# A priori predictions of type I and type V isotherms by the rigid adsorbent lattice fluid

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## Abstract

Adsorbents exhibiting non type I adsorption behaviour are becoming increasingly more important in industrial applications, such as drying and gas separation. The ability to model these processes is essential in process optimisation and intensification, but requires an accurate description of the adsorption isotherms under a range of conditions. Here we describe how the Rigid Adsorbent Lattice Fluid is capable of a priori predictions both type I and type V adsorption behaviour in silicalite-1. The predictions are consistent with experimental observations for aliphatic (type I) and polar (type V) molecules in this hydrophobic material. Type V behaviour is related to molecular clustering and the paper discusses the model parameters governing the presence/absence of this behaviour in the predicted isotherms. It is found that both the solid porosity and the adsorbate interaction energy/energy density are deciding factors for the isotherm shape. Importantly, the model, whilst thermodynamically consistent, is macroscopic and thus computationally light and requires only a small number of physically meaningful parameters.

## Keywords

Stepped isotherms Lattice fluid model Adsorption thermodynamics Type V isotherms## 1 Introduction

Sigmoidal isotherms, or type V by the IUPAC classification system, are widespread in adsorption systems involving polar molecules, such as water on activated carbon. The behaviour is typically attributed to molecular clustering of the adsorbate, due to weak adsorbate—adsorbent versus strong intermolecular interactions and hence unfavourable adsorption, with subsequent pore filling. This leads to an initially convex isotherm, with an inflection to become concave and showing saturation at high pressures. Indeed, this type of isotherm is mostly associated with adsorption in mesoporous adsorbents, where condensation of the adsorbate, that is, formation of bulk liquid, in the porous structure is possible. However, type V isotherms have also been observed for a number of microporous materials, such as (silicon) aluminium phospates (ALPO and SAPO) (Henninger et al. 2010), zeolitic imidazolates, (e.g. ZIF-8) (Cousin Saint Remi et al. 2011) and metal organic frameworks (MOFs) (Küsgens et al. 2009), for which this mechanism seems somewhat unsatisfactory as by their definition the size of the micropores should preclude the existence of bulk liquid. Molecular simulations have shown that molecular clustering in micropores is possible and particularly in interconnected pore structures, such as pore channels in zeolites, this clustering can become of a scale large enough to yield typical type V behaviour (Puibasset and Pellenq 2008; Trzpit et al. 2007). From a thermodynamic viewpoint, the simplest relationship which can yield this type of isotherm for a homogeneous surface is based on the Langmuir expression, which includes molecule—molecule interactions. This can be derived through statistical thermodynamics as done by both Frumkin and Guggenheim and Fowler (Frumkin 1925; Fowler and Guggenheim 1939; Ruthven 1984):

Non-localised effects can be introduced, as done in the Hill—de Boer equation (Hill 1946). More recently Shigetomi et al. considered both adsorbate—fluid and fluid—fluid interactions through a Van der Waals potential in a statistical thermodynamic study of water and methanol on type A zeolite (Shigetomi et al. 1982). The resulting isotherms contained additional inflections due to these molecular interactions. Using a similar approach, it was also shown how surface heterogeneity can cause steps in isotherms; this is particularly the case where large differences exist in heats of adsorption for different sites or when dealing with bulky or branched molecules (Nitta et al. 1984a, b). In addition to models rooted in statistical thermodynamics, many other relationships exist which can yield type IV and V isotherms, but many assume multilayer adsorption and not all of them are thermodynamically consistent. The Sips isotherm for instance has theoretical merit and has been used to fit sigmoidal isotherms, but has zero slope at low pressures and thus does not converge to Henry’s Law under these conditions. Similar problems arise for the Dubinin equations, which are also routinely used to fit type IV and V isotherms. For a good overview of available models the readers is referred to Buttersack (2019). For truly microscopic insights in the origins of type IV and V behaviour, researchers have resorted to computational methods, such as molecular simulation, with great success. For typical process design and simulation however, the computational effort would be prohibitive and a macroscopic model be more appropriate.

Macroscopic models describing equilibrium adsorption behaviour are a an essential tool in designing separation processes and to this end empirical equations are often used in separations involving sigmoidal isotherms, as they are simple analytical expressions and hence have small computational demand (Van Assche et al. 2016; Cousin-Saint-Remi and Denayer 2017; Hefti et al. 2016). Their obvious drawbacks are lack of physical meaning and predictive behaviour. Discretisation with linear interpolation between isotherm points, obtained during separation experiments has been proposed as an alternative approach to modelling processes, but in the presence of steps and inflections, many experiments at different concentrations would be required (Haghpanah et al. 2012). The ability to predict this type of behaviour a priori with small computational effort using a thermodynamically consistent framework, would evidently be a great asset.

_{2}isotherms for the so-called ‘large pore’ structure of MIL-53 (Al) exhibited type V behaviour. Under the same conditions, the CO

_{2}isotherms for the ‘narrow pore’ structure remained type I, as shown in Fig. 1. We hypothesised that this behaviour is a result from strong molecule—molecule interaction combined with a large pore volume. Under such conditions, after initial adsorption, additional molecules prefer forming clusters with previously adsorbed molecules, rather than interacting with the solid. A small number of studies using LF based approaches have shown similar predictions, but so far all studies involve adsorbents which expand upon molecule insertion (De Angelis and Sarti 2011; Galizia et al. 2012). Although LF models can clearly predict both type I and V isotherms from first principles, here we show that they can do so for a non-flexible adsorbent as well.

In this contribution we will explore which RALF model parameters determine the shape of the predicted isotherms using the parameters for silicalite-1 as our modelling adsorbent. The aim is to gain an understanding of the importance of porosity and the ratio of energy parameters in the RALF model that lead to the prediction of the two types of isotherm shape.

## 2 Theory

The Rigid Adsorbent Lattice Fluid and its equations have been described in great detail in Brandani (2019). In essence, the RALF model represents an equation of state based on a lattice of occupied and vacant sites, which only considers the interaction energy between occupied sites. Despite being based on a statistical thermodynamic approach, a simplification of the partition function leads to a macroscopic model with only a small number of modelling parameters. Here we will suffice by listing only the key equations of the model.

It can be seen that \(P_{j}^{*}\) is in effect an energy density, and \(T_{j}^{*}\) a measure for the interaction energy. The parameter \(r_{j}^{0}\) is related to the close-packed density and volume through \(M_{j}\), the molecular mass, and describes the number of lattice sites occupied per molecule of species \(j\). It should be self-evident, that the pure component characteristic parameters have a physical meaning, which can be determined from existing thermodynamic data.

For any mixture, e.g. a solid—adsorbate system, the corresponding characteristic parameters follow from the pure component parameters through a set of mixing rules. The RALF model utilises the same mixing rules as used by Sanchez and Lacombe (Sanchez and Lacombe 1976; Lacombe and Sanchez 1976), which conserve the molecular volume and number of pair interactions in the closed packed state (Brandani 2019). A final rule describes the mixture interaction density, \(P^{*}\):

*j*in the lattice. The binary interaction parameter, \(\kappa_{jk}\), signifies an enhanced or reduced energy density resulting from attractive or repulsive interaction, respectively between molecules

*j*and

*k*, with \(\kappa_{jj} = 0\). In a purely predictive mode, \(\kappa_{jk}\) is set to zero. As described in the introduction, a final adaptation is made in RALF as compared with earlier versions of the lattice fluid model, which allows for an increase in the close-packed volume for the adsorbed phase due to confinement constraints, through \(\xi_{jA}\). This correction concomitantly leads to a reduction in both the energy density and close-packed density, whilst leaving the characteristic temperature unaffected. As we are interested in using the RALF model in a purely predictive way, in this work, \(\xi_{jA} = 0\).

It is shown in Brandani (2019) how the LF equations for a system comprising a crystalline ‘rigid’ adsorbent lead to a corresponding expression for the Gibbs energy for the solid phase. For a system with a *single* adsorbate, the residual Gibbs energy is given by:

*residual*refers to the departure of a thermodynamic property from that of an ideal gas at the same temperature and pressure (Smith et al. 2004; Gmehling et al. 2012).

*j*to be equal in the adsorbed and fluid phases. The subscript

*A*is added for clarity in Eq. 15 to describe the adsorbed phase, but will be dropped from now on. Isotherms can be constructed by solving Eq. 15 for the number of moles adsorbed at any given combination of pressure and temperature.

*j*. For the single component (and a ‘frozen’ solid) they are:

*j*is equal to the logarithm of its fugacity coefficient, \({ \ln }\varphi_{j}\), i.e.

Finally it is worth considering that the use of a lattice fluid naturally leads to expressions for the behaviour at both infinitely high pressures yielding finite adsorbed phase concentrations, and in Henry’s law limit, thereby preserving thermodynamic consistency (Brandani 2019). The expressions under these conditions have also been given in the supporting information.

## 3 Parameterisation of the RALF model

_{2}, CO, etc. due to good availability of experimental data on these systems (Brandani 2019). The characteristic parameters for a number of molecules and silicalite-1 are listed in Table 1. The characteristic parameters for the molecules have been determined from bulk vapour-liquid equilibrium data, as outlined in Ref. (Sanchez and Lacombe 1976), carried out independently from this study or the determination of characteristic parameters for silicalite-1 (De Angelis et al. 2007). With the characteristic parameters in place, the RALF model can now be used as a purely predictive tool, without the need for adjustable fitting parameters. From Table 1 it can be seen that silicalite-1 is described by three characteristic parameters, which is equivalent to assuming that it can be described as a homogeneous solid, i.e. having a single characteristic adsorption site. It has been shown in various publications that this is a realistic assumption for small molecules (Zhu et al. 2000; Sun et al. 1998; Golden and Sircar 1994). This also means that any predictions of steps or inflections in isotherms are not due to surface heterogeneities.

Characteristic parameters for silicalite-1 and a number of molecules

Characteristic parameter | \(T_{j}^{*}\) (MPa) | \(\varvec{P}_{\varvec{j}}^{\varvec{*}}\) (K) | \(\rho_{\varvec{j}}^{\varvec{*}}\) (kg/m | \(M_{j}\) (kg mol | References |
---|---|---|---|---|---|

CH | 215 | 250 | 500 | 0.016 | De Angelis et al. (2007) |

C | 320 | 330 | 640 | 0.030 | De Angelis et al. (2007) |

C | 320 | 375 | 690 | 0.044 | De Angelis et al. (2007) |

CO | 300 | 630 | 1515 | 0.044 | De Angelis et al. (2007) |

C | 470 | 880 | 915 | 0.046 | De Angelis et al. (2007) |

CH | 510 | 1080 | 900 | 0.032 | De Angelis et al. (2007) |

H | 670 | 2400 | 1050 | 0.018 | De Angelis et al. (2007) |

\(T_{s}^{*}\) (MPa) | \(P_{s}^{*}\) (K) | \(\rho_{s}^{*}\) (kg/m | \(\varepsilon\) (−) | ||
---|---|---|---|---|---|

Silicalite-1 | 1060 | 650 | 2577 | 0.31 | Brandani (2019) |

## 4 Results

In order to understand the predictions by the RALF model in greater detail, we will now focus on some of the key parameters in the lattice fluid model and their effect on isotherm shape. These parameters are solid porosity, \(\varepsilon\) (or reduced density), and the characteristic parameters for the adsorbate molecules, i.e. interaction energy, \(T_{j}^{*}\), energy density, \(P_{j}^{*}\) and close packed density, \(\rho_{j}^{*}\).

Polar molecules have relatively high values for \(P_{j}^{*}\) and \(T_{j}^{*}\), as compared to apolar molecules, such as alkanes. The localised charges on polar molecules clearly endow them with an increased interaction energy in the lattice fluid. This in turn translates into strongly coordinating behaviour and a tendency to show adsorption behaviour which leads to molecular clustering. Table 1 shows values for \(P_{j}^{*}\) and \(T_{j}^{*}\) for various molecules. Similarly, the strength of the molecule’s interaction with the solid itself must be a critical parameter as to whether or not additional molecules adsorb onto the solid surface or coordinate to already adsorbed molecules. From a chemical point of view, this solid – adsorbate interaction can be effected by changing the solid’s polarity. For instance, in a zeolite this could be achieved by changing the silicon to aluminium ratio and introducing charge balancing cations. From a structural point of view, confinement of molecules in the solid is expected to have a similar effect. A large pore volume is more likely to cause clustering of polar molecules, whereas this behaviour is not expected for a dense solid with nanosized pores.

## 5 Solid porosity, ε

## 6 Adsorbate characteristic parameters

It is clear from the results presented so far, that solid porosity is only one deciding factor for the shape of the isotherm. We will now explore the effect of the molecules’ characteristic parameters.

*hypothetical*molecule with a molecular weight of 0.04 kg mol

^{−1}; at given \(T_{j}^{*}\) and \(P_{j}^{*}\), \(v_{j}^{*}\) now is fixed by Eq. 4, whereas \(\rho_{j}^{*}\) is determined by Eq. 20.

## 7 Solid polarity

## 8 Concluding remarks

Although the origins of type V isotherms in microporous solids have been studied for over a decade, yielding microscopic insights through molecular simulations and related computational methods, macroscopic models that can predict this behaviour a priori are rare. Predictive macroscopic models, however, are a an essential tool in designing efficient separation processes and with the advent of new exciting materials being developed that show non-trivial adsorption behaviour, the need for these models could not be higher. Here we have discussed how the simple RALF model can predict both type I and V isotherms from first principles and identified which modelling parameters are critical in determining the resulting isotherm shape. As reported previously in molecular simulation studies, the trade-off between molecule—molecule and molecule—adsorbent interaction determines the adsorption behaviour. In the LF model this is mostly described by the values for the interaction energy and energy density (\(T_{j}^{*}\) and \(P_{j}^{*}\)) of both the molecule and the solid and the porosity of the solid. The (dis)similarity between the former dictates whether molecular clustering is likely, whereas the latter acts as a geometric barrier for this behaviour. The molecule—adsorbent interaction can further be tweaked by adjusting their binary interaction parameter, \(\kappa_{js}\), but this should ultimately be a utility to match the predictions to experimentally available data.

It is impressive that a simple lattice fluid based model, where the parameters of the adsorbates can be determined from bulk fluid properties (Linstrom and Mallard), can replicate different adsorption behaviour without prior assumptions, making it a useful tool in process simulations and design.

## Notes

## Supplementary material

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