Probleemstelling:

Gas adsorption in nanoporous materials gives rise to many promising applications such as natural gas storage, carbon capture, gas detection, … To investigate the adsorption behavior, one can experimentally measure the amount of adsorbed species as function of vapor pressure, i.e. measure the adsorption isotherm, or compute it by means of Monte Carlo simulations in the grand canonical ensemble, i.e. Grand Canonical Monte Carlo (GCMC). However, there exist several Metal-Organic Frameworks (MOFs), i.e. a class of hybrid porous solids consisting of inorganic bricks connected to each other by means of organic linkers, which exhibit a high degree of framework flexibility and are also referred to as breathing MOFs. They can undergo transformations characterized by large changes in the unit cell volume under influence of mechanical pressure, guest adsorption, temperature, … This framework flexibility makes direct simulation of adsorption isotherms using GCMC much more difficult since accounting for framework vibrations in MC is not trivial. Furthermore, it remains difficult to assess to what extent the various microscopic contributions to the Hamiltonian (host-host, guest-host or guest-guest) have a large impact on the observed macroscopic adsorption behavior.

In this context, the use of (semi-)analytic models for the thermodynamic potential provides a promising approach. Such a model consists of an analytic expression for the thermodynamic potential in a given ensemble as function of the thermodynamic state variables (temperature, volume, chemical potential/number of adsorbed guest, …). The potential is usually expanded into various contributions (empty host, guest-guest, guest-host interaction) and features various system-specific parameters (eg. single particle mean interaction energy) that can be estimated from molecular simulations. Finally, all thermodynamic properties, including the adsorption isotherm, can then be computed by means of Legendre transformations and function derivatives. Moreover, the sensitivity of these properties towards the system-specific parameters in the model can be assessed, which allows to gain crucial insight into the physical origins of the adsorption processes. Such a model has already been developed in the canonical ensemble in the mean-field approximation (see figure). 

Doelstelling:

In this thesis, we will construct an analytic expression for the thermodynamic potential of guest-loaded MOFs in the grand canonical ensemble, i.e. the grand canonical potential Ω(µ,V,T). The starting point for such a model is the following generally valid expression Ω:

in which Fhost(V,T) represents the Helmholtz free energy of the empty host and N(µ,V,T) represents the grand canonical adsorption isotherm. Various models are already available to estimate the empty host contribution such as the Quasi Harmonic Approximation (QHA), i.e. free energy of 3N volume-dependent harmonic oscillators, or thermodynamic integration (TI) of the pressure P(V,T) obtained from molecular dynamics (MD) simulations. Therefore, in this thesis, we will focus on possible models for the adsorption isotherm. The most simple models that will be tested include the Langmuir and BET isotherms.

In which the saturation amount Ns, the monolayer Henry constant K and multilayer Henry constant K2 are volume and temperature dependent, while the vapor pressure Pvap depends on the chemical potential and temperature of the gas reservoir. Due to the clear physical interpretation of these parameters (Ns, K and K2), they can be computed by means of simple molecular simulations such as random particle insertions. However, to improve the accuracy, we will also investigate more complicated analytic expressions through the application of Taylor expansions or Padé approximations. In general, the parameters in these expressions will not retain their a clear interpretation. Therefore, they will be computed by comparison with direct molecular simulation in the grand canonical ensemble such as GCMC simulations or even hybrid MC/MD simulations. For various systems, such simulation data is already available at the CMM which can serve as a good starting point.

Once the analytic expression for the grand canonical potential is available, Legendre transformations will be applied to gain access to the thermodynamic potential in other ensembles. On the one hand, the potential will be Legendre transformed to the canonical ensemble and compared with a mean-field model for the Helmholtz free energy developed at the CMM to assess which contributions are lacking in either model. On the other hand, the grand canonical potential will also be Legendre transformed to the osmotic ensemble to compute the osmotic adsorption isotherm as well as the unit cell evolution which can be compared with experimental results.