Isoreticular Curves: A Theory of Capillary Condensation To Model Water Sorption within Microporous Sorbents

Metal-organic frameworks have gained traction as leading materials for water sorption applications due to precise chemical tunability of their well-ordered pores. These applications include atmospheric water capture, heat pumps, desiccation, desalination, humidity control, and thermal batteries. However, the relationships between the framework pore structure and the measurable water sorption properties, namely critical relative humidity for condensation, maximal capacity, and pore size or temperature for the onset of hysteresis, have not been clearly delineated. Herein, we precisely formulate these relationships by application of the theory of capillary condensation and macroscopic thermodynamic models to a large data set of MOF water isotherms. These relationships include a concept termed isoreticular curves that relates the critical pressure for pore condensation (α), gravimetric capacity (Q_max), and hydrophilicity (the Gibbs free energy for binding water, ΔG) as Q_max = a₁(ΔG/ln α)² + a₂(ΔG/ln α), with constants a₁ and a₂ dependent upon the density and volume occupied by the linker and secondary building unit, and framework topology. Through this analysis, we propose guidelines for the maximization of sorption capacity at a given relative humidity with minimal hysteresis and discuss the theoretical limits for capacity at low relative humidity. This model provides an explanation for the lack of high-capacity frameworks at low relative humidity, as increasing pore size also causes an increase in relative humidity. We propose a loose upper bound of Q_max = -0.25(1/ln α)² - 1.75(1/ln α) for the limit of maximal capacity at a given relative humidity in the dry regime. These guidelines are consequential for the design of new materials for water sorption applications.