3.3 Dissolution activation energy
To intuitively characterize the tendency of molecules dissolving into pore entrances on membrane surface, dissolution activation energy (E S) is proposed here, which reflects the origin and energy variation of dissolution behaviors. For molecule transport through a membrane, the total activation energy (E P) includesE S and diffusion activation energy (E μ).[48,49] TheE P is extracted from Arrhenius plot by measuring molecule permeance through membrane over a temperature range of 10 to 45 °C (Figure 3a), and the corresponding equation is:[19,50]
\begin{equation} P=P_{0}\times e^{\frac{-E_{P}}{\text{RT}}}\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ }(3)\nonumber \\ \end{equation}
In equation 3, P andP 0 (L m-2 h-1bar-1) represent the permeance of a solvent and the pre-exponential factor, respectively, R (kJ mol-1 K-1) is the gas constant, andT (K) is the temperature. Accordingly, E Pvalues of various solvents for these membranes are obtained (Tables S1 and S2). Then theE μ is calculated by the following equations:[51]
\begin{equation} \mu=\mu_{0}\times e^{\frac{E_{\mu}}{\text{RT}}}\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ }(4)\nonumber \\ \end{equation}
In equation 4, μ and μ 0 (Pa s) are the viscosity of solvent and pre-exponential factor, respectively. By combining equations 3 and 4, the obtained expression is below:[52]
\begin{equation} P\ \sim\ \mu\times e^{\frac{-E_{P}}{E_{\mu}}}\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ }(5)\nonumber \\ \end{equation}
Therefore, E μ is obtained from the slope value in the correlation plot between solvent permeance and viscosity (Figure 3b). As an example, the E μ of water for MOF-CH3@CH3 membrane is calculated to be 16.64 kJ mol-1. To vouch for the accuracy of calculated E μ values, another method based on Darcy’s law is adopted (Figure S18).[45,53] And the result is 16.96 kJ mol-1, close to that calculated by equation 5. Next,E S values of solvents for hierarchical lamellar membranes are acquired.
Note that molecules would spread on the membrane surface firstly before dissolving into membrane, where the spreading activation energy should be considered. Here, the spreading activation energy of water on MOF-CH3@CH3 and MOF-CH3@NH2 membrane surface was calculated as an example (Figures S19-22). Results show that the spreading activation energy values are comparable for MOF-CH3@CH3 (107.6 kJ mol-1) and MOF-CH3@NH2(113.2 kJ mol-1) membranes, which should be resulted from the uniform MOF skeleton structure on membrane surface.[54] To eliminate the impact of spreading activation energy value on the calculation of E Sfor molecules drilling into pore entrance, membranes are immersed into corresponding solvents to reach a saturation condition, under which the following E S values are acquired.[55]
For molecules drilling into pore entrances, theE S is mainly controlled by molecule-molecule and molecule-pore interactions. Specifically, molecules in bulk state need to adjust the configuration to drill into confined pores (from larger aggregates to smaller ones), during which the bindings between molecules should be broken, thus inevitably consuming energy. Meanwhile, molecules that contact the pore entrances would exert interactions with the groups on pore rims, which releases energy to compensate the energy consumed.[56] Therefore, these two energies should determine molecular dissolution efficiency collectively. Figure 3c displays that for MOF-CH3@CH3membrane with hydrophobic pores on surface layer, theE S values of both polar and nonpolar solvents are above 0. This indicates that the energy released by the formation of molecule-pore interactions is smaller than the energy consumed by molecule rearrangement. Thus extra energy is needed to push the molecules to drill into pore entrances. In contrast, for MOF-CH3@NH2 membrane with hydrophilic pores on surface layer, the E S values for nonpolar solvents are above 0, while that for polar solvents are below 0. This finding implies that hydrophilic groups on pore entrance tend to exert stronger interactions with polar solvent (e.g. hydrogen bond interaction), thus producing more positive energy that compensates the energy consumed by molecule rearrangement. Therefore, polar solvents tend to spontaneously drill into hydrophilic pores. As shown in Figure 3d, taking water as an example, water molecules in bulk state are bonded with each other through hydrogen bonds. Upon contacting pore entrances, new hydrogen bonds are formed between water molecules and –NH2 groups on pore rims. Meanwhile, the configuration of bulk water is adjusted to drill into confined pores, accompanied by the breakage of intermolecular hydrogen bonds. In this manner, water molecules dissolve into the hierarchical MOF lamellar membrane.