Cooperative Sorption on Heterogeneous Surfaces

Abstract

Heterogeneous adsorbents, those composed of multiple surface and pore types, can result in stepwise isotherms that have been difficult to model. The complexity of these systems has often led to appealing to empirical equations without physical insights, unrealistic assumptions with many parameters, or applicability limited to a particular class of isotherms. Here, we present a statistical thermodynamic approach to model stepwise isotherms, those consisting of either an initial rise followed by a sigmoid or multiple sigmoidal steps, founded on the rigorous statistical thermodynamic theory of sorption. Our only postulates are (i) the finite ranged nature of the interface and (ii) the existence of several different types of microscopic interfacial subsystems that act independently in sorption. These two postulates have led to the superposition scheme of simple surface (i.e., Langmuir type) and cooperative isotherms. Our approach has successfully modeled the adsorption on micro–mesoporous carbons, gate-opening adsorbents, and hydrogen-bonded organic frameworks. In contrast to the previous models that start with a priori assumptions on sorption mechanisms, the advantages of our approach are that it can be applied universally under the above two postulates and that all of the fitting parameters can be interpreted with statistical thermodynamics, leading to clear insights on sorption mechanisms.

Understanding Sorption Mechanisms Directly from Isotherms

“…Such an approach was applied to experimental isotherm data, including NH 3 adsorption on Kuf-1a (a hydrogen-bonded organic framework), water adsorption on an aluminophosphate molecular sieve, and CO 2 adsorption on PCN-53 (a metal− organic framework), from which the sorbate cluster number and the free energy of sorption were determined for mechanistic insights. 37 Second, we demonstrate that the AB isotherm can be interpreted in a manner relatable to Types I−II and IV−VI, even for the parameter range corresponding to Type III (B > 0 with C = 0). To do so, let us rewrite the AB isotherm as…”

“…Moreover, heterogeneous materials with multiple pore sizes were modeled successfully by multiple terms of the cooperative isotherm. Such an approach was applied to experimental isotherm data, including NH 3 adsorption on Kuf-1a (a hydrogen-bonded organic framework), water adsorption on an aluminophosphate molecular sieve, and CO 2 adsorption on PCN-53 (a metal–organic framework), from which the sorbate cluster number and the free energy of sorption were determined for mechanistic insights …”

“…First, we show that the vapor-to-interface transfer free energy of a sorbate has a natural link to the key quantity in cooperative sorption for Types IV–VI. We start with the statistical thermodynamic cooperative isotherm , ⟨ n 2 ⟩ = N m A 1 a 2 + m A m a 2 m 1 + A 1 a 2 + A m a 2 m where m is the number of sorbates cooperatively sorbed to a patch, N m is the number of patches, and − RT ln A ν calculated from A ν is the free energy of moving ν sorbates from saturated vapor to the interface, respectively. We can easily see that eq reduces to the Langmuir–Freundlich model under A 1 = 0, as θ = false⟨ n 2 false⟩ m N m 0.25em = A m a 2 m 1 + A m a 2 m In the literature on surface energy characterization, f C = − RT ln A m has often been referred to as “adsorption energy”.…”

“…At the same time, what makes an isotherm type different from another cannot be clarified when different isotherm models are used for different isotherm types. Building on the success of our statistical thermodynamic ABC , and cooperative , isotherms in modeling experimental sorption data with mechanistic insights, the objectives of our papers are to establish a consistent approach to model all isotherm types, attributing the difference to the sorbate–sorbate and sorbate–surface interactions in a systematic manner; to generalize the language of the traditional sorption models (such as the monolayer capacity and the BET constant, see Appendix A) to the model-free concepts of partitioning and association coefficients that can be applied across the isotherm types; to eliminate the apparent contradictions caused by applying the site-specific models alongside with cross-sectional area of sorbates for the purpose of surface area determination. …”

“…While our focus is chiefly on Types I–III, its natural connection to Types IV–VI will be established in reference to our recent work. , (Note that we focus on the multi-stepwise Type VI-like sorption on heterogeneous materials, instead of the strict definition of “layer-by-layer adsorption on a highly uniform nonporous surface.”) A comparison between the isotherm models and statistical thermodynamics will reveal and identify the stumbling block of the traditional approaches: inconsistent treatment of attractive and repulsive interactions which has confused the interpretation of biomolecular solvation and solubilization in the recent past. − We will demonstrate how clarity is attained by treating attraction and repulsion on an equal footing.…”