Novel Method To Determine Accessible Volume, Area, and Pore Size

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Novel Method To Determine Accessible Volume, Area, and Pore Size Distribution of Activated Carbon L. F. Herrera, Chunyan Fan, D. D. Do,* and D. Nicholson School of Chemical Engineering University of Queensland St. Lucia, Qld 4072 AUSTRALIA ABSTRACT: We present a new procedure to determine the geometric area, accessible pore volume, and pore size distribution of activated carbon, and we test this with a detailed computer simulation study of a number of porous solid models. For these model adsorbents with known atom configurations, we determine the “intrinsic” accessible volume, the surface area, and pore size distribution using the Monte Carlo integration method proposed by Herrera et al. (Herrera, L.; Do, D. D.; Nicholson, D. A Monte Carlo integration method to determine accessible volume, accessible surface area and its fractal dimension. J. Colloid Interface Sci. 2010, 348 (2), 529-536). The inverse problem postulates that the theoretical adsorption isotherm is a linear combination of local isotherms, and matches the theoretical isotherm to the “computer-experimental” adsorption isotherm. The results suggest that this method is a promising tool to determine structural parameters of a porous solid. As a corollary, we propose a definition for the absolute isotherm as an alternative to the excess isotherm used in the literature.

1. INTRODUCTION Adsorption of gases in porous solids has been used as the principal characterization tool to determine solid structural parameters such as pore volume, surface area, and pore size distribution.2-4 Although there are other methods used in the characterization, adsorption remains the method of choice because it is readily carried out in most laboratories. Readers can refer to a number of reviews on this subject2,3 for further details. For the purpose of probing the structure, simple gases such as nitrogen or argon are commonly used, although some researchers have reasons to suggest other adsorbates, such as carbon dioxide, n-butane, benzene, etc. For example, adsorption of carbon dioxide at ambient temperatures is chosen for analysis of solids having pores of molecular dimensions, such as char, because its adsorption at these temperatures avoid the problem of diffusional limitation suffered by nitrogen adsorption under cryogenic conditions (77 K). Benzene was also chosen by Dubinin and co-workers, as the adsorbate to probe the structure of activated carbon, because of the compatibility of the flat structure of benzene and the slit shape of micropores. No matter which gases and which conditions are used in the characterization, the derived information must be consistent in the sense that the volume and the surface area should describe the correct geometric aspects of the solid under consideration, rather than some apparent values. Unfortunately this is not the case with the present methods currently practiced in the literature. For example, the surface area is commonly determined with the BET theory.5 This is usually obtained from the analysis of low temperature argon and nitrogen adsorption data over the reduced pressure range between 0.05 and 0.2 and between 0.05 and 0.35, respectively (this range is only suitable for nonporous solids or solids with large pores; for solids containing micropores the applicable range is much lower). In the presence of pores of molecular dimension, the BET area does not truly reflect the geometric area; for example, with super activated carbon the BET surface area was reported to be around 3000 m2/g.6 Clearly this r 2011 American Chemical Society

does not carry a geometric meaning because the area, counting both sides of a single sheet of graphene layer, is only 2600 m2/g, and the super activated carbon is not a single graphene sheet. Although the overestimation of the geometric area by the BET method is well-known, it can also underpredict the area, for example, when there are ultrafine pores that can pack only a single layer of adsorbate. The consequence of these uncertainties is that serious doubts are raised about what fraction of a quoted BET area can actually be attributed to exposed adsorbent surface. In the case of volume, there are two main methods to calculate it: helium expansion and the adsorption of some probe gas at its boiling point. The helium expansion measures the void volume, which is an apparent volume of the remaining space outside the solid and the volume of the pores and cavities inside the solid (Vvoid). The other method measures the apparent pore volume (Vpore) by the adsorption of gas at its boiling point. The difference between the void volume and the pore volume is the gas phase volume. It has been shown by Do et al.7 that neither of these methods provides the correct geometric volume, although some authors have argued that this is not important, even when helium adsorbs in the expansion experiment. For example, Neimark and Ravikovitch8 modeled the process of helium expansion and derived an apparent volume. Clearly, because of the possibility of helium adsorption in fine pores, this apparent volume is much greater than the actual geometric volume. One can argue that it does not much matter what void volume is used in the calculation of the excess amount provided that the dead volume is recorded and mentioned;9 however, we believe that this is not acceptable because it can lead to an incorrect physical interpretation of the adsorption. For example, if the void volume is overestimated, the adsorption excess at high pressures under supercritical conditions Received: October 25, 2010 Accepted: February 15, 2011 Revised: January 23, 2011 Published: March 03, 2011 4150

dx.doi.org/10.1021/ie102169u | Ind. Eng. Chem. Res. 2011, 50, 4150–4160

Industrial & Engineering Chemistry Research

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Figure 1. Schematic diagram of the inverse method (MCI, Monte Carlo integration; MCOp, Monte Carlo optimization).

can be very negative, leading to the incorrect conclusion that the adsorbed phase is not as dense as the bulk phase.10,11 Here, we argue that the most consistent way to characterize a porous solid is to obtain information about the actual geometric details of the solid. In a number of recent publications,12-16 we have put forward the notion of accessibility and introduced the concepts of accessible volume, accessible surface area, and accessible pore size distribution (A-PSD). By “accessibility”, we mean that volume, area, and pore size will depend on the choice of the adsorbate since clearly adsorbates of different sizes will probe different volumes and surface areas. This is because smaller molecules can approach closer to the solid surface than larger molecules, probe the surface in finer detail than larger ones, and enter small pores that larger molecules cannot. Do and co-workers proposed that the accessible space inside a porous solid is the region in which the potential energy of interaction between a probe molecule and the solid is nonpositive. This definition provides a clear and unambiguous framework for the characterization of porous solids and also makes a difference with the classical idea of accessibility presented by some authors such as Jagiello17,18 and Bradley.19 How do we choose a gas that fits the requirements of probing the pore volume? This gas should be simple and should form a condensate under normal conditions. Argon and nitrogen are among the most suitable candidates because under moderate conditions they can condense in pores. Other gases can also serve the same purpose. Adsorbates with strong hydrogen bonding should be avoided because they tend to attract to themselves more strongly than they interact with the surface and hence they are not appropriate to probe the solid structure. Herrera et al.1 have developed a Monte Carlo integration method to determine the accessible volume and surface area for solids whose atomic configurations are known. However, for unknown solids this information is not available, and therefore the challenge is to determine these quantities experimentally. This is a classic inverse and typically ill-posed problem. The usual procedure for the characterization of unknown solid is given as follows: 1. Carry out measurements of the adsorption isotherm. 2. Develop a model for pores of specific geometry and different sizes. 3. Calculate the local isotherms for these pores. The set of these local isotherms is called the kernel. 4. Assume that the solid in question is composed of pores having the same geometry as that of model pores in the kernel such that the theoretical total isotherm for the solid is a linear combination of all the local isotherms. The set of linear coefficients is then simply the local volume per unit mass because the experimental isotherm is in moles per kilograms while the local isotherm is typically expressed as

moles per cubic meter void volume. The definition of this void volume will be presented in section 2.1. 5. Match the theoretical isotherm in part 4 to the experimental isotherm of part 1 to derive information about the volumes and surface areas of all model pores in the kernel. Finally, the pore volume of the solid is the sum of all the local volumes. For the geometric area, we can calculate it based on the knowledge of area per unit volume for each model pore. To demonstrate that the above approach is a potential tool to characterize porous solids, we apply this to a number of model adsorbents and choose argon as the molecular probe. Since we construct the model adsorbent, a Monte Carlo integration method1 (MCI) can be used to derive the geometric properties of the solid: accessible volume, accessible surface area, and accessible pore size distribution (A-PSD). These properties are the exact structural properties of the solid. For the inverse determination of these properties from the sole information of an adsorption isotherm, we use the grand canonical Monte Carlo (GCMC) simulation to derive the adsorption isotherm of argon on this solid model. These two procedures are the computer experimental stages (the left panel of Figure 1). The inverse task is to determine these structural properties from the adsorption isotherm. To achieve this goal, the model adsorbent is assumed to be a linear combination of independent unit cells. These cells are model pores of our choice, for example, uniform pores of different sizes, and their structural properties are obtained from MCI with the same probe molecule. The set of local adsorption isotherms of these unit cells (which is obtained from GCMC simulation) is the kernel. The kernel and the structural properties for all the unit cells are the theory (the right panel of Figure 1). From the matching of the adsorption isotherm of the adsorbent and the linear combination of local isotherms (F(P) = ∑mjFj(P)), we will obtain the number of each unit cell (mj) that constitutes the model solid under consideration (the central part of Figure 1). The inverse problem is effectively solved with the Monte Carlo optimization method (MCOp).20 Knowing the numbers of all unit cells, we obtain the structural properties of the solid adsorbent as follows:

∑mjvacc;j S ¼ ∑mj sj FðRÞ ¼ ∑mj facc;j ðRÞ Vacc ¼

ð1Þ

where vacc,j, sj, and fj(R) are the accessible volume, accessible surface area, and pore size distribution of the unit cell j, respectively. mj is the number of the unit cell j used to fit the 4151

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computer experiment data, and Vacc, S, and F(R) are the accessible volume, surface area, and pore size distribution of the adsorbent.

2. THEORY A volumetric adsorption experiment is carried out by dosing a known amount of gas, N, into the sample cell and after allowing a sufficient time for the system to reach equilibrium. Therefore, the direct (raw) experimental data is a relationship between the total amount dosed into the adsorption cell and the equilibrium pressure. The amount adsorbed is taken to be the difference between the dosing amount and the amount that is left in the gas phase. Nex ¼ N - NG

ð2Þ

The amount N introduced into the adsorption cell is known accurately, but this is not the case for the amount left in the gas phase, NG, which is calculated as NG ¼ Vvoid FG

ð3Þ

where FG is the density in the gas phase at equilibrium, obtained from the knowledge of equilibrium pressure and the equation of state; PG = f(FG,T). The outstanding problem in the calculation of the excess amount, Nex, is the void volume Vvoid. This is not a major problem for subcritical adsorption at temperatures well below the critical point because N . NG, but it is for supercritical adsorption and even for subcritical adsorption at temperatures close to the critical temperature. Traditionally this void volume is obtained by carrying out a helium expansion experiment, and the problems associated with this approach have been discussed in the literature, for example, by Neimark and Ravikovitch8 and other references therein. Of these, the most critical, in our opinion, is the overestimation of the actual volume occupied by the gas. Furthermore, different gases can occupy different spaces, and therefore it is important that this is recognized and one should determine the gas space that would be occupied by the gas under consideration. We call it accessible volume, which is adsorbate dependent. If the void volume, Vvoid, in eq 3 is taken to be the one obtained from a helium expansion experiment, it is likely to overestimate the actual geometric volume because of the helium adsorption in small pores and helium accessibility in fine pores where other adsorbates cannot access and the adsorption excess will consequently be underestimated. 2.1. Accessible Volume and Accessible Surface Area from MC integration. The accessible volume is the volume accessible to the center of mass of an adsorptive molecule. If the atomistic configuration of the solid atoms is known, it can be obtained from the Monte Carlo integration.1 Briefly this method is as follows: we insert a single particle at random in the simulation box and calculate its potential energy with the solid; if the solid-fluid potential is nonpositive, the insertion is counted as a success. We repeat this process many times (say 107); the fraction of success is g and the accessible volume is then simply g times the volume of the simulation box. We obtain not only the total accessible volume by using the above procedure but also the “local” accessible volume in terms of the distance from the surface. This is done by defining a bin k as one having distances from the surface falling between dk and dkþ1. For each successful insertion, we search for the shortest distance between this insertion point and the solid. Let this

Figure 2. Physical parameters derived from the proposed method.

distance be d, and if it falls between dk and dkþ1, we increase the number of success of the bin k by 1. Once all the random insertions have been completed, we determine the fraction of success of the bin k as gk. Thus the “local” accessible volume of the bin k is simply gk times the volume of the sampling box, and the total accessible volume is merely the sum of all local accessible volumes. Having the “local” accessible volume of each bin k, the interfacial area between the bin k and bin k þ 1 is Sk ¼

Vk þ 1 - Vk dk þ 3=2 - dk þ 1=2

ð4Þ

where Vk is the local accessible volume of the bin k. The distance dkþ1/2 is the distance of the bin k from the surface, which is taken to be the arithmetic average between dk and dkþ1. The first nonzero area close to the surface, which is the boundary where the solid-fluid potential is zero, is defined as the geometric surface area (ST). Given the definition of the accessible volume and accessible surface area, we define the following terms that we will use in the characterization of a porous solid: Accessible volume of the adsorption cell (Vcell) is the geometric volume occupied by the gas phase, which is the volume outside the porous solid and the accessible volume of very large pores. Accessible pore volume is the geometric volume that is accessible to the center of the adsorbate molecules inside the pores Vacc. This volume is calculated with eq 1. The accessible pore size distribution f(R) is obtained as the histogram plot of the accessible pore volume Vacc,j against the pore size Rj. Accessible external surface area is the geometric area of the surface outside the porous solid and inside the large pores (Sout). Accessible internal surface area (Sin) is the area of the pores’ surface inside the solid, and it is calculated from j

Sin ¼

∑ βi vacc, i

i¼1

ð5Þ

where βi is the ratio of the accessible area to the accessible volume of pore i, βi = Si/vacc,i. The total surface area is then ST = Sout þ Sin. The representation of the physical properties defined previously is presented in Figure 2. 2.2. Mass Balance Equation of a Volumetric Method. We have argued that the accessible volume and area are the fundamental parameters that should be used in the analysis and interpretation of adsorption, and therefore it is equally important that we propose a scheme to obtain these parameters experimentally. What that means is that, given a solid and an isotherm 4152



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of adsorption of some probe molecule on that solid, how do we derive the accessible volume and area? We tackle this question from an angle of computer simulation, and this is done stepwise as follows: 1. First, we choose a model solid whose structural details are sufficient for us to derive its exact accessible pore volume and accessible external and internal surface areas by the Monte Carlo integration method. This can be done on a unit cell of pore network (connecting pores of various sizes), and this unit cell is a representative of the solid under consideration. 2. We next conduct a GCMC simulation on this unit cell to obtain the ensemble average of the number of particles as a function of chemical potential (pressure). Readers can refer to various monographs in the literature for more details about GCMC simulation methods.21,22 For simple gases, such as argon, modeled as a single Lennard-Jones site, the equation of state obtained by Johnson et al.23 can be used to relate the chemical potential and pressure. 3. To simulate the total amount inside the adsorption cell containing the solid, we proceed as follows in order to mimic closely the adsorption cell that is used experimentally: If the void volume of the experimental adsorption cell is Vvoid0 and the pore volume of the solid is Vacc0 (the prime is used for the experimental system), we can use these parameters to scale the experimental system to the simulation box so that the volume ratio of the simulation box is the same as that of the real system: Vacc Vacc 0 ¼ ð6Þ Vvoid Vvoid 0 For a real experimental system used in collecting adsorption data, typical values of relevant parameters are Vvoid0 = 15 cm3, m = 0.1 g, ε = 0.3, and Fp = 1 g/cm3, giving a typical volume ratio for a practical adsorption system of Vvoid 0 15 cm3 ¼ 500 ¼ 0 Vacc ð0:3Þð0:1 gÞ=ð1 g=cm3 Þ Therefore, in simulation, we choose the simulated adsorption cell (Vcell) to have a volume on the order of 500 times the accessible volume of the unit cell in the simulated adsorption cell. Knowing the number of particles in the unit cell of accessible volume Vacc in step 2, we can simply obtain the total number of particles in the simulated adsorption cell as N ¼ ÆNp æ þ Vcell FG

ð7Þ

We repeat this for all values of chemical potential (pressure), and the relationship between this total number and the equilibrium pressure is the simulated “experimental” data that we are going to use in the inverse determination in the following steps. 4. Given the simulated “experimental” data obtained in step 3, we are now in a position to derive the required accessible parameters. This is the classic inverse problem, and it is done via the mass balance of the adsorption cell. Thus, at a given equilibrium pressure in the adsorption cell, the mass balance would require that the total mass in this cell must be equal to the mass in the accessible volume of the adsorption cell, plus the mass in the pores and the excess mass on the external surfaces, i.e.

obtained from the equilibrium pressure via equation of state), and m is the mass of solid. The second term in eq 8 accounts for the excess mass on the external surfaces of the solid, the surfaces of those large pores that are considered too large to constitute a pore. So what are the pores that we consider as pores whose adsorbed amounts are accounted for by the last term of eq 8? This depends on how extensive are the experimental data. As a general rule, a wider range of pressure would allow us to determine a larger pore volume in a given solid. We will discuss this further in section 3.3. Equation 8 is the theoretical mass balance, and it can only be computed when we define the model surface to compute the surface excess density, Γ, and the model pores to compute the absolute pore density, Fj. Once this is done, the theoretical total number of particles in the adsorption cell can be matched against the experimental value, Nexp0 , for all values of pressure, and the results of this matching are the derived accessible volume of the adsorption cell, the external surface area (Sout), and the specific accessible pore volume of each pore (vacc,j). How do we choose the model pores and a model surface? To this end we have to make some assumptions about the porous structure of the solid under consideration. For example, if the solid is of activated carbon type, then the usual approach is to use a graphitic slit as a model pore or we can use a defective graphitic slit for it to model the heterogeneous surface of the pore walls. If the pores of the solid are known to be cylindrical in shape, then it is appropriate to choose a cylinder as a model pore. Once we have chosen a surface model and a pore model, the accessible pore volume vacc,j and accessible internal surface area sacc,j of each pore j are derived by using the Monte Carlo integration method that is described in section 2.1. Next we apply the GCMC molecular simulation to obtain the excess surface density, Γ, for the model surface and for model pores the absolute pore density, Fj (for j = 1, 2, ..., M). The excess surface density and the absolute pore density are defined as follows: ÆNæ - Vcell FG number of particles Γ¼ ð9Þ m2 accessible area S ÆNæ number of particles ð10Þ vacc, i m3 accessible volume For pores, we use the absolute density, while for the model surface we use the excess surface density because it is not possible to define an absolute surface density for a surface. The pore densities are obtained for pores having sizes up to HM, where HM is the largest pore in the set of model pores in the kernel; any pores having size greater than HM will be treated as ones being lumped together with the external surface as the accessible external surface with an area Sout. As we indicated previously, the kernel is the set of local isotherms for the model surface and all the model pores, and for convenience it is denoted as Ψ. This kernel is known once we have defined our model surface and model pores. In abstract form we can write the mass balance equation (eq 8) for the total number of particles in the adsorption cell as follows: Fj ¼

NðpÞ ¼ f ðΨðpÞ, ΩÞ

M

N ¼ Vcell FG þ mSΓ þ m

∑

j¼1

vacc j Fj

ð8Þ

where N is the total number of particles in the adsorption cell, FG is the density of the bulk gas at equilibrium (which can be

ð11Þ

where Ω is the set of parameters that we would like to derive from the matching between the theoretical total number of particles, N, in eq 11 and the experimental value, Nexp. Thus the inverse is an optimization task to minimize the following 4153



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residual: K

Re s ¼

∑

k¼1

(

N½Ψðpk Þ; Ω - N exp ðpk Þ N exp ðpk Þ

)2 ð12Þ

Equation 12 is our fundamental working equation for the inverse determination. From the experimental measurements we have a relationship between the equilibrium pressure and the amount dosed into the system, N. The important feature of this relationship is that it is a monotonic function, in which the equilibrium pressure is always increasing with the amount dosed. Why is this important? This is so because in supercritical adsorption when one uses the supercritical adsorption isotherm (which is in the form of excess versus pressure) to fit the data, problems arise because there may be a maximum in the adsorption followed by a negative excess at higher pressures if the void volume is not chosen correctly. We also would like to stress that using the theoretical mass excess for the fitting against the experimental excess data cannot be recommended because the mass excess is a calculated value, not the measured data. Therefore, using unreliable values of calculated data in the fitting might yield unreliable and often unacceptable structural parameters for the solid. This might not be an issue when we deal with subcritical adsorption, usually at temperatures well below the critical point because the amount adsorbed is much greater than the amount left in the gas phase, due to the very low bulk gas density. Equation 11 is our working equation, but what is the maximum pore size, HM, that we should use in that equation? Clearly this depends on the range of pressure that is available from the experimental data. The larger the pressure range, the larger value of HM that can be determined, because if the pressure range is too low, many large pores are sparsely covered with and their volumes are not effectively probed by the adsorbed molecules. These large pores are thus treated as two independent surfaces, one from each wall of the pore, and therefore volumes of these pores cannot be determined. We match the experimental data of the equilibrium pressure P versus the amount dosed, N, against the theoretical value given in the right-hand side (RHS) of eq 11. The unknown parameters of the theoretical equation are the following: • the accessible volume of the adsorption cell Vcell, which also includes the accessible volumes of very large pores inside the solid (The large pores are those pores where adsorbate does not fully occupy the pores over the pressure range of interest.) • the external surface area, Sout • the specific accessible volumes of all model pores j (j = 1, 2, ..., M), vacc,j The fitting between the theoretical equation and the experimental data will give the structural parameters of the system. These are distinguishable because the bulk gas density, the surface excess, and the pore densities behave differently with pressure. 2.3. Adsorption Isotherm. There are two ways that we can present an adsorption isotherm. The first one is the absolute isotherm, which is defined as follows: " # M N - Vcell FG molecules ¼ SΓ þ aabs ¼ vacc ð13Þ j Fj kg m j¼1

∑

external surface (the first term). It has been shown by Do et al.15 that the excess density on a surface is always positive and the absolute isotherm as defined above is always positive and always increases monotonically with pressure. Alternatively, we can define it as the excess isotherm as follows: aex ¼

N - Vcell FG - mVacc FG m M

¼ SΓ þ

∑

j¼1

vacc j ðFj

"

# molecules kg

- FG Þ

ð14Þ

The excess isotherm considers the excess amounts in all pores as well as on the surface. Although the excess surface density is always positive by virtue of an open surface (no geometric constraint) and the excess pore density can be negative (at high pressures under supercritical conditions) if the three-dimensional confinement of the pore is such that the packing of molecules is inefficient, the excess isotherm as defined in eq 14 can be negative. Now comes the difficult question. Which isotherm would we recommend? Clearly, in the spirit of complying with what has been practiced in the literature, one would choose the excess isotherm. However, we would like to present here a case to support the use of the absolute isotherm. This absolute isotherm is always positive, irrespective of whether the adsorption condition is subcritical or supercritical and, most importantly, adsorption increases monotonically with pressure. On the other hand, the excess isotherm would show a maximum under supercritical conditions. 2.4. Adsorption Isotherms of Arbitrary Adsorbates. We have presented in section 2.2 a means to derive the physical properties of a porous solid from a single framework of analysis, and the basis for this is the mass balance equation of the total number of particles introduced into the adsorption cell. This is done with a reference adsorbate, say argon. Once we have obtained the set of physical parameters for a given adsorbate, we can obtain the similar set for an adsorbate B by using the following formulas: SB ¼ SAr acc, B

!

Ar ð15bÞ acc, Ar vj Vj As a first approximation, we can assume that the external surface area is probed by all adsorbates of any size. If B is larger in size than argon, then the accessible volume of adsorbate B in pore j is smaller than that for argon. There are certain pores having sizes smaller than a critical value that the adsorbate B cannot = 0. This is the sieving effect, enter, i.e., vBj = 0 because Vacc,B j resulting from a difference in adsorbate sizes. Thus, using the relations presented in eqs 15a and 15b, we can write the following equations for the absolute isotherm and excess isotherm for any adsorbate B: " # M molecules acc, B B B vj Fj ð16Þ aabs ¼ SΓ þ kg j¼J

vBj

¼

Vj

ð15aÞ

∑

M

This absolute isotherm accounts for all adsorbate molecules in M pores (the second term on the RHS of the above equation) and the excess amount (relative to the external gas phase) on the

aex ¼ SΓ þ B

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∑

j¼J

" acc, B vj ðFBj

- FBG Þ

# molecules kg

ð17Þ



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Table 1. Exact Physical Properties of Porous Solids model 1 case I 2

case II

model 2 case III

case I

case II

model 3

pore surface area (nm )

216.00

108.00

49.50

719.00

71.80

937.00

external surface area (nm2)

26.10

26.10

26.10

26.10

26.10

26.10

total surface area (nm2)

242.00

135.00

75.60

746.00

97.80

963.00

pore volume (nm3)

10.00

50.00

50.00

105.00

67.00

283.00

accessible volume (nm3)

10 000.00

25 000.00

25 000.00

52 500.00

33 500.00

142 000.00

Figure 3. Comparison between the GCMC “exact” data and the results obtained by optimization for a 0.65 nm slit pore. The contributions of the pore, external surface, and gas phase to the total isotherm are shown. The lines are the exact results, while the symbols are the results derived from the fitting. The inset shows the agreement between the computer experiment and the derived data over the low pressure region (shown in log-log plot).

where J is the critical pore of a size below which the adsorbate B cannot access. The implication of this sieving is clear, and in future work where we will discuss mixture adsorption, say between A and B, pores having sizes smaller than HJ will accommodate only the smaller adsorbate A while pores having sizes greater than HJ will contain both A and B. Mixture adsorption must account for this explicitly.

3. RESULTS AND DISCUSSION 3.1. Kernel. One of the important steps of the characterization method is the development of the kernel. This is because it has a large impact on the results obtained. There are three basic parameters in the development of a kernel: (i) the pore model, (ii) the interaction energy of the solid, and (iii) the potential model of the adsorptive. (i) The pore model: To construct the kernel of local isotherms for pores of different sizes, we need to assume a pore model. Of course, one cannot avoid this aspect of modeling, because given only the adsorption isotherm to derive information about a solid, this is a choice that has to be made. Since we are interested in activated carbon which is known to possess pores formed from graphene-like surfaces, we will assume a slit pore model. In addition to the

Figure 4. The top figure shows the comparison between the GCMC “exact” data and the results obtained by optimization for case I of model 2. The contributions of the pores, flat surface, and gas phase to the total isotherm are indicated in each figure. The lines are the derived results, while the symbols are the exact data. The inset shows the agreement between the computer experiment and the derived data over the low pressure region. The bar graphs are the exact (bottom) and derived (top) pore size distributions.

slit geometry, the pores are assumed to be homogeneous with a surface energy equal to that of a graphite surface. With these assumptions, we can construct a kernel for pores of various sizes and for an open surface. (ii) The interaction energy of the solid: The pore walls and the flat surface are simulated by using the Steele 10-4-3 potential equation.24 The solid atoms are made of carbon 4155



Figure 5. The top figure shows the comparison between the GCMC “exact” data and the results obtained by optimization for case II of model 2. The contributions of the pores, flat surface, and gas phase to the total isotherm are indicated in each figure. The lines are the derived results, while the symbols are the exact data. The inset shows the agreement between the computer experiment and the derived data over the low pressure region. The bar graphs are the exact (bottom) and derived (top) pore size distributions.

whose Lennard-Jones parameters are 0.34 nm and 28 K for the collision diameter and well depth of the interaction energy, respectively. (iii) The potential model of the adsorptive: The kernel is calculated for the adsorption of argon at 87.3 K on graphitic slit pores of sizes varying from 0.65 to 4.0 nm. An argon particle is simulated by a single Lennard-Jones site and the collision diameter and well depth of interaction energy are 0.3405 nm and 119.8 K, respectively. In addition, we choose this temperature because it is the temperature commonly used in the characterization of porous solids with argon. Besides the construction of this kernel, the density of the gas phase at each pressure point is calculated by using the Johnson equation of state for Lennard-Jones fluids.23 3.2. Solids Models. The method is tested with three different porous solid models. The models are constructed by combining a number of slit pores, and they are as follows: • Model 1 is a porous solid comprised by one single slit pore and a flat surface (unimodal solids). We provide three examples for this model. Cases I, II, and III have pore widths of 0.65, 1.5, and 2.6 nm, respectively.

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Figure 6. The top figure shows the comparison between the GCMC “exact” data and the results obtained by optimization for model 3. The contributions of the pores, flat surface, and gas phase to the total isotherm are indicated in each figure. The lines are the derived results, while the symbols are the exact data. The inset shows the agreement between the computer experiment and the derived data over the low pressure region. The bar graphs are the exact (bottom) and derived (top) pore size distributions.

• Model 2 is a model with six slit pores and a flat surface (multimodal solids). Two examples for this model are considered. Case I is a microporous solid having pores of widths 0.65, 0.8, 0.9, 1.1, 1.2, and 1.5 nm, and case II is a mesoporous solid having pores of widths 2.0, 2.2, 2.4, 2.6, 2.8, and 3.0 nm. • Model 3 is a micromesoporous solid, comprised of pores of 14 different widths: 0.65, 0.8, 0.9, 1.1, 1.2, 1.5, 1.6, 1.8, 2, 2.2, 2.4, 2.6, 2.8, and 3 nm. The exact structural parameters for these solid models, obtained from the MC integration, are given in Table 1. Having these graphitic slit pore models, we carry out the GCMC simulation to obtain the number of particles in the pore as a function of temperature and chemical potential (pressure), and then construct the experimental data of the total number of particles in the adsorption cell as follows: N 0 ¼ N þ Vcell FG

ð18Þ

where the ratio of the adsorption cell volume, Vcell, to the accessible volume is commensurate to the value that one would encounter in real experiments. Matching this “computer 4156



Figure 7. The top figure shows the comparison between the GCMC “exact” data and the results obtained by optimization for case I of model 2. The contributions of the pores, flat surface, and gas phase to the total isotherm are indicated in each figure. The lines are the derived results, while the symbols are the exact data. The inset shows the agreement between the computer experiment and the derived data over the low pressure region. The bar graphs are the exact (bottom) and derived (top) pore size distributions.

experiment” data versus the theoretical equation (eq 11), we derive (i) the accessible volume of the adsorption cell, (ii) the surface area for each pore, (iii) the pore size distribution, (iv) the total pore volume, and (v) the total surface area. 3.3. Characterization of the Pore Models at 87.3 K. We use the porous solid model 1, which is our simplest model, to test the method. Because the solid model is a single slit pore with its two walls simulated as a Steele surface and the kernel was constructed in a similar manner, we expect the derived adsorption isotherms, accessible volume, surface area, and pore size distribution to match the exact values perfectly. This is indeed the case, and we show this for the 0.65 nm slit pore in Figure 3, where excellent agreement is observed between the computer generated adsorption isotherms and the isotherms derived from eq 11 for the 0.65 nm slit pore. We also obtain excellent agreement for the cases of 1.5 and 2.6 nm slit pores (results are not shown). The physical properties of the porous solids, the accessible volume of the adsorption cell, and the surface area show perfect agreement with the target with a percentage of error less than 0.01%, while for the total pore volume a maximum error of 0.08% is observed

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Figure 8. The top figure shows the comparison between the GCMC “exact” data and the results obtained by optimization for case II of model 2. The contributions of the pores, flat surface, and gas phase to the total isotherm are indicated in each figure. The lines are the derived results, while the symbols are the exact data. The bar graphs are the exact (bottom) and derived (top) pore size distributions.

for all three cases. These results demonstrate the feasibility of using the mass balance of eq 8 to derive the physical parameters. When we calculate the BET surface area from the adsorption isotherm and compare it with the geometric surface area, the results show that the surface area is underestimated by around 39.69% for the 0.65 nm pore (case I) while in the case of 1.5 and 2.6 nm pores (cases II and III), the BET theory overpredicts the surface area by 41.92 and 69.73%, respectively. This is because for small pores only one layer can be accommodated in the pore to measure the surface area given by the two walls of the slit pore. In contrast, the 1.5 and 2.6 nm pores can accommodate more than two layers, and that is why there is an overprediction of the surface area by the BET theory. The smaller percentage of error obtained with the proposed method compared with that obtained from the BET theory shows the efficiency of the method in the derivation of the surface area. Now we discuss the results obtained for the solid model 2. Case I of this model is a microporous solid, while case II is a mesoporous solid. The comparison between the derived results and the exact parameters for these two examples are presented in Figures 4 and 5, where we see an excellent agreement in the isotherms and the derived pore size distributions. The success of the method in the derivation of the PSD for microporous and 4157



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Figure 9. Comparison between the absolute and excess adsorption isotherms for the mesoporous solid in model 2 (top) and the porous solid in model 3 (bottom). The LHS and the RHS show the experimental computer isotherms in the pores at 87.3 and 180 K, respectively.

mesoporous solids is remarkable because it was reported that the Dubinin-Stoeckli (DS), Horvath-Kawazoe (HK), and density functional theory (DFT) methods fail in the derivation of the PSD for a porous solid composed of slit pores.25 We compare the exact data and the derived results for the accessible volume of the adsorption cell of these two cases. The results show that there is a percentage of error of 0.15% for the microporous solid in case I and less than 0.01% for the mesoporous solid in case II. For the case of the total surface area both examples have a percentage of error less than 0.01%. On the other hand, the BET surface area is underestimated by 23% and overestimated by 79% for cases 1 and 2, respectively. This is in accord with what we have found with model 1; i.e., BET underestimates microporous solids while it overestimates mesoporous solids. Finally, we present the derived results for the porous solid composed of micropores and mesopores (model 3). Figure 6 shows, once again, the excellent agreement between the exact and derived isotherm and their corresponding PSDs. Quantitatively, our derived results for the accessible volume and the total surface area underestimate the exact corresponding results by 0.05 and 0.03%, respectively. On the other hand, the BET method underestimates the exact area by 11%. The proper derivation of the physical properties of this micromesoporous solid (typical of activated carbon) demonstrates the potential of this method to derive not only the accessible volume and the surface area but also the PSD covering the micropore and mesopore range within a single analysis framework.

3.4. Effect of Pressure Range in the Derivation of the Physical Properties of Porous Solids. We have shown that the

proposed method derives excellent results for the physical properties. These results were obtained when the adsorption isotherm over the range of pressures up to 100.0 kPa (which is around 0.99 the reduced pressure) was used in the fitting. To illustrate the effects of the range of pressures on the characterization of a porous solid, we calculate the physical properties of the two cases of the solid model 2 by using a smaller pressure range with pressures less than 10.0 kPa, and the results are shown in Figures 7 and 8. Figure 7 shows the adsorption isotherm and the PSD results for the microporous solid whose pores are between 0.65 and 1.5 nm. The results show that even when a low range of pressures is used, the results of the physical properties obtained are as good as those obtained when the full range of pressures is used (see Figure 4). For example, the accessible pore volume of the adsorption cell is overestimated by 1.31% while it was underestimated by only 0.15% when the wider range of pressure was used. For the surface area the percentage of error is 0.02%, compared to 0.01% when the wider range of pressure was used. For the case of mesoporous solid using pores between 2.0 and 3.0 nm, the pressure range affects the quality of the results. Figure 8 shows that when the smaller range of pressure is used the method fails in resolving the 3.0 nm pore size and yields two additional pore sizes, 1.1 and 1.9 nm, in the micropore region. Additionally, the overestimation of the surface area increases 4158


Industrial & Engineering Chemistry Research from 0.01 to 11% when this smaller range of pressures is used. This is expected because over the smaller pressure range the large pores are not filled and therefore their local isotherms behave like those for a flat surface. Despite the larger percentage of error (11%) in the derivation of the total surface area, the result is still acceptable, compared to the BET surface area with an error of around 78%. The accuracy in the determination of the accessible volume is also affected and the results show an underestimation of around 1.4%, compared to 0.01% when the full range of pressure was used. 3.5. Excess and Absolute Isotherms. We have shown that our method has great potential in the derivation of the physical properties, accessible volume of the adsorption cell, surface area, and PSD from the mass balance equation (eq 8). Let us now present the absolute and excess adsorption isotherms for some of the porous solid models. We show the adsorption isotherms for the contribution of pores for the mesoporous solid in model 2 and the micromesoporous solid in model 3. The top graphs in Figure 9 show the adsorption isotherms for the mesoporous solid of model 2. The absolute and excess isotherms at 87.3 K in the left-hand side of Figure 9 show no difference between the two. The adsorption isotherms at 180 K in the right-hand side of Figure 9 show a large difference between the absolute and the excess isotherms. The absolute amount in the pore always increases with pressure. This is because even at supercritical conditions the gas can be compressed in small pores. In contrast, the excess amount shows a maximum at 1.0 106 Pa. This maximum is due to the large density of the bulk phase26 which has a strong effect in the second term of eq 13. Similar results are reported by Do and co-workers for the adsorption of argon and nitrogen in slit pores at supercritical conditions.27,28 The two bottom figures in Figure 9 show the comparison between the absolute and excess isotherms at 87.3 and 180 K for the micromesoporous solid of model 3. The isotherms show the same features as the previous case. The absolute and excess isotherms behave similarly when the adsorption is carried out at 87.3 K. Under supercritical conditions (180 K) the absolute isotherm shows a steady increase in the amount adsorbed while the excess isotherms show maxima and then level off as the pressure increases.

4. CONCLUSIONS A new method has been presented in this paper to derive the accessible volume of the adsorption cell, external surface area, accessible pore volume, and PSD within a single framework. It is illustrated by adsorption of argon in a number of model solids covering the microporous and mesoporous regions. In comparing with the exact values, the percentage errors of accessible volume and the surface area are less than 2% and 0.05%, respectively. In contrast, the areas obtained from the BET method show deviations up to 71%. Our method also shows good agreement between the derived PSD and the exact PSD for all porous solid models studied in this paper. We have also proposed an absolute isotherm as an alternative to the excess isotherms commonly used in the literature. These isotherms are calculated for the adsorption of argon under subcritical and supercritical conditions in microporous and mesoporous solid models. The absolute isotherm shows a monotonic increase of the amount adsorbed with pressure at all conditions, while the excess isotherm shows a maximum and then decreases asymptotically at supercritical conditions.

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’ AUTHOR INFORMATION Corresponding Author

*Author to whom all correspondence should be addressed; E-mail: [email protected]; Fax: þ61-7-3365-2789.

’ ACKNOWLEDGMENT Support from the Australian Research Council is gratefully acknowledged. ’ REFERENCES (1) Herrera, L.; Do, D. D.; Nicholson, D. A Monte Carlo integration method to determine accessible volume, accessible surface area and its fractal dimension. J. Colloid Interface Sci. 2010, 348 (2), 529–536. (2) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic Press: New York, 1982; 303 pp. (3) Rouquerol, F.; Rouquerol, J.; Sing, K. Adsorption by Powders and Porous Solids Principles, Methodology and Applications; Academic Press: New York, 1999; p 465. (4) Do, D. D. Adsorption Analysis: Equilibria and Kinetics; Imperial College Press: London, 1998; 892 pp. (5) Brunauer, S.; Emmett, P. H.; Edward, T. Adsorption of gases in multimolecular layers. J. Am. Chem. Soc. 1938, 60, 309–319. (6) Otowa, T.; Yamada, M.; Tanibata, R.; Kawakami, M., Preparation, pore analysis and adsorption behaviour of high surface area active carbon from coconut shell. In Gas Separation Technology; Vansant, E., Dewolfs, R., Eds.; Elsevier: Amsterdam, 1990; pp 263-270. (7) Do, D.; Herrera, L.; Fan, C.; Wongkoblap, A.; Nicholson, D. The role of accessibility in the characterization of porous solids and their adsorption properties. Adsorption 2010, 16 (1), 3–15. (8) Neimark, A. V.; Ravikovitch, P. I. Calibration of Pore Volume in Adsorption Experiments and Theoretical Models. Langmuir 1997, 13 (19), 5148–5160. (9) Myers, A. L.; Monson, P. A. Adsorption in porous materials at high pressure: Theory and experiment. Langmuir 2002, 18 (26), 10261–10273. (10) Malbrunot, P.; Vidal, D.; Vermesse, J.; Chahine, R.; Bose, T. K. Adsorption measurements of argon, neon, krypton, nitrogen, and methane on activated carbon up to 650 MPa. Langmuir 1992, 8 (2), 577–580. (11) Malbrunot, P.; Vidal, D.; Vermesse, J.; Chahine, R.; Bose, T. K. Adsorbent Helium Density Measurement and Its Effect on Adsorption Isotherms at High Pressure. Langmuir 1997, 13 (3), 539–544. (12) Do, D. D.; Do, H. D. Appropriate volumes for adsorption isotherm studies: The absolute void volume, accessible pore volume and enclosing particle volume. J. Colloid Interface Sci. 2007, 316 (2), 317–330. (13) Do, D. D.; Nicholson, D.; Do, H. D. On the Henry constant and isosteric heat at zero loading in gas phase adsorption. J. Colloid Interface Sci. 2008, 324 (1-2), 15–24. (14) Do, D. D.; Do, H. D.; Wongkoblap, A.; Nicholson, D. Henry constant and isosteric heat at zero-loading for gas adsorption in carbon nanotubes. Phys. Chem. Chem. Phys. 2008, 10, 7293–7303. (15) Do, D. D.; Do, H. D.; Fan, C.; Nicholson, D. On the Existence of Negative Excess Isotherms for Argon Adsorption on Graphite Surfaces and in Graphitic Pores under Supercritical Conditions at Pressures up to 10,000 atm. Langmuir 2010, 26 (7), 4796–4806. (16) Fan, C.; Herrera, L. F.; Do, D. D.; Nicholson, D. New Method to Determine Surface Area and Its Energy Distribution for Nonporous Solids: A Computer Simulation and Experimental Study. Langmuir 2010, 26 (8), 5610–5623. (17) Lopez-Ram on, M. V.; Jagiezzo, J.; Bandosz, T. J.; Seaton, N. A. Determination of the Pore Size Distribution and Network Connectivity in Microporous Solids by Adsorption Measurements and Monte Carlo Simulation. Langmuir 1997, 13 (16), 4435–4445. 4159



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Novel Method To Determine Accessible Volume, Area, and Pore Size

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