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Characterization of Micro-Mesoporous Carbonaceous Materials. Calculations of Adsorption Isotherm of Hydrocarbons D. D. Do* and H. D. Do Department of Chemical Engineering, University of Queensland, St. Lucia, Queensland 4072, Australia Received February 12, 2001. In Final Form: October 3, 2001 In this paper we apply a method recently developed by Do and co-workers1 for the prediction of adsorption isotherms of pure vapors on carbonaceous materials. The information required for the prediction is the pore size distribution and the BET constant, C, of a corresponding nonporous surface (graphite). The dispersive adsorption force is assumed to be the dominant force in adsorption mechanism. This applies to nonpolar and weakly polar hydrocarbons. We test this predictive model against the adsorption data of benzene, toluene, n-pentane, n-hexane, and ethanol on a commercial activated carbon. It is found that the predictions are excellent for all adsorbates tested with the exception of ethanol where the predicted values are about 10% less than the experimental data, and this is probably attributed to the electrostatic interaction between ethanol molecules and the functional groups on the carbon surfaces.
1. Introduction Equilibrium characterization of vapor/porous solid pairs is one of the prime requirements for adsorption study and design in separation and purification processes. Two fundamental aspects affecting this characterization are the pore structure (average pore size and pore size distribution) and the surface chemistry (functional groups and their distribution). The latter is only important where functional groups are playing an important role, that is, when the density of the functional groups is high and the adsorbate molecule is strongly polar. We shall not deal with strongly polar molecules in this paper and study only the adsorption mechanism whereby the dispersive forces are exerted by the proximity of pore walls on adsorbate molecules. There are a number of methods available in the literature for the calculation of adsorption isotherm from the knowledge of pore size distribution (PSD). Sophisticated methods such as the molecular dynamics (MD), Monte Carlo simulation, and the density functional theory (DFT) are theoretically capable of describing the adsorption in the pore system exactly.2-4 The Grand Canonical Monte Carlo simulations (GCMS) were also applied successfully.5,6 What we would like to present in this paper is to apply a new and simple method developed recently by Do et al.1,7 for the calculation of adsorption equilibria in activated carbon and carbonaceous materials. A number of adsorbates will be tested in this paper: benzene, toluene, n-pentane, n-hexane, and ethanol to illustrate the potential of our predictive model. * To whom all correspondence should be addressed. Phone: 617-3365-4154. Fax: 61-7-3365-2789. E-mail: duongd@ cheque.uq.edu.au. (1) Do, D. D.; Nguyen, C.; Do, H. D. Colloids Surf., A 2001, 187, 51-71. (2) Olivier, J.; Conklin, W.; Szombathely, M. Stud. Surf. Sci. 1994, 87, 81. (3) Olivier, J. J. Porous Mater. 1995, 2, 9. (4) Ravikovitch, P.; Vishnyakov, A.; Russo, R.; Neimark, A. Langmuir 2000, 16, 2311. (5) Lastoskie, C. M.; Quirke, N.; Gubbins, K. E. In Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., et al., Eds.; Elsevier: Amsterdam, 1997; p 745. (6) Gusev, V. Y.; O’Brien, J. A. Langmuir 1997, 13, 2822. (7) Nguyen, C.; Do, D. D. Langmuir 1999, 15, 3608.
Figure 1. Schematic diagram of carbon structure and adsorption mechanisms.
2. Theoretical Development Irrespective of the pore dimension, the process of adsorption in any pores can be visualized as follows.1 For a given pressure (fugacity) in the bulk fluid phase, the interior of a pore can be viewed as a space where higher density of molecules can be found due to the external forces acting on them. These molecules move around the pores, but statistically there is a spatial distribution of these molecules across the pore. This is due to the interactive forces between those molecules and the pore surface atoms, and the strength of such forces is a function of distance from the pore surface. The density of these molecules is very high near the surface due to these forces, while away from the surface the density is lower (statistically speaking). To model this physical process we shall assume that such a distribution is a step function, that is, uniform high density near the surface and uniform lower density near the center of the pore. We use the terms adsorbed molecules for those near the surface and occluded molecules for those near the center of the pore (Figure 1). The model presented in this paper rests on the following assumptions: (i) Activated carbon has pores of slit shape. (ii) The adsorption mechanism comprises of two steps in series: multilayering followed by pore filling.
10.1021/la010232r CCC: $22.00 © 2002 American Chemical Society Published on Web 12/06/2001
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(iii) The dispersive force is the dominant adsorption force. (iv) The concentration of adsorbed phase in the multilayering step is described by a BET-type equation, with allowance for the enhancement in the C constant. (v) The pore filling step is described by the Kelvin equation. (vi) The adsorbed phase density is the same as the corresponding liquid density. 2.1. Potential Energy. In general the density of the molecules inside a pore is much higher than that in the bulk fluid phase, which is due to the attractive forces between the pore surface atoms and those molecules. Assuming the dispersive force as the dominant force in adsorption, we use the 10-4-3 Steele potential8 to describe the potential energy between a molecule and a surface as
[(
)
1 σfs φ(z) ) φw 5 z
10
( )
1 σfs 2 z
4
-
σfs4
]
6∆(z + 0.61∆)3
(1)
Figure 2. Potential energy profiles for nitrogen (H ) 0.72, 1, 1.5 nm, dashed lines) and benzene (H ) 0.88, 1, 1.5 nm, continuous lines).
where φw ) 4πFsσfs2∆fs. For a slit pore having two walls with the outermost layers separated by a distance H, the potential energy equation between an adsorbate molecule and the pore is
φP ) φ(z) + φ(H - z)
(2)
2.2. Layering Adsorption Mechanism. The adsorption energy in a pore is always greater than that of a corresponding flat surface having the same surface chemistry. What this means is that if the fluid pressure inside a pore is Pp, the adsorption mechanism in a pore is viewed as that of molecular layering, a process akin to that of BET. We take the form of the BET equation to describe the partition between molecules in the adsorbed phase and those in the pore. This equation is
Cp(H)xp t ) tm (1 - xp){1 + [Cp(H) - 1]xp}
(3)
Figure 3. Plot of the heat of adsorption versus pore width for nitrogen and benzene.
adsorbate molecules wet the surface and hence the contact angle is taken as zero.
tm ) (vM/NA)1/3
3. Results and Discussion
xp ) Pp/P0
3.1. Adsorption Energy, BET Constant and Enhancement Factor. The enhancement due to overlapping of the potential fields contributed by two opposite walls of a pore is shown in Figure 2 as plots of potential energy profiles across the pore for nitrogen (dashed lines) and benzene (continuous lines) at three different pore widths. For large pores two minima are observed, while for smaller pores two minima are collapsing into one, which occurs at the center of the pore. Such overlapping of the two minima also results in a much lower potential energy, meaning that the adsorption is stronger in small pores. For example, for benzene the adsorption is very strong when the pore width is about 0.88 nm. For pores having width smaller than 0.88 nm, the potential energy is increased due to the greater significance of the repulsive forces. The minimum of the potential energy profile is taken as the heat of adsorption. As seen in Figure 2, this heat of adsorption is a function of pore width, and in general the smaller the pore width, the greater the heat of adsorption. This functional dependence of the heat of adsorption on pore width is illustrated in Figure 3 for nitrogen and benzene. Here we see that the heat of adsorption increases with a decrease in pore width, and once the pore width is less than about one collision diameter, σfs, the heat of adsorp-
The BET constant Cp of a pore is related to the BET constant C of a corresponding flat surface given by the following equation, Cp ) C exp[(Qp - Q)/RT], where Q is the heat of adsorption of the flat surface and Qp is that of a pore. 2.3. Pore Filling Mechanism. The second stage of adsorption is the pore filling process, which is governed by
σss γvM cos θ H -t) 2 2 RT ln(Pp/P0)
(4)
which is simply the Kelvin equation for slit pore,9 but the fundamental difference here is the use of the pore pressure Pp instead of the bulk gaseous pressure. Justification of the use of eq 4, originally derived for large pores where continuum holds, for pores of all size including small pores has been addressed by Do et al.1 In the absence of information on the contact angle, we will assume that the (8) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: Oxford, 1974. (9) Innes, W. B. Anal. Chem. 1957, 29, 1069.
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Figure 4. Reduced pore filling pressure of (a) nitrogen (77 K) and (b) benzene (303 and 333 K (dashed line)) versus pore width (center surface atom to center surface atom). The symbols are DFT data of Latoskie et al.5
tion begins to decrease with further decrease in the pore size, which is due to the significance of the repulsive force. In the generation of Figures 2 and 3, we have used the molecular parameters from Reid et al.,10 surface tension from Jasper,11 and the BET constant for a flat surface from Kaneko.12,13 3.2. The Pore Filling Pressure versus Pore Width. The adsorption mechanism in pores occurs via the layering mechanism followed by the pore filling process. As mentioned before, the pressure at which pore filling occurs is lower for smaller pores. Figure 4 shows the pore filling pressure for nitrogen adsorption in pores at 77 K (Figure 4a) and benzene at two temperatures 303 and 333 K (Figure 4b) as a function of pore width. For nitrogen of Figure 4a, the continuous line is the result of our theory while the symbols are the DFT results of Lastoskie et al.5 As seen in that figure, the agreement between our theory and the DFT results is remarkable. When the pore width is greater than about 10 nm, the pore filling pressure is close to the vapor pressure as one would expect for large pores. When the pore width is decreased (less than 5 nm), the pressure required to fill such pore is smaller and this filling pressure will reach a minimum when the pore width is approximately equal to the collision diameter, σfs. Any further decrease in the pore width will result in a sharp increase in the pore filling pressure, which is due to the significance of the repulsive force as discussed earlier. The effect of temperature on the pore filling pressure is shown in Figure 4b for benzene at 303 and 333 K. The BET constant for benzene at 303 K is 47 (Kaneko12), and that at 333 K is calculated according to the formula (10) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The properties of gases and liquids; McGraw-Hill: New York, 1988. (11) Jasper, J. J. Phys. Chem. Ref. Data 1972, 1, 841. (12) Kaneko, K. Private correspondence, 1999. (13) Kaneko, K. J. Membr. Sci. 1994, 96, 59.
[
C ) exp
]
(Q1 - QL) RT
(5)
The differential heat between the heat of adsorption of the first layer and the heat of liquefaction is assumed to be constant. This is reasonable for temperatures below the critical temperature (for benzene Tc ) 562.1 K). Knowing the BET constant at 303 K to be 47, we obtain Q1 - QL ≈ 9.7 kJ/mol. Hence the BET constant at 333 K is estimated to be 33. We see from the figure that the pore filling pressure (plotted in reduced scale) does not vary much with temperature. The one at higher temperature is above the lower temperature curve, meaning that for a given pore width the reduced pressure at which the filling occurs is slightly higher for higher temperature. 3.4. Adsorption Isotherm. The theory presented in this paper provides us with a means to calculate the adsorption isotherm when the pore size distribution and the BET constant for a corresponding flat surface are known. The inverse problem to this is that when the adsorption equilibria data are available over a fairly wide range of pressure (reduced bulk pressure from about 10-5 or 10-6 to nearly unity), the theory can be used to derive the pore size distribution. To achieve either the task of predicting the adsorption capacity when the PSD is known or the task when the PSD is required from the knowledge of accurate adsorption data, we require a mathematical procedure to relate them. We let f(H) be the pore volume distribution (m3/kg/m), with f(H) dH being the volume of pores having width falling between H and H + dH. For a given bulk gas phase pressure P, the critical pore width calculated from eq 4 is denoted as HK(P), and pores having a size smaller than this critical width will be filled with adsorbate, while for pores having a size greater than HK, layering of molecules is occurring. Mathematically the total amount adsorbed (in m3/kg) in geometrical volume is given by
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2t(H,P)
∫σH (P) f(H) dH + ∫H∞ (P) H - σss f(H) dH K
ss
K
(6)
The first term in the right-hand side of eq 6 is the volume of pores filled with adsorbate molecules while the second term is volume of adsorbate molecules formed as layers in pores having width greater than HK(P). With the current knowledge of the state of the adsorbed phase in the micropore not fully understood, the adsorbed phase density is assumed to be equal to the liquid density. This assumption is reasonable in mesopores, and for micropores of molecular dimension the adsorbed phase density is expected to be either greater or smaller than the corresponding liquid density, depending on the ratio of the adsorbate size to the pore physical size. If this ratio is an integer, the adsorbed phase density is expected to be greater than the liquid density due to the stronger attraction forces of the two opposite walls. However, if the ratio is not an integer, the adsorbed phase density could be less than the liquid density due to the spatial constraint. Therefore for solids having a pore distribution, it is reasonable to assume that the adsorbed phase density is the same as the corresponding liquid density. Various workers have studied the state of the adsorbed phase.14-16 Assuming liquid density of the adsorbed phase, the overall adsorbed concentration is simply
Cµ ) W/vM
(7)
where vM is the liquid molar volume (m3/mol). Equations 6 and 7 provide the mathematical representation of the adsorption isotherm, that is, given a distribution f(H), one can simply integrate eq 6 to obtain the liquid volume adsorbed and then the molar adsorbed concentration from eq 7. In the literature, many workers have assigned a continuous distribution function (or a combination of continuous functions) to describe the pore size distribution, with continuous distribution functions such as normal, log-normal, γ, and β distribution functions. However, micropore size distribution is usually very rugged and is not properly described with a continuous function or even a combination of continuous functions. What we shall do here is to present a discrete way of presenting the pore size distribution, that is, the PSD is mathematically described by a set of
(Hi,Wi) for i ) 1, 2, ..., m
(8)
where the pore range has been divided into m subranges, and for the ith subrange the representative pore width is Hi and the pore volume of that subrange is Wi. Let HM be the pore width of the Mth subrange such that all subranges having Hi < HM will be filled with adsorbate molecules while those ranges having widths greater than HM will be layered with adsorbate molecules. This means that the volume occupied by adsorbate molecules in the solid is given by M
W)
∑ j)1
m
Wj +
∑
j)M+1
[ ] 2t(Hj,P) Hj - σss
Wj
(9)
This equation is simply the discrete version of eq 6. The width HM is the critical width determined from eq 4, and (14) Kaneko, K. Colloids Surf. 1996, 109, 319. (15) Kaneko, K. Carbon 2000, 38, 287. (16) Ustinov, E. Russ. J. Phys. Chem. 1988, 62, 797.
Figure 5. Fitting of theory with benzene adsorption on Ajax activated carbon at 303 K.
it is a function of the pressure. Knowing the volume, the molar adsorbed concentration is calculated from eq 7. For computation, the following algorithm can be used: 1. Start with i ) 1. 2. For a given pressure, calculated the bulk gas reduced pressure, x ) P/P0, and then the pore reduced pressure is calculated from
[
xp(Hi) ) x exp -
]
RE0(Hi) RT
3. If xp(Hi) > 1, all pores of the ith subrange are filled with adsorbate molecules. Otherwise, those pores could be either filled with adsorbate or layered with adsorbate molecules. We check this by investigating the inequality
γvM σss Hi > t(Hi,P) + + 2 2 RT ln[xp(Hi)] If this is true, then pores are layered with adsorbate molecules occupying a volume of
[
]
2t(Hi,P) W Hi - σss i
otherwise, adsorbate molecules will occupy the full volume W i. 4. Repeat the steps 1 to 3 until all subranges have been dealt with. 5. Calculate the total volume occupied by adsorbate molecules in all pores (eq 9). 6. Calculate the molar adsorbed concentration from eq 7. 3.5. Inverse Problem of PSD Determination. Pore size distribution is usually determined from adsorption equilibria data. This is obtained by inverting eq 9.17 Usually the distribution covering the micropore and mesopore range is sufficient in the study of adsorption equilibria as adsorption in macropores is negligible. So we first choose a range of pore width, usually from 1 to 20 nm, and then choose a number of subranges. Usually the number of subranges of 30-50 is adequate with more subranges being concentrated in the micropore range. After this has been decided, the optimization task is one to determine the volumes for each of the subranges, that is, Wj (j ) 1, 2, ..., m). Thus for a given set of adsorption (17) Nguyen, C.; Do, D. D. Langmuir 2000, 16, 1319.
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Figure 6. Plot of pore volume versus pore width for the case of fitting against benzene data at 303 K.
Figure 9. Prediction of adsorption isotherm of benzene at 293, 303, 313, and 333 K using benzene PSD.
Figure 7. Pore size distribution of activated carbon obtained from the fitting of benzene data at 303 K. Figure 10. Prediction of the adsorption isotherm for toluene at 303, 333, and 363 K.
Figure 8. Pore adsorption isotherm in pores having width 1.21 (A), 1.33 (B), 1.47 (C), 1.62 (D), 1.78 (E), 1.96 (F), 2.15 (G), 2.37 (H), 2.61 (I), and 2.87 nm (J).
equilibrium data (Pk,Cµk) for k ) 1, 2, ..., N, the optimization is carried out by minimizing the following residual: N
min R )
{
M(Pk)
∑ ∑ j)1
k)1
m
Wj +
∑
j)M+1
[
]
2t(Hj,Pk) Hj - σss
}
2
Wj - vMCµk
(10)
This optimization is effectively carried out with built-in routines in MatLab with an optimization toolbox. Optimization programs with constraint are used to acknowledge the fact that volumes are non-negative. Instead of
Figure 11. Prediction of the adsorption isotherm for n-pentane at 303, 333, 353, and 423 K.
optimizing eq 10 for all parameters, Wj (j ) 1, 2, ..., m) simultaneously, we adopt the approach to optimize for subrange volume one at a time. This is the case, because in this type of adsorption problem we recognize that at low pressure small pores are filled and when pressure is increased the small pores have already been filled so it is larger pores that are now being filled. It is this kind of progressive filling (adsorption) that we decide to optimize for the volume of the first subrange first. Once this is done, the volume of the second subrange is optimized next.
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Figure 12. Prediction of the adsorption isotherm for n-hexane at 288 K.
Figure 13. Prediction of the adsorption isotherm for ethanol at 273 and 303 K.
We repeat this until all subrange volumes have been optimized. At the end of this one-pass, residual is not necessarily minimized. We carry out the same sequence until there is no further change in the residual. We test this optimization procedure with benzene adsorption data at 303 K on a commercial activated carbon having the following properties: particle density 0.733 g/cm3, total porosity 0.71, macropore porosity 0.31, micropore porosity 0.4, BET surface area 1200 m2/g, micropore volume 0.44 cm3/g, and mesopore surface area 82 m2/g. We choose 30 subranges between 1 and 30 nm, with 25 subranges being concentrated in the micropore range. The result of the optimization gives an excellent fit between the theory and the data (Figure 5). The plot is done as a semilogarithm plot to show the excellent nature of this optimization. The result from the fitting is a set of volumes of all the subranges, which is plotted in Figure 6 as a function of pore width. Knowing the volume for each pore range, we calculate the pore volume distribution ∆W/∆H ) (Wj+1 - Wj)/(Hj+1 - Hj) versus H, and the results are shown in Figure 7. The adsorption isotherm as shown in Figure 5 does not reveal the individual contribution from each pore subrange. For a given reduced pressure in the gas phase, the fractional loading of small pores is greater than that of
larger pores due to the stronger adsorption affinity of smaller pores. But the amount of adsorbate adsorbed in a pore depends on the volume of that pore. Given pore volumes of this activated carbon as shown in Figure 6, Figure 8 shows the individual contribution of pores having widths of 1.21, 1.33, 1.47, 1.62, 1.78, 1.96, 2.15, 2.37, 2.61, and 2.87 nm. We see that even though the smallest pore range having a width of 1.21 nm has highest adsorption affinity, the amount adsorbed is quite small due to the small volume of that subrange. This information of individual contribution from all subrange can only be available with the theory. Before we discuss the potential of this theory in predicting the adsorption isotherms of various hydrocarbons at different temperatures, we would like to discuss the significance of the optimization procedure adopted in this work. As mentioned before, our procedure is a sequential one, with the optimization being done with one subrange at a time. Two aspects need to be considered here. In the first aspect, like any other optimization, the choice of the initial guess can be very critical. We shall study how this would affect the task of determining the PSD. The second aspect deals with the way subranges can be optimized one at a time. Although we have discussed that the first subrange volume should be optimized first and then the procedure repeated for the next subrange until no further change in the residual is observed, we shall investigate this procedure by optimizing the last subrange first and move backward. First let us consider the choice of the initial guess for the volumes of all subranges. The most logical choice is to choose equal volume for all subranges. By knowing the total pore volume VT (from adsorption isotherm), we can choose the initial guess for all subranges as VT/m, where m is the number of subrange. However, sometimes adsorption data are not collected up to a high enough pressure, so that the total pore volume is not known. If that is the case, we investigate the situation where the initial guesses for all subranges are taken to be zero. As comprehensively tested with our sequential optimization procedure, we have found that the convergence is always achieved with this zero initial guess. Typically 10 iterations are required to reach convergence. When the uniform initial guess is used, we also achieved convergence with
Micro-Mesoporous Carbonaceous Materials
the converged solution very close to that when zero initial guess is used. The BET-type eq 3 has been used to describe the layering process in pores. Here we study the effect of using the Aranovich equation18 to describe the layering, and it is found that there is no difference in the PSD when either the BET or Aranovich equation is used. This is because the difference in these two equations is only for reduced pressures greater than 0.3, beyond which the process of pore filling is operating. 3.7. Prediction of Adsorption Isotherms. Having used the benzene data at 303 K to derive the pore size distribution (Figure 7), we would like now to use that PSD to predict the adsorption isotherm of benzene for 293, 313, and 333 K. The results are shown in Figure 9, and it is seen that the predictions are excellent. The plots are plotted as a semilog plot to enlarge the scale of pressure. Having a successful prediction of benzene adsorption isotherm at three temperatures, we now turn to the prediction of isotherm of another adsorbate, toluene, at 303, 333, and 363 K. The results are shown in Figure 10. We see that the agreement is excellent, despite the simplicity of the model proposed in this paper. We further test the model with the paraffin, n-pentane, and n-hexane. The results are shown in Figures 11 and 12, respectively, where once again we see the very good prediction of the model. (18) Aranovich, G. L. J. Colloid Interface Sci. 1991, 141, 30.
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We finally test the model with another class of adsorbate, the alcohol. Here we use ethanol as a model compound. The predictions are shown in Figure 13 for two temperatures 273 and 303 K. The agreement between the theory and the experimental data (symbols) is excellent at pressures much less than the vapor pressure. When the pressure is high, the experimental data are consistently higher than the theoretical predictions. We attribute this to the polar nature of the ethanol molecule, and the additional adsorption could be due to the electrostatic interaction between ethanol molecules and the functional groups on the carbon surface. 5. Conclusions We have presented in this paper a simple tool for the prediction of the adsorption isotherm of hydrocarbon adsorbed on carbonaceous materials. The mechanism of adsorption is that of sequential steps of multilayering and pore filling, which is applicable to all pores. The method was applied to adsorption data of benzene at 303 K to derive PSD, from which the predictions of adsorption isotherms of toluene, n-pentane, n-hexane, and ethanol are excellent, with the exception of ethanol where some deviation at high pressure is observed. This is attributed to the polar nature of the ethanol molecule. Acknowledgment. This work is supported by the Australian Research Council. LA010232R
