# Development of a semi-analytic thermodynamic model for the grand canonical potential of flexible MOFs

# Development of a semi-analytic thermodynamic model for the grand canonical potential of flexible MOFs

Promotor(en):**18MODEV12**/ Model and software development, Nanoporous materials**V. Van Speybroeck, L. Vanduyfhuys**/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 coordinates (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 and their sensitivity towards the system-specific parameters in the model can be assessed. Such a model has already been developed in the canonical ensemble in the mean-field approximation (see figure).

**Goal**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. For the empty host, various models are already available 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:

Possible models for the adsorption isotherm include the Langmuir isotherm, BET isotherm and others.

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. 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.

**Engineering and physics aspects**Physics: development of advanced models for the thermodynamic potentials

Engineering: derivation and sensitivity analysis of adsorption isotherm and heat of adsorption of nanoporous materials

- Study programmeMaster of Science in Engineering Physics [EMPHYS], Master of Science in Physics and Astronomy [CMFYST]ClustersFor Engineering Physics students, this thesis is closely related to the cluster(s) NANO, MODELINGKeywordsThermodynamic potential, Legendre transformation, Helmholtz free energy, grand canonical potential, adsorption isotherm, grand canonical monte carloRecommended coursesSimulations and Modelling for the Nanoscale