Early Removal of Weak-Binding Adsorbates by Kinetic Separation

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Early Removal of Weak-Binding Adsorbates by Kinetic Separation Jared T. Burde and M. Mercedes Calbi* Department of Physics, Southern Illinois University, Carbondale, Illinois 62901-4401

ABSTRACT When a surface is exposed to a binary mixture of gases at the same partial pressures, it is expected that the species that binds to it more strongly will occupy a higher percentage of the surface due to its preferential adsorption. However, that is not necessarily the case during the equilibration process because the species with smaller binding adsorbs faster. Here, we demonstrate that a competition based on both binding energies and adsorption rates rules the kinetics of adsorption of a binary mixture on a homogeneous lattice of sites. This generates a selectivity process that has the opposite effect to the one that happens at equilibrium: the early removal of the weaker species by its faster adsorption. SECTION Nanoparticles and Nanostructures

(the peak value normalized by the equilibrium value) and the overshoot time (the time at which the overshoot is reached) on the temperature, chemical potentials, and binding energies of the system. Our starting point is a lattice gas model of a binary mixture of gases adsorbing on an array of sites. Assuming simple, spherical adsorbate particles, we allow only single-site occupation. Then, the total energy Ei,k for a particle of chemical species i on site k is found as X X si, k þ Jij sj, k ð1Þ Ei, k ¼ εi, k þ Jii k, NN k, NN

E

ver since carbon nanotubes were discovered, the possibility of adsorbing gases on their inner or outer surfaces has triggered numerous investigations of their storage abilities. Most of the studies of gas physisorption on nanotube bundles have been focused on equilibrium properties of the adsorbed films to identify the conditions that give rise to the formation of different phases on the various regions of the bundle.1,2 In addition, their possible use for gas separation applications was suggested early on, based on their potential to selectively adsorb different species as a consequence of disparities in their binding strength.3-12 This selectivity mechanism is based on removing the strongeradsorbing particles through its preferential adsorption.13 However, before reaching that equilibrium state, another selective process can take place based on the difference of the adsorption rates. In this case, the faster-adsorbing species could be removed from the mixture by controlling the time of exposure of the mixture to the adsorbent.13 When considering the uptake of gases that are directly in contact with exposed surfaces, faster adsorption rates occur in systems with lower binding energies.14 Therefore, a competition based on both the binding energies and the adsorption rates is expected to happen during the adsorption of a binary mixture of gases. We know, for example, that, after reaching equilibrium at the same partial pressure, the species with the higher binding energy will enjoy the greatest coverage. However, the weaker-binding species has faster adsorption kinetics and could reach a coverage value higher than its equilibrium value before the stronger species can adsorb significantly. We show in this work that that is indeed the case and that the result of this process is an “overshoot” in the coverage of the weaker species that is reached before the final equilibrium is achieved. This selectivity process can then be used to remove the weaker species (by its faster adsorption) instead of waiting to capture the stronger species at equilibrium. We analyze the dependence of the percent overshoot

r 2010 American Chemical Society

with Jij representing the interaction energies between particles of each chemical species and si,k taking on the value 1 if a particle of species i is in the adjacent site k and 0 if that site is unoccupied. The probabilities of adsorption and desorption are then given by Wads, i ¼ exp½ -βðEi, k - μi Þ ð2Þ Wdes, i Diffusion of particles along the lattice can be taken into account by “jumps” between adjacent sites with probability Wkl ¼ exp½ -βðEi, l - Ei, k Þ ð3Þ Wlk Simulations of the adsorption kinetics on the lattice can then be performed by using a Kinetic Monte Carlo (KMC) algorithm, as we have done in previous works.14,15 The lattice is in contact with an infinite reservoir of particles that contains both species at different chemical potentials, at a given temperature. Starting with an initially empty lattice, the transition probabilities for every possible event are calculated; Received Date: December 29, 2009 Accepted Date: February 1, 2010 Published on Web Date: February 05, 2010

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DOI: 10.1021/jz900468t |J. Phys. Chem. Lett. 2010, 1, 808–812

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a specific event is selected to occur following the algorithm rule, and the time is advanced.14,15 In this way, we track the time evolution of the system as it transitions from state to state, following it throughout the equilibration process while the coverage of each species is recorded as a function of time. All through the present study, we consider a single chain of adsorption sites with only one possible binding energy for each species (a one-dimensional homogeneous lattice), representing a groove or a strip along the external surface of a nanotube. In this Letter, we show results for systems in which the particle-particle interactions have been neglected. This model gives a fairly good representation of real situations when, for example, particle-particle interactions are much smaller than particle-site interactions (strong adsorbents). A purely analytical solution can be derived in this case, providing an independent verification of our simulation approach. This is because no energy barriers are present along the lattice in this case; each site is populated independently of the others, and diffusion processes do not play any role in the kinetics. We have observed that particle-particle interactions or binding energy heterogeneity modify the adsorption evolution of the species, but the weaker species overshoot, and the early selectivity process;the main findings that we want to report in this study;still occur. A detailed analysis of these effects is beyond the scope of this Letter and will be published elsewhere. Denoting each species as s (strong) or w (weak), we can postulate the following rate equation for each species dni ¼ Wads, i ð1 - ni - nj Þ - Wdes, i ni ð4Þ dt

Figure 1. Fractional coverage as a function of time for both species in the mixture (full lines). The dashed lines show the coverage evolution for each species as if they were the only species present in the gas (T = 100 K, εs = -400 K, εw = -200 K).

like that it if there were the single species present, and the weaker one adsorbs faster (see dashed lines). Eventually, as the system evolves to equilibrium, each species feels the presence of the other, and the weaker species has to surrender sites to the stronger species, which is favored at equilibrium. At that point, however, the stronger species's adsorption is considerably slowed down as the weaker species desorbs (notice the inflection in the blue curve). Beyond the existence of the peak time (which could be considerably small), we note that the weaker species's coverage remains greater than the stronger one over a fairly large period of time until both curves cross each other. The black points represent simulation data, and the solid lines show the curves calculated from eq 5. As can be seen, there is good agreement between the theoretical and simulated results. The slight deviations observed at equilibrium are an artifact of the finite nature of the lattice; by increasing the length of the lattice, it is possible to make them disappear completely, though at the cost of significant computing power. By fixing the equilibrium coverage of the stronger binding species, we can demonstrate the effect of the weaker species's equilibrium coverage on the percent overshoot experienced. We focus on the cases where the final equilibrium coverage of the stronger species is greater than that of the weaker one. As shown in Figure 2a, the overshoot increases with the coverage of the weaker species. While the peak time decreases, going to 0 as the system approaches a monolayer, the cross time increases, with the overshoot reaching values comparable to the equilibration time of the system. The cross time values for all combinations of the coverages can be found in the Supporting Information. Figure 2b shows the increase in the percent overshoot as the equilibrium coverage of the weaker species is increased. For a given value of the stronger species's coverage, an increase in the coverage is due to a higher pressure, which means more particles are moving in the gas, causing faster adsorption kinetics and higher overshoot. However, increases in the fixed coverage of the stronger species have a similar effect. In order to achieve the same coverage as the weaker species as the partial pressure of the strong species increases, we also need to considerably increase the pressure of the weaker species. This again accelerates the adsorption of this species, causing larger

with i,j = s,w and i 6¼ j. In these equations, ni represents the ratio of the number of particles adsorbed to the total number of sites of the array, and Wads(des),i gives the adsorption (desorption) transition probability for each species. By solving both differential equations simultaneously, we obtain equations for the coverage as a function of time for each species i ¼ s, w ð5Þ ni ðtÞ ¼ C1, i e -r1 t þ C2, i e -r2 t þ neq, i The coefficients C and time constants r are functions of the transition probabilities and thus depend on the thermodynamic properties of the system. Explicit expressions for these quantities can be found in the available Supporting Information. The equilibrium coverage for each species neq,i can be found directly from the system partition function as 1 ð6Þ neq, i ðT, μi , μj Þ ¼ β½ðεi - εj Þ - ðμi - μj Þ βðε μ Þ i i þ e 1þe with i,j = s,w and i 6¼ j; εi and μi are, respectively, the binding energy and chemical potential of species i, and β = 1/kBT (with kB being the Boltzmann constant and T the temperature of the system). Figure 1 shows a typical simulation run where the coverage of the weaker species experiences a large overshoot before falling back to its equilibrium coverage. While the coverage of the stronger species is clearly higher at equilibrium, the weaker species's coverage surpasses the stronger one by nearly the same amount at the peak time. As discussed earlier, this occurs because each species starts adsorbing at the rate


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Figure 2. (a) Fractional coverage of the weaker binding species for increasing values of its chemical potential (neq,s= 0.6; T = 100 K, εs = -400 K, εw = -200 K). The dashed line represents the evolution of the stronger species when neq,w = 0.3. (b) Percent overshoot for several values of neq,s.

Figure 3. (a) Fractional coverage as a function of time for the weaker binding species with binding energies of -100 (top), -150 (middle), and -200 (bottom); T = 100 K. The dotted line represents the evolution of the stronger species when εs = -400 K and neq,s = 0.6. (b) Overshoot as a function of εw for several values of εs.

Table 1. Ratio R = [(nw/ns)peak]/[(ns/nw)eq] for Several Coverages of the Lattice

species can be understood based on the results for the adsorption kinetics of a single species obtained in our previous work.14 As the binding energy decreases, the adsorption rate of the weaker species increases, causing it to adsorb even faster compared to the stronger species, producing a higher overshoot. In the same way, increases in the binding energy of the stronger species causes a reduction in the adsorption rate of the stronger species, which gives the weaker species more time to gain excess coverage, thereby increasing the overshoot of the system. In fact, the overshoot is an increasing function of the binding energy difference, as shown in Figure 3b. Finally, we look at the dependence of the overshoot on the temperature of the system. In this case, the equilibrium coverages and binding energies of both species are fixed, and the temperature is varied. The greatest overshoot is reached at the lowest temperature, which also exhibits the longest overshoot time. As the temperature increases, the overshoot becomes less pronounced and occurs much earlier in the evolution of the system. Figure 4a shows this effect, while panel (b) shows the decrease in the magnitude of the overshoot as the temperature goes up. As the temperature decreases, the adsorption kinetics of both species slows down, explaining why the overshoot occurs much later at lower temperatures. However, because of the initial exponential dependence of the kinetics on βε,14 the two species are not affected equally. Rather, the stronger species experiences a greater reduction in adsorption rate because of its higher binding energy. It is for this reason that the weaker species has more time to gain excess coverage and therefore achieves a higher overshoot at the lowest temperatures. This also shows that decreasing the temperature has the opposite effect on the peak time to increasing the energy difference, indicating that the peak time is not a single function of β(εs - εw). As an example of a real system, we consider a mixture of H2 (weak) and CH4 (strong) adsorbing on a groove of a nanotube bundle. In this case, εw ≈ 600 K16 and εs ≈ 2000 K.17 The results for the cross time are similar to the ones

neq,s neq,w

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.1 0.2

2.17

0.62 2.65

0.30 1.31

0.19 0.81

0.13 0.58

0.10 0.46

0.08 0.40

0.07 0.45

0.09

1.33

0.3 0.4

3.22

2.03

1.49

1.23

4.13

3.19

3.23

0.5

7.24

overshoots, and the true maximum of the percent overshoot is achieved for the highest coverages of the stronger species. Besides the percent overshoot values, which only involve the evolution of the weaker species, we can also compare the ratio nw/ns at the peak time to the ratio ns/nw at equilibrium. If we take the ratio R between these two fractions, values greater than 1 explicitly indicate that the early selectivity process is more effective than the one at equilibrium (always focusing on the cases of interest where ns is greater than or equal to nw at equilibrium). Table 1 shows that this ratio also increases with the coverage of the weaker species. However, in this case, the highest values occur when both species end up covering the same percentage of the surface (no difference at equilibrium), with the maximum happening when each species covers half of the lattice (0.5-0.5). Equilibrium selectivity for mixtures with this kind of composition would be completely ineffective, while the kinetic selectivity at an earlier time would allow a very efficient separation process. We also investigate the role of the binding energy by fixing the coverage of both species and changing their binding energies. Figure 3a shows that, as the magnitude of the binding energy of the weaker species decreases, the system sees an increased overshoot that occurs at an earlier time. This overshoot increases even more as the binding energy of the stronger species increases. The increase in percent overshoot stemming from decreases in the binding energy of the weaker


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SUPPORTING

INFORMATION AVAILABLE Explicit expressions for coefficients C and r in eq 5; table with cross time values corresponding to the same coverage combinations as those in Table 1. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Author: *To whom correspondence should be addressed. E-mail: mcalbi@ physics.siu.edu.

ACKNOWLEDGMENT We acknowledge the support provided by the National Science Foundation through Grants CBET-0746029 and DMR-0705077. We are also grateful to Aldo Migone for useful comments and discussions. Figure 4. (a) Fractional coverage of the weaker species for various temperatures. From top to bottom, the curves correspond to T = 100, 150, 200, and 300 K; εs = -400 K, εw = -200 K. (b) Decrease in the percent overshoot with increasing temperature.

REFERENCES (1)

Migone, A. D.; Talapatra, S. Gas Adsorption on Carbon Nanotubes. In Encyclopedia of Nanoscience and Nanotechnology; Nalwa, H. S., Ed.; American Scientific Publishers: Los Angeles, CA, 2004; Vol. 4, pp 749-767. (2) (a) Johnson, J. K.; Cole, M. W. In Adsorption by Carbons; Bottani, E. J., Tascon, J. M. D., Eds.; Elsevier Science Publishing: Amsterdam, The Netherlands, 2008; Chapter 9. (b) Calbi, M. M.; Cole, M. W.; Gatica, S. M.; Bojan, M. J.; Johnson, J. K. In Adsorption by Carbons; Bottani, E. J., Tascon, J. M. D., Eds.; Elsevier Science Publishing: Amsterdam, The Netherlands, 2008; Chapter 15. (c) Migone, A. D. In Adsorption by Carbons; Bottani, E. J., Tascon, J. M. D., Eds.; Elsevier Science Publishing: Amsterdam, The Netherlands, 2008; Chapter 16. (3) Wang, Q.; Challa, S. R.; Sholl, D. S.; Johnson, J. K. Quantum Sieving in Carbon Nanotubes and Zeolites. Phys. Rev. Lett. 1999, 82, 956–959. (4) Byl, O.; Kondratyuk, P.; Forth, S. T.; Fitzgerald, S. A.; Chen, L.; Johnson, J. K.; Yates, J. T.Jr. Adsorption of CF4 on the Internal and External Surfaces of Opened Single-Walled Carbon Nanotubes: A Vibrational Spectroscopy Study. J. Am. Chem. Soc. 2003, 125, 5889–5896. (5) Mao, Z.; Sinnott, S. B. Separation of Organic Molecular Mixtures in Carbon Nanotubes and Bundles: Molecular Dynamics Simulations. J. Phys. Chem. B 2001, 105, 6916– 6924. (6) Challa, S. R.; Sholl, D. S.; Johnson, J. K. Adsorption and Separation of Hydrogen Isotopes in Carbon Nanotubes: Multicomponent Grand Canonical Monte Carlo Simulations. J. Chem. Phys. 2002, 116, 814–824. (7) Jiang, J. W.; Sandler, S. I. Nitrogen and Oxygen Mixture Adsorption on Carbon Nanotube Bundles from Molecular Simulation. Langmuir 2004, 20, 10910–10918. (8) Arora, G.; Sandler, S. I. Air Separation by Single Wall Carbon Nanotubes: Thermodynamics and Adsorptive Selectivity. J. Chem. Phys. 2005, 123, 044705. (9) Jiang, J. W.; Sandler, S. I. Shape versus Inverse-Shape Selective Adsorption of Alkane Isomers in Carbon Nanotubes. J. Chem. Phys. 2006, 124, 024717. (10) Kahng, Y. H.; Hallock, R. B.; Calbi, M. M. Competitive Adsorption of He-3 and He-4 to Carbon Nanotube Bundles. J. Low Temp. Phys. 2005, 138, 217–222. (11) Chen, H. B.; Sholl, D. S. Rapid Diffusion of CH4/H2 Mixtures in Single-Wall Carbon Nanotubes. J. Am. Chem. Soc. 2004, 126, 7778–7779.

shown in the Supporting Information table (sizable fractions of the total equilibration time, with differences less than 10%). Conversely, since the overshoot values are much greater due to the higher energy difference, the ratio R is at least 2 orders of magnitude larger, reaching a maximum of around 1000 when neq,s = neq,w = 0.5. Typical equilibration times for H2 or CH4 when adsorbed as single species are on the order of 1-2 h;18,19 since equilibration times for the mixtures are expected to be larger than those for the single species (see Figure 1), the effect of the overshoot reported here should be easily observed in adsorption kinetics experiments of mixtures. The results obtained in this study demonstrate the presence of a selectivity process during the adsorption of a binary mixture on a homogeneous surface that could be used to remove the weaker binding species before the system reaches the final equilibrium state. Since this would happen early on during the equilibration process, this removal process could be done faster than removing the stronger species. Moreover, when the components of a mixture are at the partial pressures that would produce a similar coverage for both species at equilibrium, this kinetic selectivity process provides the only effective method to separate the components of the mixture. We show that the percent overshoot of the weaker species and the overshoot time depend on the binding energies, chemical potentials, and temperature. The highest overshoots occur when (a) there is a large difference in the binding energies of the species, (b) the competing species would cover a similar fraction of the surface, and (c) the temperature is lowered. Interestingly, while the overshoot time decreases as the overshoot increases in the first two cases, lower temperatures produce not only higher overshoots but also longer overshoot times; this could be especially important at the time of optimizing designs or conditions for practical applications. On the other hand, even if the focus is on the final equilibrium state of the adsorbed mixture, the results presented here (in particular, the cross time values) emphasize the fact that failure to wait enough time for equilibration may easily result in not getting the adsorbed film with the right amount of each species covering the surface.


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Chen, H. B.; Sholl, D. Predictions of Selectivity and Flux for CH4/H2 Separations Using Single Walled Carbon Nanotubes As Membranes. J. Membr. Sci. 2006, 269, 152–160. Yang, R. T. Gas Separation by Adsorption Processes; Butterworths Publishers: Stoneham, MA, 1987. Burde, J. T.; Calbi, M. M. Physisorption Kinetics in Carbon Nanotube Bundles. J. Phys. Chem. C 2007, 111, 5057–5063. Calbi, M. M.; Riccardo, J. L. Energy Barriers at the Ends of Carbon Nanotube Bundles: Effects on Interstitial Adsorption Kinetics. Phys. Rev. Lett. 2005, 94, 246103. Stan, G.; Bojan, M. J.; Curtarolo, S.; Gatica, S. M.; Cole, M. W. Uptake of Gases in Bundles of Carbon Nanotubes. Phys. Rev. B 2000, 62, 2173–2180. Calbi, M. M.; Cole, M. W. Dimensional Crossover and Quantum Effects of Gases Adsorbed on Nanotube Bundles. Phys. Rev. B 2002, 66, 115413. Rawat, D. S.; Calbi, M. M.; Migone, A. D. Equilibration Time: Kinetics of Gas Adsorption on Closed- and Open-Ended Single-Walled Carbon Nanotubes. J. Phys. Chem. C 2007, 111, 12980–12986. Rawat, D. S.; Krungleviciute, V.; Heroux, L.; Bulut, M.; Calbi, M. M.; Migone, A. D. Dependence of Single-Walled Carbon Nanotube Adsorption Kinetics on Temperature and Binding Energy. Langmuir 2008, 24, 13465–13469.


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Early Removal of Weak-Binding Adsorbates by Kinetic Separation

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