Chapter 1: Introduction to Computational Modelling of Microporous Materials

Porous materials are materials that have void spaces within them that can be occupied by guests to produce a range of different properties and functionalities. Depending upon the size of their pores, porous materials are classified as microporous (pores smaller than 2 nm), mesoporous (pores between 2 and 50 nm) or macroporous (pores larger than 50 nm). Here, we will focus mostly upon microporous materials, as these pores can host small molecules and atoms and therefore provide functionality across a range of applications from encapsulation, separation, catalysis, sensing and more. As a note on terminology, the term sorption is generally used in the discussion of porous materials as these materials will both adsorb guests onto surfaces as well as absorb guests into the material's bulk.

The computational modelling of materials is increasingly widespread, and porous materials modelling is no exception to this. Computational modelling in porous materials has already had decades of application, especially in the field of zeolites, with significant impact in rationalising the structure and property of the materials, frequently providing additional atomic-level insight into the materials that could not be uncovered through experimental characterisation alone. This a posteriori rationalisation of experimental observations built a foundation upon which many predictive studies have been built, for example, screening materials in advance for useful properties prior to experimental realisation. A commonly used term is computational materials design, but we urge caution in not overusing the term ‘design’ when a more appropriate term is ‘screening’ or ‘discovery’. We should leave the term ‘design’ for those occasions when our insight, knowledge and software truly have enabled design. Computational screening, with consideration of both whether a hypothetical material can be experimentally realised and have target properties, is powerful in its own right.

Ever increasing computing power, increasing computational literacy and programming ability, combined with open-source data and algorithms hold great potential for the further impact of modelling in the field of porous materials. As in all computational fields, there is significant emphasis on the potential for advances through the use of artificial intelligence, notably machine learning. There is enormous value in all porous material scientists understanding, even if not directly applying, the potential of computational modelling for the advancement of their own research. Beyond technological advancements, we can also expect an increasing focus on understanding and predicting more complex behaviour, for example, consideration of multifunctionality in a material, such as the interplay of porosity with optoelectronic properties, and the consideration of the importance and utilisation of defects as an opportunity for property tuning.

In this book, the key foundations for porous materials modelling are laid out, along with the major areas where modelling is applied, successes so far, as well as the future prospects for the field. In this chapter (Chapter 1), the key classes of microporous materials are introduced, along with the applications of porous materials. This is followed by the introduction of the key types of modelling approaches that are applied in the area of porous materials. This chapter should therefore provide the foundation for accessing the following chapters that focus on specific application areas of porous materials modelling. Chapter 2 (Addicoat) outlines the approaches and successes in the structure prediction of porous materials, covering both bottom-up approaches from building blocks and top-down approaches from libraries of topological nets. Examples are discussed from zeolites, metal–organic frameworks, polymers and porous molecules. With a known structure of a material, either from a simulation or an experimental crystal structure, many further simulations analysing properties and function can be performed. Chapter 3 (Evans) discusses the analysis of the mechanical properties of porous materials. How to analyse the strength and flexibility of porous materials is covered, along with the potential to uncover unusual properties, such as negative thermal expansion. The modelling of both sorption and diffusion behaviour in porous materials is covered next (Glover and Besley, Chapter 4). This spans sorption simulations of guest uptake, pore analysis, and molecular dynamics and enhanced sampling techniques for guest diffusion. Modelling the spectroscopic and catalytic properties of porous materials is then covered (Morales-Vidal and Ortuño, Chapter 5), including a discussion of how to choose appropriate methods, followed by examples of mechanistic investigations, focused upon metal–organic frameworks and covalent organic frameworks. Finally, the future for the field, in particular through the application of artificial intelligence techniques, is covered (Jensen and Olivetti, Chapter 6).

There are a large range of microporous materials and these can have a variety of chemical compositions, whether inorganic or organic in nature, as well as hybrid materials containing both inorganic and organic components. Microporous materials also vary by their degree of crystallinity, with some having both short- and long-range order, and amorphous porous materials lacking any long-range order at all. In this section, the key classes of porous material that are discussed in later chapters will be introduced.

Zeolites are naturally occurring crystalline aluminosilicate minerals that are typically found in regions where there has been volcanic activity. The name zeolite originates from how the rock appeared to ‘boil’ upon heating, with the ancient Greek zein for ‘to boil’ and lithus for ‘rock’, so literally ‘zeolite’ means ‘boiling rock’. It was not until the 1940s that Barrer found a method to synthesise zeolites in a laboratory. Barrer was able to synthesise zeolites via mimicking the conditions where they form naturally, in particular the high temperature and pressure, along with organic template molecules that would direct the formation to a targeted zeolite structure. There are now more than two hundred zeolite structures that are either naturally occurring or have been synthesised in the laboratory.¹ 

Zeolites are composed of [SiO₄]^{4 −} and [AlO₄]^{5 −} tetrahedra, and these primary building units, known as T-sites, are linked by the bridging oxygen atoms to produce corner-sharing tetrahedra. Through the multitude of ways that these tetrahedra can be linked and arranged to form crystalline arrays, there are millions of hypothetical zeolite topologies. There are a series of what are known as ‘secondary building units’ or SBUs that can be used to characterise a zeolite topology. Examples of SBUs include n-membered rings (where n = 3, 4, 5, 6…), and then the equivalent doubled rings, double 4-ring (D4R), double 5-ring (D5R), double 6-ring (D6R) and so on that have n bridging oxygens linking the two rings, as well as a small number of other SBUs linking different sized rings. Each zeolite has a three letter framework type code assigned by the International Zeolite Association,¹  which will be unique to a zeolite topology. An example of the zeolite with the framework type code MFI is shown in Figure 1.1.

The microporosity of zeolites originates either from the channel formed in an SBU or from the channels formed between SBUs when they are connected into a zeolite topology. The pore dimensions of zeolites vary across the full range of the definition of microporous materials. The first SBU where a small guest can feasibly pass through is the 6MR, with a diameter of ∼2 Å. Not only will the pore dimensions of different zeolite structures vary, but obviously also the shape of the channels, such as straight or curved channels, and some with larger cavities at points in a channel or side pockets connected to a main channel. Zeolites vary further by their pore dimensionality. There are zeolites with only 1-dimensional channels, those with 2-dimensional channels and those with 3-dimensional interconnected channels.

In terms of chemical composition, while an all-siliceous zeolite would only contain [SiO₄]^{4 −} tetrahedra, typically zeolites are aluminosilicates, with some [SiO₄]^{4 −} substituted for [AlO₄]^{5 −} tetrahedra. While the chemical composition may vary, all zeolites with the same framework structure (topology) will still have the same framework type code. For example, the code MFI refers to the structure that is common to both the all-siliceous silicialite-1 material (see Figure 1.1) and for ZSM-5 that contains both silicon and aluminium. In every instance where an Al^{3 +} ion substitutes a Si^{4 +} ion, there needs to be a ‘charge-compensating’ cation to maintain the zeolite structure's overall charge neutrality. These cations are generally not part of the framework itself but are more loosely bound in the zeolite's pores. The cations can be the organic templates used in the synthesis, or metal cations such as sodium or calcium, with the latter ions generally being solvated by water unless the zeolite has been dehydrated. Alternatively, the compensating cation can be a proton, and this then imparts a high acidity to the zeolite structure, that can be used in catalytic applications. Overall, the majority of zeolite applications stem from the presence of compensating cations and thus as a result of the aluminium substitution in the framework. The Si/Al ratio for a zeolite can range from 1 to infinity, with the lower limit of 1 a result of Lowenstein's rule that states that [AlO₄]^{5 −} tetrahedra will not form direct Al–O–Al bridges.

Beyond zeolites, there are a range of related materials known as ‘zeotypes’. Zeotypes can adopt the same framework structures as zeolites, but have different chemical compositions. For example, there is the family of aluminophosphates (ALPOs) consisting of Al^{3 +} and P^{5 +} ions (without the need for any compensating cation), SAPOs consisting of silicon, aluminium and phosphorus, zinc phosphates, and germanium sulfides. Through altering the chemical composition and in particular through the possibility to incorporate transition metal ions, this brings access to a further range of properties, particularly in terms of catalysis.

The applications for zeolites stem in particular from their chemical and physical stability, the presence of charge compensating cations, and the fact that zeolite pore dimensions are commensurate with small guest molecules and zeolite structures are available with a wide range of pore dimensions, shapes and topologies, allowing one to ‘pick’ an appropriate zeolite for a specific target application. Naturally occurring zeolites are typically very cheap and available on a large scale, and some synthetic zeolites are also available at a reasonably low cost. Zeolites have three major applications, several of which involve multimillion tonne productions. These applications are in ion exchange, catalysis and as molecular sieves (see Figure 1.2).

In ion exchange, the charge compensating cations in the zeolite are able to be exchanged with other cations, as the cations remain loosely bound. The selectivity of a zeolite towards a cation is determined by the topology and chemical composition, as this will affect the ability of an ion to diffuse through the pores and the strength of the binding at the sorption site. Zeolite A has been used in ion exchange on a large scale in washing detergents, where it acts as a water softener by selectively removing Ca^{2 +} ions from hard water. Zeolites have also been used to remove radioactive ions from nuclear wastes or after nuclear disasters, or toxic ions from the waste water of heavy industry.

Zeolites are highly hydrophilic materials and will absorb significant quantities of water even at raised temperatures. This has led to their use as drying agents, for example, in hospitals for drying gases used for medical treatments or on a large scale in chemical industry, such as drying liquid propane. Zeolites can perform molecular separations, thus acting as ‘molecular sieves’, by several different mechanisms:

• (i) Size exclusion: excluding molecules with dimensions larger than the zeolite's pores, and thus separating the larger molecules from a mixture.

• (ii) Sorption differences: where two different components of a mixture have different enthalpies of sorption within a zeolite. For example, in pressure swing adsorption, a mixture would be forced into a zeolite under pressure, then the pressure will be lowered slightly and only the more weakly bound guest component will be released from the zeolite. A further pressure reduction will release the other component, allowing the separate components to be collected.

• (iii) Diffusion control: due to differences in the size and shape of the guest molecules, they diffuse through the zeolite's pores at different rates and can therefore be separated from a mixture.

Molecular separations performed via these routes are much more energy efficient than the majority of molecular separations that are performed through energy intensive methods, such as distillation. It is estimated that molecular separations account for 10–15% of the world's energy usage, and that alternative separations that do not use heat can make most of these separations ten times more efficient.³  Lively and Sholl suggest seven separations that can still particularly benefit from improved energy efficiency in their processes: hydrocarbons from crude oil, uranium from sea water, alkenes from alkanes, rare-earth metals from ores, benzene derivatives (such as xylenes) from each other, trace contaminants from water and greenhouse gases from dilute emissions.³ 

The final key application of zeolites, and in fact the most significant, is in catalysis, specifically acid catalysis being used for hydrocarbon cracking. When a proton is the charge compensating cation in a zeolite, then the resulting Brønsted acid sites are extremely acidic, multiple times the strength of concentrated sulfuric acid. Furthermore, the selectivity of the zeolites, as discussed above, can be used to have selective catalytic results. This selectivity can be based on reactant size, product size or the size of the transition state for the reaction. Particularly important zeolite catalysis includes methanol to light olefin conversion, benzene alkylation and xylene isomerisation.

The computational modelling of zeolites has a long history over many decades. Many of the approaches for modelling both zeolite structure and properties have inspired the approaches later used in other types of porous materials, such as metal–organic frameworks. The scope of zeolite modelling is broad, including modelling structures, synthesis, nucleation, growth and dissolution of surfaces, catalysis, sorption and diffusion.²  Much of the early work on modelling zeolites was facilitated by their comparably simple range of interactions that needed to be described in structures containing only two or three elements (silicon, oxygen and aluminium), for which there were molecular mechanics based forcefield descriptions that required only a few parameters to reliably reproduce their behaviour. Early simulations of zeolite structures often treated the zeolite structure itself as fixed and rigid, only considering guest binding and guest flexibility, a useful tool to reduce the computational cost of the screening. This ‘rigid body approximation’ approach is still often used for other porous material characterisations, although host flexibility and pore breathing motions have been shown to be critical for fully understanding or predicting many guest behaviours, including in zeolites. As computing power has increased, the calculation possibilities upon zeolite systems have expanded, to include large-scale electronic structure calculations as well as high-throughput simulations on hundreds of thousands of materials at the forcefield level.

Metal–organic frameworks (MOFs) are a ‘hybrid’ form of porous material, that consist of both an inorganic and organic component, linked by strong coordination bonds. The IUPAC definition of MOFs is that they are a ‘coordination network with organic ligands containing potential voids’, where a coordination network itself is a ‘coordination compound extending through repeating coordination entities in two or three dimensions’. MOFs were originally synthesised and reported as coordination compounds in the 1990s by Robson,⁴  with Yaghi later giving them the name of metal–organic frameworks. MOFs are a field of intense research interest, with tens of thousands of structures reported, massively outweighing the ∼200 zeolite structures that have been synthesised and covering a much broader range of the hypothetical topologies that could be adopted. Indeed, X-ray diffraction crystal structures now make up a sizeable proportion of all crystal structures deposited in the Cambridge Structure Database. The considerable attention on MOFs stems from the great chemical diversity of the structures that can be synthesised by combining different inorganic and organic components, resulting in a wide range of interesting, and potentially tunable, performance across sorption, catalysis, optical and electronic properties.

The inorganic component of MOFs is typically a metal ion or cluster as a node, connected by organic ligands to form open frameworks that are both porous and crystalline. An example MOF structure is shown in Figure 1.3 of MOF-5, which consists of a zinc-based metal cluster (Zn₄O) as the node and 1,4-benzodicarboxylic acid as the struts. This forms a cubic topology that is known by a three-letter code, pcu, and a resultant 3-dimensional pore network. MOF-5 has an alternative name of IRMOF-1, with the ‘IR’ standing for ‘isoreticular’. The term reticular chemistry for a series of MOFs is used where they have the same topology, but different pore sizes – for example, by changing the length of the organic ligand, but keeping the same metal cluster as the node.⁵  Another example of a reticular series of MOFs is for the MIL series from the Materiaux de l'Institut Lavoisier, which consists of terephthalate ligands, but with a variety of different metal centres (e.g. V^{3 +}, Cr^{3 +}, Al^{3 +}, Fe^{3 +} and many other transition metals).⁶  Isoreticular chemistry provides a route to tuning of properties with ligand choice, such as pore size with ligand size, or sorption properties with the functionalisation of ligands to add functional groups that strongly interact with particular guest molecules. Increasing ligand length to increase pore size does not allow one to easily reach very large pore sizes, as the frameworks will often interpenetrate, removing the potential void space, once the voids become sufficiently large to accommodate a second interpenetrated repeat of the framework.

Compared to zeolite structures, which typically have surface areas up to ∼1000 m² g⁻¹, MOFs can reach significantly larger porosity, with the current record for a Brunauer–Emmett–Teller (BET) surface area of over 7000 m² g⁻¹.⁷  Another distinction between MOFs and zeolites is that, broadly speaking, MOFs have a much greater degree of structural flexibility and the ability therefore to have a structural response to external stimuli such as temperature, pressure and guest sorption. For this reason, MOFs are frequently referred to as ‘soft’ porous crystals.⁸  Often the structural response is significant, as will be discussed further in Chapter 3, with expansion and contractions that double or halve the volume, or, as has been a recent focus, with the comparatively unusual feature of negative thermal expansion.⁹ 

While MOFs have already exhibited large numbers of interesting properties, one of the hurdles to commercial applications has been issues with both chemical and physical stability, with many MOFs being kinetically unstable in the presence of air or water. A notable exception to instability comes from zeolitic-imidazolate frameworks (ZIFs), which are topologically equivalent to zeolites, but with zinc as the node (in place of silicon in a zeolite) and an imidazole as the ligand (in place of oxygen), where the O–Si–O angle is closely reproduced by the Zn–(centre of imidazole)–Zn angle. ZIFs generally have excellent chemical stability, and can even be refluxed in many solvents, water and alkaline solutions.¹⁰ 

MOFs hold great potential for commercial application, but thus far there are limited examples of these. One exception is the use of a MOF for the storage and controlled release of 1-methylcyclopropene (1-MCP), which can slow the ripening of fruit by inhibiting the adsorption of ethylene on the fruit's surface. This MOF is therefore used in the packaging of fruit to prolong the fruits’ lifetime; for apples this can increase the lifetime by many months. A second commercial application of MOFs is in the storage of hazardous gases used in the electronics industry. Beyond these commercial applications, MOFs are being investigated for a wide range of applications, including applications in storage and separations, as discussed already for zeolites. For storage and separation, MOFs have potential advantages over zeolites in their diversity and hence tunability of pore sizes and shapes, but currently struggle to compete on factors such as cost, stability and ease of synthesis, though this can be expected to change over time. MOFs show significant promise for the storage of methane or hydrogen in cars powered by fuel cells and renewable technology, where the MOFs have potential advantages in their low weight compared to alternatives. MOFs could also become components of batteries and some MOFs can conduct protons and could therefore be proton conducting membranes within fuels cells. MOFs are also being investigated for use in drug delivery, due to the possibility of loading a large number of drug molecules within the MOFs’ pores and then tuning the material for a desired drug release profile.

MOFs can be used in catalysis, either being directly involved in the reaction through the use of transition metal ions in their frameworks, or by the MOF acting as a host to support a homogeneous catalyst. MOFs are being investigated for photocatalysis, for example visible-light driven hydrogen evolution, or the reduction of carbon dioxide to hydrocarbons. While currently these are preliminary studies that typically use sacrificial donors, there are potential advantages in MOF tunability through changing the framework components, such as the fact that MOF frameworks are open, allowing easy diffusion of reactants and products into and out of the material, and that it is relatively easy to chemically functionalise a MOF so as to incorporate other molecular components, for example, through post-functionalisation methods. The potential for strong stimuli-response behaviour of MOFs (compared to zeolites) also opens the potential for application as chemical sensors. As further MOFs are synthesised and characterised, there is also the potential for finding structures with particularly promising high-temperature superconductivity or electrical conductivity or as supercapacitors. Finally, there is considerable interest in the use of defects in MOFs, via ‘defect engineering’ as another route to control and optimise their performance.

Many of the approaches earlier applied to zeolites have now been applied to MOFs, often on a large-scale, such as high-throughput studies of sorption or guest selectivity, using the increased computing power available in the last 10 years. In some cases, large-scale screening of MOFs for function outside of sorption or guest uptake is still hindered by computing power, given the inclusion of metals in their structure and a vast chemical diversity, which means that reliable forcefields (FFs) are only available for a handful of MOFs where they have been specifically parameterised. This often results in the need for the use of electronic structure methods, particularly density functional theory (DFT) calculations, which obviously come with an associated significant increase in computational cost compared to molecular mechanics-based methods. However, for many MOF properties, such as electronic, optical and catalytic properties, the simulation of these properties these will require electronic structure calculations.

Covalent organic frameworks (COFs) are crystalline and porous frameworks that only consist of organic elements and are much rarer than MOFs. The lack of strongly directional coordination bonds compared to MOFs makes the design and realisation of such materials with only covalent bonds challenging, even when trying to apply the principles of reticular chemistry. A notable challenge is in obtaining a crystalline COF solid compared to the amorphous organic polymers discussed in the next section. Common chemistries involved in the synthesis of COFs are boron condensation, triazine trimerisation and imine condensation.

Many of the COFs reported to date are 2-dimensional structures, for example, containing hexagonal layers stacked upon each other with varying degrees of order in the stacking layers. An example of a 2-dimensional COF is shown in Figure 1.4, COF-1, which is composed of boronate units connected by benzene rings and is synthesised from 1,4-benzenediboronic acid.¹¹  There are also some examples of 3-dimensional COFs, such as COF-108,¹²  another COF material composed only of C–C, C–O, C–B and B–O bonds. At the time of synthesis, COF-108 had the lowest density of any reported crystalline material.¹²  The strong covalent bonding in COFs can produce very stable materials, even at high temperatures and, again, there is an opportunity for tuning of properties through the massive possible space of potential organic components that could be used and tailored as precursors in COFs.

COFs have similar potential applications to that of MOFs, across gas storage, such as methane and hydrogen, gas separation, sensing due to the potential optical properties such as fluorescence, or electrical conduction being responsive to guests. Additionally, there is considerable interest in the use of COFs in photocatalysis, such as for water splitting or carbon dioxide reduction. The potential advantages of COFs include the opportunity to tune performance through the vast array of options for chemical components, and the fact that they are inherently lighter compared to zeolites and MOFs as they only contain light atoms and no metals.

Porous polymer networks, as with COFs, are (micro)porous and contain only light atoms connected by covalent bonds, but unlike COFs, are not crystalline, but are instead amorphous, lacking long-range order. The porosity of porous polymer networks does not originate from the topology resulting in void spaces, but rather from the inefficient packing of the polymer components. The inefficient packing is often by design, for example, through the awkward geometric shape or rigidity of the monomers, as the majority of polymers, particularly those with flexible hydrocarbon chains, are non-porous due to the ability of the polymer chains to efficiently pack so as to remove all void space. Many of these polymers have a high thermal stability, and some have exceedingly high surface areas.

There is a wide range of chemical diversity in porous polymers reported to date, including hyper-crosslinked polymers (HCPs), porous aromatic frameworks (PAFs), conjugated microporous polymers (CMPs) and polymers of intrinsic microporosity (PIMs). PIMs are synthesised typically from linear monomers that are engineered to have ‘awkward’ 3-dimensional geometries, typically including a spiro-unit, that cannot pack efficiently.¹³  PIMs are typically soluble in organic solvents, opening up potential processing opportunities, for instance, the ability to form membranes that cannot be achieved for MOFs, COFs and zeolites, or most other polymer networks. HCPs generate porosity through the high degree of cross-linking between chains to create a rigid framework with open pores.¹⁴  CMPs are a type of HCP where a large number of aromatic rings and/or alkyne groups help infer both a relatively rigid open framework as well as a conjugated network that produces useful interactions with light.¹⁵  Finally, PAFs typically have a component that has aromatic groups and a tetrahedral carbon, for example PAF-1, a model of which is shown in Figure 1.5, and which has a BET surface area above 5000 m² g⁻¹.¹⁶ 

The potential applications of porous polymers include those already mentioned for other porous materials. This includes in guest storage, as the polymers are inherently light as they consist only of light elements and in molecular separations, especially when processable into membranes or due to their high chemical and thermal stability. Porous polymers can perform catalysis, for example, through the inclusion of an active site in their precursors or via incorporation into a pore. CMPs in particular have been heavily investigated for potential use in water splitting for renewable energy.

A significant advantage of modelling porous polymers comes from their amorphous nature, which means that there will be no single crystal X-ray diffraction crystal structure that can be used to rationalise or understand the properties of the material. Therefore, structure prediction can play a big role here in producing realistic structural models, as discussed in Chapter 2. Large models with at least hundreds of atoms are required to rationalise or predict sorption and separation behaviour in systems which lack long range order, necessitating the use of molecular mechanics simulations. However, for understanding or predicting optoelectronic behaviour in these polymers, electronic structure calculations are required, therefore necessitating the use of smaller molecular or cluster models.

Thus far, all the porous materials discussed are linked by a common feature, they are extended networks that have continuous chemical bonding in three dimensions. By contrast, porous molecular materials are built from discrete individual molecules that are not chemically bonded to each other but are held together by intermolecular forces that are much weaker than coordination or covalent bonding.¹⁷  Porous molecular materials are much rarer than porous network materials, as in general molecules will form more stable, low-energy arrangements that minimise the void space between molecules and are non-porous as a result. In the solid state, there are two key ways that a porous molecular solid can be formed, either through intrinsic porosity, or through extrinsic porosity. Intrinsic porosity arises when a molecule has some internal void space, for example in a ‘cage’ molecule (as shown in Figure 1.6) or due to the geometric shape of the molecule, for instance the bowl-shaped calixarene family or belt-like cucurbiturils. Extrinsic porosity arises where there is inefficient packing between the molecules in the solid state. An example from the 1960s is Dianin's molecule, also known as an ‘organic zeolite’, which packs so as to leave hourglass-shaped 1-dimensional channels running along one axis that can host small guests.¹⁸  Even intrinsically porous systems often make use of pockets of extrinsic porosity in the solid state to form interconnected networks of intrinsic pores.

In the following sections, we firstly discuss porous molecular materials that are made solely from organic components (Section 1.2.5.1) separately from hybrid porous molecular materials that contain metals as well as organic components (Section 1.2.5.2).

Porous organic cages are porous molecular materials that are polycyclic molecules with the shape of a cage that have 3-dimensional structures with three or more molecular windows.²¹  An example organic cage, CC3,¹⁹  is shown on the left-hand side of Figure 1.6. Early research in this area by Cram and Warmuth reported nanocontainer cages formed through the condensation reaction of cavitands.^22,23  This has been followed by others, including Skowronek and Gwaronski,²⁴  Mastalerz²¹  and Cooper,²⁵  with a particular focus on the use of imine chemistry in the synthesis of the molecular cages. While originally porous molecular systems reported very low porosity (with BET surface areas below ∼400 m² g⁻¹), certainly compared with MOFs, more recent research interest in the field has resulted in significant increases in surface area, for instance a BET surface area of >3700 m² g⁻¹ from a large porous molecular cage from Mastalerz and co-workers.²⁶ 

Porous organic cages are typically synthesised via dynamic covalent chemistry, and most often imine condensation reactions, although boronic acid condensation, sulfide exchange and alkyne metathesis reactions have also been used. Dynamic covalent chemistry has the advantage of reversibility, so that these highly symmetric structures can form rather than a statistical mixture of oligomers, or an amorphous polymer structure. Organic cages have been formed in a variety of different shapes and underlying topologies (e.g. related to cubes, tetrahedra and so on).²⁷  The discrete nature of cages means they can, in general, be more easily processed than network materials due to their solubility in common solvents. Membranes can be formed directly from cages, or cages can be included in a membrane, thin-film, or coating as an additive. Considering the imine chemistry typically involved in the synthesis of organic cages, several systems have been shown to be remarkably chemically and physically stable, for example, being resistant to many hours of boiling in water²⁸  or to exposure to highly alkaline or acidic environments through chemical modification of the systems.²⁹ 

Organic cages have interesting potential uses in the liquid state, such as sensing, for example, with the liquid fluorescing until a guest explosive molecule enters the cage cavity and quenches the cage's fluorescence, allowing detection of the explosive.³⁰  Cages are an example of a system that can be designed to be a ‘porous liquid’, where a porous liquid is a material that exhibits the properties of a microporous material while also having the transport properties of a fluid.³¹  Porous organic cages were adapted to form the first example of a type II porous liquid in 2015, where a type II porous liquid is a solution of molecular pores.³²  This was achieved via two different routes, firstly through functionalisation of a cage with crown ether groups that increased the solubility of the system, but were too large to penetrate the molecular pores, or, alternatively and more successfully, by scrambling a mixture of closely related, but differently functionalised molecules that could not effectively crystallise and thus were liquid at room temperatures. Such porous liquids have potential in the separation of gas mixtures, where their unique liquid state compared to other microporous solids has the potential to be exploited from an engineering perspective.

Both crystalline and amorphous solid-state cage materials have been researched for potential applications, particularly in separations. Porous organic cages have already exhibited potential for the separation of noble gases, xylenes, hydrocarbons such as alkane/alkene mixtures, air pollutants and hydrogen isotopes.^33–36  Cages are also being explored for other applications, such as for water desalination by reverse osmosis³⁷  and as proton conduction membranes.³⁸  To compete with other porous (network) materials for commercial applications, particularly on cost, porous cages will need to take advantage of their unique aspects, in particular their modular nature for processability and tunability.

The same modelling approaches that have been used for zeolites first, and more recently MOFs, have also been applied to porous organic cages. In particular, given the small size of the pores in cage systems, the need for consideration of the cages’ flexibility for accurate guest–host modelling is to be expected.³⁹  The limited number of reported organic cages, particularly in terms of those with X-ray diffraction crystal structures, means that large-scale screening for cages has not been carried out in the way that it has for MOFs. While for MOFs, hypothetical structures can be constructed (see Chapter 2), this is much less trivial for cage systems; a small change in the cage could completely change the way in which the system would pack in the solid state, so it is not as easy to construct reasonable solid-state structures for hypothetical cage molecules. Crystal structure prediction techniques have been used to predict the solid-state structure with great success,⁴⁰  but this is not currently a computationally cheap or quick technique. However, the modular nature can also be viewed more optimistically – it opens the door to screening the systems in a different way, by analysing the individual molecules in a system and finding descriptors for their solid-state properties through analysis of the single molecules alone, which is much computationally cheaper.^41,42  Finally, the lack of metals in organic cage systems means that forcefield-based approaches are typically used in modelling them in the solid-state for behaviour such as guest sorption and diffusion. For optoelectronic properties, electronic structure based methods obviously need to be applied, but again there is the modular advantage; the analysis can be carried out on a single molecule and thus has significantly lower computational cost than a solid-state structure. However, many cage molecules themselves contain a very large number of atoms, meaning that even molecular-level excited state calculations are computationally demanding.

Metal–organic polyhedra (MOPs) are self-assembled from metal ions and ligands to form discrete molecular architectures, in contrast to MOFs, which are built from the same components, but have extended networks.^43–46  MOPs have been reported in a wide range of polygon topologies, including tetrahedra, octahedra and cubes, as well as larger polyhedra. An example of a MOP combined from ditopic ligands and palladium is shown on the right-hand side of Figure 1.6. MOPs are also termed ‘molecular flasks’ and can be used to encapsulate guests and to influence the reactivity of guests; for example, stabilising species, changing reaction selectivity or accelerating reactions. Compared to porous organic cages, the metals within the polyhedra provide additional catalytic opportunities. As with porous organic cages, MOPs are also being investigated as porous liquids.⁴⁷ 

There has been very little modelling of MOPs in the sense of what would typically be thought of as ‘porous materials modelling’, rather, most effort has focused on electronic structure calculations of the systems, in particular to rationalise the self-assembly outcomes. Computational investigation of MOPs based on geometric analysis to predict assembly,⁴⁸  guest recognition⁴⁹  and catalytic reactivity⁵⁰  has been conducted, but these studies are relatively few in number.

In the following chapters, the theory and approaches for modelling specific aspects of porous materials are discussed in detail. In this chapter, we provide a light touch overview of the main techniques used in porous materials modelling, which aims at being little more than an introduction of key methods. The key to all computational chemistry techniques is a mathematical description of the system that allows the calculation of the system's energy and how that depends on the position of the atoms in the system. From such a description, we can derive insight into a system's properties. For porous materials, this involves the calculation of the energetics driving the assembly of the material, host/guest interactions, optoelectronic, mechanical, and catalytic properties and so on. In terms of the level of theory, there are statistical mechanics simulations that use simple models to reproduce the macroscopic properties of a porous system, for instance, the flow of guest media through a material's pores. If modelling at a classical level of theory, forcefield potentials are used, and for a quantum mechanical description, there are then multiple options of electronic structure calculation.

Structural characterisation of the pore system of a porous material is key to understanding the material's properties, and there are many characterisation tools available, and an increasing number of these are now open-source. This topic will be covered in more detail in Chapter 4. Structural characterisation includes determining the dimension and shapes of the pores, and their interconnectivity. The interconnectivity of the channels must always be considered in the context of the size and shape of the guest whose ability to sorb and diffuse through a pore is being considered. Structural descriptors can include the largest cavity diameter in the pore system, as well as the pore limiting diameter, the diameter of the largest sphere that can diffuse across the model. While structural characterisation is generally performed on a rigid model taken from either an X-ray diffraction crystal structure or a simulation, it is also relevant to consider whether thermal motions should at least be kept in mind.

In computational studies of porous materials, it has been common to apply the ‘rigid host approximation’, particularly for simulations of guest sorption (Section 1.3.7). This approximation means that the porous host itself is kept completely rigid, with fixed atomic positions and no consideration of the flexibility. This approximation can be convenient for many reasons: (i) it substantially reduces the computational cost as only guest/host and guest/guest interactions need to be considered; (ii) it simplifies simulations considerably, making them possible on a reasonable timescale, for example, making it much easier to reach convergence than if thousands or more host configurations needed consideration; and (iii) an accurate mathematical description of the host structure's flexibility may not be available (for example, for a MOF structure). However, it has long been recognised, even in early zeolite studies – and arguably zeolites are the least ‘flexible’ of the porous materials discussed above – that consideration of host flexibility is often key to an accurate description of the system. Consideration of host flexibility is particularly important when the guest size is commensurate with that of the host's pores.

The term pore limiting envelope (PLE) has been introduced as a useful descriptor of a pore's diameter beyond the static pore diameter that would be considered from inspection of a crystal structure of a porous material alone, particularly considering an X-ray diffraction structure would be measured at a low temperature. The PLE is a histogram of pore diameters typically measured by sampling a dynamical simulation where the material's pores have been able to ‘breathe’. This consideration can be key to understanding experimental observations of guest diffusion that would appear to be implausible from inspection of the pores in the static crystal structure alone.³⁹ 

Molecular mechanics is an overarching term for the use of classical simulations that treat atoms and bonds as balls and springs, respectively, and do not explicitly consider the motion of electrons. The potentials that are used to describe the interaction of the atoms are known by several terms, including FF potentials, classical potentials or interatomic potentials (IP). The classical approximation allows for much larger systems to be considered than would otherwise be the case with the same computational resource if electronic motion was being considered.

Molecular mechanics works by dividing the potential energy of the system into a series of different contributions including self-energy, two-body terms, three-body terms, five-body terms and so on (although the series is normally truncated at the four-body terms). The potentials are divided into intermolecular and intramolecular contributions and the exact form of the potentials chosen is dependent on the system and properties being modelled. The typical form of a FF potential is:

where the first three terms are intramolecular interactions, and the remaining terms are intermolecular. The intramolecular terms consist of the two-body bond stretching term, normally related to Hooke's law, and where the equilibrium bond length and the ‘spring’ strength of the bond are the input parameters in the FF for a given bond pairing. An alternative bond stretching term that is frequently used is the Morse potential. The three-body term for bond bending generally has a similar form, and the final intramolecular term, the four-body term to describe torsional arrangements can take many forms to describe the potential energy surface related to the rotation about the dihedral angle of a central bond. The non-bonding intermolecular terms consist of the Coulombic electrostatic interactions and van der Waal forces, the latter including both long-range dispersion and short-range repulsion. For van der Waals forces there are a range of simple mathematical descriptions that can be used, included the commonly applied 12-6 Lennard-Jones potential or Buckingham potential. The parameters used to describe the potential must then be provided for each atom pair in the system to be modelled.

There are an enormous number of available FFs, both open-source and commercial, that have been applied to the modelling of porous materials. The exact choice of FF will depend on the porous material system, for example, for zeolites there are FFs that can describe the Si–O–Si angles and the polarisability of the oxygen atoms well (the latter through additional terms to those in eqn (1.1) above). Furthermore, the FF choice will depend on which property or properties one is trying to consider; for example, if calculating sorption properties in a material, and one is assuming the host porous material is rigid, then only host/guest and guest/guest potentials are required. Often there are multiple literature potentials for a specific guest, for example, even a diatomic gas guest can be described as a single sphere or with additional parameters to describe, for example, the molecule's quadrupole. For some systems, in particular MOFs, the development of FFs for these systems is an ongoing area of research, given the difficulty of accurately describing the metal coordination environments in particular, as well as the incredible breadth of coordination chemistry involved in the thousands of MOFs reported. As is often the case with FF selection, a trade-off must be made between accuracy for a single system and transferability across a broader range of systems. Again, the exact choice will depend upon the specific research aim. Finally, we note that while FFs by definition do not implicitly include the consideration of electronic motion, and thus cannot describe bond breaking and bond formation, there are a subset of FFs known as ‘reactive forcefields’ where the bond breaking and bond formation have been parameterised with additional terms, opening the possibility for specific studies of reactions with a FF-based approach.

Electronic structure calculations apply a quantum mechanical description of a chemical system, rather than the classical description in FF calculations. Electronic structure simulations are used particularly for the analysis of optoelectronic and catalytic properties of porous materials (as discussed in Chapter 5), but also for more reliable structural and energetic features across porous materials modelling, where computational resource allows. The most commonly used electronic structure method on porous materials is DFT. We can expect to see the continuing trend of an increasing number of porous materials studies using DFT calculations in the future with continuing access to greater computational power. It is worth noting that while molecular mechanics calculations themselves are computationally cheaper by orders of magnitude, it is not trivial to parameterise a new system if an accurate FF is not already available. Furthermore, in large-scale screening studies of thousands or hundreds of thousands of systems, the chemical diversity is likely to challenge the transferability of a FF. Therefore, the potential ability to ‘brute force’ DFT simulations does have its advantages.

DFT is a way of solving the Schrödinger equation using the electron density instead of the wavefunction. Within DFT, the energy and by extension other properties can be expressed as a functional, a function of a function, of the electron density. DFT is in principle exact; however, the exact form of the functional, generally known as the density or exchange–correlation functional, is unknown and in practice we have to choose one of a number of possible approximations for the density functional. The simplest approximation is the local density approximation (LDA), where the functional only depends on the value of the electron density for all points in space. The generalised gradient approximation, or GGA, goes beyond LDA by not only including the density at all points in space but also the gradient of the electron density. GGA is probably the most commonly used density functional when describing porous solids. Beyond GGA, there are among others, hybrid functionals, which include a contribution of exact exchange from wavefunction theory, and LDA + U/GGA + U, where LDA or GGA is combined with a Hubbard-like term acting on the d- or f-electrons. Use of hybrid functionals or LDA + U/GGA + U especially improves the description of the electronic, optical and magnetic properties of materials. Choosing the right functional is one of the biggest challenges when using DFT and might involve comparing the predictions of DFT to experimental data, where available.

In many porous materials, dispersion is a significant contribution to the materials’ cohesive energy and hence one would ideally describe the effect of dispersion accurately. However, many of the standard classes of density functionals – LDA, GGA, hybrid functionals and LDA + U/GGA + U – do not correctly describe the attractive long-range dispersion. The simplest and most commonly used approach to correct this issue is the DFT + D method by Grimme and co-workers, which adds pair-wise C₆R⁻⁶ terms to provide a description of dispersion. There are different parametrisations of this method, the latest of which is DFT + D4⁵¹  and the most commonly used one currently is DFT + D3,⁵²  which have been parameterised for a range of density functionals, e.g. PBE + D3. Beyond DFT + D, there are methods like Tkatchenko and Scheffler's TS-vdW-DFT.⁵³ 

The time-dependent extension of DFT (TD-DFT) allows for the prediction of optical absorption and, after relaxation of the excited state, luminescence spectra of porous materials. Within the linear response version of TD-DFT, which is the most routinely used and available in many DFT codes, the equivalent of the density functional in ground-state DFT is the exchange–correlation kernel, which in principle depends not only on the density but also time/frequency. However, in practice this exchange–correlation kernel is generally approximated by a ground-state density functional, ignoring the time/frequency dependence, in the so-called adiabatic approximation. Hybrid density functionals or range-separated hybrid density functionals, where the amount of exact exchange varies with the interelectron separation, are usually preferred for TD-DFT calculations as they minimise problems in the description of charge-transfer excited states, such as metal–ligand excitations.

The cost of DFT calculations can be reduced by switching to density functional tight-binding (DFTB)⁵⁴  or extended tight-binding (xTB)⁵⁵  methods in which the (exchange–correlation) energy is expanded in a Taylor series. Different DFTB and xTB methods exist, differing in the order at which the Taylor series is terminated, the basis-set used and how different terms are approximated, amongst other ways. Extensions of DFTB methods to excited states exist.

The density in DFT calculations, as well as the Kohn–Sham orbitals used to solve the density, are expanded in terms of a basis-set. The most commonly used basis-sets for DFT calculations are atom-centred Gaussian or Slater basis-sets and delocalised plane-wave basis-sets. The former traditionally find application in molecular calculations and the latter in periodic calculations on crystalline materials. The difference between Gaussian and Slater basis-sets is the exact form of the basis-functions used. The Slater basis-functions are essentially the atomic orbitals of a free atom while the Gaussian basis-functions are approximation to those atomic orbitals in terms of Gaussian functions. Even though Gaussian basis-sets require more basis-functions than Slater basis-sets for a similar description of the density and Kohn–Sham orbitals, and struggle to reproduce the cusp of the electron density at the nuclear positions, most molecular codes use Gaussian basis-sets because multi-centre integrals of basis-functions, which pop up frequently in DFT, are much easier to calculate for Gaussian basis-functions. Independent of the class of basis-set used, increasing the quality of the basis-set, i.e. using more basis-functions, gives more accurate results but increases the computational cost of the calculation. As such there is typically a trade-off between the quality of the results and the cost of the calculations, which specifically becomes an issue for calculations on large molecules (or large cluster models of a solid) and solids with large unit cells.

Molecular dynamics (MD) is a deterministic method that explores the phase space of a system over time through Newton's equations of motion. The output of the simulation is a trajectory of the system's coordinates (and forces) over time. This allows us to understand in porous materials how either the host, the guest, or the host and guest in concert evolve in a system over time. The equations of motion are not directly integrated, but rather a finite differences method is used to determine the particles’ trajectory. In porous materials, these simulations will normally be carried out using molecular mechanics approaches. However, ab initio molecular dynamics (AIMD) simulations of porous systems with modest numbers of atoms have been carried out, in particular where bond breaking and formation were of interest.

Depending upon which features in a system are of interest, MD simulations can be carried out in a variety of ensembles. An NVT ensemble, with a constant number of particles, volume of the simulation cell and temperature could be used to equilibrate a model at a particular temperature. The microcanonical or NVE ensemble conserves the energy of the system, with the volume and the number of particles also kept constant but allowing the temperature and pressure to fluctuate. An MD simulation using an NVE ensemble could be used to study the diffusion of a guest through a material's pores. A variety of thermostats can be used to keep the temperature of the system constant, and equivalently there are barostats for pressure allowing for simulations using the NVT and NPT ensembles, and so on.

While molecular mechanics based MD simulations can observe events routinely happening on a picosecond to nanosecond timescale in porous materials, this may not be sufficient to observe ‘rare events’ that may occur on timescales longer than tens of nanoseconds. As an example, in a system where the pores of the material are commensurate with a guest's dimensions, a 10 ns MD simulation may not once observe the guest diffusing through the pore neck of the material. However, the guest would diffuse on a longer timescale, and would therefore be observed to diffuse through the material on an experimental timescale – thus this could be an event that we would like to be able to rationalise, determine the energetic barriers towards it occurring, and be able to predict in advance of experimental observation. This is an example of a situation in which an enhanced sampling technique could be applied to address the timescale limitation of a regular MD simulation. Enhanced sampling methods to explore the response of porous materials to external pressure and temperature are discussed in Chapter 3. Examples of enhanced sampling techniques include metadynamics simulations, umbrella sampling and variational enhanced sampling. Broadly speaking, these approaches look to accelerate the dynamics of simulations, for instance, by applying an external bias, so that the configurations of the system related to rare events are observed more frequently and the free energy differences between the configurations can be calculated.

Grand Canonical Monte Carlo (GCMC) simulations are used in porous materials to simulate the sorption of guests within a host. GCMC simulations can be used to produce representative guest-loaded configurations and levels, for example, to predict guest uptake or selectivity, and can also reproduce adsorption isotherms. Monte Carlo (MC) simulations in general are not deterministic simulations (unlike MD), but rather make use of random numbers to perform random moves and orientation changes to generate successive configurations of a system. The algorithms involved in MC make use of importance sampling, normally via a Boltzmann factor, to ensure the majority of the simulation is spent sampling low-energy configurations. The grand canonical ensemble keeps the chemical potential, system volume and temperature fixed, with the MC moves being guest particle addition, particle deletion and particle displacement (e.g. translation or rotation). If guest mixtures are being simulated, then an additional move can be an identity swap of two particles. For larger guests, for example long chain hydrocarbons, additional move types may need to be considered to increase the likelihood of a successful guest insertion into the host. GCMC simulations almost always use the rigid host approximation due to computational cost considerations when trying to equilibrate a guest/host mixture or multiple mixtures at different loading levels, although there are hybrid-GCMC simulation approaches that allow the system to dynamically evolve during MD simulations upon guest loading. These simulations will be discussed in detail in Chapter 4.

Artificial intelligence techniques, in particular machine learning (ML), will be discussed in their application to porous materials in Chapter 6. ML is highly topical at the moment and is an approach that seeks to create data-driven predictive models. Supervised ML is the most commonly applied of ML techniques to porous materials so far and is in essence about fitting a function to data through pattern recognition. Examples of supervised ML algorithms include decision tree algorithms and random forest models. For supervised ML, one requires a sufficient quantity of high-quality data, based on either prior experimental or simulated results. An example for porous materials would be a dataset of sorption properties across a material class. Given input of a new material in this class, you would want to have trained the model to accurately predict that material's sorption behaviour, without resorting to either experiments or computational calculations of the properties. A key part of the process of training a supervised machine learning model is to find adequate structural descriptors for the material. These could, for example, be the chemical composition of the material, a 2D representation of the connectivity of the atoms in the porous material host, or a 3D representation of the structure of the material, or at least of a key environment within the material. During the training of the ML model, a function is fitted to give the best performance for that data based on an appropriate metric with which to judge the performance. Generally, to assess the performance of the model, the data will have been divided into a training set, upon which the model is trained, and a test set which can be used for validation of the model on unknown data.

Another area of ML relevant to porous materials is the unsupervised ML used in natural language processing (NLP). The use of NLP specifically applied to the extraction of useful data from the chemical literature through text mining has the potential to unlock further data-driven predictions. The chemical literature is a vast array of information that it is no longer possible to manually extract on a large scale. The data is also reported in a variety of formats – text, tables, figures in chemical journals as well as within patents etc. NLP is an approach to train algorithms to extract targeted features of the chemical literature. One example in porous materials of the application of NLP is to extract information on porous material synthesis procedures and then to use that data to predict laboratory synthesis procedure outcomes. In Chapter 6, the future potential and directions of artificial intelligence techniques in porous material modelling are discussed.

This chapter has given a general overview to the field of computational porous materials modelling. The variety of different microporous material types has been discussed and contrasted, and the key areas where porous materials modelling is of value have been highlighted. The introduction to key terms and computational methods provides a foundation for reading the remaining chapters that focus on specific computational modelling areas within the field.