Free energy landscapes for the thermodynamic understanding of adsorption-induced deformations and structural transitions in porous materials

Theoretical isotherm equation for adsorption-induced structural transition on flexible metal-organic frameworks

Flexible metal-organic frameworks (MOFs) exhibit an adsorption-induced structural transition known as "gate opening" or "breathing," resulting in an S-shaped adsorption isotherm. This unique feature of flexible MOFs offers significant advantages, such as a large working capacity, high selectivity, and intrinsic thermal management capability, positioning them as crucial candidates for revolutionizing adsorption separation processes. Therefore, the interest in the industrial applications of flexible MOFs is increasing, and the adsorption engineering for flexible MOFs is becoming important. However, despite the establishment of the theoretical background for adsorption-induced structural transitions, no theoretical equation is available to describe S-shaped adsorption isotherms of flexible MOFs. Researchers rely on various empirical equations for process simulations that can lead to unreliable outcomes or may overlook insights into improving material performance owing to parameters without physical meaning. In this study, we derive a theoretical equation based on statistical mechanics that could be a standard for the structural transition type adsorption isotherms, as the Langmuir equation represents type I isotherms. The versatility of the derived equation is shown through four examples of flexible MOFs that exhibit gate opening and breathing. The consistency of the formula with existing theories, including the osmotic free energy analysis and intrinsic thermal management capabilities, is also discussed. The developed theoretical equation may lead to more reliable and insightful outcomes in adsorption separation processes, further advancing the direction of industrial applications of flexible MOFs.

Elastic heterogeneity governs asymmetric adsorption-desorption in a soft porous crystal

Metal-organic frameworks (MOFs), which possess a high degree of crystallinity and a large surface area with tunable inorganic nodes and organic linkers, exhibit high stimuli-responsiveness and molecular adsorption selectivity that enable various applications. The adsorption in MOFs changes the crystalline structure and elastic moduli. Thus, the coexistence of adsorbed/desorbed sites makes the host matrices elastically heterogeneous. However, the role of elastic heterogeneity in the adsorption-desorption transition has been overlooked. Here, we show the asymmetric role of elastic heterogeneity in the adsorption-desorption transition. We construct a minimal model incorporating adsorption-induced lattice expansion/contraction and an increase/decrease in the elastic moduli. We find that the transition is hindered by the entropic and energetic effects which become asymmetric in the adsorption process and desorption process, leading to the strong hysteretic nature of the transition. Furthermore, the adsorbed/desorbed sites exhibit spatially heterogeneous domain formation, implying that the domain morphology and interfacial area between adsorbed/desorbed sites can be controlled by elastic heterogeneity. Our results provide a theoretical guideline for designing soft porous crystals with tunable adsorption hysteresis and the dispersion and domain morphology of adsorbates using elastic heterogeneity.

Tunable acetylene sorption by flexible catenated metal-organic frameworks

The safe storage of flammable gases, such as acetylene, is essential for current industrial purposes. However, the narrow pressure (P) and temperature range required for the industrial use of pure acetylene (100 < P < 200 kPa at 298 K) and its explosive behaviour at higher pressures make its storage and release challenging. Flexible metal-organic frameworks that exhibit a gated adsorption/desorption behaviour-in which guest uptake and release occur above threshold pressures, usually accompanied by framework deformations-have shown promise as storage adsorbents. Herein, the pressures for gas uptake and release of a series of zinc-based mixed-ligand catenated metal-organic frameworks were controlled by decorating its ligands with two different functional groups and changing their ratio. This affects the deformation energy of the framework, which in turn controls the gated behaviour. The materials offer good performances for acetylene storage with a usable capacity of ~90 v/v (77% of the overall amount) at 298 K and under a practical pressure range (100-150 kPa).

Slacking of Gate Adsorption Behavior on Metal-Organic Frameworks under an External Force

As flexible metal-organic frameworks (MOFs) and their gate adsorption behaviors are increasingly expected to be used in gas storage and separation systems, evaluating their performance by considering their usage patterns in actual processes is becoming increasingly important. Herein, we show that the shaping of the elastic layer-structured MOF-11 (ELM-11; [Cu(BF₄)₂(4,4'-bipyridine)₂]) into pellet forms using polymer binders smears its stepwise uptake associated with the CO₂ gate adsorption. This is a critical problem because the superior adsorption properties of flexible MOFs are highly dependent on the sharpness of the step. Free energy analysis by molecular simulations revealed that the slacking of the gate adsorption is natural from a thermodynamic point of view. In other words, the external force exerted by the polymer binders, which prevents the expansion of MOF particles upon the gate opening, changes the free energy landscape of the system. This causes the flexible motifs within the MOF particles to undergo a structural transition at slightly different pressures from each other. The force profile dependence of the slacking phenomenon on both adsorption and desorption isotherms was also investigated. It was revealed that controlling the force profile applied to MOF particles is important to mold MOF pellets that satisfy the robustness and sharpness of the gate adsorption. Finally, we examined the coating of pellets to verify the relationship between the force profile and the degree of slacking and discussed possible strategies to improve the sharpness of the gate adsorption on MOF pellets considering the revealed mechanism.

Water Adsorption in Soft and Heterogeneous Nanopores

ConspectusLiquids under confinement differ in behavior from their bulk counterparts and can acquire properties that are specific to the confined phase and linked to the nature and structure of the host matrix. While confined liquid water is not a new topic of research, the past few years have seen a series of intriguing novel features for water inside nanoscale pores. These unusual properties arise from the very specific nature of nanoporous materials, termed "soft porous crystals"; they combine large-scale flexibility with a heterogeneous internal surface. This creates a rich diversity of behavior for the adsorbed water, and the combination of different experimental characterization techniques along with computational chemistry at various scales is necessary to understand the phenomena observed and their microscopic origins. The range of systems of interest span the whole chemical range, from the inorganic (zeolites, imogolites) to the organic (microporous carbons, graphene, and its derivatives), and even encompass the hybrid organic-inorganic systems (such as metal-organic frameworks).The combination of large scale flexibility with the strong physisorption (or even chemisorption) of water can lead to unusual properties (belonging to the "metamaterials" category) and to novel phenomena. One striking example is the recent elucidation of the mechanism of negative hydration expansion in ZrW₂O₈, by which adsorption of ∼10 wt % water in the inorganic nonporous framework leads to large shrinkage of its volume. Another eye-catching case is the occurrence of multiple water adsorption-driven structural transitions in the MIL-53 family of materials: the specific interactions between water guest molecules and the host framework create behavior that has not been observed with any other adsorbate. Both are counterintuitive phenomena that have been elucidated by a combination of experimental in situ techniques and molecular simulation.Another important direction of research is the shift in the systems and phenomena studied, from physical adsorption toward studies of reactivity, hydrothermal stability, and the effect of confinement on aqueous phases more complex than pure water. There have been examples of water adsorption in highly flexible metal-organic frameworks being able to compete with the materials' coordination bonds, thereby limiting its hydrothermal stability, while tweaking the functional groups of the same framework can lead to increased stability while retaining the flexibility of the material. However, this additional complexity and tunability in the macroscopic behavior can occur from changes in the confined fluid rather than the material. Very recent studies have shown that aqueous solutions of high concentration (such as LiCl up to 20 mol L^-1) confined in flexible nanoporous materials can have specific properties different from pure water and not entirely explained by osmotic effects. There, the strong ordering of the confined electrolyte competes with the structural flexibility of the framework to create an entirely new behavior for the {host, guest} system.

Multi- and instabilities in gas partitioning between nanoporous materials and rubber balloons

In the two-balloon experiment, two rubber balloons are connected and allowed to exchange gas. Owing to the non-monotonic relationship between the radius of the balloon and the pressure of gas inside it, the two-balloon system presents multi- and in-stabilities. Herein, we consider a two-adsorbent system, where two different adsorbents are allowed to exchange gas. We show that, for rigid adsorbents, the thermodynamic equilibrium state is unique. Then, we consider an adsorbent-balloon system, where an adsorbent exchanges gas with a rubber balloon. This system can exhibit multiple states at thermodynamic equilibrium- two (meta)stable and one unstable. The size of the balloon, pressure of gas in the balloon, and partitioning of gas between the adsorbent and the balloon differ among the equilibrium states. Temperature changes and the addition/removal of gas into/from the adsorbent-balloon system can induce catastrophe bifurcations and show hysteresis. Furthermore, the adsorbent-balloon system exhibits a critical temperature where, when approached from below, the discrepancy of balloon size between the two (meta)stable states decreases and, beyond, bistability is impossible. Practically, our findings preclude multiple partitions of adsorbed gas in rigid, mixed-linker or stratified metal-organic frameworks and may inspire new soft actuator and sensor designs.

Predicting the Features of Methane Adsorption in Large Pore Metal-Organic Frameworks for Energy Storage

Currently, metal-organic frameworks (MOFs) are receiving significant attention as part of an international push to use their special properties in an extensive variety of energy applications. In particular, MOFs have exceptional potential for gas storage especially for methane and hydrogen for automobiles. However, using theoretical approaches to investigate this important problem presents various difficulties. Here we present the outcomes of a basic theoretical investigation into methane adsorption in large pore MOFs with the aim of capturing the unique features of this phenomenon. We have developed a pseudo one-dimensional statistical mechanical theory of adsorption of gas in a MOF with both narrow and large pores, which is solved exactly using a transfer matrix technique in the Osmotic Ensemble (OE). The theory effectively describes the distinctive features of adsorption of gas isotherms in MOFs. The characteristic forms of adsorption isotherms in MOFs reflect changes in structure caused by adsorption of gas and compressive stress. Of extraordinary importance for gas storage for energy applications, we find two regimes of Negative gas adsorption (NGA) where gas pressure causes the MOF to transform from the large pore to the narrow pore structure. These transformations can be induced by mechanical compression and conceivably used in an engine to discharge adsorbed gas from the MOF. The elements which govern NGA in MOFs with large pores are identified. Our study may help guide the difficult program of work for computer simulation studies of gas storage in MOFs with large pores.

Adsorption Contraction Mechanics: Understanding Breathing Energetics in Isoreticular Metal-Organic Frameworks

A highly porous metal-organic framework DUT-48, isoreticular to DUT-49, is reported with a high surface area of 4560 m2·g-1 and methane storage capacity up to 0.27 g·g-1 (164 cm3·cm-3) at 6.5 MPa and 298 K. The flexibility of DUT-48 and DUT-49 under external and internal (adsorption-induced) pressure is analyzed and rationalized using a combination of advanced experimental and computational techniques. While both networks undergo a contraction by mechanical pressure, only DUT-49 shows adsorption-induced structural transitions and negative gas adsorption of n-butane and nitrogen. This adsorption behavior was analyzed by microcalorimetry measurements and molecular simulations to provide an explanation for the lack of adsorption-induced breathing in DUT-48. It was revealed that for DUT-48, a significantly lower adsorption enthalpy difference and a higher framework stiffness prevent adsorption-induced structural transitions and negative gas adsorption. The mechanical behavior of both DUT-48 and DUT-49 was further analyzed by mercury porosimetry experiments and molecular simulations. Both materials exhibit large volume changes under hydrostatic compression, demonstrating noteworthy potential as shock absorbers with unprecedented high work energies.

Statistical mechanics of binary mixture adsorption in metal-organic frameworks in the osmotic ensemble

Although crucial for designing separation processes little is known experimentally about multi-component adsorption isotherms in comparison with pure single components. Very few binary mixture adsorption isotherms are to be found in the literature and information about isotherms over a wide range of gas-phase composition and mechanical pressures and temperature is lacking. Here, we present a quasi-one-dimensional statistical mechanical model of binary mixture adsorption in metal-organic frameworks (MOFs) treated exactly by a transfer matrix method in the osmotic ensemble. The experimental parameter space may be very complex and investigations into multi-component mixture adsorption may be guided by theoretical insights. The approach successfully models breathing structural transitions induced by adsorption giving a good account of the shape of adsorption isotherms of CO₂ and CH₄ adsorption in MIL-53(Al). Binary mixture isotherms and co-adsorption-phase diagrams are also calculated and found to give a good description of the experimental trends in these properties and because of the wide model parameter range which reproduces this behaviour suggests that this is generic to MOFs. Finally, a study is made of the influence of mechanical pressure on the shape of CO₂ and CH₄ adsorption isotherms in MIL-53(Al). Quite modest mechanical pressures can induce significant changes to isotherm shapes in MOFs with implications for binary mixture separation processes.This article is part of the theme issue 'Modern theoretical chemistry'.

Molecular Mechanism of Swing Effect in Zeolitic Imidazolate Framework ZIF-8: Continuous Deformation upon Adsorption

Zeolitic imidazolate framework ZIF-8 displays flexibility of its structure by rotation of its imidazolate linker. This "swing effect" has been widely described in the literature, both experimentally and theoretically, as a bistable system where the linker oscillates between two structures: "open window" and "closed window". By using quantum chemistry calculations and first-principles molecular dynamics simulations, it is shown that the deformation upon adsorption is in fact continuous upon pore loading, with thermodynamics of packing effects being the reason behind stepped adsorption isotherms experimentally observed. Finally, we study a variant of ZIF-8 with a different linker, highlighting the influence of the linker and the balance of microscopic interactions on the framework's flexibility.

Controlling relative polymorph stability in soft porous crystals with a barostat

We use Monte Carlo simulations to investigate the thermodynamic behavior of soft porous crystal (SPC) adsorbents under the influence of an external barostat. We consider SPCs that naturally exhibit polymorphism between crystal forms of two distinct pore sizes. In the absence of barostatting, these crystals may be naturally divided into two categories depending on their response to stress applied by the adsorbate fluid: those which macroscopically deform and change the volume of their unit cell ("breathing") and those which instead undergo internal rearrangements that change the adsorbate-accessible volume without modifying the unit cell volume ("gate-opening"). When breathing SPCs have a constant external pressure applied, in addition to the thermodynamic pressure of the adsorbate fluid, we find that the free energy difference between the crystal polymorphs is shifted by a constant amount over the entire course of adsorption. Thus, their relative stability may be easily controlled by the barostat. However, when the crystal is held at a fixed overall pressure, changes to the relative stability of the polymorphs tend to be more complex. We demonstrate a thermodynamic analogy between breathing SPCs held at a fixed pressure and macroscopically rigid gate-opening ones which explains this behavior. Furthermore, we illustrate how this implies that external mechanical forces may be employed to tune the effective free energy profile of an empty SPC, which may open new avenues to engineer the thermodynamic properties of these polymorphic adsorbents, such as selectivity.

Relationship between Pore-size Distribution and Flexibility of Adsorbent Materials: Statistical Mechanics and Future Material Characterization Techniques

Measurement of the pore-size distribution (PSD) via gas adsorption and the so-called "kernel method" is a widely used characterization technique for rigid adsorbents. Yet, standard techniques and analytical equipment are not appropriate to characterize the emerging class of flexible adsorbents that deform in response to the stress imparted by an adsorbate gas, as the PSD is a characteristic of the material that varies with the gas pressure and any other external stresses. Here, we derive the PSD for a flexible adsorbent using statistical mechanics in the osmotic ensemble to draw analogy to the kernel method for rigid materials. The resultant PSD is a function of the ensemble constraints including all imposed stresses and, most importantly, the deformation free energy of the adsorbent material. Consequently, a pressure-dependent PSD is a descriptor of the deformation characteristics of an adsorbent and may be the basis of future material characterization techniques. We discuss how, given a technique for resolving pressure-dependent PSDs, the present statistical mechanical theory could enable a new generation of analytical tools that measure and characterize certain intrinsic material properties of flexible adsorbents via otherwise simple adsorption experiments.

The Influence of Intrinsic Framework Flexibility on Adsorption in Nanoporous Materials

For applications of metal–organic frameworks (MOFs) such as gas storage and separation, flexibility is often seen as a parameter that can tune material performance. In this work we aim to determine the optimal flexibility for the shape selective separation of similarly sized molecules (e.g., Xe/Kr mixtures). To obtain systematic insight into how the flexibility impacts this type of separation, we develop a simple analytical model that predicts a material’s Henry regime adsorption and selectivity as a function of flexibility. We elucidate the complex dependence of selectivity on a framework’s intrinsic flexibility whereby performance is either improved or reduced with increasing flexibility, depending on the material’s pore size characteristics. However, the selectivity of a material with the pore size and chemistry that already maximizes selectivity in the rigid approximation is continuously diminished with increasing flexibility, demonstrating that the globally optimal separation exists within an entirely rigid pore. Molecular simulations show that our simple model predicts performance trends that are observed when screening the adsorption behavior of flexible MOFs. These flexible simulations provide better agreement with experimental adsorption data in a high-performance material that is not captured when modeling this framework as rigid, an approximation typically made in high-throughput screening studies. We conclude that, for shape selective adsorption applications, the globally optimal material will have the optimal pore size/chemistry and minimal intrinsic flexibility even though other nonoptimal materials’ selectivity can actually be improved by flexibility. Equally important, we find that flexible simulations can be critical for correctly modeling adsorption in these types of systems.

Tuning flexibility to control selectivity in soft porous crystals

We use flat-histogram Monte Carlo simulations to study how changing the flexibility of soft porous crystals (SPCs) affects their selective adsorption of a binary, size-asymmetric supercritical fluid. Specifically, we consider mesoporous SPCs which have multiple minima in their free energy profiles as a function of pore size such that they are capable of exhibiting polymorphism between a narrow and large pore phase. While specific fluid-pore interactions determine the shape of both pores' selectivity curve as a function of adsorbate pressure, an individual pore tends to selectively adsorb a species based on the size of the adsorbate molecule relative to itself, thereby shifting the pore's selectivity curve relative to its polymorph. By controlling the flexibility of a SPC, the relative thermodynamic stability of the two pore phases may be varied, thereby changing the overall selectivity of the SPC during adsorbate loading. We investigate this for two classes of SPCs: one representative of "gate-opening" materials and another of "breathing" materials. For gate-opening materials, this control is much more salient than in breathing ones. However, for the latter, we illustrate how to tune the free energy profile to create materials which breathe multiple times during adsorption/desorption.

Statistical mechanical model of gas adsorption in porous crystals with dynamic moieties

Some nanoporous, crystalline materials possess dynamic constituents, for example, rotatable moieties. These moieties can undergo a conformation change in response to the adsorption of guest molecules, which qualitatively impacts adsorption behavior. We pose and solve a statistical mechanical model of gas adsorption in a porous crystal whose cages share a common ligand that can adopt two distinct rotational conformations. Guest molecules incentivize the ligands to adopt a different rotational configuration than maintained in the empty host. Our model captures inflections, steps, and hysteresis that can arise in the adsorption isotherm as a signature of the rotating ligands. The insights disclosed by our simple model contribute a more intimate understanding of the response and consequence of rotating ligands integrated into porous materials to harness them for gas storage and separations, chemical sensing, drug delivery, catalysis, and nanoscale devices. Particularly, our model reveals design strategies to exploit these moving constituents and engineer improved adsorbents with intrinsic thermal management for pressure-swing adsorption processes.

Application of computational methods to the design and characterisation of porous molecular materials

Composed from discrete units, porous molecular materials (PMMs) possess properties not observed for conventional, extended solids. Molecular simulations provide crucial understanding for the design and characterisation of these unique materials.

Multicomponent adsorption in mesoporous flexible materials with flat-histogram Monte Carlo methods

We demonstrate an extensible flat-histogram Monte Carlo simulation methodology for studying the adsorption of multicomponent fluids in flexible porous solids. This methodology allows us to easily obtain the complete free energy landscape for the confined fluid-solid system in equilibrium with a bulk fluid of any arbitrary composition. We use this approach to study the adsorption of a prototypical coarse-grained binary fluid in "Hookean" solids, where the free energy of the solid may be described as a simple spring. However, our approach is fully extensible to solids with arbitrarily complex free energy profiles. We demonstrate that by tuning the fluid-solid interaction ranges, the inhomogeneous fluid structure inside the pore can give rise to enhanced selective capture of a larger species through cooperative adsorption with a smaller one. The maximum enhancement in selectivity is observed at low to intermediate pressures and is especially pronounced when the larger species is very dilute in the bulk. This suggest a mechanism by which the selective capture of a minor component from a bulk fluid may be enhanced.

Exploring the Flexibility of MIL-47(V)-Type Materials Using Force Field Molecular Dynamics Simulations

The flexibility of three MIL-47(V)-type materials (MIL-47, COMOC-2, and COMOC-3) has been explored by constructing the pressure versus volume and free energy versus volume profiles at various temperatures ranging from 100 to 400 K. This is done with first-principles-based force fields using the recently proposed QuickFF parametrization protocol. Specific terms were added for the materials at hand to describe the asymmetry of the one-dimensional vanadium-oxide chain and to account for the flexibility of the organic linkers. The force fields are used in a series of molecular dynamics simulations at fixed volumes but varying unit cell shapes. The three materials show a distinct pressure-volume behavior, which underlines the ability to tune the mechanical properties by varying the linkers toward different applications such as nanosprings, dampers, and shock absorbers.

CO2 adsorption-induced structural changes in coordination polymer ligands elucidated via molecular simulations and experiments

Molecular simulations and experiments were used to elucidate guest-induced structural changes in the coordination polymer CPL-2.

Exact matrix treatment of an osmotic ensemble model of adsorption and pressure induced structural transitions in metal organic frameworks

Here we present an exactly treated quasi-one dimensional statistical mechanical osmotic ensemble model of pressure and adsorption induced breathing structural transformations of metal–organic frameworks (MOFs).

Understanding gate adsorption behaviour of CO2 on elastic layer-structured metal–organic framework-11

We demonstrate that CO₂ gate adsorption behaviour of elastic layer-structured metal–organic framework-11 can be described by a thermodynamic model by free energy analysis with the aid of an adsorption experiment and a molecular simulation.

Softening upon Adsorption in Microporous Materials: A Counterintuitive Mechanical Response

We demonstrate here that microporous materials can exhibit softening upon adsorption of guest molecules, at low to intermediate pore loading, in parallel to the pore shrinking that is well-known in this regime. This novel and counterintuitive mechanical response was observed through molecular simulations of both model pore systems (such as slit pore) and real metal-organic frameworks. It is contrary to common belief that adsorption of guest molecules necessarily leads to stiffening due to increased density, a fact that we show is the high-loading limit of a more complex behavior: a nonmonotonic softening-then-stiffening.

Understanding adsorption-induced structural transitions in metal-organic frameworks: from the unit cell to the crystal

Breathing transitions represent recently discovered adsorption-induced structural transformations between large-pore and narrow-pore conformations in bi-stable metal-organic frameworks such as MIL-53. We present a multiscale physical mechanism of the dynamics of breathing transitions. We show that due to interplay between host framework elasticity and guest molecule adsorption, these transformations on the crystal level occur via layer-by-layer shear. We construct a simple Hamiltonian that describes the physics of host-host and host-guest interactions on the level of unit cells and reduces to one effective dimension due to the long-range elastic cell-cell interactions. We then use this Hamiltonian in Monte Carlo simulations of adsorption-desorption cycles to study how the behavior of unit cells is linked to the transition mechanism at the crystal level through three key physical parameters: the transition energy barrier, the cell-cell elastic coupling, and the system size.