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.
Hookean Voronoi Energy
A gradient flow of the Hookean Voronoi free energy with N=400 sites. The energy, presented in paper [S4], divides the into Voronoi regions and assigns energy based upon average distance of the site to the boundary via a sisimple Hooks law energy. It is motivated by packing of diblock micelles that form quasi-periodic Frank Kasper phases and periodic C15 Laves phases. The gradient flow starts from random initial data and descends to an equilibriwith 26 defects -- regions that are not 6-sided. In the image regions with 7 sides are marked with a star and 5-sides with a pentagon. Each region is shaded by its local energy. Increasing the number of sites yields an average density of defects, on the order of 8.8% of the regions, and an associated average energy per region that is higher than the
energy of the unit-area regular hexagon. Indeed, in the large N limit the probability of converging to a given average energy approachs a limiting (bulk) distrubtion that excludes ordered states.
[63] K. Promislow and Qiliang Wu, Existence of pearled patterns in the planar Functionalized Cahn-Hilliard equation, J. Differential Equations, to appear (2015).
[62] Shibin Dai and K. Promislow, Competitive Geometric Evolution of Amphiphilic Interfaces, SIAM Journal Math. Analysis, to appear (2015).
[61] A. Doelman, G. Hayrapetyan, K. Promislow, and B. Wetton, Meander and Pearling of Single-Curvature Bilayer interfaces in the Functionalied Cahn-Hilliard equation, SIAM Journal Math. Analysis, 46 (6), 3640-3677 (2014). DOI
[60] J. Duncan, Q. Wu, K. Promislow, G. Henkelman, Biased gradient-squared descent saddle point finding method, Journal Chemical Physics, 140 194102 (2014) DOI.
[59] G. Hayrapetyan and K. Promislow, Spectra of Functionalized Operators arising from hypersurfaces, to appear in ZAMP (2014).
[57] K. Promislow and L. Yang, Existence of compressible bilayers in the functionalized Cahn- Hilliard equation, SIAM J. Dynamical Systems, 13 (2) 629-657 (2014) DOI .
[56] Shibin Dai and K. Promislow, Geometric Evolution of Bilayers under the Functionalized Cahn-Hilliard equation, Proc. Roy. Soc. London, Series A, 469 : 20120505 (2013) DOI.
[52] K. Promislow and H. Zhang, Critical points of Functionalized Lagrangians, Discrete and Continuous Dynamical Systems, A 33 1-16 (2013).
[51] N. Gavish, J. Jones, Z. Xu, A. Christlieb, K. Promislow, Variational Models of Network Formation and Ion Transport: Applications to Peruorosulfonate Ionomer Membranes, Polymers 4 (2012) 630-655.
[47] N. Gavish, G. Hayrapetyan, K. Promislow, L. Yang, Curvature driven flow of bi-layer interfaces, Physica D 240 675-693 (2011).
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
[55] T. Bellsky, A. Doelman, T. Kaper, K. Promislow, Adiabatic Stability of Semi-Strong Interactions in an Activator-Inhibitor system: The Weakly Damped Regime, Indiana U. Math Journal, 62 No. 6 1809-1859 (2013) .
[46] P. van Heijster, A. Doelman, T. Kapper, and K. Promislow, Front interactions in a three-component system, SIAM J. on Applied Dynamical Systems 9 292-232 (2010).
[44] Mohar Guha and K. Promislow, Front propagation in a noisy, nonsmooth excitable media, Discrete and Cont. Dyn. Systems, Ser. A 23 (3) 617-638 (2009)
[41] R. Moore and K. Promislow, The semi-strong limit of multipulse interaction in a thermally driven optical system, J. Diff. Eqs. 245 (6) (2008) 1616-1655.
[34] A. Doelman, T. Kaper, K. Promislow, Nonlinear asymptotic stability of the semi-strong pulse dynamics in a regularized Gierer-Meinhardt model, SIAM Math. Analysis 38 (6) (2007) 1760-1787.
[38] P. A. C. Chang and K. Promislow, Nonlinear stability of oscillatory pulses in the parametric nonlinear Schrodinger equation, Nonlinearity 20 743-763 (2007).
[21] Keith Promislow, A renormalization method for modulational stability of quasi-steady patterns in dispersive systems, SIAM J. Math. Analysis 33 No. 6 (2002), 1455-1482.
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
[49] K. Promislow, J. St. Pierre, B. Wetton, A simple, analytic model of polymer electrolyte membrane fuel cell anode recirculation at operating power including nitrogen crossover, J. Power Sources 196 10050-10056 (2011).
[48] Z. Zhang, K. Promislow, J. Martin, H. Wang, B. J. Balcom, Bi-modal water transport behavior across a simple Nafion membrane, J. Power Sources 196 8525-8530 (2011).
[45] K. Promislow and B. Wetton, PEM fuel cells: A Mathematical Overview, Invited Review Paper to SIAM Applied Math. 70 369-409 (2009)
[39] I. Nazarov and K. Promislow, The impact of membrane constraint on PEM fuel cell water management, J. Electrochem. Soc. 154 B623-630 (7) (2007).
[36] P. A. C. Chang, G.-S. Kim, K. Promislow, B. Wetton, Reduced dimensional computational models of polymer electrolyte membrane fuel cell stacks, J. Comp. Physics 223 797-821 (2007).
[35] J. St-Pierre, B. Wetton, G.S.-Kim, K. Promislow, Limiting current operation of proton exchange membrane fuel cells, J. Electrochem. Soc. 154 (2) (2007) B186-B193.
[33] A. Shah, G.-S. Kim, W. Gervais, A. Young, K. Promislow, J. Li, S.Ye, The effects of water and microstructure on the performance of polymer electrolyte fuel cells, J. Power Sources, 160 (2006), 1251-1268.
[30] I. Nazarov and K. Promislow, Ignition Waves in a Stirred PEM Fuel Cell, Chemical Engineering Science 61 (10) (2006) 3198-3209.
[29] K. Promislow, J. Stockie, B. Wetton, A Sharp Interface Reduction for Multiphase flow in a Porous PEM Fuel Cell Electrode, Proc. Roy. Soc. London: Series A 462, No. 2067 (March 2006) 789-816.
[24] P. Berg, K. Promislow, J. St. Pierre, J. Stumper, B. Wetton, Water management in PEM fuel cells, J. Electrochem. Soc. 151 (No. 3) (2004) A341-A354.
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].
Related Publications
[50] T. Kapitula and K. Promislow, Stability indices for constrained self-adjoint operators, Proc. AMS 140 (2012) 865-880.
[22] R. Carretero-Gonzalez and K. Promislow, Localized breathing solutions for Bose-Einstein condensates in periodic traps. Phys. Rev. A. 66 Sept 033610 (2002).
[19] J. Bronski, L. Carr, R. Carretero-Gonzalez, B. Deconinck, J. Kutz, K. Promislow, Stability of Attractive Bose-Einstein Condensates in a Periodic Potential, Phys. Rev. E 64 Nov. 056615 (2001).
[17] J. Bronski, L. Carr, B. Deconinck, J. Kutz, K. Promislow, Stability of repulsive Bose-Einstein condensate ina period potential, Phys. Rev. E 63 Mar. 036612 (2001)
[15] Keith Promislow and J. Nathan Kutz, Bifurcation and Asymptotic Stability in the Large Detuning Limit of the Optical Parametric Oscillator, Nonlinearity 13 (2000), 675-698.
[14] Yi Li and Keith Promislow, The Mechanism of the Polarizational Mode Instability in Birefringent Fiber Optics, SIAM Journal on Math. Anal., 31 (2000), 1351-1373.
[13] Jerry Bona, Francoise Demengel, Keith Promislow, Fourier Splitting and Dissipation of Nonlinear Dispersive Waves, Proc. Roy. Soc. Edingburg, Series A, 129 (1999), 477-502.
[12] Yi Li and Keith Promislow, Structural Stability of non-ground state traveling wave of a Nonlinear Schrodinger System, Physica D 124 (1998), 137-165.
