It is vitally important to be able to calculate the free energy of mixing (ΔFmix) for polymer blends in order to predict the occurrence of phase separation. This gives insight into the microscale structure of the material, which has a great effect on both the physical and mechanical properties. It is particularly challenging to determine free energy in mixtures that include branched polymers. However, as blends containing epoxy resins are widely used in industry, we are developing methods that will enable the free energy of these to be calculated as a function of crosslinking extent.
The Flory-Huggins model is the most commonly used technique to determine ΔFmix. It treats polymers as random self-avoiding walks on a lattice, and calculates the difference between the free energies of the polymers when arranged in a randomly mixed state versus a phase separated one.
A combinatorial entropy term (ΔSmix) is determined by working out the probability that the polymers will be found in each of these conformations. This can be done by splitting the polymers up into segments, and multiplying together the probabilities of placing each segment onto the lattice. These probabilities take into account previously-placed segments via a mean field approach, where the filling up of the lattice is registered, but not the specific locations of segments other than the one placed directly before.
The energy of mixing (ΔEmix) is determined separately, using nearest neighbour interactions, and free energy is then calculated as normal: ΔFmix = ΔEmix - TΔSmix.
Despite the wide usage of the Flory-Huggins model, it has several shortcomings. Firstly, the mean field approach causes the local environment of each polymer to be ignored in favour of the overall state of the system. This drowns out any differences between linear and branched polymers, when in fact branching points indicate a more densely packed conformation.
The Flory-Huggins model also treats entropy and energy individually, and therefore neglects entropy contributions other than combinatorial. In reality, energetic interactions (in particular strong ones such as hydrogen bonding, which are very common in polymers) impact on the conformation of the polymer and therefore affect the entropy.
It is clear that a better representation is required, but it is notoriously difficult to theoretically improve upon Flory-Huggins. Instead, we are exploiting the use of Monte Carlo simulations to reduce the need for complex mathematics. Our current work extends Meirovitch’s ‘Hypothetical Scanning’ and ‘Hypothetical Scanning Monte Carlo’ methods. The original work calculated the entropy of linear polymers of various lengths, and we have improved the model to cover both free energy and branched polymers.
These simulations follow the same concept as the Flory-Huggins model in that they calculate the probabilities of placing successive polymer segments onto a lattice. This time, however, the positions of previously-placed segments are taken into account, so the local environment is considered. Additionally, nearest neighbour energetic interactions are directly involved in calculating the probabilities, rather than being added on separately at the end, which better represents the interplay between energy and entropy.
