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Langmuir 1999, 15, 3608-3615
A New Method for the Characterization Of Porous Materials C. Nguyen and D. D. Do* Department of Chemical Engineering, University of Queensland, St. Lucia, Queensland 4072, Australia Received September 1, 1998. In Final Form: February 9, 1999 A simple new method of characterization of porous materials is developed in this paper. In this method, we combine the traditional Kelvin equation with an improved statistical adsorbed film thickness to account for the finite dimension of pores. The adsorption process in a pore is viewed as a molecular layering process followed by a filling mechanism. Here, we argue that this mechanism is valid not just for mesopores but also micropores because of our allowance for the statistical film thickness to be a function of not only pressure but pore size as well. The model is tested against the data of nitrogen adsorption onto activated carbon samples: the fitting is found to be excellent, and the results are comparable with those obtained by other methods such as the DFT method. The much shorter computation time required by this method compared to that by DFT makes this method a very attractive alternative in the PSD characterization.
I. Introduction The distribution of pore size is the most important aspect in the field of adsorption because it is part of the basic information about a porous material. Other factors of importance are pore topography, surface chemistry, fractal dimension, and so forth. The pore size distribution (PSD) affects many physical properties of the adsorbent, as well as the physical adsorption of adsorbate via dispersive forces. The IUPAC classification, in which pores are classified into micro-, meso-, and macropores1 is mostly based on the difference in the adsorption mechanisms occurring in those pores: (i) the enhanced adsorption in micropores, (ii) the capillary condensation in mesopores, and (iii) the bulk liquid condensation of vapors in macropores. The adsorption in macropores becomes significant only at very high pressures (above the relative pressure of 0.95 for most subcritical vapors). At low and medium pressures, adsorption is significant only in microand mesopores. As a consequence, the term “PSD” measured by physical adsorption by and large refers to the distribution of micro- and mesopores. In the traditional methods of characterization, mesoand micropores are characterized separately with different equations to reflect the different mechanisms of adsorption in those pores. The diameters of the mesopores are normally calculated using the Kelvin equation or from the volume of mesopore if the mesopore surface area is known. They can also be obtained by using techniques such as mercury porosimetry.2 Other techniques, such as small-angle X-ray scattering or electron microscopy, can be employed to measure the mesopore size of some materials directly. The problem with micropore characterization, on the other hand, is still far from being resolved, despite the fact that there are quite a number of procedures for micropore size determination such as the DS, MP, and HK methods and so forth. These methods have been used quite frequently in the literature; however, * Corresponding author. (1) Greg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic Press: London, 1982. (2) Webb, P. A.; Orr, C. Analytical Methods in Fine Particle Technology; Micromeritics Instrument Corporation, 1997.
as pointed out by Russel et al.,3 they all suffer some degree of inconsistency and limitations. For instance, the DS equation requires an assumption of a normal distribution for micropores, and the HK method suffers from the idealization of the micropore filling process. Furthermore, the analysis of these methods may be cumbersome if the contributions of meso- and micropores cannot be delineated. It is also known that each of these methods is applicable only to a particular type of isotherms and, more specifically, to limited regions of the isotherm. Sophisticated methods such as the molecular dynamics (MD) and Monte Carlo simulation are theoretically capable of describing the adsorption in the pore system exactly. However, their application is not widespread because of the large number of calculations required, even for a limited number of particles.4 The density functional theory (DFT) has been proved to be a practical alternative to both molecular dynamic and Monte Carlo simulations. It provides an accurate description of the adsorption in inhomogeneous systems while requiring less computing time than the MD and the Monte Carlo simulation methods.2 Grand Canonical Monte Carlo simulations (GCMS) were also applied successfully to many adsorption problems. However, the calculation of the DFT as well as the GCMS methods still require some assumptions, which limit their application to adsorbates with a simple molecular structure such as nitrogen, argon, methane and so forth.5-7 Moreover, they can be very demanding in computational time and require parameters which must be obtained by matching the theory with macroscopic properties obtained experimentally such as the vapor pressure, the liquid density and the adsorption onto a nonporous material. (3) Russel, B. P.; LeVan, D. M. Carbon 1994, 32, 845. (4) Gusev, V. Y.; O’Brien, J. A. Langmuir 1997, 13, 2815. (5) Lastoskie, C. M.; Quirke, N.; Gubbins, K. E. In Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steel, W. A., Zgrablich, G., Eds.; Elsevier Science B. V.: New York, 1997; p 745. (6) Ehrburger-Dolle, F.; Gonzalez, M. T.; Molina-Sabio, M.; RodriguezReinoso, F. Characterization of Porous Solids IV; McEnaney, B., Mays, T. J., Rouquerol J., Rodriguez-Reinoso, F., Sing, K. S. W., Unger, K. K., Ed.; The Royal Society of Chemistry: Cambridge, 1997. (7) Samios, S.; Stubos, A. K.; Kanellopoulos N. K. Langmuir 1997, 13, 2795.
10.1021/la981140d CCC: $18.00 © 1999 American Chemical Society Published on Web 04/24/1999
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In this paper, we develop a new and simple method for characterization of pore size distribution. This method will utilize the combination of the Kelvin equation (to account for condensation) and the statistical adsorbed film thickness (to account for molecular layering). The key difference between our theory and the others in the literature is the allowance of the statistical film thickness to bear a dependence on the pore size. This will allow us to extend the applicability of the Kelvin equation from the meso- to the micropore region. As a result, the model unifies the mechanism of adsorption in both micro- and mesopores, and therefore, there is no need for the demarcation of these pores in the characterization for the pore size distribution. In our method, the PSD of both micro- and mesopores is calculated from the adsorption isotherm of a subcritical vapor. The model will be tested against the nitrogen adsorption data available in the literature. II. Theoretical Section The theory is developed on the basis of the assumption that the knowledge about the adsorption on a flat surface can be extended to describe the adsorption occurring in a pore. This approach assumes that the surface chemistry of the flat surface and the pore walls surface is similar, so that the pore dimension is the only factor that makes adsorption in the pore different from that occurring on a flat surface. To some extent, this principle has been employed in methods such as the t or R plot methods.1 We now first briefly describe some basic equations for a flat surface and then propose proper equations applicable for pores of finite dimension. 1. Adsorption on a Flat Surface. a. BrunauerEmmett-Teller (BET) Equation. Adsorption of a subcritical vapor on a flat surface with no limit on the thickness of the adsorbate is usually described by the BET equation of the following form.
CP V ) Vm (Po - P)[1 + (C - 1)(P/Po)]
(1)
where Vm is the volume of adsorbate required to cover the flat surface with a monolayer of adsorbate, and the coefficient C which is a measure of the adsorptive power on a unit surface is defined as follows:
C ) Ae(a-l) with a ) Q/RT and l ) λ/RT
(2)
Here λ is the heat of liquefaction, and Q is the energy of interaction between the first layer of adsorbate and the surface. For a flat surface, this energy of interaction is a constant. However, when we deal with a pore of finite dimension, this energy is a function of pore size. b. Statistical Thickness t. Knowing the amount V adsorbed onto a flat surface as given in eq 1, the statistical adsorbed film thickness can be calculated as t ) V/S where S is the area of the flat surface. Thus, the thickness of a single adsorbed layer is tm ) Vm/S, where Vm is the monolayer capacity. The statistical thickness t then can be expressed explicitly in terms of pressure as
t ) tm
CP (Po - P)[1 + (C - 1)(P/Po)]
(3)
It should be noted that for a given reduced pressure a surface having a higher C has a larger thickness of adsorbed layer. Despite the fact that the exponential factor
A is characteristic for a solid material, constant C in a pore will be different from that for a flat surface even when they are made of the same material. This is due to the enhancement of the dispersive forces in a pore as we will discuss later. 2. Adsorption in a Confined Space. Having discussed some basic equations for a flat surface, we now turn to the discussion about pores of finite size. In such pores, the number of adsorbate layers is limited by the pore size, and this number is approximately given as n ) d/tm, where d is the characteristic dimension of a pore, which can be either pore radius or pore half-width. Being statistical, the maximum number of layers n can take a noninteger value. This is not a serious problem because during the multilayering process, the adsorbate molecules do not necessarily form a complete layer before another layer can be started. Furthermore, like the statistical adsorbed film thickness t, n will be used only as a statistical parameter. Brunauer et al.8-10 have developed a number of modified BET equations to allow for the molecular layering in pores of finite dimension. The additional parameter introduced is the parameter n, which is the number of layers that can be accommodated by one wall of the pore. Because the adsorption in a pore takes place at sites opposite to each other across the pore center, the growing of the adsorbed layer with pressure is limited at the center of the pore. The merging of two opposite layers occurs for smaller pores, whereas for larger pores, the condensation occurs before the layers can meet. The modified BET equation has the following form:10
V Cx × ) Vm 1 - x 1 + (nf/2 - n/2)xn-1 - (nf + 1)xn + (nf/2 + n/2)xn+1 1 + (C - 1)x + (Cf/2 - C/2)xn - (Cf/2 + C/2)xn+1 (4) where V is the volume adsorbed by one wall of the pore. Here, x is the reduced pressure ()P/Po) and f ) exp(∆l), where ∆l is the excess of the reduced evaporation heat due to the interference of the layering on the opposite wall. When n approaches infinity, eq 4 reduces to the classical BET equation (eq 1). By allowing the parameter n to take a noninteger value due to its statistical nature, the statistical adsorbed film thickness t for adsorption in slitlike pores then will be calculated as:
Cx t × ) tm 1 - x 1 + (nf/2 - n/2)xn-1 - (nf + 1)xn + (nf/2 + n/2)xn+1 1 + (C - 1)x + (Cf/2 - C/2)xn - (Cf/2 + C/2)xn+1 (5) We will use this statistical film thickness in our theory to determine the pore size distribution. To achieve this, we need to address the coefficient C for pores of finite dimension. For the same adsorbate and adsorbent system, the parameter C for a finite pore is not the same as that in the case of a flat surface due to the enhancement of (8) Brunauer, S.; Emmet, P. H.; Teller, E. J. Am. Chem. Soc. 1938, 60, 309. (9) Brunauer, S.; Deming, L. S.; Deming, W. E.; Teller, E. J. Am. Chem. Soc. 1940, 62, 1723. (10) Brunauer, S. The Adsorption of Gases and Vapours; Oxford University Press: London, 1945.
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adsorption in a finite pore. We will discuss this in further details in Section 4. 3. Heat of Adsorption. Heat of adsorption can be calculated as the decrease in the potential energy of the adsorbate when transferred from the bulk gas phase to the adsorbed state. Thus, theoretically, the heat of adsorption Q at zero loading can be taken as the depth of the potential energy profile of the first adsorbed molecules. We now discuss the various potential energy functions commonly used in the literature. a. Potential Energy of Molecules above a Flat Surface. The potential energy of a molecule locating at a distance z from a surface can be calculated by summing the pairwise 12-6 potential energy between this molecule and all of the surface atoms. If the distance between the adsorbate molecule and the solid surface is large compared with the distance between two adjacent surface atoms, the summation of all pairwise potential energies can be replaced by a process of simple integration. The results of this process are the 10-4 potential if the surface is a single lattice layer, the 9-3 potential if it is composed of a dense layer of solid atoms, and the 10-4-3 or the sum of 10-4 potential if it is composed of parallel lattice layers.11 b. Potential Energy of Molecules in a Pore. The potential energy of interaction between a molecule placed into the interior of a slitlike pore and its pore walls is simply the summation of the potential energies between the molecule and each of the two opposite pore walls, as discussed in the last paragraph. The potential energies can be calculated by summing the two potential energy equations, one for each pore wall, and its form could be a 10-4, 9-3, or 10-4-3 potential, as discussed above. Application of these analytical potential equations is reasonable for larger pores, where the distance between the adsorbate molecule and the surface is much larger than the solid interatomic distance. For pores of small dimension, especially for those having diameters on the order of the adsorbate molecule size, this assumption is no longer valid. In this case, the summation of all pairwise 12-6 potential energy of interaction will provide a more accurate description of the potential energy between the molecule and the pore, and this is the approach we adopt in this paper. The overlapping of the force fields exerted by the two opposite pore walls changes with the pore width. As a result, the potential enhancement is highest when the pore half-width is approximately equal to the collision diameter. The exact pore half-width at which the enhancement is greatest depends on the pore configuration chosen. Knowing the potential energy of the adsorbate in a pore, the heat of adsorption Q at zero loading can be taken as the depth of the potential energy profile. 4. BET C Coefficient. Let us now discuss the behavior of the constant C in the BET equation when dealing with pores of finite dimension. The constant C is a function of the heat of adsorption, as defined in eq 2; it will therefore be a function of the pore size because Q is a function of pore size as discussed in the last section. Here, we use the subscript s and p to denote the parameters of adsorption onto a flat surface and a pore, respectively; that is, the BET C coefficient and the heat of adsorption for a flat surface will be denoted as Cs and Qs, and those for a pore will be Cp and Qp. The eq 2, rewritten here for them are as follows: (11) Steele, W. A. The Interaction of Gases with Solid Surface; Pergamon Press: Oxford, 1974.
Nguyen and Do
Cs ) A exp
( (
Cp ) A exp
Qs λ RT RT
) )
Qp λ RT RT
(6a) (6b)
The heats of adsorption are calculated as the minima of the potential energies corresponding to a flat surface and a pore, as discussed in the above sections. For adsorption onto a flat surface and a pore of the same material at a given temperature T, the preexponential factors of eq 6 are the same. Dividing eq 6b by 6a, we get the coefficient of the n-BET equation Cp of a pore written in terms of the BET coefficient Cs of a flat surface
Cp(r) ) Cse[Qp(r)-Qs]/RT
(7)
5. Pore Filling Process. a. Kelvin Equation. The capillary condensation of a vapor in a confined space is described by the Kelvin equation.1,2 This equation in conjunction with an equation for the statistical thickness t, which accounts for the layering of the adsorbate prior to the capillary condensation stage, can be used to calculate the dimension of the pore where the condensation takes place. The combination, the so-called the modified Kelvin equation, has the following form:
rK(P) - t(P) ) -
2γνm cos θ RT ln(P/Po)
(8)
This equation has proved to be useful in calculating the diameter of pores in the mesopore range. However, it fails when applied to narrower pores, especially for pores in the micropore range.12,13 The modified Kelvin equation gives a smaller pore diameter compared with that obtained by other independent methods, such as X-ray small-angle scattering. The fact that the Kelvin method is based on the bulk liquid properties is often used to explain the failure of the method. For instance, there are arguments which claim that because of the enhancement of the potential energy in narrower pores the density and the surface tension of the adsorbate are different from those of the bulk liquid at the same temperature. These arguments led to attempts to introduce empirical equations, which allow for the change in the properties of the adsorbed phase in small pores. As a result, new parameters have been introduced into the Kelvin equation to modify the surface tension or density, and the estimation has been improved significantly.13,14 b. Modified Kelvin Equation for Pores of Finite Dimension. In our view, however, the failure of the Kelvin equation in predicting the diameter of smaller pores rests mostly on the failure of the estimation of the statistical adsorbed film thickness t. In the traditional approach, the statistical thickness t is calculated by using a standard isotherm, which was measured on a reference nonporous sample. With regards to the above discussions, this is justified only when the pore is wide enough that the effect of the proximity of the pore walls is negligible. In larger pores, the assumptions for the BET equation to be applicable are by and large satisfied, and the heat of (12) Lastoskie, C.; Gubbins, K. E.; Quirke, N. J Phys. Chem. 1993, 97, 4786. (13) Maglara, E.; Kaminsky, R.; Conner, W. C. Characterization of Porous Solids IV; McEnaney, B., Mays, T. J., Rouquerol, J., RodriguezReinoso, F., Sing ,K. S. W., Unger, K. K., Ed.; The Royal Society of Chemistry: Cambridge, 1997; p 25. (14) Evans, R.; Marini Bettolo Marconi, U. Chem. Phys. Lett. 1985, 114, 415.
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Langmuir, Vol. 15, No. 10, 1999 3611
adsorption in those pores is similar to that of adsorption onto a flat surface. On the other hand, the space for adsorbate molecules in narrower pores is far from being infinite for the application of the traditional statistical t-thickness. Moreover, as explained above, the heat of adsorption in narrower pores is higher than that of adsorption onto a flat surface. Thus, the implementation of the standard statistical thickness t in the calculation of the diameter of the narrower pores is a gross simplification. This problem can be resolved by using the modified n-BET equation to account for the finite space in the narrower pores, and the effects of the potential energy enhancement are accounted for by using eq 7 for the Cp coefficient calculation. The model equation for the layering process will be Figure 1. Model of carbon slitlike pores definition of pore radius r and the statistical thickness t.
t(P, r) Cp(r)x ) × tm 1-x 1 + (nf/2 - n/2)xn-1 - (nf + 1)xn + (nf/2 + n/2)xn+1 1 + [Cp(r) - 1]x + [Cp(r)f/2 - Cp(r)/2]xn - [Cp(r)f/2 + Cp(r)/2]xn+1
(9) The modified Kelvin equation adjusted for adsorption in pores of finite diameters then has the following form:
rK(P) - t(P, r) ) -
2γνm cos θ RT ln(P/Po)
(10)
Equations 7, 9, and 10 are the three basic equations for the subsequent development of a method to determine the pore size distribution from only the knowledge of an adsorption isotherm data. III. Application on Carbonaceous Adsorbents In this section, we apply the above theory for the adsorption of nitrogen onto carbonaceous adsorbents. 1. Model of Pore Structure of Carbonaceous Adsorbents. a. Micropores and Mesopores. It has been widely accepted in the literature that micropores in adsorbents of the carbonaceous family are slitlike in shape, with two parallel pore walls consisting of many graphite layers. In many papers,5,12,15 the pore wall is modeled as a structureless semi-infinite slab, which allows for the application of the derived potential equations such as the 10-4-3 and 9-3 potentials. In our view, because to the proximity of the adsorbate molecule to the pore wall, any assumption of an homogeneous distribution of the mass centers over the graphite plane or in the space of the graphite domain will lead to the underestimation of the effect of the repulsive forces exerted by individual carbon atoms nearest to the adsorbate molecule. A more realistic model of the pore wall structure is used in this paper. Here, the micropore is visualized as the gap between the graphite sheets, which are stacked on top of each other with an interlayer spacing of 0.3354 nm. The stacking of the graphite layers is in a hexagonal arrangement, which is the most common form of staking in graphite structure.16 The number of graphite layers of the pore wall is also limited to 3 or 4, corresponding to a realistic pore wall thickness of about 1.1-1.5 nm. This model structure of the pore wall makes possible the use of the Lennard-Jones (LJ) pair potential between the individual carbon atoms (15) Seaton, N. A.; Walton, J. P. R.; Quirke, N. Carbon 1989, 27, 853. (16) Introduction to Carbon Science; Marsh, H., Eds.; Butterworths: Cornwall, 1989.
and the adsorbate molecule. A schematic model of the micropores of carbonaceous absorbents is shown in Figure 1. For mesopores, the shape of a cylindrical pore is more representative than the slit shape assumed for the micropores. This is because the size of the microcrystallites which are the constituents of the pore wall is on the order of only a few nanometers,17 making a cylindrical shape more suitable for larger pores. For simplicity, the terms “pore radius” for cylindrical and “pore half-width” for slitlike pores are used interchangeably hereafter. b. Pore Radius, Thickness t, and Maximum Number of Layers. The thickness tm of one layer of nitrogen molecule can be calculated from its liquid molar volume νm as tm ) (vm/N)1/3. The definitions of the pore radius r and the statistical thickness t are presented in Figure 1. With those definitions, the maximum number of layers of adsorbate which can be accommodated by one wall of a pore will be n ) (r - 0.17)/tm. 2. Adsorption of Nitrogen. Nitrogen is used very frequently in modeling the adsorption in pores. The reason for it is 2-fold: first, nitrogen molecules are nonpolar and can be modeled as spherical, and second, nitrogen adsorption data are readily available in the literature, but if not, they can be easily measured using standard equipment. a. Heat of Adsorption and the BET C Coefficient. The BET Cs constant of nitrogen adsorption onto nonporous carbon materials is obtained by using the data from Kaneko.18 The statistical thickness t of the adsorbate film adsorbed onto a surface obtained with such a BET Cs constant agrees well with that obtained by using an empirical equation proposed by Harkins and Jura19 over a range of pressure. The collision diameters and the interaction energy of carbon and nitrogen are taken from the literature.11,20 The collision diameter σ12 and interaction energy 12 are calculated using the Berthelot mixing rule.11 The heat of adsorption onto a surface Qs and in a pore Qp are then calculated using the LJ pair interaction between the nitrogen molecule and individual carbon atoms of the slab and the pore walls, respectively. The result of the Qp calculation is shown in Figure 2, where the plot of the BET coefficient Cp, which is calculated as (17) Setoyama, N.; Ruike, M.; Kasu, T.; Suzuki, T.; Kaneko, K. Langmuir 1993, 9, 2612. (18) Kaneko, K. J. Membr. Sci. 1994, 96, 59. (19) Harkins, W. D.; Jura, G. J. J. Am. Chem. Soc. 1944, 66, 1366. (20) Reid, R. C.; Prausnitz, J. M.; Polling, B. E. The Properties of Gases and Liquids; McGraw-Hill: New York, 1987.
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Figure 2. Heat of adsorption and the Cp coefficient as a function of the pore half-width.
Figure 3. Molecular layering process in pores of different radii.
a function of the pore half-width using eq 7, is also included. As seen in the figure, the heat of adsorption reaches a maximum that is about twice the heat of adsorption on a flat surface when the pore half-width is close to the collision diameter σ12. As the pore size increases, the heat of adsorption Qp decreases and reaches an asymptote of the heat of adsorption of a flat surface. The same pattern is observed for the plot of the Cp versus the pore halfwidth. Like Qp, Cp increases sharply when the pore halfwidth is less than the collision diameter σ12, reaching a maximum at the half-width close to the collision diameter σ12. The Cp coefficient also approaches the value of Cs when the pore size increases. b. Layering Process of Nitrogen in Pores of Different Sizes. With the Cp coefficient calculated as shown in previous section, the layering process in pores on different sizes can be evaluated by using eq 9. The excess energy of the nitrogen liquefaction ∆l needed in the equation can be obtained by using a set of experimental data. In this case, we choose the nitrogen data of the sample AX21 from the work of Lastoskie.5 Examples of the statistical thickness t of the adsorbed layer in pores of various sizes are shown in Figure 3. We can see that if the half-width is close to the collision diameter, pores are practically filled with nitrogen at very low pressures solely by the layering mechanism. It is important to recognize this because it means that pores in this range can be completely filled with the adsorbate regardless of whether or not the Kelvin mechanism is functioning. The enhanced layering process is effective only in a rather narrow pore radius range (micropore range). Above that range, the process more or less resembles the surface layering observed in adsorption on flat surfaces. On the other hand, as the pore size decreases below this range, the layering process
Figure 4. (Bottom) Filling pressure of nitrogen in carbon slitlike pores: MK ) dashed line; HK-dotted line; DFT ) symbol; and this model ) solid line. (Top) Concept of the pore filling process.
is hindered because of the increasing repulsive forces of the pore walls, and adsorption becomes less and less favorable. c. Filling Pressure. As discussed above, pores having half-widthes close to the collision diameter σ12 are filled with nitrogen at very low pressures. As pressure increases, pores having radii greater than σ12 are filled successively. The threshold pressure at which pores are filled with nitrogen is called the filling pressure. For an adsorbateadsorbent system, the filling pressure is a function of the pore size only and can be calculated by eqs 7, 9, and 10. Figure 4 shows the plot of the filling pressure versus the pore half-width for nitrogen adsorption into carbonaceous pores at 77 K. Also shown in the figure are results from the conventional Kelvin equation, the HK equation, and the DFT calculations.5 Our result is better than the HK or the conventional equations and is remarkably similar to that obtained by using the DFT theory.5 The advantage clearly lies with our method because of the very short computational time required. To give an example, using a Pentium 233 MHz and a computing platform of MatLab 5, the time required is about 1 min. The concept of the pore filling process by liquid nitrogen is also demonstrated in Figure 4, where the PSD of an arbitrary activated carbon is added on the top of the filling pressure plot. In this figure, pores that are completely filled with nitrogen at the relative pressure of 10-3 are shown as the range limited by two lines a and a′. The area of the shaded region is the volume of pores that are completely filled with adsorbate. For pores outside that range, the adsorption process still follows a layering
Characterization of Porous Materials
Figure 5. Nitrogen adsorption isotherms at 77 K in pores of reduced half-widthes off 6, 8, 10, 15, 25, and 50 (reading from left to right).
Langmuir, Vol. 15, No. 10, 1999 3613
Figure 7. Nitrogen adsorption isotherms at 77 K in pores of reduced half width of 0.88, 0.87, 0.865, and 0.86 (reading from left to right).
4. Integral Equation of Adsorption for Porous Adsorbents. Let f(r) be the pore volume distribution, with f(r) dr being the pore volume having radii between r and r + dr. The pore volume W is calculated from the following equation.
W)
Figure 6. Nitrogen adsorption isotherms at 77 K in pores of reduced half-width of 1.4, 1.6, 2, 3, and 6 (reading from left to right).
mechanism. This is in contrast to many theories such as TVFM, HK, and so on, which assume that the filling process starts from the narrowest pores onward. However, because the volume of pores in the submicropore range is usually very small, the assumption of sequential filling from the smallest pores upward is reasonable, except for solids having a large portion of volume in the submicropore range. 3. Predicted Nitrogen Adsorption Isotherm. Using the analysis shown above, the isotherms of nitrogen adsorption in uniform pores of different sizes can be generated. Examples of such isotherms are shown in Figures 5-7. The isotherms of larger pores, i.e., mesopores, presented in Figure 5 exhibit a typical type IV behavior, that is, a multilayering process followed by a sudden capillary condensation step. The enhancement of the layering process can be observed in Figure 6, where the isotherms of pores in the micropore range are shown. We can see in the figure that the smaller is the pore radius, the narrower is the pressure range within which the layering process takes effects. The isotherms of pores having half-widthes closer to those of the collision diameter exhibit a type I shape with a very sharp rise to its maximum capacity at very low pressures. As the pore size is reduced below the value of the collision diameter, the enhancement is diminished because of the strong repulsive forces. Figure 7 shows that for pores having reduced halfwidth smaller than 0.9, the layering process is hindered so much that it become significant only at higher pressure and that those pores cannot be filled with the adsorbate, even at saturation pressure.
∫0r
max
f(r) dr
(11)
where rmax is some upper limit of the pore range. For a given pressure P, pores having radii less than the critical threshold radius, rK, calculated in eq 10 will be completely filled with adsorbate. However, for pores having radii greater than rk, the molecular layering occurs such that the amount adsorbed is equal to the surface area times the statistical film thickness as calculated in eq 9. Written in terms of the volume adsorbed, we have the following adsorption isotherm equation:
V(P) )
∫0r (P) f(r) dr + ∫rr(P) art(P, r) f(r) dr k
max
k
(12)
where a/r is the specific area per unit pore volume factor and is equal to 1/r for slit shape pores and 2/r for cylindrical pores. It is worthwhile to re-emphasize that our theory does not need the demarcation size to delineate between the micropore range and the mesopore range. IV. Pore Size Distribution of Activated Carbons The developed model is tested against the nitrogen adsorption data taken from work of Lastoskie et al.5 and Ehrburger-Dolle6 et al. The four microporous carbon samples in those papers are originally coded AX21, CXV, AC610, and D52. 1. PSD by Optimization. A usual method of calculating the PSD of an adsorbent from the experimental adsorption data is to assume a distribution function for the PSD and then optimize the parameters of the distribution function to achieve the best fit to the experimental data. Basic distributions such as Gaussian, log-normal, and γ functions are the most frequently used to describe the PSD. Very often when a simple PSD function cannot provide a satisfactory fit, a combination of the basic distribution functions is used instead. The advantage of this approach is that the optimization procedure is straightforward and results are very easy to interpret and ready for comparison. However, there are also some drawbacks with this method, and they are not easy to overcome. First, if a simple distribution function is used, it may not describe the PSD closely enough, or the convergence may not be achieved in the optimization process. On the other hand, if a complex
3614 Langmuir, Vol. 15, No. 10, 1999
Figure 8. Model fitting of the data of nitrogen adsorption onto the sample AX21.
Nguyen and Do
Figure 10. Model fitting of the data of nitrogenn adsorption onto the sample CXV.
Figure 9. Pore size distribution of the sample AX21.
function is chosen, there will be too many fitting parameters, and there is a danger of strong correlation among parameters. Furthermore, the convergence of the optimization process sometimes is not achieved, and it depends critically on the choice of the initial guess. In this paper, we use a different approach in which we divide the pore range into small intervals with emphasis on the lower end, and the adsorption in the pore system is then the summation of the adsorption in individual pore intervals. The volume of each interval is optimized to achieve the best fit to the experimental data. The results of the optimization process is then a histogram of the PSD, which can be further analyzed by using any of the standard distribution functions, if necessary. The advantage of the method is that the conversion of the optimization is fast and guaranteed regardless of the initial conditions. 2. PSD of Different Activated Carbons. The fitting results and the corresponding PSDs of the microporous activated carbon samples are presented in Figures 8-15. For the purpose of comparison, the PSDs are shown here as functions of the pore width measured in angstroms. As seen in all cases, the model fits well to the experimental data. It is observed that there are kinks at the lower pressure regions of the model isotherms, especially the isotherm of the sample D52. This is a consequence of the fact that the PSD is described by histograms rather than by continuous distribution functions. It means that the smoothness of the model isotherms can be improved by further subdividing the pore intervals into smaller ones. A common feature of the obtained PSDs is that they all exhibit two distinct peaks, with one of the peak having a mean pore size of about 11-12 Å, whereas the position of the other varies from sample to sample. The peaks of the
Figure 11. Pore size distribution of the CXV sample.
Figure 12. Model fitting of the data of nitrogen adsorption onto the sample AC610.
sample AX21 are positioned at the pore sizes of about 12 and 22 Å, far apart from each other as seen in Figure 9. The distance between the peaks is reduced for samples AC610 and D52, and for the sample CXV, the two peaks partly overlap each other. Knowing that the dimension of the pore clearance is 3.4 Å smaller than the model pore width, we can see that all pores of the sample D52 (Figure 15) fall into the micropore range, whereas some pores of the samples AX21 (Figure 9) and AC610 (Figure 13) can be classified as mesopores. Unlike those samples, which can be called microporous, the sample CXV is a typical micro/mesoporous adsorbent with pores spreading out in a rather wide size range (Figure 11) and a quite significant contribution of mesopores. It is interesting to compare the above results with those obtained by the DFT.5 As seen, there are some overall
Characterization of Porous Materials
Langmuir, Vol. 15, No. 10, 1999 3615
constrained by the form of a chosen distribution, for example γ distribution in the work of Lastoskie et al. In this way, the chosen distribution function may not correctly describe certain parts of the pore size distribution, especially the region where the two peaks overlap. V. Conclusions
Figure 13. Pore size distribution of the sample AC610.
The same mechanism, molecular layering followed by condensation or filling, can be used for describing the adsorption in meso- and micropores if the layering process in pores is allowed for the enhancement caused by the overlapping of the force fields. This principle is proved by using the Kelvin equation in conjunction with the enhanced statistical film thickness to calculate the adsorption occurring in porous materials. The PSDs calculated by this method are comparable with those obtained by using the DFT. The optimization procedure used in this paper also shows some advantages over the traditional process. Although tested on the data of the nitrogen adsorption onto activated carbon samples only, the model is open for application in other adsorbate-adsorbent systems. Acknowledgment. Support from the Australian Research Council is gratefully acknowledged. Nomenclature
Figure 14. Model fitting of the data of nitrogen adsorption onto the sample D52.
Figure 15. Pore size distribution of the sample D52.
similarities between the two sets of PSDs of the samples AX21 and AC610, for example, the shapes of the PSDs, and the relative distances between the peaks of the PSDs. On the other hand, the fitting for the sample CXV is much better if our model is used. The double peak PSD of this sample (Figure 11) is also different from the result of the original work by Lastoskie,5 where the PSD calculated by using trimodal γ distribution is shown with only one peak centered at a pore width of about 17 Å. The better fit to the experimental data may lend some credibility to the PSD obtained by this method. Another point worthwhile to mention is the ways the PSDs were obtained in earlier work and in this paper. In other work, the distribution is
A ) Preexponential coefficient of eq 2 C ) Coefficient of the BET equation Cs ) Coefficient of the n-BET equation for adsorption onto a flat surface Cp ) Coefficient of the n-BET equation for adsorption in pores N ) Avogadro’s number P ) System pressure Po ) Relative pressure Q ) Heat of adsorption Qs ) Heat of adsorption onto a flat surface Qp ) Heat of adsorption into a pore R ) Universal gas constant S ) Surface area T ) Temperature V ) Pore volume Vm ) Monolayer volume d ) Characteristic pore dimension n ) Number of adsorbate layers r ) Pore radius rk ) Pore radius from the Kelvin equation rmax ) Upper limit of the mesopore range t ) Statistical thickness of the adsorbate film layer tm ) Thickness of a single layer of the adsorbate ∆l ) The Excess of the reduced liquefaction heat γ ) Liquid surface tension νm ) Liquid molar volume 12 ) Nitrogen-carbon interaction energy a ) Reduced energy of adsorption l ) Reduced heat of liquefaction σ12 ) Collision diameter of the nitrogen-carbon interaction θ ) Liquid-solid contact angle LA981140D
