4.1. Interaction Energy between Diffusion Molecules and PE Chain
The interaction energy (
Eint), indicating the intensity of interaction between the diffusion molecule and polymer, is derived according to the following equation:
where
Etotal is the total energy of the system,
EPE is the energy of PE chain,
Ediff,i is the energy of diffusion molecule
i in the PE chain. A negative
Eint value is corresponding to stable interaction between the components [
39,
40]. More negative
Eint indicates a stronger interaction in the system.
In this study, there was only physical bonding between diffusion molecules and polymer chains, and no chemical reaction and new chemical bonds involved. The non-bonding interactions between diffusion molecules and polymer chains, and the interatomic potential for the interactions among LDPE atoms and diffusion molecules were considered. The interaction energies of D-limonene, myrcene, ethyl hexanoate, 2-nonanone, linalool, O
2 and H
2O with PE chain were calculated by MD simulation and were listed in
Table 2. The stronger the interaction between diffusion molecules and polymer chains will result in the higher the energy barrier to be overcome and the smaller the corresponding diffusion coefficient. Therefore, if only take the effect of interaction energy on the diffusion into account, the diffusion coefficients of the diffusion molecules in PE should decrease in the following order:
DH2O >
DO2 >
DD-LE >
D2-NO >
DME >
DEH >
DLO. However, besides the interaction energy, molecular size and free volume of polymer cell are also important factors affecting the diffusion of molecules, which will be further discussed in a later section.
4.3. Fraction of Free Volume
The size and shape of a free volume play an important role in the diffusion behavior of diffusion molecules in polymers. Generally, the larger the free volume fraction of polymers results in the larger the diffusion coefficient. When calculating the free volume of the system, the hard sphere probe model in MS 8.0 was used to analyze the kinetic radius of the diffusion molecule. The free volume and occupied volume of the cell can be obtained when the molecular probe with a certain radius
RP moves on the van der Waals surface. The ratio of free volume to simulated cell volume is defined as
FFV [
51]:
where,
VF is the free volume of the polymer,
VO is the occupied volume of the polymer, and
VS is the total volume of the polymer.
The probe radii of 1.0 Å, 1.5 Å, 2.0 Å, 2.7 Å were selected to calculate the free volume of PE cells. Free volume morphologies with the different probe radius in PE cell are shown in
Figure 6.
Table 6 shows that with the increase of
RP, the
FFV of PE cell decreases. The free volume measured by 2.7 Å can meet the space requirement of jump diffusion for O
2 and H
2O. It also means that for larger flavor organic molecules, there are fewer free holes to jump. The polymer chains can wriggle themselves, and when their wriggling creates enough jumping space, flavor organic molecules will jump from one hole to other holes.
4.4. Diffusion of O2 and H2O in PE before and after Adsorption
The diffusion of O
2 and H
2O in PE before and after adsorption of flavor molecules were simulated by the MD method, and the related MSD curves were shown in
Figure 7 and
Figure 8. In the long-term MD simulation, the diffusion of molecules in random motion is affected by the steric hindrance of PE chain, resulting in a decrease of diffusion coefficient. Compared with
Figure 4, O
2 or H
2O compete with the flavor molecules for free holes in PE, which restricts the diffusion of O
2, H
2O and flavor molecules. After adsorption of flavor molecules, O
2 and H
2O diffuse faster than all flavors, and their MSD curves increase more rapidly. The reason is that O
2 and H
2O are smaller than flavor molecules and need less diffusion space. Moreover, the effect of flavor molecules on O
2 is small; therefore, O
2 diffusion is faster than H
2O.
The diffusion coefficients of O
2 and H
2O before and after PE adsorption are listed in
Table 7.
Table 7 indicates that the diffusion of O
2 before and after PE adsorption is stronger than that of H
2O. This is inconsistent with the results of interaction energy analysis (
Section 4.1). Because the difference between the interaction energies of O
2 and H
2O is relatively small, thus, effects of the interaction energies on the diffusion of both two molecules are not decisive. However, the size of O
2 molecule is smaller than that of H
2O molecule; therefore, O
2 molecule is easier to diffuse in PE. The diffusion coefficients of O
2 and H
2O in adsorbed PE are all smaller than those in pure PE. In order to better understand the influence of flavor organic molecules on the diffusion of O
2 and H
2O in PE, the
FFV of PE and the interaction energies of O
2 or H
2O with flavor organic molecules were further analyzed.
The
FFV of PE cells, before and after flavor adsorption, was calculated with a probe radius of 1.0 Å and listed in
Table 8. The interaction energies of D-limonene, myrcene, ethyl hexanoate, 2-nonanone and linalool with O
2 or H
2O were calculated by MD simulation and were listed in
Table 9.
Table 8 shows that the
FFV of PE cell decreased with the loading of O
2, while increased with the loading of H
2O. This is because the alkane chain has strong hydrophobicity, and the free volume becomes larger after loading H
2O. When flavor molecules were diffused in PE, the
FFV of PE cell decreased, which led to the decrease in the diffusion coefficients of O
2 and H
2O.
In addition to the
FFV, the interaction between O
2 or H
2O molecule and flavor organic molecules also affects the diffusion coefficients of O
2 or H
2O.
Table 9 shows the interaction energies between O
2 or H
2O molecule and five flavor organic molecules. All the interaction energies were negative, which indicates the presence of the flavor organic molecules will hinder the diffusion of O
2 and H
2O molecules, and thus, lead to the decrease in the diffusion coefficients (
Table 8). Besides, the interaction energies between H
2O and five flavor organic molecules were stronger than those between O
2 and five flavor organic molecules, especially for ethyl hexanoate, 2-nonanone and linalool. The equilibrium adsorption configurations of H
2O molecule with five flavor organic molecules were shown in
Figure 9. The H atom in H
2O molecule is positively charged and expected to absorb near to the negatively charged C atoms in D-limonene and myrcene (
Figure 9a,b). O atom of higher electronegativity is contained in the ethyl hexanoate, 2-nonanone and linalool molecules, which can form the hydrogen bond with the H atom in H
2O (
Figure 9c–e). Therefore, the interaction between H
2O and ethyl hexanoate, 2-nonanone and linalool molecules are stronger than those of D-limonene and myrcene. Therefore, in a word, the presence of the flavor organic molecules results in greater hindrance to the diffusion of H
2O compared with the case of O
2, which can further explain the greater decrease in the diffusion coefficient of H
2O in the presence of flavor organic molecules (
Table 7).