Asymptotic Behaviour of Time Stepping Methods for Phase Field Models.

Adaptive time stepping methods for metastable dynamics of the Allen-Cahn and Cahn-Hilliard equations are investigated in the spatially continuous, semi-discrete setting. We analyse the performance of a number of first and second order methods, formally predicting step sizes required to satisfy specified local truncation error σ in the limit of small length scale parameter ϵ → 0 during meta-stable dynamics. The formal predictions are made under stability assumptions that include the preservation of the asymptotic structure of the diffuse interface, a concept we call profile fidelity. In this setting, definite statements about the relative behaviour of time stepping methods can be made. Some methods, including all so-called energy stable methods but also some fully implicit methods, require asymptotically more time steps than others. The formal analysis is confirmed in computational studies. We observe that some provably energy stable methods popular in the literature perform worse than some more standard schemes. We show further that when Backward Euler is applied to meta-stable Allen-Cahn dynamics, the energy decay and profile fidelity properties for these discretizations are preserved for much larger time steps than previous analysis would suggest. The results are established asymptotically for general interfaces, with a rigorous proof for radial interfaces. It is shown analytically and computationally that for most reaction terms, Eyre type time stepping performs asymptotically worse due to loss of profile fidelity.

Existence, bifurcation, and geometric evolution of quasi-bilayers in the multicomponent functionalized Cahn-Hilliard equation.

Multicomponent bilayer structures arise as the ubiquitous plasma membrane in cellular biology and as blends of amphiphilic copolymers used in electrolyte membranes, drug delivery, and emulsion stabilization within the context of synthetic chemistry. We present the multicomponent functionalized Cahn-Hilliard (mFCH) free energy as a model which allows competition between bilayers with distinct composition and between bilayers and higher codimensional structures, such as co-dimension two filaments and co-dimension three micelles. We construct symmetric and asymmetric homoclinic bilayer profiles via a billiard limit potential and show that co-dimensional bifurcation is driven by the experimentally observed layer-by-layer pearling mechanism. We investigate the stability and slow geometric evolution of multicomponent bilayer interfaces within the context of an [Formula: see text] gradient flow of the mFCH, addressing the impact of aspect ratio of the amphiphile (lipid or copolymer unit) on the intrinsic curvature and the codimensional bifurcation. In particular we derive a Canham-Helfrich sharp interface energy whose intrinsic curvature arises through a Melnikov parameter associated to amphiphile aspect ratio.

Dependence of the dielectric constant of electrolyte solutions on ionic concentration: A microfield approach.

We present a microfield approach for studying the dependence of the orientational polarization of the water in aqueous electrolyte solutions upon the salt concentration and temperature. The model takes into account the orientation of the solvent dipoles due to the electric field created by ions, and the effect of thermal fluctuations. The model predicts a dielectric functional dependence of the form ɛ(c)=ɛ_{w}-ßL(3αc/ß),ß=ɛ_{w}-ɛ_{ms}, where L is the Langevin function, c is the salt concentration, ɛ_{w} is the dielectric of pure water, ɛ_{ms} is the dielectric of the electrolyte solution at the molten salt limit, and α is the total excess polarization of the ions. The functional form gives a remarkably accurate description of the dielectric constant for a variety of salts and a wide range of concentrations.

Systematic interpretation of differential capacitance data.

Differential capacitance (DC) data have been widely used to characterize the structure of electrolyte solutions near charged interfaces and as experimental validation of models for electrolyte structure. Fixing a large class of models of electrolyte free energy that incorporate finite-volume effects, a reduction is identified which permits the identification of all free energies within that class that return identical DC data. The result is an interpretation of DC data through the equivalence classes of nonideality terms, and associated boundary layer structures, that cannot be differentiated by DC data. Specifically, for binary salts, DC data, even if measured over a range of ionic concentrations, are unable to distinguish among models which exhibit charge asymmetry, charge reversal, and even ion crowding. The reduction applies to capacitors which are much wider than the associated Debye length and to finite-volume terms that are algebraic in charge density. However, within these restrictions the free energy is shown to be uniquely identified if the DC data are supplemented with measurements of the excess chemical potential of the system in the bulk state.

Biased gradient squared descent saddle point finding method.

The harmonic approximation to transition state theory simplifies the problem of calculating a chemical reaction rate to identifying relevant low energy saddle points in a chemical system. Here, we present a saddle point finding method which does not require knowledge of specific product states. In the method, the potential energy landscape is transformed into the square of the gradient, which converts all critical points of the original potential energy surface into global minima. A biasing term is added to the gradient squared landscape to stabilize the low energy saddle points near a minimum of interest, and destabilize other critical points. We demonstrate that this method is competitive with the dimer min-mode following method in terms of the number of force evaluations required to find a set of low-energy saddle points around a reactant minimum.

Spatial and temporal mapping of water content across Nafion membranes under wetting and drying conditions.

Water transport and water management are fundamental to polymer electrolyte membrane fuel cell operation. Accurate measurements of water content within and across the Nafion layer are required to elucidate water transport behavior and validate existing numerical models. We report here a direct measurement of water content profiles across a Nafion layer under wetting and drying conditions, using a novel magnetic resonance imaging methodology developed for this purpose. This method, multi-echo double half k-space spin echo single point imaging, based on a pure phase encode spin echo, is designed for high resolution 1D depth imaging of thin film samples. The method generates high resolution (<8 microm) depth images with an SNR greater than 20, in an image acquisition time of less than 2 min. The high temporal resolution permits water content measurements in the transient states of wetting and drying, in addition to the steady state.

Magnetic resonance imaging of water content across the Nafion membrane in an operational PEM fuel cell.

Water management is critical to optimize the operation of polymer electrolyte membrane fuel cells. At present, numerical models are employed to guide water management in such fuel cells. Accurate measurements of water content variation in polymer electrolyte membrane fuel cells are required to validate these models and to optimize fuel cell behavior. We report a direct water content measurement across the Nafion membrane in an operational polymer electrolyte membrane fuel cell, employing double half k-space spin echo single point imaging techniques. The MRI measurements with T2 mapping were undertaken with a parallel plate resonator to avoid the effects of RF screening. The parallel plate resonator employs the electrodes inherent to the fuel cell to create a resonant circuit at RF frequencies for MR excitation and detection, while still operating as a conventional fuel cell at DC. Three stages of fuel cell operation were investigated: activation, operation and dehydration. Each profile was acquired in 6 min, with 6 microm nominal resolution and a SNR of better than 15.