On the Thermodynamics of Framework Breathing: A Free
Energy Model for Gas Adsorption in MIL-53
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Abstract
When
adsorbing guest molecules, the porous metal–organic
framework MIL-53(Cr) may vary its cell parameters drastically while
retaining its crystallinity. A first approach to the thermodynamic
analysis of this “framework breathing” consists of comparing
the osmotic potential in two distinct shapes only (large-pore and
narrow-pore). In this paper, we propose a generic parametrized free
energy model including three contributions: host free energy, guest–guest
interactions, and host–guest interaction. Free energy landscapes
may now be constructed scanning all shapes and any adsorbed amount
of guest molecules. This allows us to determine which shapes are the
most stable states for arbitrary combinations of experimental control
parameters, such as the adsorbing gas chemical potential, the external
pressure, and the temperature. The new model correctly reproduces
the structural transitions along the CO<sub>2</sub> and CH<sub>4</sub> isotherms. Moreover, our model successfully explains the adsorption
versus desorption hysteresis as a consequence of the creation, stabilization,
destabilization, and disappearance of a second free energy minimum
under the assumptions of a first-order phase transition and collective
behavior. Our general thermodynamic description allows us to decouple
the gas chemical potential μ and mechanical pressure <i>P</i> as two independent thermodynamic variables and predict
the complete (μ, <i>P</i>) phase diagram for CO<sub>2</sub> adsorption in MIL-53(Cr). The free energy model proposed
here is an important step toward a general thermodynamics description
of flexible metal–organic frameworks