pubs.acs.org/Langmuir © 2010 American Chemical Society
New Method to Determine Surface Area and Its Energy Distribution for Nonporous Solids: A Computer Simulation and Experimental Study Chunyan Fan,† L. F. Herrera, D. D. Do,* and D. Nicholson‡ School of Chemical Engineering, University of Queensland, St. Lucia, Queensland 4072, Australia. Current address: Department of Storage and Transportation Engineering, China University of Petroleum, Qingdao, China. ‡ Current address: Theory and Simulation Group, Department of Chemistry, Imperial College, London SW72AY, U.K †
Received November 12, 2009. Revised Manuscript Received February 25, 2010 We present in this article a new method to determine the “geometrical” surface area of nonporous solids. This method is based on the total number of molecules dosed into the adsorption cell and a knowledge of the distribution of molecules between the gas phase and the surface phase. By matching this experimental amount with the corresponding theoretical equation, we can derive not only the surface area but also its energy distribution and the void volume of the adsorption cell. The method avoids the limitations of other methods presented in the literature. The BET method, for example, involves unrealistic assumptions and necessitates the choice of a molecular projection area. Our method does not suffer from these assumptions or limitations and is self-consistent, from the measurement of adsorption data to the final analysis of the surface area. The novelties of the method are the following: (i) it is valid over the complete range of reduced pressure, (ii) it does not require a molecular projection area, (iii) beside the total surface area, we also derive its energy distribution, and (iv) the helium expansion method (or any equivalent method) is not required to determine the void volume.
1. Introduction One parameter that has been quoted in almost every application of finely divided solids is the surface area.1,2 It is used as an indicator to compare samples and often to evaluate how good a solid is in terms of its adsorptive capacity or its catalytic activity. Among the many methods that have been proposed in the literature for the determination of surface area, BET remains the method of choice because of the simplicity of its application.3 However, it is equally well known that it involves many unrealistic assumptions and requires a knowledge of the molecular projection area. Unfortunately, more than one value of the molecular projection area has been suggested even for simple gases such as argon and nitrogen. For more complex gases, the variation is clearly greater.1,2 Thus, there is a need to develop a consistent method that does not require the unrealistic assumptions of the BET method and does not require the molecular projection area, which is known to vary with the range of pressure and temperature. Attempts4,5 have been made to remedy these defects, but the inconsistency remains. In this article, we will present a coherent method that is physically realistic. This method does not require the determination of the void volume, knowledge of the molecular projection *Author to whom all correspondence should be addressed. Fax: þ61-73365-2789. E-mail:
[email protected]. (1) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic Press: 1982; pp xi, 303. (2) Rouquerol, F.; Rouquerol, J.; Sing, K. Adsorption by Powders and Porous Solids: Principles, Methodology and Applications. Academic Press: New York, 1999; p 465. (3) Brunauer, S.; Emmett, P. H.; Edward, T. Adsorption of gases in multimolecular layers. J. Am. Chem. Soc. 1938, 60, 309-319. (4) Do, D. D.; Do, H. D. Adsorption of argon on homogeneous graphitized thermal carbon black and heterogeneous carbon surface. J. Colloid Interface Sci. 2005, 287, 452-460. (5) Do, D. D.; Do, H. D. GCMC-surface area of carbonaceous materials with N2 and Ar adsorption as an alternative to the classical BET method. Carbon 2005, 43, 2112-2121.
5610 DOI: 10.1021/la9043107
area or a restriction of the range of reduced pressure over which the method is applicable. Furthermore, it provides not only the total “geometrical” surface area but also the areas associated with different adsorption energies of the different regions of the surface. It is of equal importance to stress that our method involves the direct use of the measured data rather than manipulated data (i.e., the excess amount) as used by all other characterization methods. Although it has been argued that, despite its unrealistic assumptions, the BET surface area is a good approximation, its many limitations are also widely recognized.6 Moreover, the alternative methods, intended to remedy the weaknesses of the BET theory (such as the DR and HK theories) have been shown to have serious limitations.7 The DFT approach overcomes several objections but in common with nearly all statistical mechanical theories requires approximations and is restricted to application with spherically symmetrical molecules. We believe that it is important to develop methods that are both consistent for all materials and physically realistic. This is the object of this article, and we justify our new method with detailed computer simulation and experimental tests on a number of carbon blacks.
2. Theory The surface area can be determined by a number of methods, such as microscopy, the permeation method, and adsorption. Among these, adsorption from the gas phase remains the method of choice and is more reliable than the others because the surface is better probed by adsorbed molecules. Given a set of adsorption data, many analysis methods are available and the BET method3 is preferred as the principal means to obtain the surface area. (6) Sing, K. S. W. Adsorption methods for the characterization of porous materials. Adv. Colloid Interface Sci. 1998, 76-77, 3-11. (7) Do, D. D.; Nicholson, D.; Do, H. D. Adsorption in micropores (nanopores): a computer appraisal of the Dubinin equations. Mol. Simul. 2009, 35, 122-137.
Published on Web 03/18/2010
Langmuir 2010, 26(8), 5610–5623
Fan et al.
However, there are many issues associated with this method, which we shall elaborate on in section 2.1. In the development of our new method, we consider the way an adsorption experiment is conducted. More often than not, this type of experiment is carried out with a volumetric device rather than a gravimetric one. In the volumetric method, a known amount of gas is dosed into the adsorption cell and a sufficient time is then allowed for the system to reach equilibrium, at which the equilibrium pressure is recorded. This is repeated for different amounts dosed into the adsorption cell, and finally a relationship between the equilibrium pressure and the dosing amount is obtained. It must be stressed that it is the amount that we introduce into the adsorption cell that is measured whereas the excess isotherm commonly reported in the literature is calculated with the use of the void volume. Furthermore, the relationship between the dosing amount and the equilibrium pressure is a monotonically increasing function, irrespective of whether the adsorption temperature is below or above the critical point. 2.1. BET Method. The BET method has been one of the most used for all volumetric machines since its description in 1938.3 However, there are some problems associated with this method. The list of problems is shown below. (1) Constant surface energy. This is rarely satisfied by all practical solids. (2) Adsorbate interactions parallel to the surface are omitted. This seriously misrepresents the fluid structure, especially in higher layers. (3) Sites in the first layer are singly occupied. This implies localized adsorption. Physical adsorption always involves mobile adsorption, in addition to localized adsorption. This assumption is restrictive. (4) The reduced pressure range of the BET plot is from 0.05 to 0.3. Very often, this range is varied to suit the requirement of a best linear fit. This selection is arbitrary and is not justified.2 (5) The molecular projection area. There are many values that were proposed in the literature for a given adsorbate!1,2 In addition to these problems associated directly with the BET theory, there is an issue regarding the determination of the void volume. Usually this is carried out at room temperature. It should be done at high temperature to avoid the possibility of helium adsorption. If the void volume is overestimated because of helium adsorption (no matter how small its polarizability), then the excess mass is underestimated and therefore the surface area is underestimated. These problems need to be circumvented, and the method that we develop in this article will overcome these problems; this is what we will describe next. 2.2. New Method. The new method uses a volumetric device that is the most common means to determine the adsorption isotherm and hence the surface area. Let the known amount of gas dosed into the adsorption cell at pressure P be N mols. The conservation of mass requires that the amount dosed be equal to the sum of the amount in the gas phase and the excess adsorption. We will assume that the surface is composed of M patches of different surface energies (which is different from the BET theory, which considers only one surface energy). The well depth of the solid-fluid interaction energy of patch j is εsf,j. We assume that each patch is large enough so that the interaction between the (8) Ross, S.; Olivier, J. P. On Physical Adsorption; Interscience Publishers: New York, 1964.
Langmuir 2010, 26(8), 5610–5623
Article
Figure 1. Schematic diagram of the adsorption cell with a surface having regions of different surface energies (three regions are shown). The accessible volume is bounded by the dashed line. Particle A resides on the boundary and has a zero solid-fluid potential energy. Particle B is inside the accessible volume and has a negative solid-fluid potential energy.
molecule on one patch and those of the adjacent patches is negligible compared to the fluid-fluid interaction in one patch, so from this assumption,8,9 the mass balance is given by eq 1 N ¼ Vacc Ff þ m
M X
Sj Γj ðPÞ
ð1Þ
j ¼1
where Vacc is the accessible volume of the adsorption cell, Sj is the surface area of patch j, and Γj is the surface excess (mol/m2) of that patch. Figure 1 shows the schematic diagram of the adsorption cell and the accessible volume (shown as the region bounded by the dashed line). The accessible volume is defined as one in which a fluid particle has a nonpositive solid-fluid potential energy with the solid surface. To illustrate graphically the excess amount in the adsorption cell, we show in Figure 2 the three configurations of the system. The first is what is actually happening in the system (an instant snapshot of the system), the second is a hypothetical configuration in which the accessible volume is occupied by particles having the same density as that of the bulk gas phase, and the last configuration is that of the excess adsorbed phase, which is the difference between the previous two configurations. What this shows is that the excess adsorbed phase is independent of the size of the system as long as it is large enough to include the adsorbed phase and the gas phase as we have shown in Figure 2a. In other words, the interface between the adsorbed phase and the gas phase is below the top boundary of the system. 2.2.1. Kernel of Local Isotherms. To apply the mass balance equation (eq 1), we need to know the bulk gas density and the surface excess per unit area (Γj) for all patches. For a given equilibrium pressure, the bulk gas density can be calculated accurately from a proper equation of state. To obtain the surface excess of a patch, we have to assume a model for it. We assume it to be either a slab solid of constant volumetric density or a lamellar structure, which is composed of parallel layers of constant surface solid density. An example of the first type is a solid with covalent bonding among the solid atoms (for example, alumina or silica gel), and an example of the latter is graphite (9) Steele, W. A. Monolayer adsorption with lateral interaction on heterogeneous surface. J. Phys. Chem. 1963, 67, 2016-2023.
DOI: 10.1021/la9043107
5611
Article
Fan et al.
Figure 2. (a) Configurations of the system. (b) Hypothetical gas phase occupying the accessible volume. (c) Excess adsorbed phase. The accessible volume is indicated by the dashed line.
with parallel graphene layers. Once we have chosen a model for the surface, we carry out the grand canonical Monte Carlo (GCMC) simulation to obtain the surface excess per unit area (which we shall hereafter call the surface excess density). For simple probes such as argon, we use the Lennard-Jones 12-6 potential equation, ji,j = 4εff [(σff/ri,j)12 - (σff/ri,j)6], to describe the fluid-fluid interaction.10 The solid-fluid potential energy between a fluid particle and the surface of the slab solid model takes the form of the 9-3 potential11 9 3 σsf 2 σsf jSf ¼ jSf ð2aÞ 15 z z where z is the shortest distance between a particle and the slab solid and the characteristic energy jsf * is given below:
jSf ¼
2π ðF σsf 3 Þεsf 3 v
ð2bÞ
Here, Fv is the volumetric density of the slab (number/m ). The cross molecular parameters are usually calculated from the Lorentz-Berthelot rule: σsf = (σss þ σff)/2 and εsf = (εssεff)1/2, where σff and εff are the collision diameter and well depth of the interaction energy of the fluid particle. A similar set with subscript ss is that for the solid atom. For a surface having a lamellar structure, the interaction between a fluid particle and a layer of constant surface density is given by the 10-4 potential energy:12 10 4 σsf 2 σsf ð3aÞ jSf ¼ jSf 5 z z 3
where jsf * is given below:
jSf ¼ 2πðFs σsf 2 Þεsf
ð3bÞ
Here, Fs is the surface density (number/m2). For a surface composed of many parallel layers with equal spacing Δ between two adjacent layers, the fluid-solid potential energy is calculated by summing the contributions from all lattice layers: jSf ¼
jSf
" # X 2 σsf 10 σ sf 4 5 z þ jΔ z þ jΔ j¼0
ð4Þ
(10) Fan, C.; Birkett, G.; Do, D. D. Effects of surface mediation on the adsorption isotherm and heat of adsorption of argon on graphitized thermal carbon black. J. Colloid Interface Sci., 2009, in press. (11) Do, D. D. Adsorption Analysis: Equilibria and Kinetics; Imperial College Press: London, 1998; pp xxi, 892. (12) Crowell, A. D. Approximate method of evaluating lattice sums of r-n for graphite. J. Chem. Phys. 1954, 22, 1397-1399.
5612 DOI: 10.1021/la9043107
Figure 3. Solid-fluid potential energy vs distance from the surface for the slab surface and the lamellar surfaces.
The excellent approximation to the above equation is the Steele 10-4-3 potential equation:13 jSf ¼
jSf
" # 4 σsf σsf 4 2 σsf 10 5 z z 3Δð0:61Δ þ zÞ3
ð5Þ
The first two terms on the RHS are from the interaction of a fluid particle with the first layer, and the last term is the potential energy contributed by all layers underneath the top layer (which is always attractive). The difference between the interaction between a fluid particle and a slab solid surface and that with a lamellar surface is shown in Figure 3. A number of features can be noted from this Figure. The penetration of the fluid particle to the surface is deeper when the surface is a slab solid, and the minimum of the potential is deeper with the lattice layer for the same number of solid atoms per unit volume of the nonporous solid. The reason for the deeper potential in the case of the lamellar structure is that the solid atoms are concentrated on the top layers whereas in the case of the slab the solid atoms are distributed uniformly over the volume of the nonporous solid. For the lamellar structure, the potential of multiple layers is deeper than that of a single layer and the contribution of the third and lower layers is negligible compared to that of the top two layers. The parameters used in the plots of Figure 3 are those for argon adsorption on graphitelike surfaces, σff = 0.3405 nm, σss = 0.34 nm, εff/k = 119.8 K, εss/k = 28 K, Fv = 114 nm-3, Fs = 38.2 nm-2, and Δ = 0.3354 nm. (13) Steele, W. A. Physical interaction of gases with crystalline solids. Surf. Sci. 1973, 36, 317-352.
Langmuir 2010, 26(8), 5610–5623
Fan et al.
Article Table 1. Differences between the BET Method and the New Method Proposed in This Article BET method
new method
uses the void volume by helium expansion uses the molecular projection area, which varies with the temperature of adsorption and the characteristics of the surface14,15 fits the surface excess can determine only one type of surface limited to a narrow range of pressure
does not need the helium expansion does not need the molecular projection area; instead a potential model with parameters that do not vary with adsorption conditions such as temperature is used fits the amount dosed into the adsorption cell can determine more than one type of surface there is no limit in the range of pressure
The solid-fluid potential energy, given in eqs 2-5, can be evaluated only when we know about the properties of the solid atoms and their configuration. For an unknown solid whose surface properties are required, which is the objective of this article, we need to define a solid-fluid potential so that the kernel (which is a collection of local isotherms) can be constructed. Each local isotherm in the kernel is associated with one particular value of the surface energy. On the basis of the form of the potential energies given in eqs 2-5, we shall choose jsf* as the characteristic energy. Because we use argon as our molecular probe in the characterization of the surface area, we still need another parameter to determine the terms in the square brackets of eqs 2-5, namely, the collision diameter of the solid atom. Most solid atoms of practical adsorbents have their collision diameters falling in the range of 0.3 to 0.4 nm. (For example, the carbon atom in the graphene layer has a value of 0.34 nm.) The construction of a kernel is done as follows: For a given representative value of the collision diameter of the solid atom, we construct a kernel that is a collection of the local adsorption isotherms, each of which corresponds to one value of the characteristic energy, jsf*. We obtain local isotherms of argon on a lamellar surface for different values of the characteristic energy, jsf*, at 87.3K, and these are shown in Appendix 1. A number of features are the following: (i) The isotherms have a distinct nonlinear shape indicating that the adsorption is outside the Henry’s law region. (A linear shape is not suitable for parameter determination because it is impossible to delineate the contribution of patches of different energies.) (ii) The isotherms of higher characteristic energies have an early onset of adsorption, compared to those of lower characteristic energies. 2.2.2. Inverse Problem of Parameter Determination. Knowing the bulk gas density, Ff, and the kernel {Γj(P); j = 1,2,...,} in eq 1, the task is to find the unknowns in that equation, namely, the accessible volume of the system and the surface areas of all patches. This is done by matching the amount dosed into the adsorption cell in the theoretical equation (eq 1) to the corresponding experimental value for all values of the equilibrium pressure. This is an optimization problem to minimize the residual of the square of the difference between the theoretical value and the corresponding experimental value. This inverse problem is feasible if the experimental data are available over a sufficiently wide range of pressure because the surface excess of one patch behaves differently from that of another patch of different energy. This is so because the onset of adsorption on high-energy patches occurs at lower pressures whereas that of lower-energy patches will do so at higher pressures. The main assumptions used in this new method are the following: (1) Topology of the surface: The surface of a nonporous solid is energetically heterogeneous. This heterogeneity is given by the coexistence of regions with different energies (patches). Langmuir 2010, 26(8), 5610–5623
(2)
(3)
Independent surface energies: The energy patches of the surface are large enough that the interaction between molecules of one patch and those of the adjacent patches is negligible compared to the fluidfluid interaction in the molecules adsorbed in a patch. Kernel: The local isotherms of different characteristic energies are obtained by using an infinite, homogeneous surface model.
We have developed a means to determine the surface area of a nonporous solid without the need to use the molecular projection area and without the need to obtain the void volume with the helium expansion procedure. The differences between our new method and the BET method are summarized in Table 1. To summarize the methodology that we have described above, we present below an algorithm of steps that are required to derive the surface area, its energy distribution, and the accessible void volume of the adsorption cell. (1) Calculation of the kernel: The determination of the area of each patch is based on a kernel of local isotherms on homogeneous surfaces with different energies. These local isotherms are obtained from the molecular simulation of a model solid that is proper for the nonporous solid under consideration. This choice is between a slab solid or a lamellar structural model (section 2.2.1). The local isotherms are obtained at the same temperature as the experimental temperature. (2) Collection of experimental data: The experimental data are measured in the form of the total amount dosed into the adsorption cell at each equilibrium pressure point. This is direct experimental data, and they are accurately measured. (3) Calculation of the surface area: By matching the experimental data collected in step 2 with the mass balance equation (eq 1), which requires the kernel that was obtained in step 1, we can derive the surface area, its energy distribution, and the accessible void volume of the adsorption cell.
3. Results and Discussion 3.1. Computer Appraisal of the New Method. Before testing our new idea for characterization experimentally, we will justify the new method with the “computer-experiment” data obtained directly from the computer simulation of a number of test surfaces. The first model surface is an infinite homogeneous surface simulated by the Steele surface potential (eq 5). The second model surface is composed of two patches, each of which is modeled as a Steele surface. The third surface model is composed of two patches that are finite in one direction and infinite in extent in the other direction, both having two layers of different surface energies and placed on top of an infinite Steele surface. The fourth model comprises three patches of different DOI: 10.1021/la9043107
5613
Article
Fan et al.
Figure 5. Configuration of the finite strip, which is finite in the y direction and infinite in the x direction (perpendicular to the page). Figure 4. Side view of four surface models used to generate the computer-experimental data. BS and S indicate Bojan-Steele and Steele potential models, respectively.
energies;the first and the third patches have three layers whereas the central patch has four layers, giving it a higher elevation than the first and third patches. The schematic diagrams of these models are shown in Figure 4. To account for the interaction energy between two fluid particles, we use the Lennard-Jones 12-6 equation, jff = 4εff[(σff/r)12 - (σff/r)6]. This interaction is mediated by the presence of the surface, and this surface mediation has been studied by a number of authors. Here we use the empirical equation that is proposed by Do and co-workers15-17 that has been found to describe noble gas adsorption on a graphite surface well. This mediation modifies the intermolecular potential energy as jffeff = gjff, where the function g is called the damping factor and is given by the following empirical equation, g = exp(-χjij,s/kT). Parameter χ is called the damping constant, and parameter jij,s is a geometric average of the solid-fluid potential energy of particles i and j. For the interaction between a fluid particle and a strip that is finite in one direction (y) and infinite in the other (x), we use the following equation developed by Bojan and Steele18-21 to (14) Herrera, L. F.; Do, D. Effects of surface structure on the molecular projection area. Adsorption of argon and nitrogen onto defective surfaces. Adsorption 2009, 15, 240-246. (15) Do, D. D.; Do, H. D.; Nicholson, D. Effects of surface structure and temperature on the surface mediation, layer concentration and molecular projection area: adsorption of argon and nitrogen onto graphitized thermal carbon black. Adsorpt. Sci. Technol. 2007, 25, 347-363. (16) Do, D. D.; Do, H. D.; Kaneko, K. Effect of surface-perturbed intermolecular interaction on adsorption of simple gases on a graphitized carbon surface. Langmuir 2004, 20, 7623-7629. (17) Do, D. D.; Do, H. D. Effects of potential models in the vapor-liquid equilibria and adsorption of simple gases on graphitized thermal carbon black. Fluid Phase Equilib. 2005, 236, 169-177 (18) Bojan, M. J.; Steele, W. A. Computer simulation of physisorption on a heterogeneous surface. Surf. Sci. 1988, 199, L395-L402. (19) Bojan, M. J.; Steele, W. A. Computer simulations of the adsorption of xenon on stepped surfaces. Mol. Phys. 1998, 95, 431-437. (20) Bojan, M. J.; Steele, W. A. Computer simulation of physisorbed krypton on a heterogeneous surface. Langmuir 1989, 5, 625-633. (21) Bojan, M. J.; Steele, W. A. Computer simulation of physical adsorption on stepped surfaces. Langmuir 1992, 9, 2569-2575.
5614 DOI: 10.1021/la9043107
calculate the potential energy of interaction jf ;S ¼ 2πðFs σsf 2 Þεsf f½jrep ðz; yþ Þ -jrep ðz; y - Þ -½jatt ðz; yþ Þ -jatt ðz; y - Þg
ð6aÞ
where the molecular parameters are defined as before. The other variables in the Bojan-Steele (BS) equation are yþ and y-. They are the y coordinates of the right-hand edge and the left-hand edge relative to the fluid particle: yþ ¼
W -y; 2
y- ¼ -
W -y 2
ð6bÞ
The origin of the y coordinate is positioned at the center of the strip (details shown in Figure 5). The repulsive and attractive functions on the RHS of eq 6a are given by " 10 σ sf 10 y 1 σ sf 1 þ jrep ðz; yÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10 z8 ðy2 þ z2 Þ y 2 þ z2 5 z σsf 10 σsf 10 σsf 10 3 1 7 þ þ þ 2 3 40 z6 ðy2 þ z2 Þ 16 z4 ðy2 þ z2 Þ 128 z2 ðy2 þ z2 Þ4 σ sf 4 y 1 σ sf 1 jatt ðz; yÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ4 þ 2 2 4 z ðy þ z2 Þ y 2 þ z2 2 z
#
ð7aÞ ð7bÞ
The potential equation (eq 6) works for any particle above or under the strip or even on the side of the strip. In the case of a particle positioned exactly on the same level as the strip (i.e., z = 0; particles C and D in Figure 5), eq 6 is undefined. In that case, we have to take a Taylor series expansion of eqs 7a and 7b and find their limit when z approaches zero. The following equations are the limits of the repulsive and Langmuir 2010, 26(8), 5610–5623
Fan et al.
Article
Figure 6. Details of the simulation box used to simulate computerexperiment data and its boundaries. Shown is the example for model 3.
attractive terms: y 1 σsf 10 63 σ sf 10 þ Oðz2 Þ lim jrep ¼ pffiffiffiffiffi zf0 1280 y y2 5 z
ð8aÞ
y 1 σ sf 4 3 σ sf 4 þ Oðz2 Þ lim jatt ¼ pffiffiffiffiffi zf0 16 y y2 2 z
ð8bÞ
After substituting eq 8 into the solid-fluid potential equation (eq 6), we find the final solution for the potential at z = 0 8 " # < 63 σ 10 σ 10 sf sf 2 - þ jf , S ¼ Ψ2πðFs σ sf Þεsf :1280 y y 9 " # σsf 4 = 3 σsf 4 ; 16 y yþ
ð9Þ
where Ψ = 1 for positive yþ and y - and Ψ = -1 for negative yþ and y-. We see that there is no singularity in the solidfluid potential energy, as physically expected. The solidfluid potential energy in the region of small z is given in Appendix 2, and this is essential in the GCMC calculations when the z coordinate of a particle is very close to zero. 3.1.1. Simulation Box. The new method to determine the accessible surface area and volume for nonporous solids was tested with adsorption isotherms obtained from a GCMC simulation of argon adsorption at 87.3 K for the four surface models that we have described in Figure 4. In all of these models, the adsorption surfaces are at the bottom of the box in the z direction. The other boundary in the z direction is treated as a hard wall, and periodic boundary conditions are applied in the x and y directions (Figure 6). The dimensions of the simulation box in the x and y directions are 15 times the collision diameter (σ), and those in the z direction are 20 times the collision diameter. To obtain the computer-experiment data that mimics a real experiment, we need to model a system such that the geometrical ratio (the ratio of the volume of the adsorption cell to the solid surface area) of the simulation box is the same as that of the real adsorption system. This is used to describe correctly the relative contributions of the gas phase (first term on the RHS of eq 1) and the adsorbed phase (second term of eq 1) to the total number of Langmuir 2010, 26(8), 5610–5623
particles in the box. It is important to do this because if we choose a simulation box such that the ratio of the volume to surface area is smaller than that of the real experiment then the contribution of the gas phase to the total number of particles in the simulation box might be insignificant and might not correctly reflect how matters would be distributed between the two phases in the real system. However, if we choose the simulation box such that its geometrical ratio is the same as that of the real system, the box volume would be too large and the number of particles in the system would be excessive, making the computation time too long to be practical. We now provide a methodology to obtain computer-experiment data for an adsorption system whose accessible volume is very large (such that the ratio of the volume to the surface area matches that of the real system). To obtain the total number of particles in the “real” system, we simply carry out the following steps, which are designed not only to save computation time but also to make the simulation more effective. 3.1.2. Algorithm to Construct Computer-Experiment Data from the Simulation Results of a Smaller Box (1) Run a computer simulation with a simulation box of reasonable height (say 20 molecular diameters) having an accessible volume Vacc and a surface area 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 = VaccFf, and hence the excess number of particle is Nex = N - NG. (2) To construct the computer-experiment data for a much larger simulation box (for example, that with a height that is 100 times larger than the first box), we know that the excess number, Nex, of a large box must be the same as that of a small box as long as the surface areas of these two systems are the same (S0 acc = Sacc). Here, we use the prime superscript to denote the larger box. Therefore, if the accessible volume of the large box is V0 acc, then the total number in this large simulation box is simply N0 = Nex þ V0 accFf, where Nex comes from the simulation of the smaller box. This will form a set of data for the large simulation box (i.e., N0 versus P). To have an idea of the order of magnitude of the ratio of the accessible volume to the surface area, we consider the geometrical aspects of adsorption characterization systems commonly used in most laboratories: the volume is on the order of 10 cm3, and the typical specific surface area of a nonporous solid is 10 m2/g. Therefore, with 1 g of solid in the adsorption cell the ratio of the void volume per unit area is Vacc 10 -6 m3 ¼ ¼ 1000 nm Sacc 1 m2 This means that the computer-experiment data should be constructed for a system having a geometrical ratio of the order of 103 nm. For our simulation box, this geometrical ratio is only 6.8 nm. This is more than 100 times smaller than that of a real system. To obtain the adsorption data of a system that has this geometrical ratio equal to that of a real system, we proceed formally with the determination of the adsorption data in the form of the total number of particles in the small simulation box for each value of the chemical potential (pressure). Then, we construct the computer-experiment data whose volume and surface area ratio is similar to that used experimentally. However, we first present a good experimental design for the determination of surface area and volume and DOI: 10.1021/la9043107
5615
Article
Fan et al.
Figure 7. Plots of the ratio of the surface excess to the gas density for argon adsorption on a lamellar surface at 87.3 K.
then use this good design practice to obtain the adsorption data for the four surface models (as shown in Figure 4). 3.1.3. Consideration of the Choice of Mass of the Sample and the Size of the Adsorption Cell. As we have mentioned earlier, the determination of the surface area and volume from the total amount adsorbed is based on the mass balance (eq 1). Because the mass balance is calculated from the contributions of the gas phase and adsorbed phases (the first and second terms on the RHS of eq 1, respectively), the two terms must be comparable to obtain reliable results. Thus, the criterion for choosing the size of the adsorption cell and the mass of the sample is such that the following ratio is of the order of unity:
mSΓðP; jSf Þ Vacc Ff
¼ Oð1Þ
ð10Þ
Because the surface might possess a distribution of surface energies, the energy used in the above equation is the representative surface energy. This will set the recommended maximum size of the adsorption cell or, for a given adsorption cell, the minimum mass of sample required to get reliable results. Rearranging this equation as follows ! ΓðP; jsf Þ Vacc ð11Þ ¼O mS Ff we see clearly that the choice of the adsorption cell volume and the mass of the sample depends on the specific surface area (S), the energetic strength of the surface (jSf*), and the temperature. To guide us in the choice of values of these parameters, we present in Figure 7 the plots of the RHS of eq 11 with pressure for 87.3 K and three different values of the characteristic energy. Figure 7 is valuable for designing a system for the reliable determination for the surface area. For example, if we anticipate that the nonporous solid has a surface strength typical of a graphite surface, then the ratio of the accessible volume to the surface area (of eq 11) should be of the order of 500 nm for adsorption conducted at 87.3K (reading from Figure 7): V ¼ 500 nm mS 5616 DOI: 10.1021/la9043107
Figure 8. Computer-experiment data obtained by GCMC and its comparison with the optimization results for surface model 1.
Thus for a typical volume of an adsorption cell of about 10 cm3, the area of the solid should be at least equal to 20 m2. For a typical specific surface area of graphitized thermal carbon black of about 10 m2/g, the minimum mass required for a reliable determination of the surface area is 2 g. This design practice means that any choice of the ratio of the accessible void volume of the adsorption cell and the surface area of the order of 500 nm will suffice. A value of 1000 nm, for example, is acceptable; however, a value of 100 000 nm is unacceptable because the contribution of mass in the gas phase completely dominates the total amount in the adsorption cell, making the determination of the surface area and its energy distribution extremely unreliable. The ratio on the order of 500 nm is calculated for surfaces with energy similar to that of graphite. For weaker surfaces, this ratio should be smaller (i.e., more sample mass is required), and for stronger surfaces, it is larger (i.e., less sample mass would suffice). Once we have determined the proper system size (which is larger than the simulation box size), we obtain the computerexperiment data for this system using the algorithm presented in section 3.1.2 for the total amount dosed into the adsorption cell for each value of the chemical potential. To illustrate the new method, we assume that the surfaces of all four models have the same adsorption strength as that of a graphite surface. Therefore, the ratio of the void volume of the adsorption cell to a surface area of 500 nm was used in the determination of the adsorption data. Once this is done, we match this against the theoretical mass balance equation (eq 1) to derive the surface area, its energy distribution, and the accessible void volume of the adsorption cell. We use the first surface model with a characteristic energy of 1700 K, which is our simplest model of a Steele surface, to test the method that we developed in this article. Because the model is a Steele surface and the kernel was constructed with Steele surfaces, we expect the derived characteristic energy and the surface area match the values that we used to obtain the computer-experiment data perfectly. This is indeed the case as shown in Figure 8 where we plot the computer-experiment data of the total amount versus the equilibrium pressure as unfilled circles. The fit of the theoretical equation (eq 1) against this data is shown as a dashed line. Because our computer-experiment data are from the GCMC simulation, we have not only the total number of particles in the box but also the number of particles in the gas phase (g) and the excess number in the adsorbed phase (). As expected, the mass balance in eq 1 also fits these data perfectly. This demonstrates the Langmuir 2010, 26(8), 5610–5623
Fan et al.
Article
Table 2. Comparison of the Different Geometrical Parameters for Surface Model 1 computer-experiment parameter
derived parameter
11.6 1700 5793.7
11.6 1700 5799.4
surface (nm2) energy of the surface (K) accessible volume (nm3)
Figure 10. Surface-energy distribution for surface model 2 with the two patches having the same surface area. (a) Exact data and (b) optimization results.
Figure 9. Computer-experiment data obtained by GCMC and its comparison with the optimization results for surface model 2 having patches with the same surface area.
feasibility of using the mass balance (eq 1) to derive the required parameters: the surface area, the accessible volume, and the characteristic energy as shown in Table 2. However, when we apply the BET theory to the excess data over the reduced pressure range from 0.05 to 0.2 we find a BET surface area of 13.1 nm2, which is 13% greater than the exact area. Using our method, we have reproduced the exact surface area of 11.6 nm2. We also test our method with another surface with a characteristic energy of 2500 K and achieve the same success, and this is shown in Appendix 3. We now turn to the second surface model in which the surface is composed of two Steele patches having characteristic energies of 1700 and 2500 K. For illustration, we use equal areas for these two patches. The total number of particles versus the pressure for argon adsorption at 87.3 K is shown in Figure 9. Fitting the mass balance in eq 1 against these data yields excellent agreement, and this is reflected in Figure 10, where we plot the surface area versus the characteristic energy. We also have excellent agreement between the data and the derived results from the fitting when the ratio of the two areas is either 3:1 or 1:3. This is shown in Appendix 3. Not only do we have excellent agreement between the computer-experiment data and the derived results, but we also have excellent derived surface areas of all patches and their characteristic energies. With the two surface models that we have dealt with so far, the agreement between the data and the derived results is very good and so are the derived parameters. This is because the patches used in the generation of the computer-experiment data are from the Steel surface and the kernel was constructed from Steele surfaces. Let us now test the theory when the patches of a surface follow the Bojan-Steele model (i.e., the patches are finite in one direction (y) and infinite in the other (x)). This will test the theory to the full extent because the patch is the Bojan-Steele type and the kernel is constructed from the Steele surfaces. We would expect the agreement to be very good if the width of the Bojan-Steele patch is large (as Langmuir 2010, 26(8), 5610–5623
Figure 11. Representation of the boundary effect for model 3.
demonstrated in the last two surface models). However, the method is tested when the width of the Bojan-Steele patches is small. This is shown next. We examined two cases: In one case, the width of the Bojan-Steele surface is 10 times the collision diameter, and in the other case, the width is 20 times the collision diameter. The boundary effect (Figure 11) in the first case will be expected to influence the results. The computer-experiment data of the total number versus the pressure for these two cases are shown in the top plots of Figure 12. The LHS is for the smaller patches (the first case), and the RHS is for the large patches. Also shown in these Figures are the number of particles in the gas phase and the excess number in the adsorbed phase. Fitting of the mass balance in eq 3 and the total number data yields excellent agreement, and the derived surface areas and their characteristic energies are shown in the bottom plots of Figure 12. Even with the small width used in the first case, the derived parameters agree fairly well with the exact values. We note that the derived characteristic energy of the weaker patch is larger than the exact value, which is due to the influence of the stronger adjacent patch. Because the width of the patch is larger as in the second case, the agreement between the derived characteristic energy and the exact counterpart is much better. DOI: 10.1021/la9043107
5617
Article
Fan et al.
Figure 12. Comparison between the GCMC adsorption data and the results obtained by optimization for model 3 having surface areas of 46.4 (LHS) and 185.5 nm2 (RHS). The bottom plots show the specific surface areas of each energy patch for (a) the exact data and (b) derived results.
Figure 13. Comparison of the GCMC adsorption data with the results obtained by optimization for model 4. The RHS plots show the specific surface areas of each energy patch for (a) the exact data and (b) derived results.
Finally we test the method with the last model, and the results of the fitting between the computer-experiment data and the derived results are shown in Figure 13. The agreement is fairly good, and the derived surface area of all three patches and their 5618 DOI: 10.1021/la9043107
corresponding characteristic energies show many interesting properties. (i) The total surface area is well produced by the method proposed. Langmuir 2010, 26(8), 5610–5623
Fan et al.
Article
Figure 14. Local density distribution for all surface models studied and snapshots for surface models 3 and 4. The results are shown for the adsorption of argon at 87.3 K at a reduced pressure of 0.2.
(ii) The surface areas of individual patches do not agree well with the exact values, but they are of the right order of magnitude. This is due to the boundary effects that we have seen earlier in surface model 3. (iii) The derived characteristic energies are slightly lower than the exact counterparts, and this is due to the boundary effects. (iv) We observe an additional characteristic energy, and this is contributed by the steps around the elevated middle patch. We now turn our attention to the BET surface areas derived for the four surface models that we have studied. These areas differ from the exact areas, and to explain the difference, we study the local density distribution, Fz ¼
ÆΔNz þ Δz æ Lx Ly Δz
ð12Þ
where Fz is the local density, ÆΔNzþΔzæ is the ensemble average number of particles in the region from z to z þΔz, with z measured from the center of the fluid particle and the closest model surface and Δz taken to be 0.0034 nm. Figure 14 shows the local density distributions for the four surface models at a reduced pressure of 0.2. From the Figure, it is seen that the adsorption of argon on all surfaces shows second-layer formation at the maximum pressure recommended for the calculation of the BET surface area, but this second-layer coverage is clearly overestimated by this model, resulting in a higher BET surface area than the exact area. In other words, the BET attempt to calculate a statistical monolayer fails here. The representation of the second layer is shown in the snapshots of the simulation box on the RHS of the Figure. So far, we have described the results obtained for the total surface area and the individual surface areas of the different patches of all surface models. Let us now show the results obtained for the accessible volume for these models. Table 3 Langmuir 2010, 26(8), 5610–5623
Table 3. Volume Result for the Four Different Surface Models surface model
exact volume (nm3)
derived volume (nm3)
model 1 model 2 model 3
5793.7 5793.7 23 143.5S 92 574.1L 13 014.6
5799.4 5800.6 23 166.7S 92 688.6L 13 842.8
model 4
shows the comparison between the derived and exact volumes. From the Table, it is seen that the derived results are similar to the expected results. Superscripts S and L indicate the results for surface model 3 using the small and large patches, respectively. 3.2. Experimental Evaluation of the New Method with Carbon Black Surfaces. The adsorption isotherms of argon at 87.3 K on Carbopack F (Supelco) and carbon black BP280 are used to test the characterization method proposed. The first isotherm is measured using an ASAP 2020 static volumetric adsorption analyzer from Micromeritics, and the second is taken from the data obtained by Gardner et al.22 Both of these sets of data are reported in the form of the excess amount. These are indirect experimental data calculated on the basis of the given total dosing amount and the void volume (measured during the experiment). By knowing the excess amount and the void volume, the total dosing amount can be obtained by eq 13 NgasI ¼ NadsI þ
PsamI ðVFC þ PsamI VABT CÞ PSTD
ð13Þ
where NgasI is the total amount of gas dosed into sample tube for the ith point, expressed in the IUPAC recommendation as moles of adsorptive gas at NTP, NadsI is the amount of gas adsorbed at the ith pressure point, VFC is the volume of free space, VABT is the portion of cold free space at the analysis bath (22) Gardner, L.; Kruk, M.; Jaroniec, M. Reference data for argon adsorption on graphitized and nongraphitized carbon blacks. J. Phys. Chem. B 2001, 105, 12516-12523.
DOI: 10.1021/la9043107
5619
Article
Fan et al.
Figure 15. Comparison of the adsorption data with the results obtained by optimization for Carbopack F and BP 280. The RHS shows the surface-energy distribution for these samples.
temperature, and C is the nonideality correction factor.23 Details of this calculation can be found in ref 23. For Carbopack F, the excess amount and the void volume are available, whereas in the case of BP280, we constructed the total amount dosed into the adsorption cell using a typical volume of the void volume of the Micromeritics ASAP 2020. Using different values of the void volume does not change the derivation of the surface area and its energy distribution because a larger number of molecules in the void volume increases the total number by the same amount. We report the experimental adsorption isotherms as the total amount in the adsorption cell versus pressure (Figure 15). The Figure shows that the results obtained from optimization agree well with the total amount dosed. The RHS of the Figure shows the individual surface areas of the different patches. The results agree with the expectation of a single surface energy for Carbopack F, suggesting that this carbon black is very homogeneous. In contrast, BP280 shows a wider range of surface energies, and this result is in agreement with the adsorption potential distribution reported by Gardner et al.22 The comparison between the BET and the derived surface areas is also shown in the Figure. The BET surface area is smaller than the derived area for Carbopack F, and the opposite is true for BP280. It is no surprise that the BET theory overestimates the surface area of BP280 because, as seen (23) Micromeritics, ASAP 2020 Accelerated Surface Area and Porosimetry System Operator’s Manual. 2008.
5620 DOI: 10.1021/la9043107
for the four different theoretical models, patches of high energy (2050 K in Figure 15) show a second layer already formed at 0.2 reduced pressure.
4. Conclusions We have presented in this article a new and effective method to determine the surface area of a nonporous solid. This method does not suffer from many of the inherent limitations of the BET theory. It is self-consistent, and not only does it derive the surface area, but it also determines the areas of the various patches, their corresponding surface energies, and the volume occupied by the gas phase in equilibrium with the adsorbed phase. We justify this method with a detailed computer simulation and test it with experimental measurements from two carbon blacks. The computer simulation results show that this method gives better and more informative results than those obtained by the BET method. The experimental results show that this method identifies the heterogeneity of the surface and gives good results for the strength of the solid surface. This was tested with Carbopack F, which has a homogeneous surface energy of the order of the graphite surface energy and Cabot BP 280, whose surface is reported in the literature to be heterogeneous. Acknowledgment. This project was funded by the Australian Research Council. C.F. acknowledges financial support from the China Scholarship Council (CSC) in the form of a scholarship. Langmuir 2010, 26(8), 5610–5623
Fan et al.
Article
Appendix 1: Kernel of Surface Excess Isotherms for Argon at 87.3 K
Appendix 2: The Taylor Series Expansion of the Bojan-Steele Solid-Fluid Potential Energy Equation 8 " # " #9 !2 8 " # < 63 σ 10 σ 10 < 231 σ 12 σ 12 jf , S σ sf 4 = 3 σ sf 4 z sf sf sf sf ¼ - þ - þ - þ ; :1280 y 16 y σsf :1024 y Ψ2πðFs σsf 2 Þεsf y y y " # 9 !4 8 " # " #9
