A computer appraisal of BET theory, BET surface area

ARTICLE IN PRESS Chemical Engineering Science 65 (2010) 3331–3340

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A computer appraisal of BET theory, BET surface area and the calculation of surface excess for gas adsorption on a graphite surface D.D. Do , H.D. Do, D. Nicholson School of Chemical Engineering, University of Queensland, St. Lucia, Qld 4072, Australia

a r t i c l e in fo

abstract

Article history: Received 21 September 2009 Received in revised form 8 February 2010 Accepted 12 February 2010 Available online 18 February 2010

In this paper we address problems associated with the calculation of the surface excess as traditionally carried out for gas adsorption on surfaces. This calculation is done through a series of steps, one of which is the application of the BET equation for the determination of surface area. The BET theory, despite its popularity, has many unjustified assumptions, requires a choice of the molecular projection area and arbitrary adjustment of the relative pressure range of the BET plot to suit the requirement of a linear best fit. The surface excess, in terms of mol per unit surface area, is therefore subject to compounded errors resulting from these factors. We quantify this with a detailed computer appraisal of adsorption of argon on a graphite surface under sub-critical and supercritical conditions. Careful consideration is paid to the bulk gas volume in the adsorption cell, and it is found that an incorrect estimate of the void volume can lead to a significant error in the surface excess per unit area. This is more pronounced with supercritical adsorption because for a given mass of adsorbent even a 0.1% overestimation of the void volume can change a correct positive surface excess to a negative one at high pressures. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Adsorption Interface Computational Chemistry Simulation Surface excess BET theory

1. Introduction Description of adsorption of gases on a surface is presented by the relationship between the surface excess (usually expressed as mole adsorbed per unit mass or mole per unit surface area) and the pressure at a given temperature (Gardner et al., 2001; Kaneko, 1994; Steele, 1974; Thommes, 2004; Nicholson and Parsonage, 1982; Sing et al., 1985). For the latter representation of mol per unit area, it is more often than not that the BET surface area is used and is the basis for a standard protocol where nitrogen at 77 K is the recommended adsorbate (Sing et al., 1985). The BET method, since its inception in 1938 (Brunauer et al., 1938), has been extremely popular because of its simplicity, despite the fact that its derivation depends on many unjustified assumptions and other problems associated with the use of the molecular projection area in the calculation of surface area. Because of its popularity (mainly due to its simplicity) the BET method has received great attention and many papers have appeared in the literature, both supporting and raising issues about the limitations of the method. Among many researchers, Sing (1973), Rouquerol et al. (1977), Grillet et al. (1979), Sing (1998), Rouquerol et al. (2007), Dollimore et al. (1976) have published over the span of many decades to promote this method as well as

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E-mail address: [email protected] (D.D. Do). 0009-2509/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2010.02.023

to highlight its limitations. Other researchers, recognising these limitations, have refined and extended the method (mostly on the account that the omission of lateral interaction is seen to be the major weakness of the BET method), at the expense of the simplicity, for example Seri-Levy and Avnir (1993) have used a two-dimensional Monte Carlo simulation to study the lateral interaction, Nikitas (1996) used a simple statistical mechanical approach to study the same, while some have extended the method to account for the surface heterogeneity, for example Dormant and Adamson (1972). One important parameter required in the BET theory is the molecular projection area. The value for this molecular area has a wide range, even for simple gases such as argon and nitrogen; for example various values for argon have been reported in the literature: typically 0.138 and 0.142 nm2/molecule (Gregg and Sing, 1985). If argon, being a spherical atom, has more than one value for the projection area, polyatomic adsorbates that can exhibit different orientations would be expected to have even a wider range. Indeed projection areas for nitrogen have been reported in the range 0.14–0.277 nm2/molecule (Arnell and Henneberry, 1948; Chung and Dash, 1977). To determine the excess amount of the adsorbed phase, a void volume is required so that the excess amount is calculated as the difference between the total amount in the system and the amount that is in the void volume occupied by gas having the same density as the bulk phase. This void volume is usually obtained by the helium expansion method. Helium is known to adsorb (although its

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D.D. Do et al. / Chemical Engineering Science 65 (2010) 3331–3340

adsorption on surfaces is very small) and its approach to the surface is closer by virtue of its small size, and therefore the void volume probed by helium could be over-estimated. Thus the determination of the surface excess, in terms of mol per unit area, has three potential sources that could contribute to the error of the surface excess. The first one is the choice of the void volume, the second source comes from the unjustified assumptions associated with the BET theory (from which a surface area is estimated) and the last is the choice of the proper molecular projection area. We will address and quantify these sources by carrying a detailed computer simulation of argon on a number of model surfaces under both sub-critical and supercritical conditions.

2. Theory Adsorption isotherm of gases on a surface is usually given as the surface excess per unit area, and it is given in the following equation:

G¼

NVvoid rf S

ðmol=m2 Þ

ð1Þ

where the numerator is the excess amount with N being the total amount in the system, Vvoid being the void volume and rf the bulk fluid density. The parameter S is the surface area, which is usually obtained with the BET method (or many of its variants) and a choice of a value for the molecular projection area. Let us now evaluate Eq. (1). It is known that the void volume, as measured by helium expansion method, is over-estimated because of the possibility of helium adsorption (helium does have polarizability, although it is small) and the fact that helium atom can approach closer to the surface. Hence, if this void volume is over-estimated, the excess amount (in the numerator of Eq. (1)) is under-estimated. If this excess amount (as a function of pressure) is used in the BET theory to obtain the amount required to form a monolayer of adsorbate above the surface, this monolayer concentration will be under-estimated. If a proper value for the molecular projection area is used (assuming that there is no error associated with this molecular area), the BET surface area, S, will be under-estimated. Interestingly, the excess amount in the numerator of Eq. (1) is under-estimated and the BET-surface area is also underestimated. The net result may be that the surface excess, in terms of mol per unit area, might be correctly, though fortuitously calculated. The objectives of this paper are to evaluate this with a detailed computer simulation, together with the evaluation of the BET method and the choice of the molecular projection area. Another problem with the BET equation (which is often overlooked in the literature) is that the equation was developed to describe the absolute adsorption (not excess), but since it fails to account for the gas phase above the adsorbed layers, it has been a common practice to use the excess amount in the application of the BET theory. We shall postpone further discussion on this matter until Section 3. 2.1. Computer simulation of adsorption on a surface In the computer simulation, we use Grand Canonical Monte Carlo (GCMC) simulation with a Metropolis algorithm to determine the amount adsorbed in a simulation box of constant volume, temperature and chemical potential. This box is a rectangular in three dimensions. The surface is located at one end of the box in the z-direction, and the opposite end is taken to be a hard wall. The height in the z-direction is large enough so that the gas phase and adsorbed phase are properly accounted for in the simulation box. Here we choose a height of 20 times the

collision diameter of the adsorbate (for argon this height is 6.81 nm). The lengths of the box in the x- and y-directions are chosen large enough to avoid finite size effects. We choose the box length of 10 times the collision diameter (for argon, Lx = Ly = 3.405 nm), and the cut-off radius in the calculation of the intermolecular potential energy is half of the box length. A limited numbers of simulations were carried out with larger simulation boxes, and these results were comparable to those obtained with the parameters that we have just described. Periodic boundary conditions are applied in the x- and y-directions to simulate a surface of infinite extent in those directions. We use argon as a model adsorbate and graphite surface as a model adsorbent. For the molecular simulation with grand canonical ensemble, the potential energies of interaction between two argon particles and that between an argon particle and the solid surface are required. For the fluid-fluid interaction, we use the Lennard-Jones 12-6 potential equation, with the collision diameter of 0.3405 nm and the reduced well-depth of interaction energy of 119.8 K. This set of molecular parameters has been shown in produce reasonably accurate vapour-liquid equilibrium of argon (Do and Do, 2005a). For the fluid–solid interaction, the potential equation takes the form of Steele 10-4-3 equation (Steele, 1973). The molecular parameters for a carbon atom in the graphene layer are 0.34 nm and 28 K for the collision diameter and the reduced well depth of the interaction, respectively. 2.1.1. Surface mediation It has been shown in a number of publications from the authors (Do et al., 2004, 2007; Do and Do, 2006a–c, 2005b, 2007a) as well as from others (Rouquerol et al., 1977; Grillet et al., 1979; Sing, 1998; Rouquerol et al., 2007; Dollimore et al., 1976; SeriLevy and Avnir, 1993; Nikitas, 1996; Dormant and Adamson, 1972; Gregg and Sing, 1985; Arnell and Henneberry, 1948; Chung and Dash, 1977; Do and Do, 2005a, 2006a–c, 2005b, 2007a; Steele, 1973; Do et al., 2004, 2007; de Boer and Kruyer, 1958; Barker and Everett, 1962; Everett, 1965; Kim and Steele, 1992; Kim and Cole, 1988; Chung et al., 1985) that the use of surface mediation in the simulation is essential in the correct description of adsorption on surfaces. The surface affects the way two adsorbed molecules interacts. The mechanisms of this mediation have been attributed to either surface polarization or multi-body interactions (Do and Do, 2007a). This results in weaker interaction between two adsorbed molecules, especially those close to the surface. For noble gases, it has been shown that the following empirical equation is adequate to describe this surface mediation (Do et al., 2004):

jeff ¼ g ji;j i;j

ð2Þ

where ji;j is the intermolecular interaction energy between two adsorbate molecules far away from the surface (i.e. no surface is the effective intermoleinterference on their interaction), jeff i;j cular interaction energy when two particles are close to the surface, and g is called the damping factor, due to the surface mediation. For argon as the adsorbate, the intermolecular interaction energy is calculated from the Lennard-Jones 12-6 potential equation, ji;j ¼ 4ei;j bðs=ri;j Þ12 ðs=ri;j Þ6 c, and the damping factor is calculated from the following empirical equation (Do et al., 2004): g ¼ expðwjij;s =kTÞ

ð3Þ

where the parameter w is called the damping constant, and jij;s is the geometric mean between the solid–fluid potential energy of pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the particle i and that of the particle j, jij;s ¼ jji;s jjjj;s j. For argon adsorption on a graphite surface, the damping constant was found to be 0.005 (Do et al., 2004).

ARTICLE IN PRESS D.D. Do et al. / Chemical Engineering Science 65 (2010) 3331–3340

3. Results and discussion For a given simulation box described in Section 2, we obtain the accessible volume and the accessible area, and a general procedure to do this has been described in Do and Do (2007b), Do (2008). The accessible volume is one in which the fluid–solid potential energy is non-positive, and the accessible area is the area of the boundary of the accessible volume at which the fluid– solid potential energy is zero. For the parameters that we specify for our simulation box the accessible area is 100s2 and the accessible volume is 1915s3, where s is the argon collision diameter ( = 0.3405 nm). 3.1. Adsorption of Ar on graphitic surface at 87.3 K We start with the simulation and analysis of adsorption of argon on a homogeneous graphitic surface under sub-critical conditions. For a given temperature, 87.3 K, we performed GCMC simulations to obtain the number of particle in the simulation box for each value of the chemical potential. The chemical potential was converted to pressure using the equation of state (EOS) of Johnson et al. (1993). We repeat this for a range of chemical potentials to give the number of particles in the box as a function of pressure. This total number of particles includes the number of particles in both the gas phase and the adsorbed phase. Having the total number of particles in the simulation box, N, we calculate the excess amount as follows: Nex ¼ NVvoid rf

ð4Þ

where Vvoid is the void volume occupied by the gas phase. This excess is taken to be the amount associated with the adsorbed phase. In this sense the adsorbed phase is merely in excess of the gas phase. 3.1.1. Computer ‘‘exact’’ solution When the accessible volume is used as the void volume ðVvoid ¼ Vacc Þ and the accessible area is used, the adsorption isotherm is the computer ‘‘exact’’ solution which is used as the reference for the others to compare with.

Gexact ¼

NVacc rf Sacc

ðmol=m2 Þ

ð5Þ

pressure. This is to show the importance of the use of the void volume in the description of isotherm per unit particle (or per unit mass). In the second representation, we present the isotherm as mol per unit surface area (surface density). Fig. 1 shows the ‘‘exact’’ solution in terms of particles and we present the surface excess per unit area in Fig. 2. In the inset of Fig. 1 we show the isotherm over the region of very low pressures, where we see a distinct sigmoid behaviour. This is an indication of a homogeneous surface and the presence of adsorbate interactions, and is fairly well known in the literature (Avgul and Kiselev, 1970; Mulero and Cuadros, 2002; Thomy et al., 1981). The excess number of particles (shown in Fig. 1 as cross symbols) is very close to the total number of particles because the gas density at 87.3 K is very small and because of the small accessible volume used in the simulation. We will discuss later (Section 3.1.3) a methodology for dealing with a system when the accessible volume is much larger in order to mimic the real adsorption system where the void volume is very large. The ‘‘exact’’ excess isotherms in Figs. 1 and 2 are our reference, against which the subsequent derived isotherms will be compared. 3.1.2. Testing the BET theory and the molecular projection area The next step in the analysis is to use the excess number of particles in the BET theory to determine surface area in order to evaluate this theory. First, we use the excess data obtained with the correct void volume, which is the accessible volume. This excess amount is then used in the BET-theory by using it in the linearized BET equation: ðP=P0 Þ 1 ðC1Þ P ð6Þ ¼ þ Nex ð1P=P0 Þ Nm C Nm C P0 where P0 is the vapour pressure, which was obtained for the potential model that we used for argon. The parameter C is the affinity equilibrium constant (related to the intermolecular interactions in the first layer (Gregg and Sing (1985)) and Nm is the amount required to form a statistical monolayer. Before we proceed further it is worthwhile to note the inconsistency in the BET equation. The data is the excess amount, i.e. the amount in excess above the amount that would be if the space is occupied with gas at the same density as the bulk gas, while Nm derived from the BET equation is the absolute amount that is necessary to form a single layer on the surface. Although the consequence of this inconsistency is minor for adsorption at

10 3

250

10 2

200

Number of Particle

Number of Particle

We now present the adsorption in two ways. In the first we describe the isotherm as a plot of the number of particles versus

10 1

10 0

10 -1

3333

150

100

50

10 -2

10 -3

0 10 0

10 1

10 2

10 3

P (Pa)

10 4

10 5

0

10

20

30

40

50

60

P (kPa)

Fig. 1. Plot of particle number versus pressure for argon adsorption on graphite at 87.3 K. Fig. 1a shows the isotherm in log–log plot and Fig. 1b in linear plot. The dashed line is the number of particle in the gas phase. The circle symbols are from the total number of particle and the solid line is the excess number of particle.



10 2

35

Surface Excess (µmol/m )

2

2


30

101

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25

Second Layer 20

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First Layer

5

10-1

0

10 1

10 2

10 3

10 4

0

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30

P (Pa)

40

50

60

P (kPa)

Fig. 2. Surface excess (mol/m2) (calculated from Eq. (5)) versus pressure for argon adsorption on graphite surface at 87.3 K. Accessible volume and accessible area are used in the calculation of the surface excess.

0.0035

1e-4

LHS of Linearized BET equation


0.0030 8e-5

6e-5

4e-5

0.0025

0.0020

0.0015

0.0010

0.0005

2e-5 0.000

0.001

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0.0000 0.00

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0.007


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0.000 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fig. 3. Plot of the linearized-BET equation versus the reduced pressure: (a) low pressure range from 0 to 0.005; (b) reduced pressure range from 0.05 to 0.3; (c) reduced pressure range from 0 to 0.6.



0.0030

0.0025

0.0020

0.0015

0.0010

0.0005

0.0000 0.00

0.05

0.10

0.15

0.20

0.25

P/P0 (-) Fig. 4. Linearized BET plot in the range of reduced pressure from 0.05 to 0.2.

14

Isosteric Heat (kJ/mol)

12

A

10

B

8

6

4

2

0 0

10

20

30

40

50

60

P (kPa) Fig. 5. Isosteric heat of argon adsorption on graphite at 87.3 K, plotted against pressure.

8

6

LJ-Local Density (-)

temperatures well below the critical point because of the very low gas density compared to the adsorbed phase density, this point has been either neglected or overlooked in the literature. But experimentally the excess quantity is obtained and used in the determination of surface area. To plot the linearized BET equation (Eq. (6)) we need to know the vapour pressure corresponding to the potential model that we use for argon. This can be obtained with the Gibbs ensemble Monte Carlo simulation, and we have found that the simulated vapour pressure is 100.2 kPa (Do and Do, 2005a) (compared to an experimental value of 101.3 kPa). Fig. 3 shows the linearized BET plot in three different ranges of the reduced pressure: the very low pressure range, the recommended range of reduced pressure between 0.05 and 0.3 and the wider range of pressure. If the BET theory is correct in the description of data on a flat surface (like graphite), we would expect linear behaviour in those plots over the three ranges. Figs. 3a and c indicate that the BET theory does not work over the low pressure range (where sub-monolayer is occurring) and the wider range of pressure because we do not observe good linearity in those plots. So the BET theory fails to account for submonolayer coverage as well as multiple layers, where the BET theory is really meant to describe. It is interesting that the authors of the BET theory (Brunauer et al., 1938) remarked that their equation is valid in the reduced pressure range from 0.05 to 0.3 because over the lower pressures the isotherm is influenced by the surface structure and heterogeneity. Here we have a homogeneous surface, and we should expect the BET should apply but it does not as evidenced in Fig. 3. Even in the recommended range of reduced pressure (0.05–0.3), we do not observe good linearity (Fig. 3b). From what we have seen with Fig. 3 where no linearity was observed, we show in Fig. 4 the linearized BET plot for a narrower range of reduced pressure from 0.05 to 0.2, and we observe an excellent linearity, from which the slope and the intercept can be reliable obtained. So it is seen that the choice of the relative pressure range in which the BET theory is linear is somewhat ad-hoc. Why do we observe good linearity for argon at 87 K in the range of reduced pressure between 0.05 and 0.2? On observing the behaviour of the surface excess in Fig. 2, we see that in this range, the first layer has been almost completely formed and the second layer is being progressively filled with argon particles but the fluid–fluid interaction in the second layer in this pressure range is not yet significant. As the reduced pressure is increased beyond 0.2, we see that the second layer is filling and therefore

3335

4

2

0

0

2

4

6

8

10

12

14

Distance (A) Fig. 6. Local density distribution for argon adsorption at 87.3 K on a graphite surface. The pressures for the solid lines are 5, 7, 10, 12, 14, 16, 18 and 20 kPa (corresponding to the range of reduced pressure from 0.05 to 0.2). The local density distribution for a much larger pressure of 60 kPa is also shown as dashed line for comparison.

the fluid–fluid interaction is becoming more important (as seen in the upward bend in the isotherm of Fig. 2; the portion between 20 and 30 kPa), which the BET theory does not properly account for. This is the reason why we do not see linearity when the reduced pressure range is extended beyond 0.2 for this adsorptive (Fig. 3). We can support this assertion with the investigation of the local density distribution and the isosteric heat. Fig. 5 shows the isosteric heat versus pressure (where the range of reduced pressure from 0.05 to 0.2 is shown in the Box labelled as A). The heat is decreasing in this range, indicating that the molecules are depositing on the second layer and the fluid–fluid interaction is not very significant. However, as we increase the reduced pressure beyond 0.02 (shown in the box labelled as B), we see an increase in the isosteric heat, an indication of the contribution from the fluid–fluid interaction. This interaction is not accounted for in the BET theory, and therefore it fails. The local density distributions versus distance from the graphite surface for reduced pressures from 0.05 to 0.2 are shown



as solid lines in Fig. 6. For comparison we show the density distribution for the reduced pressure of 0.6 as dashed line in the same figure to show the completion of the second layer. In this pressure range (0.05–0.2) we see that the second layer is being progressively filled and the density of the second layer is low enough to have any significant fluid–fluid interaction. Let us now accept the reduced pressure range of applicability of the BET theory between 0.05 and 0.2 for argon adsorption on a homogeneous graphite surface. We then proceed with the BET theory in a formal manner. The equilibrium constant C and the number of particles that would be required to form a monolayer of adsorbate (Nm) are obtained from the slope and intercept as Nm ¼ ðS þ IÞ1 C ¼ 1þ S=I From Fig. 4, we find the slope and the intercept from the best linear fit, and thence the equilibrium constant and the monolayer number are 122 and 85, respectively. Given this value of Nm, we calculate the BET surface area as 11.9 nm2, using the molecular projection area of 0.138 nm2. This BET-surface area is in close agreement with the accessible (geometrical) area of 11.6 nm2. If we use the range from 0.05 to 0.3 that is recommended in the literature, we find a BET surface area of 14.3 nm2, which is 23% greater than the true geometrical area. This is due to the increasing importance of the fluid-fluid interaction in the reduced pressure range between 0.2 and 0.3. We can conclude at this point that for a homogeneous surface, like the graphitic surface, the BET method can be successfully applied using argon at 87 K in the reduced pressure range from 0.05 to 0.2, rather than the usual recommended range of 0.05 and 0.3. 3.1.3. Incorrect use of void volume In characterization of real systems, the void volume per unit area is very large, and hence the contribution from the gas phase can be significant. It is, therefore, important that the void volume is determined correctly. Take a typical adsorption apparatus for characterization as an example, the volume is of the order of 10 cm3, and the typical specific surface area of a non-porous solid is 10 m2 per gram. Therefore with 1 g of solid the ratio of the void volume per unit area is 1 mm. In the simulation that we just performed this ratio is only of the order of 0.01 mm. If one wishes to carry out simulations with a large simulation box whose geometrical factor (volume to surface area ratio) mimics that of a real adsorption cell, a very large box with a height of 2000 times the collision diameter is required. This procedure is not only expensive in terms of computation time, but also extremely ineffective as practically most of the volume in the box is occupied by molecules in the gaseous phase. We suggest a different procedure to achieve our goal to obtain the total number of particle in the large simulation box. We take advantage of the fact that the excess number of particles in the small simulation box must be the same as that in the larger box (no matter how large it is), provided that these two boxes have the same surface area for adsorption. The height of the smaller box must be chosen such that the local density must approach the bulk gas density at distances well below the top boundary of the box to ensure that we have sufficient volume space for the gas phase. To prove that the surface excess of the smaller box is the same as that of a larger box, we carry out a simulation with a box whose height is 200 times the collision diameter (H= 68.1 nm), compared to the height of 6.81 nm of the smaller box. Results of the simulation of this large box are shown in Fig. 7 as plots of the number of particle versus pressure. As expected the number of particles in the gaseous phase becomes more significant,

Fig. 7. Plots of the total number of particle in the box, the gas phase number and the excess amount for the case of large simulation box having height equal 200 times the collision diameter for argon adsorption on graphite at 87.3 K.

Fig. 8. Plots of the surface excess versus pressure for the small box (height is 20 times the collision diameter) and very large box (height is 200 times the collision diameter).

compared to what we have seen earlier in Fig. 1 for much smaller simulation box (whose height is only one tenth of the height of the larger box), and therefore the correct determination of the void volume is important in the determination of the excess amount. Fig. 8 shows the plots of the surface excess (mol/m2) versus pressure for the smaller box (of Fig. 2) and the larger box that we are dealing with now, and it comes as no surprise that the surface excess is independent of the size of the simulation box, which it should be because we correctly determine the contribution from the gas phase. This result means that no matter what volume box we use, we get the same excess number from the simulations. This provides us with a methodology to obtain simulation data for an adsorption system whose accessible volume is very large, i.e. there is no need to carry out a simulation such that the geometric ratio is exactly the same as the real system. To get the total number of particles in the real system, we simply carry out the


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following steps, which are designed not only to save computation time but also to make the simulation far more effective: 1. Run a computer simulation with a reasonably sized simulation box, having an accessible volume Vacc and a surface area of Sacc. The result from this simulation is the total number of particles N. The number of particles in the gas phase is calculated from NG ¼ Vacc rG , and hence the excess number of particles is Nex ¼ NNG . 2. Now to construct the data for a much larger simulation box, we know that the excess number, Nex, of a large box must be the same as that of the small box, as long as the surface areas of these two systems are the same. So if the accessible volume 0 , then the total number in this large of the large box is Vacc 0 rG , where Nex comes simulation box is simply N0 ¼ Nex þVacc from the simulation of the smaller box. This will form a set of data for the large simulation box. Fig. 10. Plot of the excess number of particle for correct void volume and void volume over-estimated by 10%. 0.0030 CORRECT VOLUME WRONG VOLUME BY 10%

LHS of Linearized BET Equation

Once we have the data of the ‘‘large’’ simulation box, mimicking the real system, we can study the effects of wrongly estimating the void volume. Very often the void volume is over-estimated, and we will study the effects of this over-estimation on the excess number of particles, on the derivation of the BET surface area and finally on the calculation of the surface excess in terms of mol per unit area. First we use the data of the GCMC simulation with the simulation box size that we have described earlier (these data were presented in Figs. 1 and 2). We then construct the simulation data for the much larger box, using the procedure that we have just described above. The results for the large box with an accessible volume of 100 times greater than that of the smaller box are shown in Fig. 9, where we show the plot of the numbers of particles versus pressure. We see the significant contribution from the gas phase due to the large accessible volume. Taking this system, assuming that we over-estimate the void volume by 10%, the excess number of particle will be underestimated. This is shown in Fig. 10 as cross symbols; also plotted in this figure is the excess number using the correct accessible volume (grey symbols). We see a significant under-prediction of the excess amount. To see the impact of this under-estimation of the excess amount, we present in Fig. 11 the BET plot using this under-estimated excess amount (cross symbols). The reduced pressure range that we present the BET plot is from 0.05 to 0.2,

0.0025

0.0020

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0.0010

0.0005 0.04

0.06

0.08

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0.12

0.14

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0.18

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0.22

P/P0 (-) Fig. 11. BET plot for the case of over-estimating the void volume by 10%.

Fig. 12. Plots of the surface excess per unit area for the case of incorrect void volume (cross symbols) and the case of correct volume (grey symbols).

Fig. 9. Number of particles in a much larger simulation box, mimicking the real adsorption system.

and as before the BET plot for other ranges does not give good linearity. Also shown in this figure is the plot using the correct excess data (grey symbols).


1. To get good data on the excess amount, or the adsorbed amount per unit mass, we must obtain a reasonably correct estimate of the void volume. 2. To get good data on the surface excess per unit area, it seems that the incorrect determination of the void volume could be acceptable if the error of the void volume is o5%.

8

6

2

As expected, because of the under-estimated excess amount, the BET theory gives a lower BET surface area of 10 nm2, compared to the correct surface area of 11.6 nm2. Thus having an under-estimated excess amount and under-estimated BET surface area, we calculate the surface excess per unit area and the results are shown in Fig. 12. The difference between the correct surface excess (grey symbols) and the incorrect surface excess (cross symbols) is clearly noticeable. To show the propagation of error in the incorrect estimation of the excess amount and the surface excess, we consider the cases where the void volume is over-estimated by 5% and 20%. The results are shown in Fig. 13 for the excess amount and Fig. 14 for the surface excess per unit area. Based on what we have seen so far, we can draw a number of conclusions for adsorption of argon on a graphite surface at 87.3 K:


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4

2

0 0.1% -2

0.2%

CORRECT VOID VOLUME INCORRECT VOLUME BY 0.2% INCORRECT VOLUME BY 0.1% INCORRECT VOLUME BY 0.3%

0.3%

-4

0

20

40

60

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100

P (MPa) Fig. 15. Surface excess of argon adsorption on graphite at 298 K. The grey symbols are from incorrect solution while the cross symbols are from the exact solution. The simulation data are from the data of a box whose volume to area ratio is 2000 times the collision diameter.

Fig. 13. Plot of the number of particle versus pressure when the void volume is over-estimated by 5% (LHS figure) and 20% (RHS figure).

Fig. 14. Plot of the surface excess per unit area versus pressure when the void volume is over-estimated by 5% (LHS figure) and 20% (RHS figure).


10

10 CORRECT VOID VOLUME INCORRECT VOLUME BY 0.1% INCORRECT VOLUME BY 0.2% INCORRECT VOLUME BY 0.3%

6 4 2 0 0.1%

-2

0.2%

0

20

6 4 0.1%

2 0

0.2%

-2 0.3%

0.3%

-4

CORRECT VOID VOLUME INCORRECT VOLUME BY 0.1% INCORRECT VOLUME BY 0.2% INCORRECT VOLUME BY 0.3%

8

Surface Excess (µmol/m2)

8

Surface Excess (µmol/m2)

3339

-4 40

60

80

100

0

20

40

60

80

100

P (MPa)

P (MPa)

Fig. 16. Surface excess of argon adsorption on graphite at 298 K. The grey symbols are from incorrect solution while the cross symbols are from the exact solution. (a) The simulation data are from the data of a box whose volume to area ratio is 1000 times the collision diameter; (b) the simulation data are from the data of a box whose volume to area ratio is 400 times the collision diameter.

3.1.4. The significant impact of incorrect void volume in the calculation of the surface excess under supercritical conditions To show the effects of the over-estimation of the void volume in the calculation of the surface excess of argon adsorption at 298 K (supercritical conditions), we follow the procedure described in Section 3.1.3 to obtain the number of the particles in a large simulation box whose height is 2000 times the collision diameter and the accessible area is 100s2 (where s is the collision diameter). The geometrical ratio of this box is of the same order as the real adsorption apparatus that is used in practice. Fig. 15 shows the results of surface excess per unit area from the ‘‘exact’’ solution (correct void volume as the accessible volume) and from the solutions where the void volume is overestimated by just 0.1%, 0.2% and 0.3%. We see that the surface excess per unit area of the exact solution never shows negative value although it does possess a maximum. The maximum is simply due to the fact that the rate of change of the adsorbed concentration with pressure is less than that in the bulk gas. On the other hand, the surface excess from the incorrect solution becomes negative at high pressure. We find it hard to comprehend that an adsorbed density can be less than the bulk gas density; the adsorbed density on a planar surface should be greater because of the presence of the surface forces. What is remarkable here is that the surface excess per unit area is extremely sensitive to the void volume. Even when the void volume is over-estimated by merely 0.1% for a given mass of adsorbent, the change in the surface excess is extremely substantial. If we reduce the size of the adsorption cell such that the ratio of accessible volume to surface area is 1000s (instead of 2000s that we have just dealt with), we see a better improvement in the calculation of the surface excess per unit area as seen in Fig. 16a. A further reduction of the geometrical ratio to 400s, we can see further improvement (Fig. 16b). Therefore, it is strongly recommended that supercritical adsorption should be carried out with the minimum dead volume possible to avoid the contribution of errors in the estimation of the void volume.

4. Conclusions We have presented in this paper a careful evaluation of the BET theory for argon adsorption on a graphite surface. It is concluded

from this thorough computer simulation experiment that: 1. The range of BET plot should be between 0.05 and 0.2. 2. The molecular projection area of argon at 87.3 K is 0.138 nm2. 3. For sub-critical adsorption, the incorrect determination of the void volume by 5% is acceptable if the surface excess per unit area is the required solution. However, for surface excess per unit mass, a correct determination is essential. 4. For super-critical adsorption, the correct determination is essential, more so than that under sub-critical conditions, no matter we are interested in surface excess per unit area or surface excess per unit mass.

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