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Improving the models used for calculating the size distribution of micropore volume of activated carbons from adsorption data
Carbon Vol. 36, No. 10, pp. 1469–1472, 1998 © 1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0008-6223/98 $—see front matter
PII: S0008-6223(98)00139-0
IMPROVING THE MODELS USED FOR CALCULATING THE SIZE DISTRIBUTION OF MICROPORE VOLUME OF ACTIVATED CARBONS FROM ADSORPTION DATA J P. O Micromeritics Instrument Corporation, Inc., Norcross, GA 30093, U.S.A. (Received 10 October 1997; accepted in revised form 5 February 1998) Abstract—A widely used model for analyzing micropore distribution in activated carbons assumes an array of semi-infinite, rigid slits of distributed width whose walls are modeled as energetically uniform graphite. Adsorption isotherms can be simulated for this system using GCMC or DFT. Micropore size distributions determined from experimental isotherms using such models usually show minima near two and three molecular diameters effective pore width, regardless of the simulation method used. This may prove to be a model-induced artifact, rather than the real situation. © 1998 Elsevier Science Ltd. All rights reserved. Key Words—A. Activated carbon, C. adsorption, molecular simulation, D. microporosity.
of pore surface, and f(H ) is the desired pore surface area distribution function with respect to H. Equation (1) represents a Fredholm integral and its inversion is well known to present an ill-posed problem. Since we are usually only interested in the numerical values of f(H ), we can rewrite eqn (1) as a summation.
1. INTRODUCTION
It is increasingly common to study adsorption processes, whether on free surfaces or in confined spaces such as pores by modeling or simulation techniques. The goal of these studies is frequently to develop an understanding that will better enable adsorption measurements to be used to characterize various adsorbents in terms of their surface properties or pore structure. Modeling methods include Grand Canonical Monte Carlo (GCMC ) and density functional theory (DFT ), using non local prescriptions such as that of Tarazona et al. [1–3]. For instance, suitably modified non-local density functional theory (MDFT ) has been shown to provide an excellent description of the physical adsorption of nitrogen or argon on the energetically uniform surface of graphite [4]. This and other formulations of DFT have been used to model adsorption in narrow slit pores and provide the basis for a method of estimating pore size distribution from experimental isotherms. Similarly, GCW modeling has been used for the same purpose [5].
Q( p)=∑ q(p, H )f (H ), (2) i i i where Q(p) is an experimental adsorption isotherm interpolated onto a vector p of pressure points, q(p, H ) is a matrix of values for quantity adsorbed i per square meter, each row calculated for a value of H at pressures p, and f(H ) is the solution vector i whose terms represent the area of surface in the sample characterized by each pore width H . The i solution values desired are those that most nearly, in a least squares sense, solve eqn (2). Certain other constraints, such as non-negativity, may be imposed to stabilize the solution. 3. PROBLEMS WITH CURRENT MODELS
Figure 1 presents graphically some of the model isotherms calculated by MDFT for slit pores in a model graphite. Note that pore width, H, as used here is the available pore width, defined by
2. APPLICATION OF MODELS TO EXPERIMENTAL DATA
The integral equation of isothermal adsorption for the case of pore size distribution can be written as the convolution
P
Q( p)= dH q( p, H ) f (H ),
H=w−s , (3) cc where w is the distance between the centers of carbon atoms of opposite model pore walls and s is the cc Lennard–Jones diameter of a carbon atom in graphite. When these rigid, graphite-based slit pore models are applied to adsorption data for activated carbons, the resulting pore width distributions rather consistently show a relatively low population of pores near ˚ width and a near absence of pores in the coopera6A ˚ . Figures 2 tive filling range centered around 9–10 A
(1)
where Q( p) is the total quantity of adsorbate per gram of adsorbent at pressure p, q( p, H ), the kernel function, describes the adsorption isotherm for an ideally homoporous material characterized by pore width H as quantity of adsorbate per square meter 1469
1470
J. P. O
by our modeling assumptions. The latter case seems the more probable. 4. EFFECTS OF SURFACE HETEROGENEITY
Fig. 1. Model isotherms representing the kernel function, q(p, H ), in eqn (2). Calculated for argon at 87 K. i
and 3 illustrate micropore size distributions obtained by this method for two typical carbons. The minimum ˚ is especially obvious. in the distribution near 10 A The question arises as to the meaning of these multimodal features, that is, are they a true characteristic of activated carbons or an artifact introduced
One observes in the model isotherms that the ˚ micropores, near relative pressure for filling 9–10 A 1×10−3, coincides with that for abrupt monolayer formation on the free surface of graphite. The assumption of energetic uniformity plays a key role in this feature of the model, and, if not a valid assumption, will certainly lead to some underestimation of pores in this size range. One is led, therefore, to explore the effect of including energetic heterogeneity of the pore walls on the derived model isotherms. A simple way to do this is by a weighted combination of the isotherms calculated by MDFT for a given pore size and a distribution of adsorptive potentials. Another model calculable by DFT that directly incorporates heterogeneity can be created by assigning different adsorptive potentials to opposite walls of a slit pore. The adsorptive potential can also be made heterogeneous by simply assigning a finite extent to a slit pore, i.e. allowing for edge effects and closed sides. A yet more sophisticated model can be constructed by assigning a patchwise heterogeneity to the pore walls, as was successfully done in modeling a nanoporous silica [6 ]. These last two models are best investigated by simulation techniques such as GCMC. 5. PORE SHAPE AND PACKING EFFECTS
Fig. 2. Micropore distribution for Calgon BPL carbon from models of Fig. 1 and adsorption isotherm data for argon at 87 K.
Fig. 3. Micropore distribution for Carbosieve-G from models of Fig. 1 and adsorption isotherm data for argon at 87 K.
Inspection of Fig. 1 reveals that the pore capacity per unit pore wall area is quite non-linear with pore ˚ and again width, showing abrupt increases near 6 A ˚ . Since geometric pore volume is strictly linear at 10 A with width, these increases are clearly due to packing effects, corresponding to the transitions from pore widths accommodating one adsorbed layer to two, and from two layers to three. Because pronounced packing effects will persist even for pores modeled with energetically heterogeneous walls, it seems likely that this is the dominant pore model feature causing the double minima in the derived pore size distributions. Figure 4 shows how pore capacity per unit pore wall area changes with pore width. The effects can be seen more clearly in Fig. 5, where the data have been transformed to mean pore fluid density according to r=QH/2 where Q is the mols adsorbed per unit wall area. If we dismiss multimodal micropore ˚ as unlikely distributions with minima at 6 and 10 A to be the general case for activated carbons, we might assume alternatively that the true distributions are smoothly monotonic over this region. With current models, this would predict that the adsorption isotherms for such carbons would show at least two
Micropore volume of activated carbons from adsorption data
1471
a dilation or swelling of the structure, which in turn would reduce observable packing effects. An estimate of the driving force for pore dilation can be derived by considering a micropore of width w exposed to adsorptive at a fixed low pressure. We imagine that the pore walls resist expansion with a separation dependent potential w(w). We can write as a condition for dilation m
Fig. 4. Pore capacity at saturation for pores modeled as in Fig. 1. The line is drawn to guide the eye.
dn dw
−
dQ(w) dw
≥0,
(4)
where n is the mols of adsorptive transferred to the pore during dilation and m is the chemical potential of the bulk adsorptive. Since n=rwa and we may take the pore area, a=1, we have dn dw
=p+w
dr dw
.
(5)
The quantity dr/dw is the derivative of the curve in Fig. 5. The quantity dn/dw has been calculated according to eqn (5) and is plotted in Fig. 6 for the situation at saturation. It can be seen to vary widely ˚ , but in magnitude for pore widths less than 20 A little else can be inferred at this time. 7. CONCLUSIONS
6. DISCUSSION
The current widely used model for analyzing micropore distribution in activated carbon assumes an array of semi-infinite, rigid slits of distributed width whose walls are modeled as energetically uniform graphite. Adsorption isotherms can be simulated for this system using GCMC or DFT. Inversion of the integral equation of adsorption to determine micropore size distribution from experimental isotherms using such models usually produces results showing ˚ effective pore width, regardminima near 6 and 10 A less of the simulation method used. This is assumed to be a model-induced artifact. The inclusion of surface heterogeneity in the model, while more realistic, does not change this observation significantly. The strong packing effects exhibited by a rigid parallel
The slit pore model that is currently widely used in simulating activated carbons is in part supported by physical evidence from electron microscopy, X-ray diffraction and other techniques. An alternative geometry for modeling activated carbon micropores might involve non parallel walls. Such a wedgeshaped pore model could reduce packing effects if the assumed wedge angle were large enough to blur the transitions between small integer numbers of layers of adsorbate as pore size is changed. It would be important to find independent physical evidence to support such a model. A second possibility is that the explicit modeling assumption concerning the inertness of the adsorbent may not be valid. If the in vacuo pore structure of a carbon is relaxed by the relatively large quantity of adsorbate uptake at low pressure, one would expect
Fig. 6. Rate of uptake of pore fluid with pore dilation based on DFT calculation for slit-like pores.
Fig. 5. Data of Fig. 4 shown as mean pore fluid density, r, at saturation based on amount adsorbed and available pore width. Packing effects are seen through four layers.
inflections in the low relative pressure region of their isotherms as pores capable of high packing density were filled. This does not seem to be the common observation.
1472
J. P. O
wall model seems likely to be the dominant feature causing the double minima in the derived pore size distributions. While alternative geometries for pore shape may be worth pursuing, it is speculated that relaxing the assumption of adsorbent inertness may also be fruitful. It is further speculated that the calculated packing effects may provide a driving force for sample dilation ˚ . Pore dilation of a in the pore size range below 20 A magnitude sufficient to mask pore fluid density oscillations should be experimentally observable.
REFERENCES 1. Tarazona, P., Phys. Rev. A, 1985, 31, 2672. 2. Tarazona, P., Phys. Rev. A, 1985, 32, 3148. 3. Tarazona, R., Marconi, U. M. B. and Evans, R., Molec. Phys., 1987, 60, 573. 4. Olivier, J. P., J. Porous Mater., 1995, 2, 9. 5. Lopez-Ramon, V., Jagiello, J., Bandom, T. J. and Seaton, N. A., Characterization of Porous Solids (COPS-IV), in press. 6. Maddox, M., Ofivier, J. P. and Gubbins, K. E., Langmuir, 1997, 13, 1737.
PII: S0008-6223(98)00139-0
IMPROVING THE MODELS USED FOR CALCULATING THE SIZE DISTRIBUTION OF MICROPORE VOLUME OF ACTIVATED CARBONS FROM ADSORPTION DATA J P. O Micromeritics Instrument Corporation, Inc., Norcross, GA 30093, U.S.A. (Received 10 October 1997; accepted in revised form 5 February 1998) Abstract—A widely used model for analyzing micropore distribution in activated carbons assumes an array of semi-infinite, rigid slits of distributed width whose walls are modeled as energetically uniform graphite. Adsorption isotherms can be simulated for this system using GCMC or DFT. Micropore size distributions determined from experimental isotherms using such models usually show minima near two and three molecular diameters effective pore width, regardless of the simulation method used. This may prove to be a model-induced artifact, rather than the real situation. © 1998 Elsevier Science Ltd. All rights reserved. Key Words—A. Activated carbon, C. adsorption, molecular simulation, D. microporosity.
of pore surface, and f(H ) is the desired pore surface area distribution function with respect to H. Equation (1) represents a Fredholm integral and its inversion is well known to present an ill-posed problem. Since we are usually only interested in the numerical values of f(H ), we can rewrite eqn (1) as a summation.
1. INTRODUCTION
It is increasingly common to study adsorption processes, whether on free surfaces or in confined spaces such as pores by modeling or simulation techniques. The goal of these studies is frequently to develop an understanding that will better enable adsorption measurements to be used to characterize various adsorbents in terms of their surface properties or pore structure. Modeling methods include Grand Canonical Monte Carlo (GCMC ) and density functional theory (DFT ), using non local prescriptions such as that of Tarazona et al. [1–3]. For instance, suitably modified non-local density functional theory (MDFT ) has been shown to provide an excellent description of the physical adsorption of nitrogen or argon on the energetically uniform surface of graphite [4]. This and other formulations of DFT have been used to model adsorption in narrow slit pores and provide the basis for a method of estimating pore size distribution from experimental isotherms. Similarly, GCW modeling has been used for the same purpose [5].
Q( p)=∑ q(p, H )f (H ), (2) i i i where Q(p) is an experimental adsorption isotherm interpolated onto a vector p of pressure points, q(p, H ) is a matrix of values for quantity adsorbed i per square meter, each row calculated for a value of H at pressures p, and f(H ) is the solution vector i whose terms represent the area of surface in the sample characterized by each pore width H . The i solution values desired are those that most nearly, in a least squares sense, solve eqn (2). Certain other constraints, such as non-negativity, may be imposed to stabilize the solution. 3. PROBLEMS WITH CURRENT MODELS
Figure 1 presents graphically some of the model isotherms calculated by MDFT for slit pores in a model graphite. Note that pore width, H, as used here is the available pore width, defined by
2. APPLICATION OF MODELS TO EXPERIMENTAL DATA
The integral equation of isothermal adsorption for the case of pore size distribution can be written as the convolution
P
Q( p)= dH q( p, H ) f (H ),
H=w−s , (3) cc where w is the distance between the centers of carbon atoms of opposite model pore walls and s is the cc Lennard–Jones diameter of a carbon atom in graphite. When these rigid, graphite-based slit pore models are applied to adsorption data for activated carbons, the resulting pore width distributions rather consistently show a relatively low population of pores near ˚ width and a near absence of pores in the coopera6A ˚ . Figures 2 tive filling range centered around 9–10 A
(1)
where Q( p) is the total quantity of adsorbate per gram of adsorbent at pressure p, q( p, H ), the kernel function, describes the adsorption isotherm for an ideally homoporous material characterized by pore width H as quantity of adsorbate per square meter 1469
1470
J. P. O
by our modeling assumptions. The latter case seems the more probable. 4. EFFECTS OF SURFACE HETEROGENEITY
Fig. 1. Model isotherms representing the kernel function, q(p, H ), in eqn (2). Calculated for argon at 87 K. i
and 3 illustrate micropore size distributions obtained by this method for two typical carbons. The minimum ˚ is especially obvious. in the distribution near 10 A The question arises as to the meaning of these multimodal features, that is, are they a true characteristic of activated carbons or an artifact introduced
One observes in the model isotherms that the ˚ micropores, near relative pressure for filling 9–10 A 1×10−3, coincides with that for abrupt monolayer formation on the free surface of graphite. The assumption of energetic uniformity plays a key role in this feature of the model, and, if not a valid assumption, will certainly lead to some underestimation of pores in this size range. One is led, therefore, to explore the effect of including energetic heterogeneity of the pore walls on the derived model isotherms. A simple way to do this is by a weighted combination of the isotherms calculated by MDFT for a given pore size and a distribution of adsorptive potentials. Another model calculable by DFT that directly incorporates heterogeneity can be created by assigning different adsorptive potentials to opposite walls of a slit pore. The adsorptive potential can also be made heterogeneous by simply assigning a finite extent to a slit pore, i.e. allowing for edge effects and closed sides. A yet more sophisticated model can be constructed by assigning a patchwise heterogeneity to the pore walls, as was successfully done in modeling a nanoporous silica [6 ]. These last two models are best investigated by simulation techniques such as GCMC. 5. PORE SHAPE AND PACKING EFFECTS
Fig. 2. Micropore distribution for Calgon BPL carbon from models of Fig. 1 and adsorption isotherm data for argon at 87 K.
Fig. 3. Micropore distribution for Carbosieve-G from models of Fig. 1 and adsorption isotherm data for argon at 87 K.
Inspection of Fig. 1 reveals that the pore capacity per unit pore wall area is quite non-linear with pore ˚ and again width, showing abrupt increases near 6 A ˚ . Since geometric pore volume is strictly linear at 10 A with width, these increases are clearly due to packing effects, corresponding to the transitions from pore widths accommodating one adsorbed layer to two, and from two layers to three. Because pronounced packing effects will persist even for pores modeled with energetically heterogeneous walls, it seems likely that this is the dominant pore model feature causing the double minima in the derived pore size distributions. Figure 4 shows how pore capacity per unit pore wall area changes with pore width. The effects can be seen more clearly in Fig. 5, where the data have been transformed to mean pore fluid density according to r=QH/2 where Q is the mols adsorbed per unit wall area. If we dismiss multimodal micropore ˚ as unlikely distributions with minima at 6 and 10 A to be the general case for activated carbons, we might assume alternatively that the true distributions are smoothly monotonic over this region. With current models, this would predict that the adsorption isotherms for such carbons would show at least two
Micropore volume of activated carbons from adsorption data
1471
a dilation or swelling of the structure, which in turn would reduce observable packing effects. An estimate of the driving force for pore dilation can be derived by considering a micropore of width w exposed to adsorptive at a fixed low pressure. We imagine that the pore walls resist expansion with a separation dependent potential w(w). We can write as a condition for dilation m
Fig. 4. Pore capacity at saturation for pores modeled as in Fig. 1. The line is drawn to guide the eye.
dn dw
−
dQ(w) dw
≥0,
(4)
where n is the mols of adsorptive transferred to the pore during dilation and m is the chemical potential of the bulk adsorptive. Since n=rwa and we may take the pore area, a=1, we have dn dw
=p+w
dr dw
.
(5)
The quantity dr/dw is the derivative of the curve in Fig. 5. The quantity dn/dw has been calculated according to eqn (5) and is plotted in Fig. 6 for the situation at saturation. It can be seen to vary widely ˚ , but in magnitude for pore widths less than 20 A little else can be inferred at this time. 7. CONCLUSIONS
6. DISCUSSION
The current widely used model for analyzing micropore distribution in activated carbon assumes an array of semi-infinite, rigid slits of distributed width whose walls are modeled as energetically uniform graphite. Adsorption isotherms can be simulated for this system using GCMC or DFT. Inversion of the integral equation of adsorption to determine micropore size distribution from experimental isotherms using such models usually produces results showing ˚ effective pore width, regardminima near 6 and 10 A less of the simulation method used. This is assumed to be a model-induced artifact. The inclusion of surface heterogeneity in the model, while more realistic, does not change this observation significantly. The strong packing effects exhibited by a rigid parallel
The slit pore model that is currently widely used in simulating activated carbons is in part supported by physical evidence from electron microscopy, X-ray diffraction and other techniques. An alternative geometry for modeling activated carbon micropores might involve non parallel walls. Such a wedgeshaped pore model could reduce packing effects if the assumed wedge angle were large enough to blur the transitions between small integer numbers of layers of adsorbate as pore size is changed. It would be important to find independent physical evidence to support such a model. A second possibility is that the explicit modeling assumption concerning the inertness of the adsorbent may not be valid. If the in vacuo pore structure of a carbon is relaxed by the relatively large quantity of adsorbate uptake at low pressure, one would expect
Fig. 6. Rate of uptake of pore fluid with pore dilation based on DFT calculation for slit-like pores.
Fig. 5. Data of Fig. 4 shown as mean pore fluid density, r, at saturation based on amount adsorbed and available pore width. Packing effects are seen through four layers.
inflections in the low relative pressure region of their isotherms as pores capable of high packing density were filled. This does not seem to be the common observation.
1472
J. P. O
wall model seems likely to be the dominant feature causing the double minima in the derived pore size distributions. While alternative geometries for pore shape may be worth pursuing, it is speculated that relaxing the assumption of adsorbent inertness may also be fruitful. It is further speculated that the calculated packing effects may provide a driving force for sample dilation ˚ . Pore dilation of a in the pore size range below 20 A magnitude sufficient to mask pore fluid density oscillations should be experimentally observable.
REFERENCES 1. Tarazona, P., Phys. Rev. A, 1985, 31, 2672. 2. Tarazona, P., Phys. Rev. A, 1985, 32, 3148. 3. Tarazona, R., Marconi, U. M. B. and Evans, R., Molec. Phys., 1987, 60, 573. 4. Olivier, J. P., J. Porous Mater., 1995, 2, 9. 5. Lopez-Ramon, V., Jagiello, J., Bandom, T. J. and Seaton, N. A., Characterization of Porous Solids (COPS-IV), in press. 6. Maddox, M., Ofivier, J. P. and Gubbins, K. E., Langmuir, 1997, 13, 1737.