Polymer phase behavior

Chi parameter by “morphing”

“How can we use simulations to predict polymer miscibility?”

Polymers are notoriously difficult to blend.  A tiny repulsive interaction between unlike monomers, when summed along an entire long chain, can be enough to overcome the entropic tendency of chains to mix uniformly.  So predicting the repulsive interaction per monomer (the “chi parameter” χ) is very challenging, because of the sensitive required.

We have developed an amazingly sensitive method for obtaining χ from molecular dynamics simulations.  In essence, we measure the thermodynamic work required to slowly “morph” half the chains in a pure-A melt into B chains, and compare this to the work to morph all the chains in a melt from A to B.  The difference is the excess free energy of mixing, which determines chi.   

The method is sensitive for two reasons.  First, the work of morphing is computed along a path on which nothing horrible happens to the polymers.  (For example, we do not boil the mixed and pure states, and try to infer the tiny free energy of mixing by taking the difference between the heats of vaporization.)  Second, we average the tiny changes in energy as the system morphs, over a large number of monomers, and thereby get very good statistics.

Chi for semiflexible chains, as a function of the ratio of stiffness for A and B chains (points, simulation results; dashed curve, analytical theory).

We first applied the method to blends of bead-spring chains of identical beads but different chain stiffness.  Chi for this system is purely entropic, arising from differences in how stiff and flexible chains pack.  Chi for such blends was predicted long ago using a field-theoretic approach;  we have obtained this chi to within 10-4 per bead, with results that compare very well to theoretical predictions.

We have also obtained chi for the simplest coarse-grained model of nonpolar polymer blends: flexible bead-spring chains with different Lennard-Jones interactions between A and B monomers. Using these chi values and self-consistent field theory (SCFT), we predict the interfacial profile for phase-separated binary blends. Our SCFT predicts agree with simulation results for the profile shape, which serves as an important check on our approach. 

Left: predicted concentration profiles for interfaces between immiscible polymers (solid curves), compared to average profiles from simulations. Right: snapshots of interface for the two cases shown above (0.85 weakly immiscible, 0.8 more strongly immiscible).

Our method is extremely general, and can be even be applied to “morphing” atomistic chains from one chemical structure to another, finally providing a way to predict miscibility of chemically realistic polymers.  This approach naturally incorporates realistic packing of monomers, and properly includes effects of configurational entropy, energetic attractions, and steric repulsion between monomers.

However, useful insights into how chain architecture influences chi can also be achieved from bead-spring chain models.  We are now exploring variations in chi for blends of propylene-like structures, in which either the main chain, side group, or branch point bead has different interactions from the rest. This work will illustrate how much chi can vary from purely architectural effects with the exact same monomers. 

Kozuch, D. J., Zhang, W., and Milner, S. T. “Predicting the Flory-Huggins Parameter for Polymers with Stiffness Mismatch From Molecular Dynamics Simulations” Polymers 8, no. 6 (2016): doi:10.3390/polym8060241

Zhang, W., Gomez, E. D., and Milner, S. T. “Predicting Flory-Huggins Chi From Simulations” Physical Review Letters 119, no. 1 (2017): doi:10.1103/PhysRevLett.119.017801

Nematic coupling by “pulling”

“How strongly do stiff polymer chains align with their neighbors?”

Semiconducting polymers for photocell materials are rather stiff, and can potentially form nematic phases, in which chains tend to orient in a common direction without forming a crystal. Nematic order can be helpful for making thin-film devices with superior properties.  We have developed a new simulation method to measure the orientational coupling for real polymers, which quantifies the tendency for nearby chain segments to align.

Above: simulation snapshots of a melt of P3HT semiconducting polymer, for (a) no forces applied, and (b) chains under tension. Below: P3HT phase diagram. Blue circles are from our simulations; red triangles are published data.

In our method, we apply a modest tension to the chains.  The tension stretches the chains, and weakly aligns the monomers along the direction of tension.  Intuitively, if the chains have orientational coupling, the mean alignment induced by the tension will be amplified, which causes the chains to orient and stretch more than would result from the effect of tension on each chain individually.  We determine the orientational coupling, by comparing simulation results for the alignment under tension to predictions from self-consistent field theory.

With a value for the orientational coupling, we use self-consistent field theory to predict the nematic-to-isotropic transition temperature TNI.  For common semiconducting polymer P3HT, we estimate TNI to be in an accessible range, well above the freezing temperature for shorter chains.  

The isotropic to nematic transition temperature depends on chain stiffness.  Locating this first-order transition in simulations by lowering the temperature is difficult because of hysteresis.  We have recently developed a new way to accurately locate the phase boundary, by observing the propagation of nematic order from a flat surface.  The surface promotes local chain alignment even in the isotropic phase, and thus nucleates the nematic phase.  

We have used this method to identify the isotropic to nematic transition in bead-spring chains as a function of  chain stiffness.  The volume fraction for formation of a nematic phase varies inversely with the persistence length Lp for chains of modest stiffness, in agreement with previous analytical predictions.  For very stiff chains with Lp greater than about six repeat units, a weaker scaling with chain stiffness is found, and is still unexplained by analytical arguments.  

Zhang, W., Gomez, E. D., and Milner, S. T. “Predicting Nematic Phases of Semiflexible Polymers” Macromolecules 48, no. 5 (2015): 1454–1462. doi:10.1021/acs.macromol.5b00013

Zhang, W., Gomez, E. D., and Milner, S. T. “Using Surface-Induced Ordering to Probe the Isotropic-to-Nematic Transition for Semiflexible Polymers” Soft Matter 12, no. 28 (2016): 6141–6147. doi:10.1039/c6sm01258b