Flat-Histogram Monte Carlo Simulation of Water Adsorption in Metal-Organic Frameworks

Abstract

Molecular simulations of water adsorption in porous materials often converge slowly due to sampling bottlenecks that follow from hydrogen bonding and, in many cases, the formation of water clusters. These effects may be exacerbated in metal-organic framework (MOF) adsorbents, due to the presence of pore spaces (cages) that promote the formation of discrete-size clusters and hydrophobic effects (if present), among other reasons. In Grand Canonical Monte Carlo (MC) simulations, these sampling challenges are typically manifested by low MC acceptance ratios, a tendency for the simulation to become stuck in a particular loading state (i.e., macrostates), and the persistence of specific clusters for long periods of the simulation. We present simulation strategies to address these sampling challenges, by applying flat-histogram MC (FHMC) methods and specialized MC move types to simulations of water adsorption. FHMC, in both Transition-matrix and Wang-Landau forms, drives the simulation to sample relevant macrostates by incorporating weights that are self-consistently adjusted throughout the simulation and generate the macrostate probability distribution (MPD). Specialized MC moves, based on aggregation-volume bias and configurational bias methods, separately address low acceptance ratios for basic MC trial moves and specifically target water molecules in clusters; in turn, the specialized MC moves improve the efficiency of generating new configurations which is ultimately reflected in improved statistics collected by FHMC. The combined strategies are applied to study the adsorption of water in CuBTC and ZIF-8 at 300 K, through examination of the MPD and the adsorption isotherm generated by histogram reweighting. A key result is the appearance of non-trivial oscillations in the MPD, which we show to be associated with water clusters in the adsorption system. Additionally, we show that the probabilities of certain clusters become similar in value near the boundaries of the isotherm hysteresis loop, indicating a strong connection between cluster formation/destruction and the thermodynamic limits of stability. For a hydrophobic MOF, the FHMC results show that the phase transition from low density to high density is suppressed to water pressure far above the bulk-fluid saturation pressure; this is consistent with results presented elsewhere. We also compare our FHMC simulation isotherm to one measured by a different technique but with ostensibly the same molecular interactions and comment on observed differences and the need for follow-up work. The simulation strategies presented here can be applied to the simulation of water in other MOFs using heuristic guidelines laid out in our text, which should facilitate the more consistent and efficient simulation of water adsorption in porous materials in future applications.

Graphical Abstract

1. Introduction

Water adsorption in porous materials is a topic of immense interest due to the wide variety of potential applications, such as dehumidification,^1,2 adsorptive cooling,² purification, and water harvesting.^3–5 In other applications, the presence of water can present specific challenges. For example, in direct-air capture of CO₂ using adsorbent-based processes,^6,7 humidity must be considered in the choice of adsorbent material.^8–10 More generally, water may compete (adversely or beneficially) with a target adsorbate, physically or chemically alter the adsorbent material, or affect the stability or useful lifetime of the material.^11–14

One class of materials that has been proposed for applications involving water adsorption is that of metal-organic frameworks (MOFs), which are porous materials composed of metal or metal-oxide clusters and organic linkers assembled to form multidimensional crystalline structures, often with high porosity and/or internal surface area.^15–17 The simple cluster-linker architecture of MOFs allows chemists to propose (and, possibly, synthesize) diverse populations of MOFs,¹⁸ with widely varying chemistry due to available cluster and linker options. As a material class, MOFs are consequently considered to be adsorbents with greater tunability than historical adsorbent types, which may allow for application-specific material design. Potential benefits deriving from the diversity of MOFs are, of course, balanced by practical limitations of actual chemical synthesis. Consequently, computational modeling, including molecular simulation, has played a prominent role in evaluating and screening MOFs for all kinds of applications.^19–21

Because of its ability to form hydrogen bonds, water exhibits anomalous behavior in bulk. Well-known examples include the liquid-phase density maximum at 4°C and expansion upon freezing, which means the melting temperature decreases with increasing pressure.²² Other examples include nonmonotonic behavior of the isothermal compressibility and isobaric heat capacity with temperature. In confined environments or tight spaces consistent with those in an adsorbent material, the behavior of water is just as interesting. Isotherms for water adsorption often differ qualitatively from those of similarly-sized gases that show otherwise common characteristics such as capillary condensation and evaporation, hysteresis and layering transitions.²³ The resultant isotherms may resemble Type V isotherms in the IUPAC classification scheme:²³ the isotherms exhibit low-loading with upward concavity at low pressure and high-loading (liquid-like density) at high pressure, sometimes with hysteresis. Experimentally, water adsorption isotherms often show high variability²⁴ compared to isotherms of simpler adsorbates.^25,26 These unique observations for water adsorption are usually attributed to the hydrogen-bonding property of water²⁷ and a resultant major difference between water and simpler adsorbates: at low pressure, adsorbing water molecules show a tendency to cluster (perhaps at preferential adsorption sites) and those clusters eventually grow and coalesce to fill pores in the adsorbent as pressure or relative humidity increases.

Molecular simulation of water adsorption in porous materials traces back over 20 years, with an early focus on carbonaceous adsorbents, mostly using the Grand Canonical (GC) Monte Carlo (MC) technique. Those early simulations supported the “coalescence” explanation of water’s unique adsorption characteristics, which were qualitatively consistent across various models of water.²⁸ Simulation of water adsorption was not, however, without challenges that are related to hydrogen bonding. The strong and directional nature of the hydrogen bond can lead to long-lived hydrogen-bonded networks in both bulk water and clusters in confinement. Even at ambient conditions, sampling these structures sufficiently can be difficult because they represent local minima in the free energy landscape. In such cases, the simulation system may become trapped in a nonequilibrium state and not transition to a lower-energy state with a different number of water molecules or simply to a different cluster configuration. The challenges related to water clustering may be more prominent for confined water, as the confinement shape can serve as a template for a water cluster of particular shape and/or size and stabilize certain clusters. The end result is that conventional simulations of water adsorption often converge very slowly and have high uncertainty in measured thermophysical properties, leaving the reliability of these simulations in question.

Water adsorption in MOFs has been examined by molecular simulation,^29–35 but not to the extent of more conventional adsorbates such as CO₂, N₂, or Ar, and even less so for low temperatures. For low temperatures, water adsorption in MOFs exhibited similar challenges encountered in simpler adsorbents, often to an exaggerated extent. Zhang and Snurr³⁶ highlighted the challenges of simulating water adsorption in a hydrophobic MOF, where the simulations converged very slowly (sometimes requiring 70 days of simulation time per point); this was attributed to water clustering in the MOF cages and the authors consequently examined several acceleration strategies such as grid-based energy calculations and continuous fractional component MC trial moves. More recently, Datar et al³⁷ simulated water adsorption in MOFs, using a sampling strategy^38,39 that estimates the probability of observing a certain number of water molecules in the MOF. In these and other examples of the simulation of water adsorption in MOFs, extensive effort was devoted to increasing the convergence speed of the simulations.

These recent investigations of simulated water adsorption in MOFs identify both the vast opportunity and need for improving such simulations.⁴⁰ Two existing options that may aid simulations of water adsorption are flat-histogram MC (FHMC) techniques (e.g., Transition-matrix MC or Wang-Landau MC) and the use of various biased MC trial moves (e.g., configurational bias (CB)⁴¹ and aggregation-volume bias moves.⁴²) To our knowledge, no existing work has examined either of these approaches in the context of water adsorption in MOFs. In particular, FHMC techniques offer a particular advantage over conventional simulations: the use of a self-consistent macrostate weight that encourages the simulation to visit states with different numbers of molecules. This by itself targets a key concern of previous water simulations: the tendency of the simulation to get “stuck” in configurations that contain a certain number of water molecules. Simulations of adsorption in simple models of pores with purely repulsive interactions showed that FHMC with grand canonical ensemble trial moves may be up to 3 orders of magnitude more efficient than simulations with only canonical ensemble trials.⁴³ Biased MC trial moves separately address low trial acceptance ratios by generating higher-quality trial moves.

The purpose of the present work is to discuss and demonstrate the use of FHMC and biased MC trial moves to improve the reliability and accuracy of simulations of water adsorption in MOFs, with the ultimate aim being the generation of high-quality adsorption isotherms. The notable sampling challenges are most prevalent at low temperatures and, consequently, the present work focuses on simulations and results exclusively for 300 K, a temperature (for the water models we use) where aggregation significantly affects simulation efficiency. In this work, the issue of cluster formation in MOF cages will arise repeatedly; while we address the issue of clusters in the pores since it is central to increasing FHMC simulation efficiency, we leave many aspects of cluster formation in MOFs unexplored. Our purpose here is to discuss the FHMC simulation strategies themselves and will leave a deeper examination of cluster formation to future work.

This paper is organized as follows. Section 2 addresses background information, including more details concerning the challenges of water simulations, a brief overview of FHMC methods, and a presentation of typical FHMC strategies. In Sec 3, we describe the molecular models used in our simulations of water adsorption, present preliminary results, and then use those results to identify specialized MC strategies that accelerate sampling in water simulations. Section 4 presents the results of our FHMC simulations that utilize those specialized strategies, discusses the particular analysis needed to convert FHMC results to an adsorption isotherm, and compares some of our results to previous simulations. Finally, we conclude the paper in Sec 5 with a summary of the present work and a discussion of potential follow-up work. The Supporting Information (SI) of this manuscript provides additional discussion on the simulation techniques introduced here as well as expanded results, software scripts, and additional visual aids.

2. Background

2.1. Simulation Challenges Due to Water Clustering

As briefly mentioned in the introduction, the propensity of model water molecules to aggregate into clusters is an important characteristic of water simulations that cause conventional simulations to converge slowly, whether the simulated system is bulk or confined water. Conventional MC simulations use the simplest move types (translations, rotations, and grand canonical insertions and deletions) that are generated in an unbiased manner. The root of this sampling inefficiency is the large negative potential energy associated with the formation of hydrogen bonds, which presents a significant energy barrier to the removal of a water molecule from a cluster. This energy barrier is manifested by a low acceptance probability for MC trial moves that reduce the size of a water cluster, which may keep the simulation in nonequilibrium configurations for long periods, ultimately requiring very long simulations to sample enough unique configurations to yield satisfactory ensemble averages.

As an example, we present the following illustration: consider a simulation configuration with two water clusters, one containing 10 water molecules and one containing 5 water molecules, where the lowest energy configuration is a single cluster of 15 water molecules. For the system to transition from the two-cluster nonequilibrium state to the single-cluster state, the two clusters must merge, and the possible paths to that equilibrium state depend on the statistical mechanical ensemble. For a canonical ensemble simulation, water molecules could move individually, by a chain of simple MC translation moves, from the small cluster to the larger cluster; the energy barrier to remove one water from the small cluster must be overcome by each molecule. In the grand-canonical ensemble, the same canonical path can move molecules from one cluster to the other, but a combination of random deletion and subsequent random insertions can generate an indirect route that moves a molecule from the small cluster to the large cluster. Deletions from the cluster must overcome the same energy barrier as translation, however. Furthermore, due to the stochastic nature of MC simulations, opposite paths that reform the small cluster are also likely. Ultimately, the clusters merge through low-probability pathways and include large, repeated energy barriers that are overcome only through long simulation runs.

There exists a variety of biased sampling methods to generate higher-efficiency MC trial moves that target water clusters.^42,44,45 One strategy is the family of aggregation-volume-bias (AVB) moves,^42,45 which preferentially move molecules into or out of aggregated (e.g., hydrogen-bonded) configurations. Another is the use of CB⁴¹ to improve the quality of attempted trial moves; CB may also be incorporated into AVB moves. Additionally, the Rosenbluth weights in CB may be computed via shortcut methods or using lookup tables that reduce the computational penalty of CBMC. The common characteristic of these biased MC moves is their purposeful constructions that address sampling bottlenecks due to water aggregation.

2.2. FHMC Methods

We now briefly discuss FHMC methods and some key features that aid the presentation of our simulation method and analysis of our results. For a full discussion of FHMC, readers may consult previous work in refs 46–48 and, for specific applications to adsorption, refs 49,50. A streamlined summary of FHMC methods is presented in Section SI.1 of the SI.

An important characteristic of FHMC for the present discussion is that the probability of accepting a transition between macrostates (in the grand canonical ensemble, the macrostate is the number of particles in the system, N; see Section SI.1) includes a biasing weight. The weight is the inverse of an estimate of the macrostate probability distribution (MPD), Π(N; μ, V, T), where μ is the chemical potential, V is the volume, and T is the temperature. For ease of notation, we henceforth drop the V and T dependence in Π. The MPD is estimated from statistics accumulated during the simulation, using either the Transition-matrix Monte Carlo (TMMC)^{46,47,51–53} or Wang-Landau (WL) techniques.^54–56 Ultimately, the weight encourages the simulation to sample all macrostates with uniform probability, eventually achieving a “flat histogram” of visited macrostates and generating a converged Π(N; μ). This is particularly useful in simulations of aggregating molecules, since the weight counteracts the tendency of the system to become stuck in one macrostate.

The most important result of FHMC simulation is Π(N; μ). Once this distribution is known, histogram reweighting can be used to determine Π(N; μ) at a different μ without additional simulation effort (see eq SI.11 in the SI and associated discussion). Average properties of the system can then be recomputed at different thermodynamic states, provided that the relevant macrostate-dependent quantities were measured during the simulation (cf. eq SI.9 of the SI). For example, histogram reweighting allows one to generate an adsorption isotherm from a pair of FHMC simulations, one for the adsorbed gas and a second for the bulk gas.⁴⁹ The converged Π(N; μ) also yields additional information about the simulated system, such as free energies, phase coexistence, and metastability.

Ultimately, the advantages of FHMC derive from these coupled characteristics: the macrostate weight encourages the system to sample all macrostates, surmounting barriers in the free energy landscape, and the converged weight provides many thermophysical properties of the system across a wide range of thermodynamic conditions. A final advantage of FHMC is the option to divide the simulation into “windows” that sample restricted macrostate ranges; the windows may be run in parallel and, provided that adjacent windows overlap, the Π(N; μ) covering the full macrostate range may be reconstructed from Π(N; μ) for the windows. (See Section SI.1 of the SI for additional discussion.)

3. Methods

3.1. Simulation and Molecular Models

To demonstrate the use and advantages of FHMC in simulating water adsorption in MOFs, we present results from the simulation of two MOF adsorbents: CuBTC⁵⁷ and ZIF-8.⁵⁸ Importantly, we will later show that these materials exhibit common characteristics of water adsorption that are reflective of water clustering, despite the difference in hydrophilicity/hydrophobicity, and both benefit from specialized sampling strategies that we describe later. We describe the simulation models of CuBTC and ZIF-8 in Section SI.2 of the SI. To briefly summarize, both MOFs were generated from previously reported crystal structures and we use existing force fields to set the Lennard-Jones (LJ) parameters and atomic partial charges; we note that the ZIF-8 force field⁵⁹ is identical to that used by Zhang and Snurr.³⁶ Electrostatic interactions were modeled using the Ewald summation method.^60,61 Both materials are microporous with the three cages of CuBTC having diameters from 0.45 nm to 1.29 nm and the single cage of ZIF-8 having a diameter of 1.1 nm. The MOFs were held rigid as is typical for most molecular simulations of crystalline materials.¹⁹ Water was represented by the SPC/E model,⁶² with some select results presented for the TIP4P model.⁶³ For demonstration purposes, we also ran simulations of TraPPE CO₂⁶⁴ (see Sec 3.2) adsorption. Pair potentials used a 10 Å cutoff and the LJ potentials used a linear-force shift,⁶¹ unless otherwise specified.

Our simulations were run using FEASST,⁶⁵ an open-source MC engine developed at NIST, using custom-written Python scripts. The SI contains example FEASST scripts and particle files (force field parameters and MOF structures) that facilitate the reproduction of our results.

3.2. Preliminary Results

We begin by presenting TMMC results for adsorption in ZIF-8 with a basic simulation setup to highlight the sampling challenges when water is the adsorbate. Two simulations were set up, one with SPC/E water as the adsorbate and the other with CO₂ as the adsorbate, at 300 K, using macrostate domains appropriate to each adsorbate in the replicated MOF (N_max = 630 and 300 for SPC/E water and CO₂, respectively). The chemical potentials were chosen to yield local maxima in ln Π(N; μ) that aid explanatory discussion. The macrostate domains were divided into windows using an exponential strategy,⁴³ with exponent 1.5, 48 windows for CO₂, and 80 windows for SPC/E water (roughly the same number of windows relative to N_max). The simplest MC move set was used (unbiased translations, rotations, and GC transfer moves, with relative weights 3:2:5) and no lookup tables or cell lists were utilized to speed up calculations for the preliminary simulations; this is a basic implementation of TMMC in FEASST. Lastly, the simulations were run on identical batches of cluster computation nodes. The preliminary results presented in this section are only intended to identify sampling challenges of different adsorbates by presenting actual acceptance statistics and configurational snapshots; broad conclusions regarding TMMC should not be drawn from these specific results. For example, the SPC/E simulations should run more slowly relative to CO₂ due to the larger N_max. Additionally, since 300 K is near the critical temperature of CO₂, but is well below the critical temperature of SPC/E water,⁶⁶ the CO₂ simulations will likely converge more quickly than SPC/E simulations.

After 24 h of run time, the simulations had completed between 73 million and 99 million trial moves for CO₂ and between 62 million and 98 million trial moves for SPC/E water (the lowest number of trials corresponds to the largest N window). The two simulations are markedly different in terms of the acceptance ratios of the GC transfer moves over the final 10⁶ trial moves: CO₂ had acceptance ratios from 2.6 × 10⁻⁵ to 0.12, while SPC/E had acceptance ratios from 1.4 × 10⁻⁵ to 0.002, with the lowest acceptance ratios at the highest N and vice versa. It is immediately obvious that GC transfer moves for SPC/E water are markedly more inefficient than the same moves for CO₂, especially at low N. The sweep characteristics (a metric that quantifies TMMC convergence, see Section SI.1 of the SI) of the two adsorbates are also noticeably different: In the 24-h run, CO₂ simulations had already swept up to 781 times at low N, though the highest-N windows had not yet swept. For SPC/E, however, most windows had not swept at all with only a few windows at mid-N having completed four or fewer sweeps. We urge caution in comparing the acceptance ratios of the two simulations: The CO₂ simulation windows have higher acceptance ratios due to both the characteristics of CO₂ and the positive feedback effect of the TMMC weight.

Figure 1 displays ln Π(N; μ) for the two simulations, with the y-axis truncated to highlight the important features. The full ln Π(N; μ) is shown in Figure SI.5 of the SI. Additionally, the figure contains two other ln Π(N; μ) traces that will be discussed later. A number of revealing features are shown in the figure. First, ln Π(N; μ) for CO₂ is smooth, which indicates that the TMMC collection matrix contains sufficient information about all macrostate transitions, and is quantitatively reflected in the high number of sweeps completed by the simulation windows. The jagged features of ln Π(N; μ) for SPC/E follow from the exact opposite situation: few sweeps and inadequate information content in the TMMC collection matrix. Second, ln Π(N; μ) for SPC/E exhibits possible oscillatory features for N/N_max < 0.2; at higher N the data trace is too noisy for further interpretation. The four local maxima in Figure 1 appear at regular intervals, approximately ΔN = 29, which is likely indicative of a repeating structural motif. Macrostates between the local maxima are lower probability (higher free energy), representing states where the additional structural feature is incomplete. Based on the results of Zhang and Snurr,³⁶ this should be related to the formation of water clusters in the cages of ZIF-8, and examination of configurational snapshots confirms this. Figure 2 shows selected snapshots of configurations of SPC/E water in ZIF-8 for macrostates that correspond to the four apparent local maxima in ln Π(N; μ), and the clusters are immediately obvious. In animations of the relevant windows (available in the SI), the clusters can be seen to remain essentially intact throughout the entire simulation run. Occasional fluctuations yield short-lived, smaller clusters or move molecules out of the clusters, but the system repeatedly returns to whole-cluster configurations, which is completely consistent with the results of Zhang and Snurr³⁶ where ZIF-8 cages were either empty or essentially filled with water; the adsorption mechanism was one in which the cages filled through rapid growth of a water cluster in one cage at a time. This is in contrast to adsorption by layering and eventual pore filling.

Figure 1:

Demonstration of macrostate distributions that may not be fully converged for simulation of SPC/E water and TraPPE carbon dioxide in ZIF-8 at 300 K. For the CO₂ simulation, N_max = 300 with 48 windows; for H₂O N_max = 630 with 80 windows. Black circles (spaced at intervals of ΔN = 29) denote apparent local maxima that may correspond to fully formed cluster configurations. The traces labeled “Base” used the basic simulation setup described in Sec 3.2 and the windows of the “Base” simulations were run for 24 h of wall time. The traces labeled “Improved” used the improved simulation setup described in Sec 3.3 and are based on results after 24 h and 48 h of run time as indicated in the legend.

Figure 2:

Snapshots of configurations of SPC/E water adsorbed in ZIF-8 at 300 K that highlight cluster formation in MOF cages. The MOF material is shown by thin lines and the atoms of SPC/E water are shown by large spheres (oxygens are red, hydrogens are white). a) N = 29 as a single cluster, b) N = 58 as two clusters, c) N = 87 as three clusters, and d) N = 116 as four clusters. Some clusters straddle the periodic boundary of the MOF simulation cell; molecules in the snapshot were moved to show the complete cluster.

The close examination of short, benchmark simulations highlights two main concerns: overall low acceptance ratios for the GC transfer moves and the persistence of water clusters. Additionally, there is the issue of slow convergence of the TMMC weight, but this is a side effect of low acceptance ratios. Recognition of these simulation challenges points toward strategies designed to improve the efficiency of the simulations and eliminate sampling bottlenecks that slow convergence of the FHMC results.

3.3. Specialized Strategies To Address Inefficient Sampling

We now discuss strategies for improving the simulation of water adsorption in MOFs, building on the preliminary results in the previous section. These strategies include FHMC weight types and specialized MC moves. The efficacy of specialized MC moves and FHMC weight both depend on the number of particles, N, and here we attempt to assign these strategies to particular macrostate ranges, summarized in Figure 3. We will first discuss the choice of FHMC weight shown in the upper half of Figure 3, and then specialized MC moves in the lower half.

Figure 3:

Efficient use of flat-histogram bias and specialized MC trials requires different approaches depending on the number of fluid particles (the macrostate), N. The graphic illustrates of our assignment of flat-histogram bias schemes and specialized MC trial moves based on N.

The FHMC weight strategy depends on the number of particles as follows. In previous studies, WL sampling was often found to aid FHMC simulation most for low N relative to the maximal loading of the simulation cell.⁵⁰ Hence, we used WL initialization for windows with N/N_max < 0.5. Otherwise, for N/N_max ≥ 0.5, TMMC was used without WL initialization. We used windows of uniform size instead of an exponential division, which does penalize the highest-N windows, but greatly aids the low-N windows where clusters predominate. Additionally, we used narrow simulation windows, which divide the simulated macrostates into many parallel execution threads. (This strategy does, however, need to be evaluated in light of the clustering effect as window size could interact with cluster formation. We show in Section SI.6 of the SI that window placement in our simulations did not affect ln Π(N; μ) or the isotherm.) Lastly, the simulations used WL for 18 decreases of ln g (starting at ln g = 1) and then collected transition statistics in the collection matrix for two additional decreases after which the weight switched to TMMC.

We will now discuss the macrostate ranges where various specialized MC trials were utilized, as summarized in the lower half of Figure 3. The first issue to address is the low acceptance ratio for GC transfer moves at high density (e.g., the lower right portion of Figure 3), which is eventually reflected in slow convergence of the TMMC weight and poor sweeping (compared to an “easier” adsorbate). A common strategy for addressing low GC transfer efficiency is to incorporate CBMC through stepwise growth of the adsorbate with multiple proposals for each constituent part; for SPC/E water (or TIP variants) this practically means the proposal of multiple trial locations for the oxygen site followed by multiple trial orientations of the hydrogen point charges. Such a CBMC approach to GC insertion (and the parallel deletion) effectively searches for a cavity to insert the oxygen, and then subsequently inserts the hydrogen point charges in a favorable orientation. Continuing, to mitigate the additional computational demand of CBMC (versus simple GC transfers), one can use a dual-cut strategy (DCCB), where the Rosenbluth weights are based on a short-range reference potential.⁶⁷ The short-range potential may also allow for the use of a cell list for compatible system dimensions. For our implementation in a water simulation, a natural choice of the reference potential is a sum of the oxygen-oxygen LJ pair potential and the Ewald realspace contribution, though truncated at 6 Å, which also allowed us to use a cell list for both MOFs. (The cutoff distance was chosen after running exploratory simulations that tested different values.) We also used the DCCB strategy to replace simple canonical rotations with regrowth moves where one or both hydrogen point charges are moved about the fixed oxygen. In this work, DCCB (without AVB, see next paragraph) was utilized for N/N_max ≥ 0.5 since DCCB becomes less efficient at lower densities when cavities are easier to find and the overhead costs of DCCB outweigh any sampling improvements.⁶⁷

The second issue to address is the clustering of water at low to intermediate densities, as shown in the lower middle portion of Figure 3. As mentioned previously, a typical strategy for dealing with molecules that form hydrogen bonds is to add AVB moves that preferentially create and destroy bonded configurations, both through canonical translations or rotations and GC transfers. To further improve AVB moves, we also incorporate DCCB (largely identical to the implementation described above), where an AVB strategy handles the oxygen site (either through an AVB translation into or out of a bonded configuration or by a GC transfer) and DCCB-based regrowth places the hydrogen point charges. Efficient use of AVB also relies on the use of a neighbor list that accelerates the identification of bonded pairs; our AVB implementation defined a bonded pair as two water molecules whose interoxygen distance falls between 2.5 Å and 5.0 Å. Lastly, AVB with DCCB incurs computational overhead that can outweigh gains from the preferential building and breaking of bonded configurations, so we only utilized AVB with DCCB in the range of 0.1 < N/N_max < 0.5.

Simulation details of the AVB and DCCB trials are summarized as follows. We utilized three types of AVB moves: The first move is the AVBMC2 translation move,⁴⁵ which for our application preferentially attempts to move an SPC/E molecule into or out of the bonded region of a randomly-selected target SPC/E molecule. The second move is another canonical move, based on an extension of the AVBMC3 move,⁴⁵ which we label AVBMC4. A full description of this AVB move is given in Ref 43 and described in Section SI.3 of the SI, but may be summarized as a move that selects two target, nonbonded SPC/E molecules and then attempts to move a third bonded molecule from one target molecule to the other. The AVB GC transfer move is simply a GC transfer move that preferentially inserts (deletes) an SPC/E molecule in (from) the bonded region of a target molecule.⁴² In our simulations, all AVB trial moves incorporated DCCB with four trial locations for each hydrogen. The full move set for the windows with AVB supplement moves was: unbiased translations, unbiased rotations, unbiased GC transfers, AVBMC2 translations, AVBMC4 translations, and AVB transfers, with relative weights 3:2:5:1:1:1. Windows that include DCCB without AVB used a move set that included unbiased translations (i.e., no DCCB), rotation by regrowth of one hydrogen, rotation by regrowth of both hydrogens, and GC transfers by DCCB growth (multiple oxygen insertions, multiple hydrogen orientations). The relative weights were 1:0.6:0.4:8, which shows preference for GC transfer moves relative to the unbiased and AVB windows.

We did not precisely optimize the macrostate boundaries for these strategies shown in Figure 3, but instead chose these macrostate boundaries based on general characteristics of the MC move types and bias types and a few manual iterations of trial and error. Precise optimization would either require many simulations with different macrostate boundaries or the development of a new, automated approach that is beyond the scope of this work. Future work could include further improvements to the strategy summarized in Figure 3, such as varying the number of steps in DCCB with N or utilizing AVB without DCCB at the lowest densities. We also note that the macrostate boundaries reported in Figure 3 are approximate because the MC moves sets are assigned for an entire FH window, and window boundaries may not precisely coincide with the stated macrostate ranges.

4. Results and Discussion

The following section presents and discusses simulation results for the two MOF systems described above, beginning with the MPD for each system. Due to interesting features in the MPDs, we then discuss the analysis of Π(N; μ) necessitated by those features, so that adsorbed phases are identified consistently and intuitively. Following, we use that specific analysis technique to generate the associated adsorption isotherms for the model systems. Lastly, we present selected results for a different water model, TIP4P, adsorbed in the ZIF-8, to make specific comparisons with the results of Zhang and Snurr.³⁶

We also ran FHMC simulations of bulk SPC/E water at 300 K, using the same cutoff and Ewald strategies as for the adsorption simulations, and the resultant ln Π(N; μ) is used to convert μ to bulk fluid pressure and identify the bulk fluid saturation pressure (p₀). For SPC/E water at 300 K as simulated here, p₀ = 1313 ± 6 Pa; we note that this is about 30 % larger than the saturation pressure of SPC/E water at 300 K using long-range corrections.^66,68 Finally, for reasons that are explained in Sec 4.4, we will present some simulation results in terms of the fugacity (f) of the SPC/E adsorbate. Here, we use the conventional definition of the fugacity:

$$f=k_{\text{B}}T\frac{\exp (\beta \mu )}{{\text{Λ}}^3}$$ (1)

where Λ is the thermal de Broglie wavelength. For reference, the saturation fugacity (f₀) of bulk SPC/E water at 300 K with the specified cutoff was computed to be 1261 ± 6 Pa.

4.1. Preliminary Results Revisited

Figure 1 contains two yet undiscussed ln Π(N; μ) for SPC/E water adsorption in ZIF-8, where the simulations were 24-h and 48-h runs using the improved FHMC strategy described in Sec 3.3. For comparison with the simple simulations, the windows of the improved FHMC simulation ran between 63 million and 157 million MC trial moves in 24 h and approximately double that in 48 h; the highest-N windows ran roughly the same number of trials in the allotted time, but the smallest-N windows ran in excess of 50 % more trials due to the altered window assignment approach. Additionally, all the windows save the three with the highest N completed at least one sweep in 24 h; in 48 h all windows had swept at least once. (We do not state the acceptance ratios of AVB and DCCB trial moves, as those ratios are not directly comparable to those of the simple MC trial moves.) For both traces, the improved FHMC strategy has resulted in improved sampling of the macrostates, which is reflected in the quality of ln Π(N; μ). The new traces show much smoother oscillations up to approximately 0.2 N_max, with six local maxima now clearly identifiable (versus four based on the simple simulation). The highest N local maximum has become more smooth after 48 h, but ln Π(N; μ) between 0.2 < N/N_max < 0.9 is still improving. While the two new simulation ln Π(N; μ) are not converged at 48 h, the improved FHMC strategy has facilitated the completion of more MC trials and sweeping has begun (albeit slowly) in all windows, both of which indicate that the proposed FHMC strategy has improved the sampling of macrostates versus that of the simple simulation’s approach.

4.2. Macrostate Probability Distributions

After confirming the improvements of the FHMC strategy for water adsorption, we now present ln Π(N; μ) for the two adsorption systems described above. We begin with FHMC simulation results for the adsorption of SPC/E water in the hydrophilic MOF CuBTC at 300 K. The full macrostate domain (0 < N < 500) was divided uniformly into 64 overlapping windows, and the MC moves for each window were assigned as described in Sec 3.3. We ran four replicate simulations that were identical except for the random-number sequence to check for run-to-run variation, and then combined the TMMC collection matrices to yield a single ln Π(N; μ) for further analysis. The four replicate runs yielded ln Π(N; μ) that were visually indistinguishable from both each other and the combined ln Π(N; μ) (see Section SI.6 of the SI). Each window of the four simulations was run for seven days of wall time or was terminated early at the completion of 500 sweeps. Figure 4 displays ln Π(N; μ) reweighted to five values of p/p₀, which correspond to different situations that are described throughout the following discussion. At the lowest pressure, the low-N macrostates have highest probability while at the highest pressure, there is a prominent high-N local maximum that is absent for the low-pressure state. For the middle pressures, there are many local maxima with nontrivial magnitude, which are a key point of discussion in the following sections. The y-axis has been truncated to highlight the important features; Figure SI.13 of the SI shows the full vertical range. In Figure 4, all ln Π contain oscillatory features for N ≤ 300 similar to the preliminary results in Figure 1. The extreme points are, however, not spaced uniformly, in contrast to the preliminary results for water adsorption in ZIF-8, and the spacing increases with pressure. As for the preliminary results, the local extrema are associated with the formation of clusters in the cages of CuBTC, as confirmed by configurational snapshots (see Section SI.4 of the SI). The sole exception is the first local maximum near N = 5, which configurational snapshots show to be macrostates with isolated water molecules dispersed throughout the MOF. The uneven intervals between clusters follow from the different cage types in CuBTC (see Sec 3.1) and, consequently, different cluster sizes. We examined ln Π(N; μ) reweighted to μ covering conditions below bulk saturation and found that the global maximum is either the first local maximum (i.e., not associated with a cluster) or the highest N local maximum. For some intermediate pressures, several local maxima have similar magnitude, such as at p/p₀ = 0.315 where the magnitude of the first three local maxima differ by less than 0.5 (dimensionless).

Figure 4:

Macrostate probability distributions for simulation of SPC/E water in CuBTC at 300 K for the noted p/p₀. The dashed lines connect selected local maxima in ln Π(N; μ).

For simulations of SPC/E water adsorption in ZIF-8, we used 80 overlapping windows with N_max = 630 and the same MC move assignment strategy described earlier. We ran six replicate simulations for this system since we found larger run-to-run variation than for the CuBTC simulations. Each window of the simulations was run for 28 days of wall time or was terminated at the completion of 500 sweeps. Figure 5 displays the MPD for SPC/E water in hydrophobic ZIF-8 at 300 K reweighted to the five noted f/f₀. (The purpose of stating the simulation conditions in terms of fugacity will become clear in Sec 4.4.) The original ln Π(N; μ) for the six replicate runs are shown in Figure SI.15 of the SI, along with the ln Π(N; μ) produced by recombination. The individual runs show noticeable run-to-run variation, especially at high N, but the recombinant ln Π(N; μ) show better agreement. Reproducibility of ln Π(N; μ) and uncertainty in the eventual isotherm are discussed in Section SI.6 of the SI.

Figure 5:

Macrostate probability distributions for simulation of SPC/E water in ZIF-8 at 300 K for the noted f/f₀. The dashed lines connect selected local maxima in ln Π(N; μ).

As anticipated in Figures 1 and 2, the ln Π(N; μ) in Figure 5 show oscillations corresponding to clusters; in fact, for the f/f₀ = 1.737 trace there are exactly 16 distinguishable local maxima prior to the final local maximum near N = 520, matching the 16 cages in the simulated ZIF-8 supercell. Additionally, those local maxima appear consistently at intervals of 29 to 35 molecules. The final local maximum, when present, is associated with a state where the MOF cages and connecting apertures are completely filled with liquid-like water. For the low fugacity ln Π(N; μ), the cluster oscillations appear exclusively as curvature changes whereas for the highest fugacity trace, there are both local maxima and curvature changes. There is a notable difference between the cluster-associated oscillations in Figures 5 versus those in Figure 4: The maximum Π(N; μ) is may be at N = 0, not at a finite number of water molecules (cf. the N ≈ 5 local maximum in Figure 4). Furthermore, the maximum Π(N; μ) is always at either N = 0 or the high N > 520 and, in fact, the first cluster has a higher probability than N = 0 only for very high μ, where the global maximum is already at high N. This implies that the stable state of the adsorbed fluid is either very low ⟨N⟩ (e.g., the MOF is nearly empty) or relatively high (e.g., the MOF is filled).

The appearance of oscillations in ln Π(N; μ) is anticipated by earlier work studying water adsorption in MOFs. For example, the ln Π(N; μ) plotted in Figure 5 of Ref 37 (TIP4P-Ew water adsorbed in MOF-806, simulated by a GCMC technique that estimates a MPD) has faint oscillations at low N and low pressure. The authors of that work did not explore those possible oscillations in ln Π(N; μ) or any connection to cluster formation in the MOF cages. In another work,³⁶ simulations of TIP4P in ZIF-8 indicated that water adsorption for this system occurs through a cage-filling mechanism, i.e., where water does not wet the pore surface but fills one cage at a time through rapid growth of a single cluster in each cage. Later experiments of water adsorption in the similarly shaped, but hydrophilic, ZIF-90 MOF⁶⁹ report a similar adsorption mechanism. An alternative description is that adsorption of water in these adsorbents takes on three possible states as it fills: 1) essentially zero adsorption (at low pressure), 2) as intact metastable clusters, or 3) in a liquid-like state that fills the accessible volume. In a statistical interpretation, the mechanism implies that incomplete clusters are low probability compared to the fully-grown clusters that fill a cage. In our work this is reflected in ln Π(N; μ), where a local maximum associated with a cluster is higher probability than adjacent macrostates that correspond to incomplete clusters. Essentially, it is intact clusters of water molecules that adsorb in the MOF cages. We will show in the following section how the recognition of clusters affects the identification of adsorbed phases during postprocessing of ln Π(N; μ) to yield an adsorption isotherm.

4.3. Analysis of the Macrostate Distribution

Adsorption isotherms are obtained by plotting the average number of molecules in the system ⟨N⟩ versus the pressure of the bulk fluid at the same chemical potential of the system. Both quantities are calculated from Π(N; μ) of the confined fluid and the bulk fluid. As implied in eqs SI.9 and SI.10 of SI, this calculation requires assigning portions of the macrostate range to a phase when Π(N; μ) is multimodal. For most situations exhibiting macroscopic phase coexistence between two phases, this is relatively straightforward because Π(N; μ) is bimodal at conditions of phase equilibrium. Thus, the local minimum between the peaks can be used to assign macrostates to each phase (cf. Refs 46,47,49,70 and many others). The situation is complicated slightly for systems exhibiting aggregation or cluster formation under dilute conditions (e.g., low overall number density) where dispersed finite-sized clusters are formed, while still exhibiting a macroscopic phase transition (e.g., a vapor-liquid or capillary phase transition). The complication lies in that the macrostate distribution possesses several intermediate local maxima. As confirmed by configuration snapshots, each of these local maxima corresponds to a system that contains an integer number of clusters. Because the clusters are finite in size, it would not be appropriate to consider each local maxima as a separate macroscopic “phase.” Those local maxima found at values of N corresponding to overall dilute number densities can be considered as belonging to a dilute fluid phase. Cluster phases can also be observed at higher overall number densities, and they should be characterized as belonging to a denser (e.g., liquid-like) fluid phase. Ultimately, because the local maxima encountered here are only metastable, macroscopic phase coexistence in these systems involves equilibrium between (1) a dilute fluid composed of a mixture of dispersed fluid particles and finite-size clusters and (2) a dense, liquid-like fluid, that can also contain clusters. By including these local maxima in one of the two fluid phases, we generate adsorption isotherms that are consistent with conventional simulation and adsorption measurements, which simply report the overall amount of fluid adsorbed, regardless of morphology or structural character of the confined fluid.

Specifically, we propose the following criteria for assigning macrostates to adsorbed phases. First, a local minimum in ln Π(N; μ) must exist, otherwise, there is a single adsorbed phase. Second, when ln Π(N; μ) contains multiple local minima at regular intervals corresponding to cluster formation, the boundary between the two phases is placed at the lowest local maximum in ln Π(N; μ) that corresponds to a cluster. The phase boundary is consequently associated with the lowest probability cluster state. This is based on a similar phase decomposition used by Hatch et al⁴⁴ for trimeric molecules that self-assemble into repeating aggregate structures. The end result is that Π(N; μ) is divided between a dilute phase of discrete clusters and a dense liquid-like phase. In the remainder of this work, we will use the terms “low-N” and “high-N” to identify the dilute and dense phases, respectively. We avoid labeling the low-N branch of the isotherm as “vapor” due to the centrality of cluster formation in the adsorption mechanism. The high-N phase might be termed “liquid” since the MOF cages and pore apertures are filled with dense adsorbate (like a liquid), but we avoid that usage to not imply a contrast with a “vapor” phase.

Figure 6 shows ln Π(N; μ) for SPC/E water adsorption in CuBTC, with annotations that identify features of the phase decomposition scheme. The figure marks the local maxima associated with clusters, connects those local maxima with a dashed line to create a pseudoprobability distribution of the clusters, and identifies the lowest cluster local maximum that divides the phases in our analysis. Macrostates to the left of the phase boundary are in the low-density cluster phase and those to the right are in the high-density phase. The data for p/p₀ = 0.272 show a high probability for the low-N macrostates, indicating that the cluster phase is the stable phase. The data for p/p₀ = 0.308 have a global maximum at a noncluster macrostate, where the MOF cages and apertures are filled with liquid-like water; the high probability of those macrostates indicates that the stable phase has filled cages and apertures. The ln Π(N; μ) data for p/p₀ = 0.296 correspond to the μ where the low- and high-density phases have equal probability, i.e., equal grand free energy and coexistence of the two adsorbed phases.

Figure 6:

Macrostate distributions for simulation of water in CuBTC at 300 K at three chemical potentials corresponding to the bulk-fluid pressures noted in the legend. The data for p/p₀ = 0.296 and 0.308 have been shifted vertically by 15 and 20 units, respectively. Solid traces indicate the ln Π(N; μ) reweighted to the noted pressure, for all macrostates in the simulation. The solid circles identify local maxima associated with cluster states and dashed lines connect the local maxima creating a conceptual probability distribution for the cluster macrostates. Vertical dash-dotted lines identify the phase boundary for each pressure. p/p₀ = 0.296 is the coexistence pressure for the adsorbed phases.

When decomposing Π(N; μ) as described above, the isotherms occasionally contained branches where ⟨N⟩ decreased with an increase in pressure (μ) and vice versa. For a rigid system, this represents a violation of thermodynamic stability criteria⁷¹ and, hence, we remove such points from the isotherm. We found that these unphysical points on the isotherm are associated with changes in ln Π(N; μ) that transfer a particular cluster from one phase to the other due to a shift in the phase boundary. For example, consider the two ln Π(N; μ) traces for SPC/E water in ZIF-8 in Figure 7. As shown, the lower-fugacity ln Π(N; μ) yields phases with ⟨N⟩ = 0.083 and 517 and the higher-fugacity (higher-pressure) ln Π(N; μ) yields phases with ⟨N⟩ = 0.087 and 506; the average loading of the high-N branch of the isotherm decreased with an increase in fugacity (or μ or pressure). Figure 7 also immediately shows why this occurred: the lowest local maximum moved from N = 486 to 413 with the increase in fugacity, which adds lower N macrostates at sufficiently high Π(N; μ) (relative to the local maximum at N = 520) to reduce the computed ⟨N⟩. This close examination also reveals that this additional processing step only affects the analysis of a ln Π(N; μ) that has a sequence of cluster local maxima with similar probability. Section SI.6 of the SI shows an example isotherm prior to the removal of unphysical points.

Figure 7:

Macrostate distributions for simulation of SPC/E water in ZIF-8 at 300 K at pressures near the limit of stability of the high-N (liquid-like) phase. Analysis of ln Π(N; μ) yielded high-N phases with ⟨N⟩ = 517 for f/f₀ = 1.729 and ⟨N⟩ = 506 for f/f₀ = 1.737. We note that the phase boundary shifted from N = 486 to N = 413 with the increase in fugacity.

Other macrostate division methods can be proposed, such as placing the phase boundary at the lowest local minimum or an inflection point.⁴⁷ We examined different options for the phase boundary and found them to mainly affect the boundaries of the hysteresis loop, that is, at the apparent limits of stability (LOS). (See Section SI.6.4 of the SI for a comparison of two options.) Given that there are system-size effects at spinodals and that the definition or identification of a LOS is dependent on subjective factors such as imposed barrier height, it is difficult to strictly identify a LOS.⁴⁷ Thus, our decision regarding the phase boundary mainly affects the isotherm at thermodynamic conditions where phase identification is already subjective.

4.4. Adsorption Isotherms

Figure 8 contains the 300 K isotherm of SPC/E water adsorbed in CuBTC as a function of the relative pressure, obtained by histogram reweighting of the ln Π(N; μ) shown in Figure 4. The isotherm shows low-N and high-N branches, as anticipated by the previous section, with a shape and narrow hysteresis loop reminiscent of the IUPAC Type V isotherm typically associated with weak adsorption. Phase coexistence between the two adsorbed phases is at approximately p/p₀ = 0.296.

Figure 8:

Isotherm of SPC/E water adsorption in CuBTC at 300 K plotted versus p/p₀, where p₀ is the bulk fluid saturation pressure. The red trace is the low-N (cluster) branch of the isotherm and the blue trace is the high-N (liquid-like) branch. Dotted lines are vertical connectors at the apparent LOS. The vertical dashed line identifies the condition of coexistence between the low-N and high-N phases (the equilibrium transition).

The low-N branch shows very low adsorption, with only ⟨N⟩ = 8.2 at the coexistence transition, followed by a rapid increase to ⟨N⟩ = 53.8 at the apparent LOS; at that point, the MOF unit cell contains on average more than one cluster (the first cluster is at roughly N = 45). Thus, the metastable points on the low-N branch of the isotherm are averages of states that contain one or two clusters. The high-N branch continuously increases from ⟨N⟩ ≈ 288 to ⟨N⟩ ≈ 394 (with ⟨N⟩ = 297 at the coexistence transition) as the phase densifies with increasing pressure. Figure 8 does yield an isotherm with hysteresis, which is not known experimentally for water adsorption in CuBTC. However, we do not argue that our simulation results contradict experimental results but instead state that, given the force field for CuBTC and water model used here, Π(N; μ) yields a first-order phase transition at 300 K and metastable states that form a narrow hysteresis loop. The contribution of metastable water clusters in CuBTC to the isotherm is a point we revisit later.

Figure 9 plots the 300 K isotherm of SPC/E water in ZIF-8, generated in the same manner as Figure 8. We note that the relative pressure is computed in the exact same manner as that for CuBTC in Figure 8, i.e., it is based on the pressure of the stable bulk SPC/E phase for the μ to which ln Π(N; μ) (of adsorbed SPC/E water) was reweighted. However, in distinction to Figure 8, the ZIF-8 isotherm includes points for p > p₀ because the analysis of Π(N; μ) according to Sec 4.3 yielded multiple adsorbed phases only for μ that correspond to p > p₀. The result is that the dilute phase appears at pressures above and below bulk saturation, while the dense phase appears exclusively above (and, in this case, far above) the bulk saturation pressure. An isotherm plotted with p > p₀ is unfamiliar, but in this case the higher pressures are essential for displaying the full isotherm. We revisit this behavior more fully in Sec 4.6, but for now we point out that other simulations have observed identical behavior for water adsorption in ZIF-8.⁷² The isotherm could be plotted differently (versus fugacity or pressure of the vapor SPC/E phase, see Section SI.6.5 of the SI for examples), but we plot it versus the stable-phase pressure of the model fluid to retain an equivalence with experimental measurements.^37,68 Fortunately, the remainder of our discussion is not dependent on the form of the pressure axis.

Figure 9:

As Figure 8, for adsorption of SPC/E water in ZIF-8 at 300 K. The isotherm is plotted versus the pressure of the stable bulk SPC/E phase in chemical equilibrium with the adsorbed phase relative to the bulk saturation pressure. We note that the bulk SPC/E vapor phase is stable for p/p₀ < 1 and the bulk liquid phase is stable for p/p₀ > 1. The Supporting Information contains isotherms of SPC/E water in ZIF-8 plotted versus fugacity and unscaled pressure.

The key feature of the 300 K isotherm of SPC/E water adsorbed in ZIF-8 is, as for that of CuBTC, a shape similar in form to the IUPAC Type V isotherm with low-N and high-N branches and a hysteresis loop. The low-N points are very low uptake: At the coexistence pressure, ⟨N⟩ = 0.67 and despite the sharp upturn thereafter, that branch only reaches ⟨N⟩ = 36.9 where it terminates. At that point, the system has, on average, less than two clusters. The high-N branch shows nontrivial compressibility, rising approximately 20 % from ⟨N⟩ = 506 to ⟨N⟩ = 604, with ⟨N⟩ = 530 at the coexistence transition. For reference, the pressure at which the adsorbed phases coexist is p/p₀ = 69 776, which corresponds to f/f₀ = 1.933, whose ln Π(N; μ) is shown in Figure 5. Low loading on the dilute branch of the isotherm is similar to experimental results⁷³ for water adsorption in ZIF-8 that report effectively no uptake for p/p₀ < 0.9. In that work, low adsorption is attributed to the hydrophobicity of ZIF-8.

4.5. Relationship between Metastability and Local Maxima in ln Π

The adsorption amounts shown in the isotherms prompt us to revisit a particular aspect Π(N; μ): the probability of local maxima relative to each other. As we pointed out earlier for both CuBTC and ZIF-8, the maximum probability macrostate was always at either a noncluster, low-N state or a high-N state where the MOF is filled; we found no μ for which a cluster local maximum is the global maximum. This general characteristic of the two water-MOF systems is reflected in the path of stable states in the isotherm: the dilute-phase branch shows low adsorption (less than one cluster) and the dense-phase branch is liquid-like (high ⟨N⟩), corresponding to the two dominant macrostates in Π(N; μ). The low- and high-pressure traces in Figures 4 and 6 show exemplar ln Π(N; μ) for those cases (the red and blue, respectively). In both situations, the average loading of the stable phases is affected by only a few clusters. For example, at low pressure (a single, dilute phase), one or two clusters have a sufficiently high probability of contributing to the average loading. At high pressure, the clusters are low probability and do not affect the average loading.

The ln Π(N; μ) for states in the hysteresis loops, however, illustrate cases where more clusters affect the ensemble average. To aid the following discussion, we point out that the three middle-pressure (fugacity) traces in Figures 4 and 5 correspond to states near the low-pressure LOS (black), the phase coexistence pressure (orange), and the high-pressure LOS (magenta). At the low-pressure LOS (where the dense phase is metastable), and for other metastable states in the isotherm on the dense-phase branch, the local maximum corresponding to the filled MOF configuration is comparable in value to the nearest cluster local maxima. (Note that those local maxima are for configurations where all but one or two cages are filled.) At the high-pressure LOS (where the dilute phase is metastable), at least three local maxima are of comparable probability. For these cases that bound the hysteresis loop, the metastable phases are characterized by a nontrivial probability of certain numbers of clusters appearing in the MOF cages. Stated otherwise, the system at those metastable or near-unstable conditions is composed of a mixture of different cluster phases or states that correspond to the intermediate local maxima (i.e., clusters occupying only a few cages or clusters occupying all but a few cages). The relative composition follows from the relative free energy of the relevant intermediate local maxima which, as we pointed out in Figures 4 and 5, are comparable in value. Thus, ln Π(N; μ) reveals that certain cluster configurations are closely tied to the metastability of the adsorbed phases.

This discussion also validates our phase decomposition scheme, that is, using the lowest local maximum in Π(N; μ) as the phase boundary. We point out that in previous work,^47,49 it is shown that a local maximum in ln Π(N; μ) becomes less prominent, eventually disappearing, near a LOS; this could be an inflection point (e.g., see two examples in Figure 2 of Ref 47). There is an analogous situation in the ln Π(N; μ) shown here, when the clusters are considered as the “relevant” configurations. For example, consider ln Π(N; μ) for the two LOS in Figure 5 (at f/f₀ = 1.737 and 2.176), where a pseudo-MPD (the dashed lines) has been created by connecting the local maxima. In those pseudo-MPDs, a “local maximum” is disappearing: At high pressure, the pseudo MPD is becoming flat, while at low pressure the filled MOF local maximum is merging into the adjacent cluster local maximum.

4.6. TIP4P Adsorption in ZIF-8

We now present results from simulations of TIP4P water⁶³ adsorption by ZIF-8 to facilitate comparison with previous work by Zhang and Snurr.³⁶ Our intent here is to provide a meaningful comparison of the results yielded by the methods described in this work to those that have been generated by other GCMC methods. The results in the work of Ref 36 serve as an important reference for this purpose. In brief, they simulated the adsorption of TIP4P water in ZIF-8 using GCMC combined with various strategies to accelerate sampling and closely examined the convergence of ⟨N⟩. Our simulations of TIP4P in ZIF-8 were run at 300 K using the same simulation strategy and the same ZIF-8 force field parameters described earlier, though the pair potentials in these simulations were cut (no tail correction or force shift) at 12.8 Å to replicate the setup of Zhang and Snurr as closely as possible. The only other notable change was to the CB moves which were necessarily modified to handle the TIP4P structure. Four replicate simulations were performed to check run-to-run reproducibility. As for SPC/E water, we also simulated bulk TIP4P water at 300 K using the same cutoff to provide Π(N; μ) that converts μ to pressure. We note that the saturation pressure of TIP4P with its pair potentials cut (no tail correction) at 12.8 Å was estimated to be 5743 Pa. (Similar to SPC/E water using the 10 Å linear-force-shifted cutoff, this is larger than the saturation pressure for TIP4P water with long-range corrections reported in Ref 66.) Additional TIP4P simulation results using the 10 Å force-shifted cutoff are in Section SI.7.1 of the SI.

Figure 10 shows ln Π(N; μ) from those simulations at five values of f/f₀, with all save one above the bulk saturation condition. The ln Π(N; μ) for TIP-P in ZIF-8 are qualitatively identical to those of SPC/E water in ZIF-8, with consistently-spaced oscillations (16 in total, at ΔN ≈ 30) and a final local maximum at high N. Configurational snapshots of the TIP4P simulations showed the same clustering behavior previously identified in Figure 2 and, hence, we similarly attribute the oscillations in ln Π(N; μ) to the formation of water clusters in the cages of ZIF-8 and the maximum at large N to a condition where the MOF cages and apertures are filled with liquid-like water. As for SPC/E water adsorption in ZIF-8, the phase transition involves a change from a dilute mixture of finite-size clusters and water molecules to a dense fluid of liquid-like water.

Figure 10:

Macrostate distributions for simulation of TIP4P water in ZIF-8 at 300 K for the noted f/f₀. f/f₀ = 1.134 is the coexistence fugacity for the adsorbed phases.

Figure 11 shows the isotherm generated from the macrostate distribution in Figure 10, plotted versus p/p₀. As for ln Π(N; μ), the shape of the isotherm is qualitatively identical to that of SPC/E water in ZIF-8 at 300 K. The hysteresis loop appears solely for p > p₀, again meaning that the adsorbed phase transition is at a pressure above bulk saturation. A semilog version of Figure 11 is in Section SI.7.1 of the SI, which more clearly shows the hysteresis loop above p/p₀ = 1. The low-N branch of the isotherm terminates with Type V upward-concavity at approximately ⟨N⟩ = 85; as for SPC/E water, this is less than three intact clusters in the MOF supercell. The entire isotherm for TIP4P in ZIF-8 is, however, at lower relative pressure compared to the isotherm for SPC/E water; the LOS are at p/p₀ ≈ 35 and p/p₀ ≈ 5600. This important difference is due primarily to the longer-range pair interaction between TIP4P molecules and the MOF and secondarily due to differences in phase behavior of the two water models (i.e., TIP4P water has a higher saturation pressure than SPC/E water⁶⁸). The longer-range potential (both LJ and electrostatic terms) strengthens the interaction with the MOF and consequently shifts the hysteresis loop to lower pressure. Ultimately, the most important conclusion to draw from Figures 10 and 11 is that adsorption of TIP4P in ZIF-8 is qualitatively identical to that of SPC/E water: cluster macrostates are clearly identifiable in ln Π(N; μ) and the phase transition to the dense phase occurs at a pressure above the bulk fluid saturation pressure.

Figure 11:

Adsorption isotherm of TIP4P water in ZIF-8 at 300 K and with the isotherm plotted versus relative pressure. Line colors and styles are identical to those in Figure 9.

The isotherm for TIP4P adsorption in ZIF-8 at 300 K is similar to the isotherm reported by Zhang and Snurr (see Figure 2 in Ref 36), in that both resemble the IUPAC Type V form, but there are nontrivial differences: First, the hysteresis region of their isotherm appears at significantly lower pressure than the isotherm we report. For example, in Figure 11, the high-pressure LOS is near p/p₀ = 5600 which corresponds to 32.5 MPa; in Ref 36 the hysteresis loop closes near 2800 Pa, many orders of magnitude lower than estimated by our simulations. Second, the low-N portion of the adsorption branch of their isotherm shows a quasi-linear region where the adsorbed amount increases from essentially zero to approximately 17 water molecules per unit cell (136 molecules per 8 unit cells, or slightly more than four clusters on average) before the MOF fills in a nearly vertical step. This is in contrast to our isotherm, whose low-N branch terminates with the rapid, concave-up rise to the LOS at which point the adsorption amounts to less than three intact clusters.

On the first point concerning the pressure ranges of hysteresis and, in particular, the range relative to the bulk fluid saturation conditions, we can point to other results for additional information. A work by Ortiz et al⁷² separately reported a GCMC isotherm for TIP4P adsorption in ZIF-8 at 300 K (though using a different force field for the MOF) which shows a hysteresis loop entirely above the bulk saturation pressure (see Figure 2 therein). Specifically, the hysteresis loop of their isotherm was between (approximately) 15 MPa and 140 MPa, whereas the bulk saturation pressure was reported as 3800 Pa. Our isotherm is more consistent with those results, though with higher adsorption and upward concavity in the metastable low-N region. We also note that some of the same authors elsewhere reported hysteresis above the bulk saturation pressure for a different hydrophobic MOF.³³ The salient point is that other simulations have reported water adsorption hysteresis above the bulk saturation pressure for the same TIP4P/ZIF-8 system. Ultimately, though, the source of the stark difference in the hysteresis pressure range between our results and those of Zhang and Snurr remains unclear and we suggest the need for follow-up work.

The second point, regarding the quasi-linear portion of the isotherm in Ref 36, is an isotherm feature that can be conceptually addressed given our approach and simulation results that ultimately derive from Π(N; μ). Given that water exists in clusters in ZIF-8, the nonzero points on the low-N branch of their isotherm imply that there are thermodynamic states (i.e., particular μ given fixed V and T) where the probability of observing several clusters is sufficiently high that the macrostates associated with those clusters meaningfully contribute to the ensemble average of N for that phase. This would be visible in a graphical depiction of ln Π(N; μ), which would contain a local maximum corresponding to multiple clusters that is noticeably higher than the other local maxima associated with clusters. We note that the relevant local maximum or maxima must also be at N smaller than that of the lowest local maximum so that those clusters are assigned to the low-N phase. A sequence of local maxima with similar magnitude, all higher than other local maxima for the low-N phase, could also generate this condition. In our examination of ln Π(N; μ) for adsorption of TIP4P in ZIF-8, we found no thermodynamic states that exhibit such local maximum (or maxima) and, hence, the low-N branches of our isotherms do not contain a region where adsorption increases continuously over a broad pressure range. This is another discrepancy that should be revisited in future work.

5. Summary and Conclusions

As described in the introduction and background, conventional methods to simulate water adsorption using GCMC suffer from inefficient sampling due to the intrinsic aggregation properties of model water molecules and, hence, simulation of water adsorption may be difficult, require long simulation times, or yield isotherms of uncertain quality. The purpose of the present work is to assemble several advanced techniques to facilitate MC simulations of water adsorption in MOFs that address sampling challenges encountered by conventional GCMC methods. Our approach includes two separate strategies: One is the use of biased MC moves that 1) specifically target the aggregation properties of water (i.e., AVB moves) and 2) generate higher quality moves (i.e., CB moves), where both advanced move types are accelerated using short-range reference potentials (dual-cut). The other is the use of FHMC (both TMMC and WL), which counteracts the tendency of water simulations to become stuck in particular N states and has the added benefit of yielding Π(N; μ). These two strategies could be implemented and evaluated independently, but our approach uses them simultaneously and additionally exploits the windowing option provided by FHMC to divide the macrostate range into separate simulations that utilize MC moves tailored to the sampling challenges of each window. Thus, there is a synergy between the advanced MC moves and FHMC brought about by their simultaneous implementation. The improvement in sampling efficiency is highlighted both in Figure 1, which shows the effect of biased MC moves, and through the reduced run time of our simulations compared to previous work.³⁶ Our move assignment scheme, based on general characteristics of water adsorption simulations and summarized in Figure 3, can be applied to simulations of other MOFs. Additional tuning of the MC move set is, of course, possible, but our approach can be utilized as a starting point for future simulations.

The key result of our simulations is the MPD, Π(N; μ), which in turn provides the adsorption isotherm. However, Π(N; μ) also provides a more detailed perspective of the behavior of the adsorbed fluid than is available from conventional MC simulations. For example, while the presence of water clusters in MOFs is known from prior work, the clusters are very clearly associated with local maxima in Π(N; μ), which also quantifies the relative probabilities of the clusters. Through reweighting Π(N; μ), we can observe the system evolving from a particular cluster state to another with changes in μ (pressure) through shifts in the relative probability of the cluster states. From Π(N; μ) we find a phase transition involving phase equilibrium between a dilute fluid composed of water molecules and clusters and a dense fluid where the pores and apertures are filled with liquid-like water.

The two MOF adsorbents examined here yielded isotherms with particular features related to the characteristics of each material. For CuBTC, the SPC/E water adsorption simulations at 300 K yielded a conventional Type V isotherm with a very narrow hysteresis loop, with all of the important isotherm features below p₀. (As we noted earlier, the hysteresis shown is a feature of the model system that should not be taken as contradictory to previous experiments that have not found hysteresis.) For ZIF-8, the simulations yielded isotherms for the SPC/E and TIP4P adsorbates with essentially the same Type V shape, but where the hysteresis loops appear entirely at p > p₀; the dense phase of adsorbed water is suppressed until pressures well above bulk saturation.

The probability of the cluster macrostates is very important in understanding the isotherm that is ultimately obtained from Π(N; μ). As we pointed out in ln Π(N; μ) itself, the stable phases are near zero loading and near maximum loading (filled MOF), with only a few clusters affecting the average loading. On the metastable branches of the isotherm, however, those clusters greatly impact the ensemble average loading. For the dilute phase, the probabilities of several cluster macrostates (usually the first two or three clusters) become sufficiently similar that all of those clusters contribute to ⟨N⟩. The result is that the metastable dilute phase is composed of a mixture of discrete cluster states, with overall low ⟨N⟩. The situation is reversed for the dense phase, where the probabilities of configurations where clusters occupy most cages (high total N) are commensurate with that of the filled-MOF state. Those statepoints along the metastable dense-phase branch of the isotherm are statistical mixtures of some configurations where the pore space is filled with liquid-like water and other configurations where most, but not all, cages contain clusters. This is a revealing perspective on the role of metastable clusters in the adsorption mechanism, where the presence of some number of clusters (in contrast to the stable phase) greatly affects the thermophysical properties of the adsorption system.

We also presented FHMC results for TIP4P water adsorption in ZIF-8, to make a more direct comparison to recent results by Zhang and Snurr.³⁶ Our results show that the adsorption of TIP4P water in ZIF-8 is qualitatively and mechanistically identical to that of SPC/E water, in that the ln Π(N; μ) and isotherms are effectively identical, though at different relative pressures. This difference is related to the strength of the LJ interaction between the water models and the MOF, rather than fundamental differences in the adsorption thermodynamics. However, our isotherm departs from the results of Zhang and Snurr in two important features: The first is that the hysteresis loop of our isotherm is entirely at p > p₀, versus the subsaturation hysteresis they showed; in that respect, our isotherm is consistent with other simulation results⁷² and experimental studies of water intrusion^74,75 that identify filling of the MOF pores at conditions far above the bulk saturation pressure. The second is that our isotherm shows no region where the loading increases linearly with pressure. Additionally, the Π(N; μ) we obtained by FHMC do not contain features necessary to yield such a linear region. These discrepancies suggest the need for follow-up work, which would hopefully reconcile the results from differing simulation approaches.

In closing, here we have described specialized techniques that facilitate the simulation of water in MOFs, through the combined use of FHMC and biased MC moves, and revealed interesting details regarding the adsorbed structures in two MOFs. While the specialized technique is useful on its own, as it makes possible the simulation of adsorption systems that have historically proven challenging, the key result of FHMC, Π(N; μ), provides a unique view of water adsorption in MOFs through its identification of clusters and quantification of the clusters’ probability. Moreover, examining the evolution of Π(N; μ) with increasing pressure reveals how these clusters affect the adsorbed phases’ properties, with particular effect in the metastable portions of the adsorption isotherm. Lastly, the recognition of metastable cluster states in our FHMC results prompts new questions about the (relative) stability of such states and how sequential cluster growth could be observed experimentally. We hope that future experimental work will revisit these questions and further investigate water clusters in MOFs or other similarly confined spaces.