Langmuir 2008, 24, 4853-4856
4853
Characteristic Heats of Adsorption for Slit Pore and Defected Pore Models G. R. Birkett and D. D. Do* Department of Chemical Engineering, UniVersity of Queensland, St. Lucia, QLD 4072, Australia ReceiVed NoVember 15, 2007. In Final Form: January 20, 2008 The characteristics of the heat of adsorption from a slit pore model of carbon are presented. This is shown to have a few key features that are always present, regardless of the pore size distribution used, as long as there is a reasonable range of pore sizes considered. The adsorption in a slit pore model is compared against the adsorption for a defected pore model. The isotherms of the defected pore model are qualitatively different from those of the slit pore and similar to those of amorphous carbon models presented in the literature. The heat of adsorption of the defected pore model is qualitatively different from the slit pore model, and its behavior falls between those of the slit pore model and the amorphous carbon models in the literature.
1. Introduction Characterization of porous solids using adsorption has traditionally been conducted by comparing isotherms from experiment and theory. This can, and does, provide valuable information about adsorbents and their interactions with the adsorbates used. However, isotherms can be insensitive to some of the features of the adsorbents’ properties, meaning that they can be hard to elucidate using isotherms alone. An additional property that has been used is the heat of adsorption. In this paper we investigate the heat of adsorption behavior for a common model of carbon adsorbents, the infinite slit pore, and a common adsorbate used for carbon characterization studies, argon.
2. Details of the Simulation Simulations were performed for the adsorption of argon at 77 K. The molecular model used for was a single site Lennard Jones (LJ) model with σff ) 0.3405 nm and ff/kb ) 119.8 K1 (where kb is the Boltzmann constant). The pores of the model activated carbon are treated as infinite slit pores. The potential energy between a pore wall and a fluid interaction site is calculated using the Steele potential,2 with the fluid-solid interaction parameters obtained using the Lorentz-Berthelot mixing rule.3 Simulations were performed in the standard GCMC ensemble.3 The simulation box was bound in one direction by the finite width, H, of the pore and was given a box length in the remaining two directions of 3.0 nm. Periodic boundary conditions were used in the lateral directions of the pore with fluid-fluid interactions having a lateral cutoff equal to half the simulation box length. The pressure range for the simulations was between 1 × 10-9 and 10 kPa (reduced pressure range from 3.7 × 10-11 to 0.37) with chemical potentials set to that of an ideal gas at that pressure. Simulations were started at the lowest pressure from a grid configuration with subsequent isotherm points taking their starting configuration from the final configuration of the previous pressure. Isotherm points were equilibrated until the number of particles was deemed to be at equilibrium. The equilibrium period ranged between 1 × 106 and 50 × 106 * To whom correspondence should be addressed. Phone: +61 7 3365 4154. Fax: +61 7 3365 2789. E-mail:
[email protected]. (1) Michels, A.; Wijker, H.; Wijker, H. Physica (The Hague) 1949, 15, 627. (2) Steele, W. A. Surf. Sci. 1973, 36, 317. (3) Allen, M. P.; Tildesley, T. P. Computer Simulation of Liquids; Clarendon Press: Oxford, 1987.
attempted Monte Carlo moves, depending on the number of particles in the simulation and the stage of adsorption. Collection of ensemble averages was conducted over a range from 15 × 106 to 60 × 106 attempted Monte Carlo moves, depending on the number of particles in the simulation. It was attempted, during the equilibration stage, to set the number of insertions such that a successful insertion was achieved, on average, every five attempted particle moves. This was not always possible, and where acceptance ratios of insertions were very low, the maximum number of insertions was set to 5 times the number of attempted moves. In this study, we will use the isosteric heat of adsorption, qst, as the heat of adsorption for ease of comparison with other studies. The isosteric heat of adsorption for the adsorption of an ideal gas is defined in this study as
qst ) RT - qd ) RT -
(∂U ∂N)
T,V
(1)
where R is the gas constant, T is the temperature, U is the configurational energy of the adsorption system, N is the absolute number of particles in the adsorption system, V is the volume of the adsorption system, and qd is the differential heat of adsorption. From a simulation standpoint, the system is the simulation cell used. For a GCMC simulation, the differential in eq 1 can be calculated directly using the fluctuation theory4
(∂U ∂N)
T,V
)
f(N,U) f(N,N)
(2)
where f(X,Y) is the cross-correlation function of ensemble properties X and Y defined as4
f(X,Y) ) 〈XY〉 - 〈X〉〈Y〉
(3)
where the angled brackets denote the ensemble average obtained from simulation. Equations 1-3 can be combined to calculate the isosteric heat for a system defined by a single simulation. If an adsorption system can be modeled using a single simulation, this is all that is required. If the link between simulation and experiment is made through a pore size distribution (PSD), then we need to use a method of combining the results of the (4) Nicholson, D.; Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: London, 1982.
10.1021/la703565m CCC: $40.75 © 2008 American Chemical Society Published on Web 03/26/2008
4854 Langmuir, Vol. 24, No. 9, 2008
Birkett and Do
Figure 1. Three general types of pore size distribution considered. Pore size is the carbon-carbon distance, based upon the carbon centers, of the slit pore.
Figure 2. Isotherms from simulation calculated using eq 4 and the PSDs plotted in Figure 1. Note that the amount adsorbed is reduced by the amount adsorbed at P/P0 ) 0.37.
simulations for each pore size considered. To calculate the total amount adsorbed, Ntot, the calculation is straightforward and familiar
Ntot )
∫0∞F(H) f(H) dH
(4)
where F(H) is the density of a pore with width H and f(H) dH is the volume of pores having a width between H and H + dH. For the practical use of eq 4, the integral is replaced by a sum over discrete intervals using the PSD of the solid. While the use of eq 4 in the calculation of the amount adsorbed is wellestablished, the method for calculating the heat of adsorption for a porous solid has been incorrectly presented in the literature.5-7 Recently, we presented the correct form of the equation to calculate the heat of adsorption for a porous solid when combining individual simulation results with a PSD.8 In that study, we failed to acknowledge similar works completed by Jaroniec9 in 1977, who considered a similar problem with heats of adsorption on a heterogeneous surface. In our previous paper, we presented a number of ways of combining the results of the individual simulations. In this paper, we use the following equation, which is derived in Appendix A:
f(N,U)H f(H) dH V(H) ) T,Vtot ∞f(N,N)H f(H) dH 0 V(H)
( ) tot
∂U ∂Ntot
∫0∞ ∫
(5)
where the cross correlation functions f(X,Y)H are defined by eq 3 to give the fluctuation quantities of a pore with characteristic size H (carbon to carbon separation of a slit pore) and V(H) is the volume of the same pore.
3. Results and Discussion Simulations have been conducted for a range of pore sizes from 0.58 to 2.0 nm. To demonstrate the features of the isosteric heat of microporous carbons with a slit pore geometry, three different PSDs have been used. These are plotted in Figure 1 as a fraction of the maximum value of the PSD function f(H). So (5) Pan, H.; Ritter, J. A.; Balbuena, P. B. Ind. Eng. Chem. Res. 1998, 37, 1159. (6) He, Y.; Seaton, N. A. Langmuir 2005, 21, 8297. (7) Nicholson, D. Langmuir 1999, 15, 2508. (8) Birkett, G. R.; Do, D. D. Langmuir 2006, 22, 9976. (9) Jaroniec, M. J. Colloid Interface Sci. 1977, 59, 371.
Figure 3. Heats of adsorption from simulation calculated using eq 5 for the PSDs plotted in Figure 1. Lines are a guide for the eye only.
the three model PSDs are characterized as being flat (PSD A), skewed toward the smaller pores with a median pore size of 0.8 nm with no pore larger than 1.1 nm (PSD B), and skewed toward the larger pores with a median pore size of 1.3 nm and no pores smaller than 0.8 nm (PSD C). Figure 2 shows expected trends in the adsorption with the PSDs skewed to smaller pore sizes having a higher relative amount adsorbed with PSD B highest at all pressures and PSD A giving a higher relative amount adsorbed then PSD C at P/P0 < 1 × 10-4. At higher pressures, the PSDs A and C give very similar adsorption. Now using eq 5 it is possible to look at the behavior of the heat of adsorption for the different PSDs, which is plotted in Figure 3. All three PSDs show some similar trends. They all start, at zero coverage, at a local minimum before rising to a local maximum. The maximum is due to an increase in the fluidfluid interactions in the micropores where the adsorption initially occurs. For argon, this is in pores with a width of approximately 0.68 nm or the smallest pores for the case of PSD C, where the smallest pore size considered is 0.8 nm. This maximum at low loading for the slit pore model of microporous carbon was observed by Coasne et al.,10 but their simulation results for the isosteric heat seem to contain an error, with the zero-loading (10) Coasne, B.; Gubbins, K. E.; Hung, F. R.; Jain, S. K. Mol. Simul. 2006, 32, 557.
Heat of Adsorption from a Slit Pore Model of C
Langmuir, Vol. 24, No. 9, 2008 4855
Figure 5. Isotherm of defected pore in Figure 4 together with the isotherm for the slit pore model with PSD A.
Figure 4. Plot of carbon positions for the three defected graphite sheets used in the defected pore simulations. Note that z-axis is not to the scale of the x- and y-axes to show details of defected graphite sheets.
heat estimated to be about 10 kJ/mol. That value is closer to the heat of adsorption on a surface rather than a pore. After the maximum in the isosteric heat, it decreases to a local minimum before increasing again. The decrease is due to adsorption occurring in progressively larger pores, where the fluid-solid interaction eventually approaches those of the surface and then decrease even further as adsorption starts to occur away from the surfaces (i.e., in the second and subsequent layers). Off-setting the contribution of fluid-solid interactions at higher loadings is the increase in the fluid-fluid interaction as densities in large pores approach those of liquid argon. PSDs A and B have sharply decreasing heats of adsorption as the maximum capacity is approached and it becomes difficult to fit additional argon molecules in the micropores. Due to its larger pores with greater packing freedom, PSD C does not see such a dramatic decrease in the isosteric heat. Now the change in the adsorption as the pores deviate from perfect slit pores to those containing defects can be studied. The effect of defected graphite surfaces on adsorption isotherms and isosteric heats has been studied for nitrogen at 77 K by Do and Do,11 who showed that defects caused deviation from Henry law behavior and increased heats of adsorption at low loading when compared to graphite. Their simple model of a defected surface involved removing random patches of carbons within a certain defect radius until a desired fraction of the carbon atoms were removed. To demonstrate the effect of defected graphite structures on the adsorption and isosteric heat in a slit pore, a defected pore is created by inserting three graphite sheets at a spacing of 0.3354 nm into a slit pore with a width of 1.3416 nm. The graphite sheets have defects created in them using the same method as Do and Do11 with a defect percentage of 50% and a defect radius of 0.492 nm. A plot of the three defected graphite sheets in the slit pore is given in Figure 4. Now the pores for the configuration in Figure 4 are created by the defects with pores of varying size, from 0.67 to 1.34 nm, approximately, created depending on how the defects of the different sheets are aligned. This defect type model can be seen as a point between the traditional perfect slit (11) Do, D. D.; Do, H. D. J. Phys. Chem. B 2006, 110, 17531.
Figure 6. Heats of adsorption for the defected pore in Figure 4 together with the heat of adsorption for the slit pore model with PSD A.
pore and the amorphous type carbon models presented by Gubbins and co-workers from reverse Monte Carlo studies.10,12-14 Now the adsorption isotherm and heat of adsorption for the defected pore are presented in Figures 5 and 6, respectively, together with the results for the slit pore model with PSD A. Now the adsorption isotherm for the defected pore is clearly shifted to the left compared with the isotherm for PSD A and has a uptake pressure range similar to that of PSD B in Figure 2. This is due to the microporous structure of the defected pore. Besides the uptake pressure, the notable feature of the defected pore’s isotherm is its shape being different from that of the perfect slit pore model with a continuous convex shape (on the log pressure scale) up to a reduced pressure of 1 × 10-6, where it goes through a point of inflection and rises smoothly to its maximum amount adsorbed. This is similar to the isotherms generated by Gubbins and co-workers with their amorphous carbon models obtained from reverse Monte Carlo.10,12-14 So this isotherm shape is not exclusive to amorphous carbons but is also a feature of the defected pore. (12) Pikunic, J.; Clinard, C.; Cohaut, N.; Gubbins, K. E.; Guet, J. M.; Pellenq, R. J. M.; Rannou, I.; Rouzaud, J. N. Langmuir 2003, 19, 8565. (13) Pikunic, J.; Llewellyn, P.; Pellenq, R.; Gubbins, K. E. Langmuir 2005, 21, 4431. (14) Jain, S. K.; Pikunic, J. P.; Pellenq, R. J. M.; Gubbins, K. E. Adsorption 2005, 11, 355.
4856 Langmuir, Vol. 24, No. 9, 2008
Now the heat of adsorption behavior for the slit pore and defected pore models, in Figure 6, shows some qualitative differences. The first is that the heat of adsorption for the defected pore is higher for all loadings except when N/N0 approaches 1. At zero loading, the heat of adsorption is greater due to the argon particles adsorbing in the interstices of the defected graphite layer. This also means that, at low loading, N/N0 < 0.1, the heat of adsorption for the defected pore is fairly constant as the adsorption occurs at a range of energetically similar sites at this low loading. This is in contrast with the slit pore model, which goes though a maximum in the heat of adsorption at low loading. Amorphous carbons10,13 start at a maximum in the heat of adsorption at zero loading with the heat of adsorption decreasing monotonically with loading. So, the behavior of the heat of adsorption at low loading is related to how close to the slit pore model the graphite is. If truly graphitic slit pores exist, there will be a maximum in the heat at low loading. When a graphitic structure is still dominant but with defects, the maximum becomes less evident, and when the carbon is amorphous, the maximum is only at zero loading. So, this might be a way to probe the graphitic structure using experimental measurements of the heat of adsorption. The heat of adsorption of argon and nitrogen on an amorphous carbon has been compared to experiment by Pikunic et al.13 with a good fit between simulation and experiment. Both simulation and experiment show a decreasing heat of adsorption with loading. However, this type of comparison with experiment needs to take into account the limitations of the experiment. The experiments conducted by Pikunic et al.13 had an evacuation pressure, or dynamic vacuum pressure, of 5 × 10-3 mbar, which is equal to a reduced pressure of approximately 1.8 × 10-5. Now, a solid whose heat of adsorption is on the order of 20 kJ/mol at low loading will have very significant adsorption at this reduced pressure. Even at a reduced pressure of 10-6, a common limitation in adsorption experiments, there will still be significant adsorption. This means that the heat measured from experiment, whether by calorimetry or equivalently by isosteres, does not give the zero loading heat of adsorption but rather the heat from the lowest pressure point measurable by the experiment. If experimental methods are improved and lower pressure adsorption can be measured by experiment, this may yield useful information about the microporous structure of carbon.
4. Conclusions Several characteristics of the heat of adsorption of subcritical argon on microporous carbon comprised of slit type pores have been identified using GCMC simulations. These have been calculated by combining the results of the simulations from individual pores using several pore size distributions. It was seen that the heat of adsorption displayed a distinct maximum at low loading for all pore size distributions. A defect pore model was also studied that gave a smoother adsorption isotherm similar to those of amorphous type carbon structures presented in the literature. The heat of adsorption of the defect pore has a greatly reduced, almost negligible, peak at low loading. The dependence of the heat of adsorption, at low loading, on pore structure might be a possible method of testing different pore models but requires experimental data at a suitably low pressure to be able to measure the heat of adsorption at close to zero loading. Acknowledgment. This research was made possible by the Australian Research Council, whose support is gratefully acknowledged. Thanks also to the University of Queensland High Performance Computing facility for a generous allocation of computing time.
Birkett and Do
Appendix Equation 5 is derived starting with the differential of the total energy, Utot, with respect to the total amount adsorbed, Ntot
( ) ( ) ( ) ∂Utot ∂Ntot
∂Utot ∂R
)
T,Vtot
/
T,Vtot
∂Ntot ∂R
(A1)
T,Vtot
where R ) -µ/RT. Now we can write the total energy of the adsorption system as
Utot )
∫0∞u(H) F(H) f(H) dH
(A2)
Now, since uF ) uN/V ) U/V, we get
Utot )
U(H)
∫0∞ V(H) f(H) dH
(A3)
Substituting eq A3 into the differential for energy on the righthand side of eq A1 gives
( ) ∂Utot ∂R
)
T,Vtot
(
∂ ∂R
)
U(H)
∫0∞ V(H) f(H) dH T,V
(A4)
tot
Since this is a fixed volume system, meaning H and f(H) are invariant with respect to R, eq A4 becomes
( ) ∂Utot ∂R
∫0∞
)
T,Va
(
∂U(H) ∂R
)
1
f(H) dH
T,VaV(H)
(A5)
Now, using eq 4 we can write
Ntot )
N(H)
∫0∞F(H) f(H) dH ) ∫0∞V(H)f(H) dH
(A6)
Substituting eq A6 into the differential for the amount adsorbed on the right-hand side of eq A1 gives
( ) ∂Ntot ∂R
)
T,Vtot
(
∂ ∂R
)
N(H)
∫0∞V(H)f(H) dH T,V
(A7) tot
Since this is a fixed volume system, meaning H and f(H) are invariant with respect to R, eq A7 becomes
( ) ∂Ntot ∂R
)
T,Vtot
( )
∫0∞
∂N(H) ∂R
1
f(H) dH
T,VtotV(H)
(A8)
Substituting eqs A8 and A5 into eq A1 gives
( ) tot
∂U ∂Ntot
)
∫0∞(
∫0 ( ∞
T,Vtot
) )
∂U(H) ∂R ∂N(H) ∂R
1
f(H) dH
T,VV(H)
1 f(H) dH T,VV(H)
(A9)
Now we can introduce the fluctuation formulas15 (∂X/∂R)T,V ) f(X,N), and substituting into eq A9 gives
f(N,U)H f(H) dH V(H) ) T,Vtot ∞f(N,N)H f(H) dH 0 V(H)
( ) tot
∂U ∂Ntot
∫0∞ ∫
(A10)
LA703565M (15) Garrod, C. Statistical Mechanics and Thermodynamics; Oxford University Press: New York, 1995.
