adsorption changes from subcritical to supercritical. 4.2. Differential Molar Enthalpy of Adsorption (Isosteric Heat). Because we are dealing with adsorption over a wide range of temperature, it is essential that we apply the appropriate equation to compute the isosteric heat for conditions far away 15858 DOI: 10.1021/la1024857

from the ideal gas. Do et al.30 have recently presented a general equation for the isosteric heat, which is defined as the negative enthalpy change of the system per unit change in the number of particles in the excess (adsorbed) phase at a constant total number of particles. This is the same as the difference between the partial molar enthalpy of the gas phase and the partial molar enthalpy of the adsorbed phase.31-34 The enthalpy change per unit change in the excess number of particle is given by   DH ð3Þ Δhex ¼ DNex T , N where the upper bar denotes the partial quantity, H is the enthalpy of the system, and Nex is the excess amount. The traditional (30) Do, D. D.; Do, H. D.; Nicholson, D. Molecular simulation of excess isotherm and excess enthalpy change in gas-phase adsorption. J. Phys. Chem. B 2009, 113, 1030–1040. (31) Nicholson, D.; Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: London, 1982. (32) Hill, T. L. Statistical mechanics of adsorption. V. Thermodynamics and heat of adsorption. J. Chem. Phys. 1949, 17, 520–535. (33) Karavias, F.; Myers, A. L. Isosteric heats of multicomponent adsorption: thermodynamics and computer simulations. Langmuir 1991, 7, 3118–3126. (34) Myers, A. L.; Calles, J. A.; Calleja, G. Comparison of molecular simulation of adsorption with experiment. Adsorption 1997, 3, 107–115.

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Figure 8. Local density plots of argon adsorption at (left) 180 and (right) 200 K.

Figure 9. Plots of I(z) vs z for the adsorption of argon at (a) 260 and (b) 180 K.

isosteric heat is qst = -Δhex to make it a positive quantity. Equation 3 can be written in terms of the fluctuation variables30 Δhex ¼ -

NG kT f ðU, NÞ þ f ðNG , NG Þ f ðN, NÞ - f ðNG , NG Þ

ð4Þ

where U is the configurational energy (sum of gas-solid and fluid-fluid interaction energies) of the system, N is the number of particles in the system, NG is the number of particles occupying the accessible volume of the system at the same density as the bulk gas, and f(X, Y) is the fluctuation variable that is defined as f(X, Y) = ÆXYæ - ÆXæÆYæ. This enthalpy change as derived in eq 4 behaves well when the excess isotherm increases monotonously with pressure. When adsorption occurs under supercritical conditions, however, the denominator of the second term on the RHS of eq 4 can become zero, making the isosteric heat infinite. This occurs at the pressure where the excess concentration versus pressure is at (35) Myers, A. L.; Monson, P. A. Adsorption in porous materials at high pressure: theory and experiment. Langmuir 2002, 18, 10261–10273. (36) Grillet, Y.; Rouquerol, F.; Rouquerol, J. Two-dimensional freezing of nitrogen or argon on differently graphitized carbons. J. Colloid Interface Sci. 1978, 70, 239–244. (37) Pace, E. L.; Siebert, A. R. Heats of adsorption and adsorption isotherms for low boiling gases adsorbed on graphon. J. Phys. Chem. 1960, 64, 961–963.

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its maximum,35 as is apparent from the following equation for the change in the excess amount in terms of pressure:30 DNex f ðN, NÞ - f ðNG , NG Þ ¼ ðNG =VG ÞkT Dp

ð5Þ

Let us now show the behavior of the isosteric heat of argon adsorption on graphite under sub- and supercritical conditions. Figure 10a shows the isosteric heat at 77 and 87.3 K (subcritical) as a function of loading. Also plotted in this figure are the experimental data at these two temperatures. As can be seen, both sets of data agree fairly well with the experimental results, confirming the potential of the molecular model as a descriptive one. The isosteric heat curves at 77 and 87.3 K are typical heat curve for noble gases on a flat surface.38 The curve at 77 K has a spike (occurring at a monolayer concentration of about 12 μmol/m2). The reason for the heat spike has been given,39 and at this heat (38) Do, D. D.; Nicholson, D.; Do, H. D. On the anatomy of the adsorption heat versus loading as a function of temperature and adsorbate for a graphitic surface. J. Colloid Interface Sci. 2008, 325, 7–22. (39) Wongkoblap, A.; Do, D. D.; Nicholson, D. Explanation of the unusual peak of calorimetric heat in the adsorption of nitrogen, argon and methane on graphitized thermal carbon black. Phys. Chem. Chem. Phys. 2008, 10, 1106–1113.

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Figure 10. Isosteric heat of argon adsorption on a graphite surface at various temperatures. The circles and squares in graph a are the experimental data at 7736 and 87.3 K,37 respectively.

spike, some particles squeeze into the first layer while the adsorption on the second layer is still progressing. This would result in a greater solid-fluid interaction (because the newly inserted particle is closer to the surface) as well as a greater fluid-fluid interaction because of the better rearrangement of particles in the first layer. Another interesting observation is that if we extrapolate the heat curve in the linear region to the monolayer concentration where the heat spike occurs then the extrapolated heat is exactly the same as the heat at the heat spike. This simply means that if adsorption had continued to occur in the first layer at loadings beyond 10 μmol/m2 instead of starting to form the second layer then the heat curve would have increased monotonously until it reaches that maximum. At 87.3 K, the heat spike is no longer visible, the isosteric heat shows a smoother behavior than that at 77 K, and the maximum at 87.3 K is lower. This behavior is followed by other temperatures below the critical point. We now turn our attention to the behavior of the isosteric heat under supercritical conditions. Two temperatures, 180 and 260 K, are chosen here as examples of supercritical conditions, and the isosteric heat curves are shown in Figure 10b as a plot of isosteric heat versus pressure instead of loading that we have done for subcritical conditions. The reason for this is that the excess surface concentration is not a monotonically increasing function with respect to pressure but rather it has a maximum (Figure 4). For supercritical conditions, we see that the isosteric heat is infinite at the pressure where the surface loading is at its maximum, Pm. Beyond this pressure, the isosteric heat starts from minus infinity and then increases as the pressure is further increased. Similar behavior of this heat curve under supercritical conditions was also reported by Myers and Monson.35 Such complex behavior of the isosteric heat is a result of the way that we define the isosteric heat. Although infinite heat is hard to understand physically, it does not cause any difficulties in the overall energy balance. Let us illustrate this by considering the change in the enthalpy of the system with pressure, (∂H/∂p)T, in the next section, and we will show that the integral molar enthalpy of adsorption is a better measure for studying heat under supercritical conditions. 4.3. Integral Molar Enthalpy of Adsorption. Because the behavior of isosteric heat under supercritical conditions has an unusual behavior at the maximum in the excess concentration, we consider an alternative by first deriving the enthalpy change for a unit change in pressure, ∂H/∂p. By applying the chain rule of 15860 DOI: 10.1021/la1024857

differentiation of this variable and using eqs 3 and 5, we get -

  kT DH f ðU, NÞ ½f ðN, NÞ - f ðNG , NG Þ ¼ þ kT VG Dp NG f ðNG , NG Þ

ð6Þ

This equation does not have any singularity like the isosteric heat in eq 4. The LHS of eq 6 versus pressure for argon adsorption on graphite is shown by the solid line with unfilled circles in Figure 11 for various temperatures. We chose 87.3 K to represent subcritical adsorption and 260 K to represent supercritical adsorption. The common denominator of these plots is the absence of any singularity. Our observation of these plots is summarized as follows: (1) The magnitude of the differential enthalpy change with pressure decreases as the temperature increases. This is understandable because the pressure scale of subcritical adsorption is much lower than that of supercritical adsorption. (2) The plot of differential enthalpy with pressure is constant at low pressure, indicating that the heat increases linearly with loading (or pressure) at low loadings where Henry’s law is obeyed. This is applied to all temperatures. (3) For subcritical adsorption, the heat curve increases to its maximum at a pressure where the first layer is nearly complete. Beyond this maximum, the heat curve shows a decrease, indicating the onset of the second layer. (4) For supercritical adsorption, we see no maximum but, on the contrary, a decrease to a minimum, followed by a small shoulder at the pressure where the excess loading is at its maximum, Pm. From eq 6 we can obtain the enthalpy change from p = 0 to some arbitrary pressure p. Given the surface excess amount at this pressure, we can calculate the integral molecular enthalpy of adsorption, which is based on the excess amount, as follows: ΔH 1 ¼ ÆNex æ ÆNex æ

Z p" 0

# f ðU, NÞ f ðN, NÞ - f ðNG , NG Þ NG dp f ðNG , NG Þ FG kT FG

ð7Þ Unlike the differential molecular enthalpy of adsorption in eq 4, the integral molecular enthalpy does not have a singularity. As an illustration, we show in Figure 12 the integral molar enthalpy for argon adsorption on graphite under sub- and supercritical conditions (solid lines with unfilled circles). Also plotted in this graph Langmuir 2010, 26(20), 15852–15864

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Figure 11. Differential enthalpy change versus pressure of argon adsorption on a graphite surface at 87.3 and 260 K.

Figure 12. Integral molar enthalpy of argon adsorption on graphite at 87.3 and 260 K.

are the conventional isosteric heats (eq 4) for comparison with the integral molar enthalpy. The common denominator among all these plots is that the magnitude of the integral molar enthalpy is the same order of magnitude as that of the isosteric heat. With the integral molar enthalpy, we do not encounter any singularity, and its magnitude is as expected for typical physical adsorption. To gain a deeper understanding of how the integral molar enthalpy behaves, we start our discussion with subcritical adsorption. Being integral in nature, the integral molar enthalpy (solid line with circles) shows less variation than the isosteric heat (dashed line). For supercritical adsorption, this is where the integral molar enthalpy is a useful quantity for characterizing the behavior of the energy of the system. The integral molar enthalpy behaves very differently from the isosteric heat (Figure 12b), except of course at very low loadings, and most importantly, it remains finite at all loadings whereas the isosteric heat becomes infinite at a pressure where the excess concentration is at its maximum. 4.4. Differential Molecular Enthalpy of the Excess Phase. Instead of dealing with the enthalpy change of the system (which includes the adsorbed phase and the gas phase) per unit change of the excess amount, as is done in eq 6, we consider the change in the excess enthalpy per unit change in the excess amount. It is given by the following equation, written in terms Langmuir 2010, 26(20), 15852–15864

of the fluctuation variables.30 

DHex DNex

 ¼ T

f ðU, NÞ - f ðUG , NG Þ f ðN, NÞ - f ðNG , NG Þ

ð8Þ

Like the enthalpy of the system that we dealt with earlier, this equation also has a singularity at the pressure where the excess concentration is at its maximum. However, as before, what we are interested in is the change in the excess enthalpy with a change in pressure because pressure is a better independent variable to deal with in supercritical adsorption:   kT DHex ½f ðU, NÞ - f ðUG , NG Þ ¼ VG Dp NG

ð9Þ

To compare this differential enthalpy change of the excess phase and that of the whole system (eq 6), we plot it in Figure 11 as solid lines with filled circles. The difference between these two is the enthalpy change contributed by the gas phase, which is negligible for subcritical adsorption (87.3 K) but significant under supercritical conditions (260 K). One distinct feature that we observe in the case of supercritical adsorption is the absence of the small shoulder in the case of the differential enthalpy of the excess phase. We recall that this shoulder for the case of the differential DOI: 10.1021/la1024857

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Figure 13. Behavior of the second factor for 87.3 and 260 K on the RHS of eq 10.

enthalpy of the system occurs at a pressure where the excess concentration is at its maximum. The behavior of eq 9 will give us an idea of how the adsorbed phase changes with pressure. The physical meaning of this variable can be seen by rearranging eq 9 as follows:   kT DHex ½f ðU, NÞ - f ðUG , NG Þ f ðN, NÞ f ðNG , NG Þ ¼ VG Dp f ðN, NÞ f ðNG , NG Þ NG ð10Þ The RHS of this equation contains three factors. The first factor is the fluctuation of excess energy per unit fluctuation of the number of particles. We expect this to have the same order of magnitude as a typical heat of adsorption. The second factor describes how fast the number of particles in the system increases with pressure compared to the gas phase. For subcritical conditions, this factor is very large but becomes smaller under supercritical conditions. The last factor is the compressibility of the gas phase. For moderate conditions (pressure of less than 10 atm), this factor is on the order of unity. In Figure 13, we plot the second factor on the RHS of eq 10 for 87.3 and 260 K to show the ratio of the fluctuation of the number of particles in the system to that in the gas phase. From eq 9, we can obtain the integral molecular enthalpy of the adsorbed phase as ΔHex 1 ¼ ÆNex æ ÆNex æ

Z p" 0

# f ðU, NÞ - f ðUG , NG Þ dp FG kT

ð11Þ

This integral molecular enthalpy of the adsorbed phase does not suffer from the singularity encountered with the corresponding differential variable in eq 8. In Figure 12, we show the integral molar enthalpy change of the adsorbed phase, ΔHex/Nex, as solid lines with solid circles. For all cases, ΔHex/Nex is smaller than ΔH/ Nex, and this difference is the contribution made by the enthalpy change in the gas phase. For subcritical adsorption, this difference is a constant over the whole pressure range because it is simply the molar enthalpy of the gas phase, which is independent of pressure for ideal gases. However, this is no longer true for supercritical adsorption in which the difference between ΔHex/Nex and ΔH/Nex is larger with increasing pressure. 4.5. Compression of the Adsorbed Phase and the Bulk Phase. To understand how the number of particles in the system 15862 DOI: 10.1021/la1024857

varies relative to the number in the gas phase, we consider the change in particle number with respect to pressure for the two phases:40 DN f ðN, NÞ ¼ Dp ðNG =VG ÞkT

DNG f ðNG , NG Þ ¼ ðNG =VG ÞkT Dp

ð12Þ

Equation 12 reduces to the ideal gas equation because f(NG, NG) = NG at low pressure. Thus, by comparing the magnitudes of the fluctuation variables f(N, N) and f(NG, NG), we know the relative compression of the two phases as a function of pressure. The results of f(N, N) over f(NG, NG) have been discussed as the second factor of eq 10 (Figure 13). Here we plot these two fluctuation variables separately to show the behavior of each phase. As it can be seen in Figure 14a, under subcritical conditions the fluctuation of the whole system is always greater than that of the gas phase because the adsorption system is dominated by the adsorbed phase. Quite interestingly, the fluctuation variable f(N, N) behaves in a manner corresponding to the layering adsorption mechanism. To show this clearly, we also plot the isotherm for each temperature in the top panel of Figure 14. In the curve of f(N, N) versus pressure, a hump can be observed when one layer is being completed and then it decreases as the next layer starts to form. However, the fluctuation f(NG, NG) increases linearly with pressure and it is consistent with the ideal gas behavior. Under supercritical conditions in the low-pressure region, both f(N, N) and f(NG, NG) behave linearly with pressure and f(N,N) is higher than f(NG, NG), but the difference is not as great as in the subcritical adsorption (Figure 14b). As pressure is increased, the difference is decreased and it becomes zero at the pressure where the excess concentration is at its maximum, Pm. This is what we explained earlier in section 4.1, where we state that the maximum in the excess concentration as a result of the rate of change of the adsorbed concentration with pressure is the same as that of the bulk phase. Beyond Pm, we observe the opposite: f(NG, NG) becomes greater than f(N, N), and both of them decrease with pressure. One point that we note is that the fluctuation f(N, N) is never zero, despite the fact that it continually decreases with (40) Specovius, J.; Findenegg, G. H. Study of a fluid-solid interface over a wide density range including the critical region. 1. Surface excess of ethylene-graphite. Ber. Bunsen-Ges. Phys. Chem. Chem. Phys. 1980, 84, 690–696.

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Figure 14. Fluctuation variables f(N, N) and f(NG, NG) of argon adsorption on a graphite surface at 87.3 K (subcritical condition) and 260 K (supercritical condition). The top plot in each graph is the isotherm at each temperature.

Figure 15. Behavior of f(NG, NG)/NG under sub- and supercritical conditions.

pressure. This suggests that the adsorbed phase can be compressed under supercritical conditions. Equation 12 also gives us a means to study the compressibility of the gas phase. We do this by studying the ratio f ðNG , NG Þ NG

ð13Þ

Figure 15 shows the behavior of this ratio under subcritical (87.3 K) and supercritical conditions (260 K). For subcritical conditions, it is unity and increases slightly when the pressure approaches the saturation vapor pressure. However, for supercritical conditions, it is unity at low pressure, increases at moderate pressure, and finally decreases at extremely high pressure. Langmuir 2010, 26(20), 15852–15864

5. Conclusions We have presented a comprehensive molecular simulation study of argon adsorption on a graphite surface over a wide range of temperature, under sub- to supercritical conditions. The behavior of the adsorption isotherm in the supercritical range is different from that in the subcritical range. Specifically, the supercritical isotherm generally has a maximum, except when the adsorption temperature is very high. We have identified that the maximum in the excess concentration corresponds to the situation in which the minimum local density between the first and second layers is equal to the bulk gas density. The net excess concentration under supercritical conditions is less than the monolayer concentration, except when the temperature is very close to the critical point, despite the fact that the local density distribution exhibits DOI: 10.1021/la1024857

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a number of layers above the surface. Finally, the difference between the subcritical and supercritical adsorptions is in the behavior of the isosteric heat. For supercritical adsorption, the isosteric heat is infinite at the pressure where the excess concentration is at its maximum, and for this case, we have proposed the use of the integral molar heat as an

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alternative because this has no singularity for supercritical adsorption. Acknowledgment. This work is supported by the Australian Research Council. C.F. acknowledges financial support from the CSC in the form of a scholarship.

Langmuir 2010, 26(20), 15852–15864