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]