3.4 Dissolution model equation
To further quantize the key parameters that affect molecular dissolution behaviors, the critical surface tension (γ C) of membrane is extracted by Zisman Plot.[57,62,63] Here, contact angles for polar (water, methanol, ethanol, acetone, acetonitrile and DMF) and nonpolar solvents (cyclohexane, n-octane, toluene, n-hexane) were measured to determine the γ C of membranes. Zisman Plots in Figures 5a and b show that γ C values of MOF-CH3@NH2 and MOF-CH3@CH3 membranes are 19.7 and 17.6 mN m-1, respectively. This defines the wetting threshold for these two membranes, which means that solvents with surface tensions less than the γ C are predicted to wet the membrane surface completely. Here, two phenomenological model equations that well correlate to E S and intrinsic parameters of membrane surface and molecule are proposed for MOF-CH3@CH3 (equation 6) and MOF-CH3@NH2 (equation 7), respectively:
\(E_{S}=K_{m}\ln\left[\left(\gamma_{L}-\gamma_{C}\right)\mu d^{2}\right]\)(6)
\(E_{S}=K_{a}\ln\left[\left(\gamma_{L}-\gamma_{C}\right)\delta_{e}\mu d^{2}\right]\)(7)
where γ L and γ C (mN m-1) are the surface tension of solvent and the critical surface tension of membrane surface, respectively, d (m) is the kinetic diameter and δ e(Pa0.5) is the Hansen solubility parameter of solvent,K m and K a are the coefficient constant of MOF-CH3@CH3 and MOF-CH3@NH2 membranes, respectively.
These two equations indicate that molecular dissolution capacity is positively correlated with the difference between solvent surface tension and membrane critical surface tension. Therefore, the interfacial properties (γ L-γ C) of membrane surface and molecule both count significantly in the equation. Besides, the parameters that correlate with molecule transport in pores (μd 2) also affect molecular dissolution efficiency (Figure 5c). Interestingly, molecular dissolution behaviors on MOF-CH3@NH2 membrane do not obey this dissolution model equation, but adding another parameter ofδ e, which represents the cohesive energy density of molecules (Figures 5d and S23). This should be ascribed to the formation of strong molecule-pore interactions, which promote the breakage of molecule-molecule interactions and then the thorough rearrangement of molecules, thus doubling the effect of molecule-molecule interaction on dissolution efficiency. According to the dissolution equations, molecular dissolution process should include two main steps: wetting the membrane surface and then entering into pores. Concretely, the wetting step is controlled by the difference of surface tension between molecule and membrane surface (γ L-γ C), which reflects the wettability of membrane surface for molecules. Subsequently, the step of molecules entering into pores should relate to the μ and d for hydrophobic pore entrances, while relating to δ e,μ and d for hydrophilic pore entrances (Table S4). Different from the strategy of calculating interfacial resistance, this viewpoint of dissolution activation energy systematically reveals the influence of molecule-molecule and molecule-pore interactions on dissolution behaviors on molecule-level.[12,57,58]