Diffusion of small molecules in a polymer matrix: Polymer packaging materials prevent the diffusion of small molecules

Polymers are often used as packaging materials. As such they need to be designed to prevent the diffusion of small molecules such as water either from the inside to the outside or vice versa. Molecular simulations can help in predicting the diffusion coefficients of small molecules in a polymer matrix.
 
Borealis was interested in the diffusion of water and methanol in a polymer matrix consisting of a mixture of polyethylene and a diblock copolymer of ethylbutylacrylate (EBA) and polyethylene (PE) at various temperatures.
 
The diffusion of small molecules can be understood by performing molecular dynamics simulations of the system of interest and looking at the movement of the molecules diffusing. Usually, this movement is characterized by the diffusion coefficient which describes how far a molecule diffuses in a given time.
 
To study the diffusion of water and methanol in the polymer matrix Borealis was interested in, a polyethylene molecule with 100 monomers and a diblock copolymer of 20 ethylbutylacrylate monomers and 80 ethylene monomers were built using MAPS' polymer builder. Five chains of each polymer were used to construct an amorphous box of the polymer matrix by means of MAPS' amorphous builder. The construction of the amorphous box made use of the Dreiding force field [1].
 
As a first step of the simulation the polymer matrix was optimized to get rid of potential close contacts in the system introduced by the build process. As the next step a molecular dynamics simulation in the NPT ensemble was carried out for 200 ps at 298 K to equilibrate the polymer matrix. After about 75 ps the system reached a density of 0.8 g/cm3 which stayed constant for the rest of the simulation.
 
MAPS' amorphous builder was used again to place five water or methanol molecules into the equilibrated polymer matrix (making use of the option to build onto a matrix). A molecular dynamics simulation in the NPT ensemble was run for 5 ns at 298, 323 and 373 K to study the diffusion of the water and methanol molecules. For the molecular dynamics simulations the Dreiding and the PCFF [2] force fields were used.
 
The simulations results were analyzed by calculating the mean square displacement of the water and methanol molecules. The mean square displacements were fit to a linear equation of the form:

〈∣r(t)−r(0)2∣〉=6Dt+B

which gave access to the diffusion coefficient D. The fit was done for the mean square displacements between 0.5 and 5 ns to give the system time to equilibrate after adding water or methanol.
 

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Figure 1: A frame of the simulation of water at 298 K (PCFF) showing the position of the water molecules in the polymer matrix (oxygen atoms in blue)

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Table 1: The diffusion coefficients calculated

Since water molecules can form hydrogen bonds between each other and liquid water possesses an entire network of hydrogen bonds the molecular dynamics simulation was repeated for the water case by using a water cluster in the polymer matrix as starting point. The water cluster was manually built from five water molecules using the sketching tools in MAPS and manually placed in a cavity of the polymer matrix. The simulation was then run for 5 ns as shown above using the Dreiding forcefield.
 
Results and Discussion: Figure 1 shows a frame from the simulation of water at 298 K. Most of the water molecules are located in the part of the polymer matrix which contains the ethylbutylacrylate monomers.
Table 1 lists the diffusion coefficients obtained and compares them with the available experimental data.
The diffusion coefficients presented in Table 1 are averages over all fives molecules. The Dreiding force field clearly overestimates the diffusion coefficient while the PCFF force field results in a value which is very close to experiment although the experimental diffusion coefficients have been determined in a pure polyethylene matrix. Therefore, the study of the temperature dependency of the diffusion coefficient and further analyses have been performed using the PCFF force field.
 
Figure 2. shows the temperature dependency of the diffusion coefficients. When the temperature is increased the diffusion coefficients also get larger although the effect is more prominent in the case of methanol.
 

c13-f2-temperature-dependent-diffusion-coefficients
Figure 2: The temperature dependency of the diffusion coefficient

The individual molecules show quite significant differences in the distance they travel during the simulation. This can be understood by looking at the surroundings of each molecule. Some molecules might slip through the polymer chains and travel a long distance while other molecules are trapped inside of a pocket of the polymers and oscillate around an equilibrium position.
 
To analyze the behavior of the individual molecules the mean square displacement has been calculated for each molecule and plotted as a function of the simulation time. Figure 3 shows the result.
 

c13-f3-water-diffusion
Figure 3: The mean square displacement of the five water molecules at 298 K (PCFF)

It is clearly visible that the water molecules 1 and 3 are far more distant from their start position at the end of the simulation than the other three molecules. To find a correlation with the chemical surrounding of each water molecule MAPS Python scripting capabilities were used to find the shortest distance between the oxygen atom of the water and any oxygen atom of the polymers for each frame of the trajectory. These distances were averaged over all frames of the trajectory. The results are as follows: H2O(1): 334.7 pm, H2O(2): 301.1 pm, H2O(3): 342.1 pm, H2O(4): 385.1 pm and H2O(5): 288.4 pm. These numbers clearly show that the water molecule which travels the least is on average closest to an oxygen atom of the polymer. The situation for the water molecules 1 and 3 which travel the longest distance is not so clear. Their average distance to an oxygen atom of the polymer is similar, but neither particularly short nor long.
 
All water molecules are closer to the oxygen containing part of the polymer chain than to the hydrocarbon end.
 
The situation is similar in case of the methanol molecules. The mean square displacement of some of the methanol molecules is significantly larger than the one for the other three molecules. If one analyses the average distance of the oxygen atom in the methanol molecule and an oxygen atom in the polymer as in the water case the following results are obtained: CH3OH(1): 387.1 pm, CH3OH(2): 376.7 pm, CH3OH(3): 289.8 pm, CH3OH(4): 354.3 pm and CH3OH(5): 378.9 pm. For methanol molecule 3 the average distance to an oxygen atom in the polymer chain is the shortest. This corresponds to the mean square displacement being the smallest for most of the trajectory. For the other four methanol molecules no clear correlation between the mean square displacement and their proximity to the ethylbutylacrylate monomers exists.
 
Since water molecules in liquid water form hydrogen bond networks the effect of such a network on the diffusion of the water molecules has been studied as well as described in the previous section. Although the simulation was started with an oxygen-oxygen distance between the water molecules of between 287 and 315 pm the water molecules move during the simulation to distances between 1635 and 2754 pm. These values are very similar to the simulation where the water molecules were placed at random start positions. There the water molecules are between 1743 and 3184 pm apart at the end of the simulation.
 
Conclusions: Molecular Dynamics simulations have been used to study the diffusion process of Water and Methanol in a polymer matrix. Detailed insight of the diffusion process has been gained, and it became obvious from this understanding, that the polymer composition, for example the inclusion of polar polymers in the matrix, has a severe impact on the diffusion process.
 
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  3. H. Yasuda, V. Stannett: J. Polym. Sci. 57 (1962) 907
  4. A. W. Myers, J. A. Meyer, C. E. Rogers, V. Stannett, M. Szwarc, Tappi 44 (1961) 58