How can we predict, and control, the rate at which polymers crystallize?
A melt of long polymer molecules is hard to crystallize, even if cooled below the freezing temperature. Polymer crystallization proceeds by nucleation. Small ordered regions appear by thermal fluctuations. If they are too small, they quickly melt away. If they are large enough, they grow by adding chain segments on the boundary. How large is large enough, depends on a competition between the interior of the nucleus, which has a lower free energy than the melt, and the surface of the nucleus, which has an interfacial tension — a free energy cost.
So to predict how readily a given polymer will nucleate, we need good values for the interfacial tension between a crystal and its melt, and for the heat of fusion of the crystal (which tells us how much lower is the free energy of the crystal than the supercooled melt). And to control nucleation, an important strategy is to introduce nanoparticles that have a favorable interfacial energy with the crystal versus the melt…
Slab melting
“How can we get melting temperatures and heats of fusion from simulations?”
Simulations most often use periodic boundary conditions, in which the left side of the simulation box is regarded as immediately adjacent to the right side of an identical copy of the system, and so forth. This is done so that the simulated volume has no surfaces, the better to represent bulk behavior.
Simulation snapshots from melting of a slab in periodic boundary conditions (shown here with the gap in the middle). As the temperature rises, the slab melts from the free surfaces inwards.
For melting studies of polymers, this causes problems. Heating a simulated polymer crystal with periodic boundary conditions leads to large superheating, in which the crystal survives as much as 100C above the nominal melting point. This makes it hard to get reasonable values for the melting temperature and heat of fusion, key parameters in nucleation theory.
Typical slab melting data, energy versus time for a fixed heating rate. The inward velocity of the melting front is proportional to T-Tm above the melting temperature, leading to a characteristic quadratic rise in energy. When the melting is done, a linear rise in energy with temperature resumes, corresponding to a constant heat capacity.
To overcome this, we melt a slab. The slab surface nucleates the melt, avoiding superheating. At finite heating rate, we detect the melting transition by the point at which the melting front begins to advance into the crystal. This gives a characteristic quadratic dependence of the system energy with time.
Once we know the true melting temperature, we can compare the energy versus temperature above and below melting, extrapolated to the melting point, to obtain the heat of fusion.
Chen, Q., Sirota, E. B., Zhang, M., Chung, T. C. M., and Milner, S. T. “Free Surfaces Overcome Superheating in Simulated Melting of Isotactic Polypropylene” Macromolecules 48, no. 24 (2015): 8885–8896. doi:10.1021/acs.macromol.5b02030
“Plunger” method for surface tensions
“How can we simulate the interfacial tension between polymer melts, crystals, and substrates?”
Simulation snapshot of “plunger” setup.
Interfacial tensions between a polymer crystal and its melt are a key ingredient in predicting the nucleation barrier. Also, nucleating agents act by providing a favorable substrate to grow a crystal from the surrounding melt. But these interfacial tensions are very challenging to measure directly. We have developed new approaches to obtain interfacial tensions from atomistic simulations of melts adjacent to crystal surfaces. These methods can also be used to screen nucleating agents.
We measure the force on a simulated nanoscale “plunger”, that restrains a melt from flowing into the gap between two crystals cleaved along a given plane. Interactions between the “plunger” (a graphene-like sheet of atoms) and the crystalline walls of the gap are turned off in the simulations.
The average force on the plunger gives the difference between the crystal−vacuum and crystal−melt interfacial free energies. Effectively, we are measuring the “capillary rise” force on the plunger, which restrains the melt from wetting the crystal-vacuum interface.
Separately, we obtain the crystal−vacuum interfacial free energy by measuring the force required to hold two crystals apart at a given separation, and integrating that force with respect to the separation. We obtain the crystal−melt interfacial free energy by subtracting the above values.
Crystal-pulling simulation setup. A crystalline slab is cleaved, and the two halves held at some separation; the force between the two halves is measured.
Chen, Q., Kozuch, D., and Milner, S. T. “‘Plunger’ Method for Simulating Crystal-Melt Interfacial Free Energies” Macromolecules 50, no. 12 (2017): 4797–4806. doi:10.1021/acs.macromol.7b00421