Charting the Complete Thermodynamic Landscape of Gas Adsorption for a Responsive Metal-Organic Framework.

New nanoporous materials have the ability to revolutionize adsorption and separation processes. In particular, materials with adaptive cavities have high selectivity and may display previously undiscovered phenomena, such as negative gas adsorption (NGA), in which gas is released from the framework upon an increase in pressure. Although the thermodynamic driving force behind this and many other counterintuitive adsorption phenomena have been thoroughly investigated in recent years, several experimental observations remain difficult to explain. This necessitates a comprehensive analysis of gas adsorption akin to the conformational free energy landscapes used to understand the function of proteins. We have constructed the complete thermodynamic landscape of methane adsorption on DUT-49. Traversing this complex landscape reproduces the experimentally observed structural transitions, temperature dependence, and the hysteresis between adsorption and desorption. The complete thermodynamic description presented here provides unparalleled insight into adsorption and provides a framework to understand other adsorbents that challenge our preconceptions.

Advanced porous materials can deliver exciting developments to industrial processes[1] and the flexibility of porous adsorbents could address complex separations similar to biological systems.[2] For example, extreme changes in adsorption behavior produce improved selectivity, storage capacity, and thermal management.[3−5] Responsive adsorbents, such as metal–organic frameworks (MOFs), can significantly change pore structure and volume in response to external stimuli such as pressure, temperature, gas adsorption or light absorption.[6]

One particularly exciting MOF that exhibits negative gas adsorption (NGA) is DUT-49. This MOF undergoes a structural transition associated with a 50% contraction of the unit cell upon increasing methane gas pressure, resulting in spontaneous desorption of methane.[7] This leads to pressure amplification in a closed system, opening the door to new applications.[8] The lack of thermodynamic understanding of the NGA phenomenon has motivated a number of computational studies.[9,10] However, efforts have mostly relied on two-state models that are unable to capture the complete phase-space during adsorption and desorption.[11]

The most appropriate description of the thermodynamics of responsive frameworks under gas adsorption is provided by the osmotic ensemble. Simulations in this ensemble impose a chemical potential of the adsorbed fluid (μ), a mechanical pressure (p), and a temperature (T). Under these conditions, the equilibrium framework minimizes the osmotic potential (Ω) through structural relaxation, changes in cell shape and volume, and adsorption of guest molecules.[12] However, atomistic simulations within the osmotic ensemble remain challenging because dynamics must be combined with particle insertion/deletion, limiting its application to a handful of systems.[13,14] Descriptions of the osmotic potential are often achieved through analytical treatment of gas adsorption.[15] This can ascertain the relative stability of two distinct and porous phases.[16] For example, the process of “breathing” between an open (op) and contracted porous phase (cp), such as that observed for MIL-53,[17] can be understood by their difference in osmotic potential, Ω – Ω (Figure ).[12] The difference in osmotic potential reveals which phase represents the global thermodynamic equilibrium as a function of gas pressure.[18] However, this methodology is not sufficient to describe NGA as it must involve metastable states.[7] Moreover, the experimentally observed hysteresis and temperature dependence of NGA cannot be reproduced by simply considering the difference in osmotic potential.[19] Experiments demonstrate that the op to cp transition at 90 K produces a much smaller NGA step than 120 K, whereas at 150 K, no transitions are observed.[19] To explain this temperature dependence and other complex adsorption processes, knowledge of not only the osmotic potential but also the osmotic barriers that exist between phases that act to prevent transformations, are required.

Figure 1

Osmotic potential (a) applied to transitions between two phases (op and cp) of an adsorbent. This can describe the “breathing” adsorption phenomenon (b).

Free-energy landscapes of chemical reactions and conformational dynamics are expressed in terms of collective variables; similarly, barriers of “breathing” MOFs in the osmotic ensemble appear at specific cell volumes. We employed molecular simulations to treat the cell volume (V) as a “collective variable” in the osmotic ensemble. The osmotic potential, as a function of volume, reveals the (meta)stable states of interest and the barriers between them for any set of thermodynamic parameters μ, p, and T. With this method, detailed landscapes of the osmotic potential, as a function of cell volume of DUT-49 and methane gas pressure, have been computed at 90, 120, and 150 K. This complete thermodynamic analysis provides comprehensive insight into phase stability under adsorption.

We employed a force field derived from ab initio data of the inorganic and organic units of the DUT-49 framework[10] with the TraPPE[20] parametrization of methane. In our hybrid MC[14,21−23] simulations, molecular dynamics (MD) trajectories in the (N, V, σa = 0, T) ensemble are new moves in a grand canonical Monte Carlo scheme. The MD simulations impose a fixed volume V and anisotropic contribution σa to the total stress tensor σ = P1 + σa,[24] allowing the cell shape to adapt to the fluctuating number of adsorbates resulting in flexible-host adsorption isotherms, Nguest(Nhost, μ, σa = 0, T; V). Isotherms were calculated for a series of unit cell volumes between 40 and 110 nm3 with a spacing of 2 nm3 (Figure ). Subsequently, the flexible-host isotherms were used to construct the osmotic potential as eq .The empty-host free energy Fhost was obtained by thermodynamic integration of the internal pressure, extracted from MD simulations of the empty framework in the (N, V, σa = 0, T) ensemble at different unit cell volumes at 90, 120, and 150 K. The free energy was chosen to be 0 kJ mol–1 in the op phase at a unit cell volume of 104 nm3, comparable to the experimentally observed unit cell volume of 100 nm3.[7] This produces a meaningful comparison of the empty-host free energy (and thus osmotic potential). Simulations were performed between 46 and 75 values of gas pressure and the chemical potential at each temperature and gas pressure was calculated with the Peng–Robinson equation of state.[25] The above approach produces the osmotic potential at different methane gas pressures and cell volumes. The osmotic landscape is shown relative to the op phase at each gas pressure in Figure to highlight differences between the phases. In contrast to previous approaches (Figure a), this surface offers a tremendous amount of additional information.

Figure 2

Computed osmotic surface of methane adsorption on DUT-49 at 120 K, as a function of unit cell volume (a) and methane gas pressure (b). Examples of the 1D osmotic surface at specific gas pressures (c).

To clearly outline the effect of osmotic barriers on gas adsorption, we can chart paths[26] on the osmotic landscape that are taken with adsorption and desorption along with the associated amount adsorbed. Figure a–c shows the path along which the system is in global thermodynamic equilibrium at each gas pressure, i.e., the path of lowest osmotic potential. Alternatively, Figure d–f shows the kinetic path along which transitions can occur only when the osmotic barrier between phases disappears. However, as thermal fluctuations can allow the crossing of barriers, a transition was noted when the barrier was reduced to 15 kBT. The same transitions are obtained for the computed 120 K surface when varying this value of “crossable” barriers between 0 and 25 kBT. The appropriate value of barrier heights that can be crossed in typical time spans of experiments at a given temperature is difficult to obtain. For example, a fluctuation energy of 6 kBT per unit cell was proposed for a gate-opening MOF.[27]

Figure 3

Osmotic potential (a) and the quantity adsorbed (b) surfaces of methane adsorption on DUT-49 at 120 K, with the thermodynamic equilibrium path, resulting in the equilibrium adsorption isotherm (c). The adsorption (solid line) and desorption (dashed line) path for kinetically accessible states on the same surfaces (d and e) that produce the adsorption isotherm (f).

At zero gas pressure, the system begins at equilibrium in the op phase, where there is also a metastable cp phase (at 50 nm3). By increasing gas pressure, the cp phase is stabilized with respect to the op phase, until the difference in osmotic potential between both phases vanishes at a gas pressure of 3.25 kPa. At this point, a transition occurs along the equilibrium path to the cp phase (Figure 3a). This contraction produces an increase in amount adsorbed, a step of 236 molecules per unit cell (molecules UC–1). Conversely, no transitions occur along the kinetic path at this pressure, as a barrier between the op and cp phase of 591 kJ mol–1 remains. This barrier only vanishes at a gas pressure of 17.5 kPa, at which the op → cp transition occurs, associated with a NGA step of 86 molecules UC–1. This predicted transition pressure is remarkably close to the observed value of 25 kPa. Although the predicted NGA step is underestimated compared to experiments (102 molecules UC–1), the observation that the NGA phenomenon can be captured by atomistic simulation is remarkable in itself. On further increasing the gas pressure to 25 kPa, the op phase becomes the equilibrium phase, shown by the second transition in Figure b. However, the barrier between these phases only disappears at a higher gas pressure of 70 kPa (Figure d).

Just as a trajectory can be walked toward increasing gas pressures, the same process can be done in reverse to track desorption. The same path is obtained for the equilibrium path, the experimentally observed hysteresis is absent with this approach. Kinetic paths are different in adsorption and desorption. Osmotic potential energy barriers during desorption, which prevent an op → cp transition, only disappear when the gas pressure decreases to 17.5 kPa. This prediction of an op → cp transition during desorption at a gas pressure similar to that of the op → cp transition during adsorption is in agreement with experiments, attributed to the flat osmotic potential landscape over the entire volume range at this pressure. Interestingly, on decreasing the gas pressure further, it is predicted from this surface that the framework remains locked into the cp phase, as observed experimentally. This is due to the barrier surrounding the cp local minimum of the osmotic potential landscape at low gas pressures.

The structural transitions derived from the kinetic pathway during adsorption and desorption at 90 K agree quantitatively with reported in situ powder X-ray diffraction experiments.[7] The presence of intermediate phases between the op and cp phases at approximately 2 kPa are reproduced, evidencing the flatness of osmotic landscape at this gas pressure.

Recently, we reported methane adsorption on DUT-49 for the range 91–190 K, observing a non-monotonic trend of NGA amount with temperature and a distinct temperature window.[19] To provide insight into this trend, simulations were completed for three temperatures (90, 120, and 150 K). The difference in osmotic potential energy between the op and cp phases (corresponding to 104 and 50 nm3, respectively) is directly computed from these simulations, as well as the osmotic potential energy barrier from op to cp (Figure a). The pressure window for stability of the cp phase (where Ω – Ω > 0) is significantly affected by temperature. This results in the op → cp phase transitions becoming unfavorable at 150 K, in line with the absence of experimentally observed phase transitions at this temperature and beyond.[19] We can alternatively consider these trends with respect to the difference in adsorbed amount in the op and cp phases (Figure b), representing the amount of NGA (NNGA). The barrier present at 90 K disappears before the isotherm crossing point, implying that NGA is suppressed, whereas at 120 K, the barrier only disappears after the op state has more methane adsorbed than the cp state, necessitating the release of gas and the NGA step. By considering the amount adsorbed in each of the phases when the osmotic potential barrier for the op → cp transition disappears, we can predict the amount of NGA (Figure b, inset). This qualitative description outlines how NGA must be a function of not only the difference in osmotic potential but also the barrier to produce a temperature window.

Figure 4

Difference in osmotic potential between op and cp phases (solid lines) and the barrier height between these phases (dashed lines) (a). The same difference in osmotic potential and barriers displayed with respect to the difference in amount adsorbed (b). The point where the osmotic potential barrier disappears is marked by a square. Predicted amount of NGA in units of molecules UC–1 (b, inset).

We have outlined an approach to investigate a responsive gas adsorption process using atomistic simulations to create an osmotic potential surface. Roaming this surface using equilibrium or kinetic trajectories produces very different isotherms. It is only by considering a kinetic pathway that accounts for osmotic potential barriers that the salient features of complex adsorption processes are produced. We used this model to understand the effects of temperature, demonstrating the previously poorly understood non-monotonic trend of NGA magnitude. The presented methodology enables ab initio prediction of NGA and permits investigation of similar counterintuitive adsorption processes. This is an important step toward predicting the evolution of conformational energy landscapes of flexible porous materials upon guest adsorption.[28] In particular, this new insight into the barriers present in responsive adsorption processes is of utmost importance for the intended application of these materials in kinetic selectivity and recognition. Many separation process are emphasized for flexible porous materials, but hysteresis is rarely addressed and poorly understood.[29] This approach highlights how desorption can access an entirely different pathway than adsorption. Flexible systems can wander beyond equilibrium, and it is only by mapping the complete energy surface that we can understand these unique adsorption phenomena and pave the way for new adsorbent technologies.