### Functionalized Cahn-Hilliard Free Energy

The Functionalized Cahn-Hilliard (FCH) free energy incorporates amphilicity, counter-ion entropy, and chain packing entropy
into an energy landscape describing the interaction of charged (functionalized) polymers and solvent. The minimizers of the FCH are comprised of
* network morphologies * which are typically * saddle points * of the more traditional Cahn-Hilliard free energy. Newtork morphologies
include co-dimension one bilayers, co-dimension two pores, and co-dimension three micelles, as well as many assocaited defect structures. We have analyzed
the geometric evolution of spatially distributed interfaces, the pearling bifurcations which yield side-band modulated equilibria or which may flow
through pinch-off to higher co-dimensional structutres. In particular we have determined that the evolution of distinct morphologies couples through
the far-field density of amphiphilic units, and that this coupling can drive extinction and pearling bifurcations. The functional form of the
FCH free energy has also motivated a novel class of saddle-point searching algorythms, see [60] and the review article
An Overview of Network Bifurcations in the Functionalized Cahn-Hilliard Free Energy.
The image at left shows a 2+1 mass-preserving gradient flow on the FCH free energy which coarsen towards a network structure.

#### Recent Talks

*Network Formation in Amphiphilic Polymer Mixtures, 74th MidWest PDE, Oct 18-19, 2014*.

#### Related Publications

### Renormalization Group Dynamics

Frequently, coherent structures interact on time scales which are slow compared to relaxation rates. Consequently the infinite dimensional PDE often contains lower dimensional dynamical system with a slow sub-flow described by the parameters of the individual coherent structures. Examples of such system arises in the semi-strong interaction of pulses in singularly perturbed reaction diffusion equations, as well as in dispersively driven optical systems. A rigorous analysis of the stability of the low-dimensional system within the full infinite dimensional system requires generation of semi-group estimates for the weakly time-dependent linearizations of the full system about the evolving coherent structures. We obtain these estimates through a renormalization group proceedure, which is particularly challenging in the regime where the decay rates become comparable to the rate of evolution of the coherent structures.

#### Related Publications

### PEM Fuel Cells/Device Level Models

In conjunction with industrial partners, including Ballard Power Systems, and engineering colleagues, we have developed an analyzed multi-phase models of water, heat, and reactant transport at both the device and material level in polymer electrolyte membrane (PEM) fuel cells. This includes analysis of parasitic reactions which can corrode carbon support materials, lead to Platinum dissolution, and ionomer degradation. The most highly coupled element of PEM fuel cell dynamics is the distribution and phase change of water at the device level. Ionic transport within the functionalized PEM membrane is very sensitive to its hydration level, simultaneously the ionic transport plays a key role in determining local reaction rates which yield water as an end product. The resulting positive feed-back loop, with wet parts of the cell getting wetter, leads to a spatially distributed system which evidences rich, hysteretic dynamics and whose descritizations result in stiff numerical systems.

#### Related Publications

### Nonlinear Optics/Nonlinear Waves

Nonlinear optical systems are a fertile hunting ground for dynamic behavior. My collaborators and I have investigated the existence and stability of breathers in optical lattices, the stability of Boise-Einstein condensates in multi-dimensional optical traps, polarizational mode bifurcations in long-haul optical fibers, and the long-time behavior of systems of pulses in dispersively driven systems. The image presents the temporal oscillations of a breather in a one-space dimension lattice -- the oscillations are concentrated on the central pulse, with weaker oscillations in adjacent pulses. Pulses with symmetric behavior on the right-side of the central pulse are omitted to show detail, from [22].