Nonlinear Dynamics of Capacitive Charging and Desalination by Porous Electrodes

The rapid and efficient exchange of ions between porous electrodes and aqueous solutions is important in many applications, such as electrical energy storage by super-capacitors, water desalination and purification by capacitive deionization (or desalination), and capacitive extraction of renewable energy from a salinity difference. Here, we present a unified mean-field theory for capacitive charging and desalination by ideally polarizable porous electrodes (without Faradaic reactions or specific adsorption of ions) in the limit of thin double layers (compared to typical pore dimensions). We illustrate the theory in the case of a dilute, symmetric, binary electrolyte using the Gouy-Chapman-Stern (GCS) model of the double layer, for which simple formulae are available for salt adsorption and capacitive charging of the diffuse part of the double layer. We solve the full GCS mean-field theory numerically for realistic parameters in capacitive deionization, and we derive reduced models for two limiting regimes with different time scales: (i) In the "super-capacitor regime" of small voltages and/or early times where the porous electrode acts like a transmission line, governed by a linear diffusion equation for the electrostatic potential, scaled to the RC time of a single pore. (ii) In the "desalination regime" of large voltages and long times, the porous electrode slowly adsorbs neutral salt, governed by coupled, nonlinear diffusion equations for the pore-averaged potential and salt concentration

Dynamics of Counterion Condensation

Using a generalization of the Poisson-Boltzmann equation, dynamics of counterion condensation is studied. For a single charged plate in the presence of counterions, it is shown that the approach to equilibrium is diffusive. In the far from equilibrium case of a moving charged plate, a dynamical counterion condensation transition occurs at a critical velocity. The complex dynamic behavior of the counterion cloud is shown to lead to a novel nonlinear force-velocity relation for the moving plate.Comment: 5 pages, 1 ps figure included using eps

Quantum Kinetic Theory of Condensate Growth---Comparison of Experiment and Theory

In a major extension of our previous model (C.W. Gardiner, P. Zoller, R.J. Ballagh and M.J. Davis, Phys. Rev. Lett. 79, 1793 (1997)) of condensate growth, we take account of the evolution of the occupations of lower trap levels, and of the full Bose-Einstein formula for the occupations of higher trap levels. We find good agreement with experiment, especially at higher temperatures. We also confirm the picture of the ``kinetic'' region of evolution, introduced by Kagan et al, for the time up to the initiation of the condensate. The behavior after initiation essentially follows our original growth equation, but with a substantially increased rate coefficient W^{+}.Comment: RevTeX, 4 pages and 4 eps figure

Forces on a boiling bubble in a developing boundary layer, in microgravity with g-jitter and in terrestrial conditions

Terrestrial and microgravity flow boiling experiments were carried out with the same test rig, comprising a locally heated artificial cavity in the center of a channel near the frontal edge of an intrusive glass bubble generator. Bubble shapes were in microgravity generally not far from those of truncated spheres,which permitted the computation of inertial lift and drag from potential flow theory for truncated spheres approximating the actual shape. For these bubbles, inertial lift is counteracted by drag and both forces are of the same order of magnitude as g-jitter. A generalization of the Laplace equation is found which applies to a deforming bubble attached to a plane wall and yields the pressure difference between the hydrostatic pressures in the bubble and at the wall, p. A fully independent way to determine the overpressure p is given by a second Euler-Lagrange equation. Relative differences have been found to be about 5% for both terrestrial and microgravity bubbles. A way is found to determine the sum of the two counteracting major force contributions on a bubble in the direction normal to the wall from a single directly measurable quantity. Good agreement with expectation values for terrestrial bubbles was obtained with the difference in radii of curvature averaged over the liquid-vapor interface, (1/R2 − 1/R1), multiplied with the surface tension coefficient, σ. The new analysis methods of force components presented also permit the accounting for a surface tension gradient along the liquid-vapor interface. No such gradients were found for the present measurements

The profile of a narrow line after single scattering by Maxwellian electrons: relativistic corrections to the kernel of the integral kinetic equation

The frequency distribution of photons in frequency that results from single Compton scattering of monochromatic radiation on thermal electrons is derived in the mildly relativistic limit. Algebraic expressions are given for (1) the photon redistribution function, K(nu,Omega -> nu',Omega'), and (2) the spectrum produced in the case of isotropic incident radiation, P(nu -> nu'). The former is a good approximation for electron temperatures kT_e < 25 keV and photon energies hnu < 50 keV, and the latter is applicable when hnu(hnu/m_ec^2) < kT_e < 25 keV, hnu < 50 keV. Both formulae can be used for describing the profiles of X-ray and low-frequency lines upon scattering in hot, optically thin plasmas, such as present in clusters of galaxies, in the coronae of accretion disks in X-ray binaries and AGNs, during supernova explosions, etc. Both formulae can also be employed as the kernels of the corresponding integral kinetic equations (direction-dependent and isotropic) in the general problem of Comptonization on thermal electrons. The K(nu,Omega -> nu',Omega') kernel, in particular, is applicable to the problem of induced Compton interaction of anisotropic low-frequency radiation of high brightness temperature with free electrons in the vicinity of powerful radiosources and masers. Fokker-Planck-type expansion (up to fourth order) of the integral kinetic equation with the P(nu -> nu') kernel derived here leads to a generalization of the Kompaneets equation. We further present (1) a simpler kernel that is necessary and sufficient to derive the Kompaneets equation and (2) an expression for the angular function for Compton scattering in a hot plasma, which includes temperature and photon energy corrections to the Rayleigh angular function.Comment: 29 pages, 17 figures, accepted for publication in ApJ, uses emulateapj.sty, corrects misprints in previous astro-ph versio

Kinetics of Bose-Condensation

The process of condensation in the system of scalar Bosons with weak $\lambda \phi^4$ interaction is considered. Boltzmann kinetic equation is solved numerically. Bose condensation proceeds in two stages: At the first stage condensate is still absent but there is non-zero inflow of particles towards $\vec{{\bf p}} = 0$ and the distribution function at $\vec{{\bf p}} = 0$ grows from finite values to infinity. At the second stage there are two components, condensate and particles, reaching their equilibrium values. We show that the evolution in both stages proceeds in a self - similar way and find the time needed for condensation, which is finite.Comment: 12 pages, LaTeX RevTeX 3.0, includes 4 eps figure

Quantum Kinetic Theory V: Quantum kinetic master equation for mutual interaction of condensate and noncondensate

A detailed quantum kinetic master equation is developed which couples the kinetics of a trapped condensate to the vapor of non-condensed particles. This generalizes previous work which treated the vapor as being undepleted.Comment: RevTeX, 26 pages and 5 eps figure

Thermal Properties of Two-Dimensional Advection Dominated Accretion Flow

We study the thermal structure of the widely adopted two-dimensional advection dominated accretion flow (ADAF) of Narayan & Yi (1995a). The critical radius for a given mass accretion rate, outside of which the optically thin hot solutions do not exist in the equatorial plane, agrees with one-dimensional study. However, we find that, even within the critical radius, there always exists a conical region of the flow, around the pole, which cannot maintain the assumed high electron temperature, regardless of the mass accretion rate, in the absence of radiative heating. This could lead to torus-like advection inflow shape since, in general, the ions too will cool down. We also find that Compton preheating is generally important and, if the radiative efficiency, defined as the luminosity output divided by the mass accretion rate times the velocity of light squared, is above sim 4x10^-3, the polar region of the flow is preheated above the virial temperature by Compton heating and it may result in time-dependent behaviour or outflow while accretion continues in the equatorial plane. Thus, under most relevant circumstances, ADAF solutions may be expected to be accompanied by polar outflow winds. While preheating instabilities exist in ADAF, as for spherical flows, the former are to some extent protected by their characteristically higher densities and higher cooling rates, which reduce their susceptibility to Compton driven overheating.Comment: 18 pages including 4 figures. AASTEX. Submitted to Ap