Written by Don Monroe
In his talk accepting the Materials Theory Award on Tuesday, Glenn H. Fredrickson of the University of California at Santa Barbara described a “methodology of molecularly-inspired simulations, not by pushing particles around, but by evolving fields in a stochastic way.” These field-theoretic simulations allow the design of polymer formulations and soft materials that are structured on the nanometer to micron scale, which are extremely difficult to simulate by traditional methods because of the wide range of lengths scales and, for polymers, time scales.
The field theories are expressed in the formal mathematical framework of statistical mechanics. The starting point for defining the field is a coarse-gained particle-based model. This underlying model can include discrete or continuous polymer chains, which can be branched in arbitrary ways, and can include both contact and long-range interactions between different polymer segments.
To generate a statistical field theory from the ensemble of configurations from the particle-based model, Fredrickson has mostly employed the auxiliary-field method proposed by Edwards. This procedure defines a field that must be complex (in the mathematical sense) to allow decoupling different polymer molecules. The self-consistent field theory version of this method is a mean-field approximation that is most accurate for dense, high–molecular-weight melts, in which each polymer molecule is penetrated by many others.
For situations where this approximation is not appropriate, Fredrickson and his colleagues adapted “complex Langevin dynamics” to sample the exact field theory, which sidesteps the oscillating signs that bedevil simulations of such complex fields. He showed two applications. The first was the phase separation (“coacervation”) in polyelectrolyte complexes that result from mixing polyanion and polycation solutions. “Mean field theory cannot predict the segregation,” Fredrickson said, but his group produced detailed phase diagrams for these systems.
Another example concerned thermoplastic elastomers, traditionally consisting of triblock copolymers with stiff styrene chain ends sandwiching a softer central polymer. The properties of these materials depend on how the different regions organize, and the material is only elastic when the soft component forms a continuous structure through the material. Fredrickson’s team explored a branched block copolymer structure that can be elastic even when up to 70% of the material is the stiffer styrene. Their field-theory simulations also provided insight into an unusual “bricks-and-mortar” phase that these materials exhibit for some compositions.
An alternative method for generating a field theory, called “coherent states,” also goes back to Edwards. This representation includes not just spatial dimensions but a dimension corresponding to position along the chain. “We have learned how to build complex Langevin simulation methods that work in this language,” Fredrickson said. The results agree with the auxiliary field approach, but are noisier and thus require longer simulation times.
However, this formulation also has a close relationship with the quantum field theories that inspired it in that the stochastic excursions of virtual quantum particles in imaginary time are analogous to the geometry of a ring polymer. In unpublished work, Fredrickson’s team has used this coherent-states formalism to explore the behavior of bosons, specifically by reproducing the phase diagram of superfluid Helium-4. They are aiming to extend the results to nonuniform systems, such as the cold bosons in optical traps that are a very active area of experimental low-temperature physics. “We think these simulations have some real potential,” Fredrickson concluded.
The Materials Theory Award, endowed by Toh-Ming Lu and Gwo Ching Wang, recognizes exceptional advances made by materials theory to the fundamental understanding of the structure and behavior of materials. Fredrickson received the award “for pioneering the development of field-theoretic computer simulation methods and their application to investigate and design self-assembling polymers and soft materials.”