Variational Methods and Planar Elliptic Growth

A nested family of growing or shrinking planar domains is called a Laplacian growth process if the normal velocity of each domain's boundary is proportional to the gradient of the domain's Green function with a fixed singularity on the interior. In this paper we review the Laplacian growth model and its key underlying assumptions, so that we may consider a generalization to so-called elliptic growth, wherein the Green function is replaced with that of a more general elliptic operator--this models, for example, inhomogeneities in the underlying plane. In this paper we continue the development of the underlying mathematics for elliptic growth, considering perturbations of the Green function due to those of the driving operator, deriving characterizations and examples of growth, developing a weak formulation of growth via balayage, and discussing of a couple of inverse problems in the spirit of Calder\'on. We conclude with a derivation of a more delicate, reregularized model for Hele-Shaw flow

Mineralogy of the Martian surface from Mariner 6/7 infrared spectrometer data

The Mariner 6/7 Infrared Spectrometer (IRS) experiment data from the 1969 Mars flyby encounters represent a valuable source of information about the IR reflection/emission spectrum of Mars in the 1.9 and 14.4 micron region. During 1990, the wavelength calibration of the IRS data was completely redone, using information from inflight spectra of Mars taken through a polystyrene film and from the locations of Martian CO2 bands. The response functions of the two instruments were then rederived using laboratory black body spectra. Also, a particular approach was taken to study the IRS data in which the effects of uncertain wavelength and intensity calibration can be minimized. This involves doing ratios of spectra. These are of particular value when applied to study contrasts between various albedo domains on the Mars surface and between spectra with differing emission angle. The latter provide a means of assessing the contribution of the atmosphere and airborne dust to the spectra. Given here are graphs showing two near infrared radiance spectra for very different albedo regions and a ratio of two spectra obtained at very different total solar pathlength (airmass)

Theory of voltammetry in charged porous media

We couple the Leaky Membrane Model, which describes the diffusion and electromigration of ions in a homogenized porous medium of fixed background charge, with Butler-Volmer reaction kinetics for flat electrodes separated by such a medium in a simple mathematical theory of voltammetry. The model is illustrated for the prototypical case of copper electro-deposition/dissolution in aqueous charged porous media. We first consider the steady state with three different experimentally relevant boundary conditions and derive analytical or semi-analytical expressions for concentration profiles, electric potential profiles, current-voltage relations and overlimiting conductances. Next, we perform nonlinear least squares fitting on experimental data, consider the transient response for linear sweep voltammetry and demonstrate good agreement of the model predictions with experimental data. The experimental datasets are for copper electrodeposition from copper(II) sulfate solutions in a variety of nanoporous media, such as anodic aluminum oxide, cellulose nitrate and polyethylene battery separators, whose internal surfaces are functionalized with positively and negatively charged polyelectrolyte polymers.Comment: 39 pages, 12 figures, 5 tables; clarified where other parameters are taken from and fixed typo

Homogenization of the Poisson-Nernst-Planck Equations for Ion Transport in Charged Porous Media

Effective Poisson-Nernst-Planck (PNP) equations are derived for macroscopic ion transport in charged porous media under periodic fluid flow by an asymptotic multi-scale expansion with drift. The microscopic setting is a two-component periodic composite consisting of a dilute electrolyte continuum (described by standard PNP equations) and a continuous dielectric matrix, which is impermeable to the ions and carries a given surface charge. Four new features arise in the upscaled equations: (i) the effective ionic diffusivities and mobilities become tensors, related to the microstructure; (ii) the effective permittivity is also a tensor, depending on the electrolyte/matrix permittivity ratio and the ratio of the Debye screening length to the macroscopic length of the porous medium; (iii) the microscopic fluidic convection is replaced by a diffusion-dispersion correction in the effective diffusion tensor; and (iv) the surface charge per volume appears as a continuous "background charge density", as in classical membrane models. The coefficient tensors in the upscaled PNP equations can be calculated from periodic reference cell problems. For an insulating solid matrix, all gradients are corrected by the same tensor, and the Einstein relation holds at the macroscopic scale, which is not generally the case for a polarizable matrix, unless the permittivity and electric field are suitably defined. In the limit of thin double layers, Poisson's equation is replaced by macroscopic electroneutrality (balancing ionic and surface charges). The general form of the macroscopic PNP equations may also hold for concentrated solution theories, based on the local-density and mean-field approximations. These results have broad applicability to ion transport in porous electrodes, separators, membranes, ion-exchange resins, soils, porous rocks, and biological tissues

Tuning the stability of Electrochemical Interfaces by Electron Transfer reactions

The morphology of interfaces is known to play fundamental role on the efficiency of energy-related applications, such light harvesting or ion intercalation. Altering the morphology on demand, however, is a very difficult task. Here, we show ways the morphology of interfaces can be tuned by driven electron transfer reactions. By using non-equilibrium thermodynamic stability theory, we uncover the operating conditions that alter the interfacial morphology. We apply the theory to ion intercalation and surface growth where electrochemical reactions are described using Butler-Volmer or coupled ion-electron transfer kinetics. The latter connects microscopic/quantum mechanical concepts with the morphology of electrochemical interfaces. Finally, we construct non-equilibrium phase diagrams in terms of the applied driving force (current/voltage) and discuss the importance of engineering the density of states of the electron donor in applications related to energy harvesting and storage, electrocatalysis and photocatalysis.Comment: 10 pages, 6 figure