Characterization and Comparison of the Performance of IRMOF-1

Oct 12, 2012 - Using Expanded Wang–Landau simulations, we have determined with great accuracy the grand-canonical partition function of CO2 adsorbed...
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Characterization and Comparison of the Performance of IRMOF-1, IRMOF-8, and IRMOF-10 for CO2 Adsorption in the Subcritical and Supercritical Regimes Jason M. Hicks, Caroline Desgranges, and Jerome Delhommelle* Department of Chemistry, University of North Dakota, Grand Forks, North Dakota 58202, United States ABSTRACT: Using Expanded Wang−Landau simulations, we have determined with great accuracy the grand-canonical partition function of CO2 adsorbed in IRMOF-1, IRMOF-8, and IRMOF-10. Then, following a solution thermodynamics approach to adsorption, we use the partition function so obtained to calculate the immersion and desorption free energy, enthalpy, and entropy, which provide a complete thermodynamic characterization of the three systems. Excess and immersion functions are found to be monotonic functions of pressure in the subcritical regime and to present an extremum in the supercritical regime. However, for both regimes, our results show the predominant contribution of the immersion enthalpy to the excess enthalpy of adsorption and the lesser contribution of the other two immersion functions (Gibbs free energy and entropy) to the corresponding excess functions. The results also indicate that the desorption free energy, corresponding to the minimum isothermal work required to regenerate the adsorbent, is directly correlated with the size of the IRMOF pore.

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INTRODUCTION The understanding of adsorption phenomena has long relied on the concepts of 2-D surface thermodynamics.1−4 However, the emergence of promising new materials, with complex pore geometries, has prompted the development of new ways to characterize and assess the performance of these porous materials. In the past decade, Myers and co-workers5−7 have proposed to analyze adsorption in terms of solution thermodynamics rather than in terms of surface thermodynamics. This approach consists of considering that the thermodynamics of adsorption is similar to that of a binary mixture, in which the adsorbate plays the role of the solute while the solid adsorbent is regarded as the solvent. The thermodynamic quantities provided by this analysis include the immersion and desorption thermodynamic functions. These are key quantities in the characterization of the adsorbent. More specifically, as discussed by Myers et al.,6 the desorption free energy, enthalpy, and entropy provide a complete thermodynamic description of the system. Moreover, the desorption free energy is a measure of the performance of the porous material, since it can be interpreted as the minimum isothermal work required to regenerate the adsorbent. This of great significance from a practical standpoint since most of the operating cost of adsorptive separations is associated with degassing the adsorbent in preparation for the next cycle. The aim of this paper is to show how Expanded Wang− Landau simulations provide a direct route for the determination of the thermodynamic functions of immersion and desorption. The principle of these simulations differs from that of © 2012 American Chemical Society

conventional molecular simulations, which consist of carrying out a series of simulation runs in the grand-canonical ensemble (constant μ, V, and T) by gradually varying the imposed chemical potential μ at a selected temperature T.8−24 The recently developed25,26 Expanded Wang−Landau simulations rely on determining, with great accuracy, the grand-canonical partition function of the adsorbed fluid. This is particularly advantageous, since once the partition function has been evaluated, all thermodynamic properties, including the desorption free energy and entropy, can be directly evaluated. In recent work, we applied the Expanded Wang−Landau simulations to the bulk25 and to the predictions of adsorption isotherms of argon and CO2 in IRMOF-1. In this work, we use Expanded Wang−Landau simulations to predict the immersion and desorption thermodynamic functions. This will allow us to provide a complete thermodynamic characterization of the adsorption of CO2 in a series of isoreticular27,28 metal−organic frameworks29−39 (IRMOF-1, IRMOF-8, and IRMOF-10) and to assess the relative performance of the three IRMOFs for this application. The paper is organized as follows. In the next section, we present the Expanded Wang−Landau simulations used in this work, as well as the molecular models for the adsorbate (CO2) and the adsorbents (IRMOF-1, IRMOF-8, and IRMOF-10). We then discuss how the results from Expanded Wang−Landau Received: July 24, 2012 Revised: October 12, 2012 Published: October 12, 2012 22938

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out in the presence of the gas phase and, as a result, yield excess properties. For any absolute thermodynamic property M, the excess property Me can be calculated as

simulations, i.e., a high-accuracy estimate for the grandcanonical partition function of the adsorbed fluid, provide a direct access to the immersion and desorption functions. We then present the results for the three adsorbents considered here, for both subcritical and supercritical adsorptions, and draw the main conclusions from this work.

M e = M − V void M g ρ g

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in which Vvoid is the void volume of the porous material5 and ρg and M g are the molar density and the molar property M, respectively, of the gas phase in equilibrium with the adsorbed fluid. Both ρg and M g can be determined directly from the grand-canonical partition function for the gas phase Θg(μ,Vg,T) as shown in prior work.25,26 From a practical standpoint, the Monte Carlo (MC) steps61 used in an Expanded Wang−Landau simulation consist of either the translation of a single molecule of CO2 (37.5% of the MC steps) or the rotation of a single molecule (37.5% of the MC steps) or a change in (N, l) (25% of the MC steps). Expanded Wang−Landau simulations are based on an iterative process to estimate Q(N,V,T,l). At the start of the simulation, we set Q(N,V,T,l) = 1 for all N and l. Every time the system visits a configuration with N full molecules and a fractional molecule at stage l, we update the value of Q(N,V,T,l) by multiplying its value by f (f is a convergence factor with an initial value such that ln f = 1). During the simulation, a histogram of the number of visits for each N and l is collected. Once all sets of (N,l) values have been visited at least 1000 times, the convergence factor is reduced by √f, the histogram for the number of visit is initialized, and the simulation is run again with this new convergence factor. This process is repeated until f reaches a minimum threshold value (here, ln f = 10−8, which means that the iterative process is repeated 27 times), leading to a highly accurate numerical estimate for Q(N,V,T,l). In terms of computational cost, the determination of the adsorption properties was carried out by Expanded Wang− Landau simulations of approximately 5 × 109 MC steps for each system. Considering prior GCMC simulation work on the adsorption of CO2 in IRMOF-1,62 the simulation of the entire adsorption isotherm was carried out with approximately 25 GCMC runs of 40 × 106 MC steps each, amounting to a total of 1 × 109 MC steps. In prior work, Chen and Sholl63 compared the statistical efficiency of a flat histogram method, the Transition Matrix Monte Carlo approach,64 to that of GCMC simulations for the determination of adsorption isotherms by molecular simulation. Both methods yielded results in excellent agreement at low pressures. At high pressures, Chen and Sholl found the flat histogram method to be more accurate than GCMC, as GCMC data exhibited more scattering. The Expanded Wang−Landau approach we apply in this work combines the advantage of a flat histogram method (here, the Wang−Landau sampling method instead of the Transition Matrix Monte Carlo approach of Chen and Sholl) with an efficient method for the insertion of molecules (here, the gradual insertion of additional molecules through the Expanded Ensemble scheme). Error bars for the thermodynamic quantities predicted by Expanded Wang−Landau simulations are obtained by carrying out independent simulation runs.25,26 In terms of the resulting adsorption isotherms, the statistical uncertainty was of 0.3% for the amount adsorbed. Models. We study the adsorption of CO2 molecules. CO2 molecules interact through the TraPPE potential,65 which consists of 3 LJ sites and 3 point charges per molecule (i.e., 1 LJ site and 1 charge per atom). The interaction between two

SIMULATION METHOD AND MODELS Simulation Method. We briefly outline how we carry out Monte Carlo Expanded Wang−Landau simulations of CO2 in IRMOFs (more details on the method may be found in prior work25,26). Expanded Wang−Landau simulations provide a highly accurate estimate for the grand-canonical partition function of atomic and molecular fluids. This is of great interest, since once the partition function for the adsorbed fluid is known, all the thermodynamic properties of the system (such as, e.g., the entropy and free energy) can be calculated through the formalism of statistical mechanics. We validated this approach by showing that it yielded results in excellent agreement with the experimental data for the bulk and at the vapor−liquid transition.25 Moreover, we showed that the grandcanonical partition function obtained from Expanded Wang− Landau simulations predicted adsorption isotherms in excellent agreement with prior grand-canonical Monte Carlo (GCMC) simulations for argon and carbon dioxide in IRMOF-1. Expanded Wang−Landau simulations rely on the expanded ensemble technique.40−47 This technique consists of dividing the insertion and deletion of molecules into a large number of steps. The simulated system is therefore composed of N molecules and of a fractional molecule. The fractional molecule is allowed to grow or shrink during the course of the simulation, resulting in variations of N in the system. As previously shown,25,26 combining this technique with the Wang−Landau sampling48−60 allows a uniform sampling of all values for the number of molecules. This, in turn, yields a high-accuracy estimate for the canonical partition function Q(N,V,T,l), in which T is the temperature and V the volume of the porous material. The grand-canonical partition function can then be calculated as ∞

Θ(μ , V , T ) =

∑ Q (N , V , T , l = 0) exp(βμN ) N =0

(2)

(1)

in which Q(N,V,T,l = 0) is the canonical partition function for a system containing N molecules and a void fractional molecule (l = 0). Carrying out Expanded Wang−Landau simulations also has several advantages. First, since the chemical potential μ does not appear in the Metropolis criteria,25,26 we do not need to specify a value for μ to run an Expanded Wang−Landau simulation. Second, since the simulation samples uniformly all values of N, a single expanded Wang−Landau simulation run at a given temperature T provides the whole adsorption isotherm, and more generally, the thermodynamic properties, for all values of the pressure. We study CO2 adsorption in IRMOF-1, IRMOF-8, and IRMOF-10 for temperatures ranging from 220 K to 360 K. For each temperature, we run four Expanded Wang−Landau simulations, one for each of the three IRMOFs considered in this work and an additional Expanded Wang−Landau simulation for the bulk gas phase of CO2. The latter simulation allows us to determine the excess thermodynamic properties measured in experiments.5 While simulations provide a direct access to absolute thermodynamic data, experiments are carried 22939

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atoms i and j, belonging to two different CO2 molecules, is given by ⎡⎛ ⎞12 ⎛ ⎞6 ⎤ qiqj σij σij ϕ(rij) = 4εij⎢⎢⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ ⎥⎥ + r ⎝ rij ⎠ ⎦ 4πε0rij ⎣⎝ ij ⎠

Ω = −kBT ln[Θ(μ , V , T )] ⎡ ∞ ⎤ = −kBT ln⎢ ∑ Q (N , V , T ) exp(βμN )⎥ ⎢⎣ N = 0 ⎥⎦

In this equation and in the rest of the paper, we drop l = 0 and note Q(N,V,T) as the canonical partition function for a system with N full molecules and a void fractional molecule. From the knowledge of Θ(μ,V,T), we determine the molecule number distribution p(N) for the adsorbate as

(3)

The interaction of an atom i belonging to the fractional CO2 molecule with an atom j, belonging to one of the N full CO2 molecules, is calculated according to 12 ⎡⎛ ⎛ σij , ξ ⎞6 ⎤ qi , ξ qj σij , ξ ⎞ l ϕ(rij) = 4εij , ξl ⎢⎢⎜⎜ l ⎟⎟ − ⎜⎜ l ⎟⎟ ⎥⎥ + πε r r 4 ⎝ ij ⎠ ⎦ 0rij ⎣⎝ ij ⎠

p(N ) =

Q (N , V , T ) exp(βμN ) Θ(μ , V , T )

(6)

and the absolute amount adsorbed from eq 6 as

(4)

∞

where ξl denotes the coupling parameter, rij represents the distance between atoms i and j, εij,ξl = (l/M)1/3εij, σij,ξl = (l/ M)1/4σij, and qi,ξl = (l/M)1/3qi. Here, M denotes the number of stages in which the insertion and deletion of a full molecule is divided (we choose M = 100) and l is the current stage value for the fractional molecule. A homothetic transformation is also applied to the skeleton of the fractional molecule. For a fractional molecule at a given stage l, the distance between the carbon and one of the oxygens is set to (l/M)1/4deq C−O, where deq C−O = 1.16 Å is the distance between C and O in a full molecule of carbon dioxide. We model the three IRMOFs as rigid cubic structures, with lattice constants and atomic coordinates taken from Yaghi et al.27 In line with prior work,9,10,14,26,66 we use the DREIDING force field67 to model the van der Waals interactions between the IRMOFs and the adsorbed CO2, with the parameters for the Lennard-Jones interactions between unlike atoms calculated using the Lorentz−Berthelot mixing rules. This approach was shown to yield adsorption isotherms of light gases in IRMOFs in very good agreement with the available experimental data (see, e.g., prior work on CH4,14 H2,9 and CO210,66,26).

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(5)

Nads =

∑ Np(N ) (7)

N =0

This quantity can then be converted into the excess adsorption Ne according to N e = Nads − V voidρ g

(8)

Absolute thermodynamic properties of the adsorbate are calculated from the Expanded Wang−Landau simulation results as U = Nads ∑ ⎡⎣ Epot(N ) + 5/2kBT ⎤⎦ p(N ) ∑ p(N ) N

N

A = −kBT ln Θ(μ , V , T ) + μNads S = U − AT G = A + P gV H = U + P gV

(9)

for the internal energy U, the Helmoltz free energy A, the entropy S, the Gibbs free energy G, and the enthalpy H of the adsorbate. In the equation for U, Epot (N) is the average potential energy per adsorbed molecule, collected during the Expanded Wang Landau simulation, for a system containing N molecules, and (5/2)kBT is the kinetic (ideal gas) contribution for a rigid linear molecule (i.e., the model used here for CO2). In the equations for G and H, we use the external pressure in the gas-phase Pg, calculated during the Expanded Wang− Landau simulations of the gas phase at μ and T. As discussed in the previous section, excess thermodynamic properties can then be readily evaluated from the properties of the external gas phase as follows

THERMODYNAMIC ANALYSIS OF CO2 ADSORPTION IN IRMOFS

The thermodynamic analysis we carry out for the adsorption of CO2 in IRMOFs is based on a solution thermodynamics approach.5−7,68,69 While the understanding of adsorption phenomena had long been relying on the concepts of surface thermodynamics, the application of these concepts to nanoporous materials like IRMOFs is made difficult by the complex geometry of the pores. The solution thermodynamics approach consists of studying the adsorption of CO2 in IRMOFs as a binary mixture, in which CO2 is the solute and the IRMOF as the solvent.5−7 This formalism provides a route to determine immersion and desorption thermodynamic function, which are key quantities in the characterization of the adsorbent. In this section, we focus on explaining how Expanded Wang−Landau simulations can be used to determine the thermodynamic functions of immersion and desorption. The output of Expanded Wang−Landau simulations of CO2 adsorbed in IRMOFs is a highly accurate estimate for the grand-canonical partition function of the system Θ(μ,V,T). This provides direct access to the grand potential through the following relation70

Ge = G − V void G g ρ g H e = H − V void H gρ g S e = S − V void S gρ g

(10)

Once the excess properties have been calculated, we calculate the excess properties relative to an ideal gas reference state as ΔGe = Ge − N eμ0 ΔH e = H e − N e H0 ΔS e = S e − N e S0 22940

(11)

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Figure 1. Logarithm of the reduced canonical partition function Q*(N,V,T) against the number of atoms adsorbed per unit cell: (a) IRMOF-1, (b) IRMOF-8, and (c) IRMOF-10. The canonical partition function is reduced with respect to the partition function of an ideal gas composed of linear molecules.

in which μ0, H0 = (7/2)RT and S0 are the chemical potential, the molar enthalpy, and the molar entropy of an ideal gas composed of linear rigid molecules. The adsorption process can then be interpreted as a two-step process.5 The first step corresponds to the isothermal compression of the gas from its ideal gas reference state to the equilibrium pressure P. For each thermodynamic function, we define the corresponding compression function associated with the isothermal compression of the gas from the ideal gas reference state defined above to the actual pressure of the system P

ΔGimm = ΔGe − ΔGcomp = ΔGe − N e(μ − μ0 ) ΔH imm = ΔH e − ΔH comp = ΔH e − N e( H g − H̅ 0) ΔS imm = ΔS e − ΔS comp = ΔS e − N e( S g − S0̅ ) (13)

We finally define the desorption Gibbs free energy |Ωe| = −ΔGimm, which can be interpreted as the minimum isothermal work required to regenerate the adsorbent. Since most of the operating cost of adsorptive separations is associated with degassing the adsorbent in preparation for the next cycle, the desorption Gibbs free energy is particularly useful in comparing the performance of different adsorbents.

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ΔGcomp = N e(μ − μ0 )

RESULTS AND DISCUSSION We first present the main output of the Expanded Wang− Landau simulations. We show in Figure 1 the canonical partition function Q(N,V,T) for systems of N CO2 molecules adsorbed in each of the IRMOFs considered in this work. In this work, temperatures range from 220 K to 360 K, which allows us to sample both the subcritical and the supercritical regimes (Tc = 304 K25,71). We find that, for a given IRMOF and a given N, Q(N,V,T) strongly depends on the temperature. When N becomes high, corresponding to an almost saturated IRMOF, Q(N,V,T) sharply decreases with temperature, with ln Q(N,V,T = 360 K) being roughly a third of its value at T = 220 K. We also observe that the features of the canonical partition

ΔH comp = N e( H g − H0) ΔS comp = N e( G g − S0)

(12)

The second step corresponds to the isothermal−isobaric immersion of the clean IRMOF in the compressed gas. While the first step (compression) only depends on the properties of the bulk gas phase, the second step (immersion) is a function of the physical properties of the adsorbent. As a result, the immersion thermodynamic functions allow for a thermodynamic characterization of the porous material. The immersion functions can be calculated as 22941

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Figure 2. Absolute amount adsorbed per unit cell (in mol/kg) against pressure: (a) IRMOF-1, (b) IRMOF-8, and (c) IRMOF-10. We also include the results from GCMC simulations66 at 225 K for IRMOF-1 (circles).

Figure 3. Absolute amount (solid line) and excess amount (dashed line) adsorbed per unit cell of IRMOF-8 (in mol/kg) against pressure: (a) 220 K and (b) 360 K.

function, plotted as a function of N, are qualitatively the same for all three IRMOFs. Once Q(N,V,T) is known, the grand-canonical partition function and the grand potential can be calculated using eqs 1 and 5. Then, through eq 7, this gives the absolute adsorption

isotherm. We present in Figure 2 the absolute isotherms obtained for the three IRMOFs studied in this work. Figure 2a shows that the results obtained from Expanded Wang−Landau simulations at 220 K for IRMOF-1 are consistent with the results from prior GCMC simulations at 225 K.66 As a result of 22942

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1, to IRMOF-8 and to IRMOF-10. Moreover, the value reached by ΔGe at saturation increases from IRMOF-1 to IRMOF-10 as a result of the greater excess amount adsorbed for the larger IRMOFs. For all IRMOFs, the predominant contribution to ΔGe comes from the compression Gibbs free energy ΔGcomp. For instance, at P = 5 bar, the immersion Gibbs free energy ΔGimm only represents 10%, 8%, and 6% of ΔGe for IRMOF-1, IRMOF-8, and IRMOF-10, respectively. In the case of supercritical adsorption (T = 360 K), Figure 4 shows that ΔGe exhibits a different behavior and reaches a minimum for a pressure of about 90 bar (IRMOF-1), 98 bar (IRMOF-8), and 107 bar (IRMOF-10). We also observe that this minimum is shifted toward lower values of ΔGe as the pore size increases, which reflects the fact that the excess amount adsorbed is greater for the larger IRMOFs. Similarly to the case of subcritical adsorption, the contribution of the compression Gibbs free energy ΔGcomp largely exceeds that of ΔGimm. We show in Figure 5 the excess enthalpy (ΔHe), the compression enthalpy (ΔHcomp), and the immersion enthalpy

the increased degree of confinement, the steps in the isotherms for IRMOF-1 are shifted toward lower pressures, compared to those for IRMOF-8 and IRMOF-10. Similarly, the maximum loading increases with the size of the pores of the IRMOFs with the following ranking, in ascending order, IRMOF-1 < IRMOF8 < IRMOF-10. We now move on to the determination of the excess adsorption isotherms. We present in Figure 3 the absolute and excess adsorption isotherms for CO2 in IRMOF-8 for the lowest and highest temperatures considered here. At T = 220 K (Figure 3a), the absolute and excess adsorption isotherms are roughly the same. This is due to the fact that, at low temperatures, the density of the gas-phase ρg is very low, implying that the second term in the equation defining Ne is negligible (see eq 8). However, at T = 360 K (Figure 3b), the two adsorption isotherms differ qualitatively. The excess adsorption isotherm exhibits a maximum for a pressure of 98 bar, while the absolute adsorption isotherm is increasing with pressure. The maximum in the excess adsorption isotherm stems from the variations of the second term in the equation defining Ne with pressure. This term is negligible at low pressures, and the excess adsorption isotherm closely follows the absolute adsorption isotherm. Since the temperature (360 K) is well above the critical point of the fluid (304 K25,71), the density of the bulk steadily increases with pressure. However, the amount adsorbed in the IRMOF increases more slowly, since the porous material starts to become saturated. This leads to the maximum and to the eventual decrease observed for the excess adsorption isotherm. These two sets of results illustrate the markedly different behavior observed for the excess amount adsorbed during subcritical (Figure 3a) and supercritical (Figure 3b) adsorption. The excess Gibbs free energy (ΔGe), the compression Gibbs free energy (ΔGcomp) and the immersion Gibbs free energy (ΔGimm) are shown in Figure 4. We recall that immersion properties are all negative quantities, since adsorption is an exothermic, spontaneous process. We start by discussing the results obtained for T = 220 K. ΔGe exhibits a sharp drop for all three IRMOFs. This drop in ΔGe coincides with the step observed for the isotherm. In particular, this drop is shifted toward higher pressure as the pore size increases from IRMOF-

Figure 5. ΔHe (solid line), ΔHcomp (circles), and ΔHimm (triangles) (in kJ/kg) against pressure at 220 K (left panel) and at 360 K (right panel). On both panels, results are shown for IRMOF-1 (top), IRMOF-8 (middle), and IRMOF-10 (bottom).

(ΔHimm). Similarly to ΔGe, the variations of ΔHe can be understood in terms of the excess amount adsorbed for both the subcritical and supercritical regimes. However, unlike the results on the Gibbs free energy, we find that the immersion enthalpy ΔHimm is the main contribution to ΔHe and largely exceeds the compression enthalpy ΔHcomp, most strikingly at 220 K. ΔHcomp is proportional to the enthalpy difference between the bulk gas phase and an ideal gas composed of linear molecules. This difference is very small at pressures of the order of 1 bar and below, leading to the small contribution of ΔHcomp shown in Figure 5. However, this enthalpy difference increases with pressure, leading to larger contributions from ΔHcomp at high pressures. This is best shown by the results obtained at 360 K, as for all IRMOFs, the contribution from ΔHcomp represents 20% of ΔHe close to the minimum in ΔHe. We present in Figure 6 the excess entropy (ΔSe), the compression entropy (ΔScomp), and the immersion entropy (ΔSimm). In line with the results obtained for ΔGe and ΔHe, the variations of ΔSe with pressure follow the trends exhibited by the excess adsorption isotherms in the subcritical and supercritical regimes. In opposition with the behavior seen for the enthalpy, the compression and the immersion entropy

Figure 4. ΔGe (solid line), ΔGcomp (dotted line), and ΔGimm (dashed line) (in kJ/kg) against pressure at 220 K (left panel) and at 360 K (right panel). On both panels, results are shown for IRMOF-1 (top), IRMOF-8 (middle), and IRMOF-10 (bottom). 22943

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qualitatively similar for the three IRMOFs, with the desorption functions increasing with pressure in the subcritical regime and exhibiting a maximum in the supercritical regime. We add that their relative magnitude, i.e., H > TS > G, follows the trend generally obtained for desorption properties.5 We summarize in Figure 8 the results obtained for the desorption free energy, or |Ωe|, for the three IRMOFs studied in this work. |Ωe| represents the minimum isothermal work required to regenerate the adsorbent. The results for the three IRMOFs outline two different behaviors associated with the subcritical and the supercritical (above 300 K) regimes, |Ωe| exhibiting a maximum for the latter. In line with results obtained for the other thermodynamic quantities, the sharp rise in |Ωe| is shifted toward the high pressures as the pore size increases. To assess the performance of the IRMOFs, we examine more closely the values taken by |Ωe| for the three porous materials considered in this work. In the subcritical regime (220 K), if we consider the maximum reached by the desorption free energy, we find the following ranking: |Ωe| (IRMOF-1) = 114 kJ/mol < |Ωe|(IRMOF-8) < 121 kJ/mol < |Ωe|(IRMOF-10) = 131 kJ/mol. In the supercritical regime (360 K), the maximum desorption free energy is in the same order: |Ωe|(IRMOF-1) = 67 kJ/mol < |Ωe|(IRMOF-8) < 72 kJ/ mol < |Ωe|(IRMOF-10) = 79 kJ/mol. In both regimes, the cost of regenerating the adsorbent is correlated with the size of the IRMOF pores. This increase in the desorption free energy (roughly 18% from IRMOF-1 to IRMOF-10 for both subcritical and supercritical adsorptions) is, however, very moderate, when compared to the increase in the loading capability (the maximum amount adsorbed in IRMOF-1 is approximately half of the maximum amount adsorbed in IRMOF-10). This quantifies the extent of the trade-off between regeneration costs and loading capability for CO2 adsorption in IRMOF-1, IRMOF-8, and IRMOF-10.

Figure 6. ΔSe (solid line), ΔScomp (circles), and ΔSimm (triangles) (in kJ/kg) against pressure at 220 K (left panel) and at 360 K (right panel). On both panels, results are shown for IRMOF-1 (top), IRMOF-8 (middle), and IRMOF-10 (bottom).

both make a significant contribution to the excess entropy, with ΔScomp playing a slightly more predominant role in both the subcritical and supercritical regimes. The significance of the contribution of ΔScomp can be attributed to the sharp entropy loss as the gas is isothermally compressed from its ideal gas reference state to its equilibrium pressure. From a quantitative standpoint, e.g., at P = 5 bar and T = 220 K, ΔScomp accounts for approximately 60% of ΔSe for the three IRMOFs. Similarly, at T = 360 K and close to the minimum in ΔSe, ΔSimm is around −0.5 kJ/kg/K for the three IRMOFs. This amounts to 20%, 16%, and 12% for IRMOF-1, IRMOF-8, and IRMOF-10, respectively. The desorption Gibbs free energy, enthalpy, and entropy are plotted in Figure 7 in the subcritical and supercritical regime for the three IRMOFs. As previously discussed, since adsorption is an exothermic and spontaneous process, immersion properties are all negative quantities. This implies that desorption properties are all positive quantities, as shown in Figure 7. At a given temperature, the features of these functions are

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CONCLUSION

In this work, we analyze the adsorption of carbon dioxide in IRMOF-1, IRMOF-8, and IRMOF-10 using a solution thermodynamics approach. To this end, we carry out Expanded Wang−Landau simulations to determine the grand-canonical partition function of the adsorbed fluid. The partition function so obtained provides a direct access to the thermodynamic properties of the adsorbed fluid, including the excess, immersion, and desorption thermodynamic functions. Qualitatively different behaviors are obtained in the subcritical and supercritical regimes. In the subcritical regime, the excess, immersion, and desorption functions vary monotonically with pressure in the subcritical regime, while all functions present an extremum in the supercritical regime. To refine our thermodynamic analysis, we examine the relative significance of the contribution of compression and immersion in the adsorption process and identify the large contribution of the immersion to the excess enthalpy and its lesser contributions to the excess entropy and Gibbs free energy. We finally assess the relative performance of the IRMOFs in terms of their desorption free energy, which measures the energetic cost to regenerate the adsorbent. Our analysis also characterizes the trade-off between the moderate increase in regeneration costs and the larger increase in loading capability as the size of the IRMOF pore increases.

Figure 7. Desorption functions (in kJ/kg) against pressure at 220 K (left panel) and at 360 K (right panel). On both panels, results are shown for IRMOF-1 (top), IRMOF-8 (middle), and IRMOF-10 (bottom). G is shown with a solid line, H with a dotted line, and TS with a dashed line. 22944

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Figure 8. |Ωe| (in kJ/kg) against pressure: (a) IRMOF-1, (b) IRMOF-8, and (c) IRMOF-10.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Drs. A. H. Fuchs and F.-X. Coudert for sharing their GCMC data on CO2 adsorption in IRMOFs. Partial funding for this research was provided by NSF through CAREER award DMR-1052808.

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