# Accurate estimation of the temperature-dependent entropy of flexible Metal-Organic Frameworks

# Accurate estimation of the temperature-dependent entropy of flexible Metal-Organic Frameworks

Promotor(en):**18NANO14**/ Nanoporous materials**V. Van Speybroeck, L. Vanduyfhuys**/Metal-Organic Frameworks (MOFs) are a class of hybrid nanoporous crystalline materials consisting of inorganic bricks connected to each other by means of organic linkers. Many of these MOFs possess a flexible framework that can transform between various states with large variations in the unit cell volume, which is also denoted as breathing. Two examples of such MOFs are MIL-53(Al), the textbook example of breathing MOFs that can cycle between a large pore and narrow pore state under influence of temperature, and DUT-49(Al), which can absorb large amounts of mechanical energy when subject to high pressures and hence serve as a nano shock absorber. In both cases, the transition between the various states can be modeled once the Helmholtz free energy F is available as function of temperature T and volume V. Furthermore, from the thermodynamic definition F = E - TS, it is clear that an accurate estimate of the temperature dependence of F requires an accurate estimate of the entropy S. Unfortunately, computing the entropy accurately is not a trivial task, which is due to the fact that the entropy cannot be written as a simple ensemble average, as is the case for the internal energy E. Instead, approximations of the potential energy surface (PES) or advanced sampling schemes are required to estimate the entropy.

**Goal**

In this thesis, various methods will be used to accurately estimate the entropy, and by extension the Helmholtz free energy, of MOFs as function of unit cell volume V and temperature T. The resulting free energy profiles will afterwards be used to investigate the temperature-induced breathing of MIL-53(Al) as well as the mechanical energy storage capacity of DUT-49. A first method that will be considered is the Quasi Harmonic Approximation (QHA) in which the free energy is written as a sum of contributions from 3N uncoupled but volume-dependent harmonic oscillators.This method includes nuclear quantum effects (NQE) but neglects anharmonicity of the normal modes. A second method uses path-integral molecular dynamics simulations (PI-MD), which extends classical MD simulations to include NQE using the path integral formulation of quantum mechanics, and computes the internal pressure P and energy E as function of V and T to arrive at the free energy by means of thermodynamic integration (TI):

As such, it includes NQE as well as anharmonicity. A third method computes the entropy directly by means of thermodynamic integration of the heat capacity and the thermal derivative of the pressure:

If again PI-MD is used to compute the heat capacity and pressure derivative, NQE and anharmonicity are both taken into account. Finally, by applying and comparing all these methods for MIL-53(Al) and DUT-49, we aim at answering the following questions:

• Is internal energy E and entropy S of MOFs strongly dependent on temperature, or can the Helmholtz free energy be written as F(V,T) = E(V) - T∙S(V)

• How important are NQE and anharmonicity to estimate entropy and internal energy?

• What is the role of entropy in the breathing of MIL-53(Al)? Can the experimental transition temperatures be reproduced?

• How does the energy storage capacity and transition pressure of DUT-49 change as function of temperature?**Engineering and physics aspects**

Physics: use of advanced thermodynamics and statistical physics

Engineering: investigation of the sensitivity of properties of MOFs towards temperature

- 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, MODELINGKeywordsThermodynamics, Statistical physics, entropy, Helmholtz free energy, temperature, anharmonicityRecommended coursesSimulations and Modelling for the Nanoscale