Scaled Debye Length Research Articles

Boundary asymptotics for a non-neutral electrochemistry model with small Debye length

This article addresses the boundary asymptotics of the electrostatic potential in non-neutral electrochemistry models with small Debye length in bounded domains. Under standard physical assumptions motivated by non-electroneutral phenomena in oxidation–reduction reactions, we show that the electrostatic potential asymptotically blows up at boundary points with respect to the bulk reference potential as the scaled Debye length tends to zero. The analysis gives a lower bound for the blow-up rate with respect to the model parameters. Moreover, the maximum potential difference over any compact subset of the physical domain vanishes exponentially in the zero-Debye-length limit. The results mathematically confirm the physical description that electrolyte solutions are electrically neutral in the bulk and are strongly electrically non-neutral near charged surfaces.

Effects of overlapping electric double layer on mass transport of a macro-solute across porous wall of a micro/nanochannel for power law fluid.

Effects of overlapping electric double layer and high wall potential on transport of a macrosolute for flow of a power law fluid through a microchannel with porous walls are studied in this work. The electric potential distribution is obtained by coupling the Poisson's equation without considering the Debye-Huckel approximation. The numerical solution shows that the center line potential can be 16% of wall potential at pH 8.5, at wall potential -73 mV and scaled Debye length 0.5. Transport phenomena involving mass transport of a neutral macrosolute is formulated by species advective equation. An analytical solution of Sherwood number is obtained for power law fluid. Effects of fluid rheology are studied in detail. Average Sherwood number is more for a pseudoplastic fluid compared to dilatant upto the ratio of Poiseuille to electroosmotic velocity of 5. Beyond that, the Sherwood number is independent of fluid rheology. Effects of fluid rheology and solute size on permeation flux and concentration of neutral solute are also quantified. More solute permeation occurs as the fluid changes from pseudoplastic to dilatant.

Quasi-neutral limit of the full Navier–Stokes–Fourier–Poisson system

The quasi-neutral limit of the full Navier–Stokes–Fourier–Poisson system in the torus Td (d≥1) is considered. We rigorously prove that as the scaled Debye length goes to zero, the global-in-time weak solutions of the full Navier–Stokes–Fourier–Poisson system converge to the strong solution of the incompressible Navier–Stokes equations as long as the latter exists. In particular, the effect of large temperature variations is taken into account.

Numerical approximation of the Euler–Maxwell model in the quasineutral limit

We derive and analyze an Asymptotic-Preserving scheme for the Euler–Maxwell system in the quasi-neutral limit. We prove that the linear stability condition on the time-step is independent of the scaled Debye length λ when λ → 0. Numerical validation performed on Riemann initial data and for a model plasma opening switch device show that the AP-scheme is convergent to the Euler–Maxwell solution when Δ x/ λ → 0 where Δ x is the spatial discretization. But, when λ/Δ x → 0, the AP-scheme is consistent with the quasi-neutral Euler–Maxwell system. The scheme is also perfectly consistent with the Gauss equation. The possibility of using large time and space steps leads to several orders of magnitude reductions in computer time and storage.

Sherwood number in porous microtube due to combined pressure and electroosmotically driven flow

Mass transfer of a neutral solute in a porous microtube is quantified in this study. An analytical expression of the Sherwood number is developed from first principles for combined flow of pressure driven and electroosmotic flow. Similarity solution method is adopted for solution of convective-diffusive species balance equation with coupled velocity profile, within the mass transfer boundary layer. It is observed that the Sherwood number increases with decrease in the Debye length (as the electric double layer becomes more compact) and it becomes constant beyond scaled Debye length of 60. Effects of the Reynolds number, dimensionless suction velocity, ratio of driving force and scaled Debye length have been investigated in detail. The analysis is useful for efficient design of microfluidic devices and flow through porous media.

Sherwood number in flow through parallel porous plates (Microchannel) due to pressure and electroosmotic flow

AbstractAn expression for Sherwood number is developed from first principles for combined pressure‐driven and electroosmotic flow in a porous rectangular microchannel. This quantifies the mass transfer of an electrically neutral solute in the microchannel and is useful for designing microfluidic devices and porous media flows. The convective‐diffusive species balance equation, coupled with the velocity field, is solved within the mass transfer boundary layer utilizing similarity method. From the simulations, it is observed that the Sherwood number increases as the electric double layer near the channel wall becomes more compact (as manifested through a decrease in the Debye length), and it reaches a constant value around the scaled Debye length of 40. The Sherwood number becomes constant at higher Debye lengths as electrokinetic effects become negligible. A detailed analysis of dependence of Reynolds number, dimensionless permeation velocity, ratio of driving force and scaled Debye length on Sherwood number is presented. © 2011 American Institute of Chemical Engineers AIChE J, 58: 1693–1703, 2012

QUASI-NEUTRAL LIMIT OF THE MULTIDIMENSIONAL DRIFT-DIFFUSION MODELS FOR SEMICONDUCTORS

This paper is devoted to the rigorous justification of the quasi-neutral limit of bipolar transient drift-diffusion models for semiconductors with p–n junctions in the multidimensional space. The general initial data and smooth sign-changing doping profiles with good boundary conditions are considered. The limit is performed rigorously by using multiple scaling asymptotic analysis, in which one main point is the construction of a more accurate approximate solution involving the effect of initial layer. The uniform estimates with respect to the scaled Debye length are obtained through the elaborate energy method and the relative entropy functional method.

Quasi-neutral limit to the drift–diffusion models for semiconductors with physical contact-insulating boundary conditions and the general sign-changing doping profile

The quasi-neutral limit in a bipolar drift–diffusion model for semiconductors with physical contact-insulating boundary conditions, the general sign-changing doping profile and general initial data which allow the presence of the left and right boundary layers and the initial layers is studied in the one-dimensional case. The dynamic structure stability of the solution with respect to the scaled Debye length is proven by the asymptotic analysis of singular perturbation and the entropy-energy method. The key point of the proof is to use sufficiently the fact that the ‘length’ of the boundary layer is very small in a short time period.

Quasineutral limit of the electro-diffusion model arising in electrohydrodynamics

The electro-diffusion model, which arises in electrohydrodynamics, is a coupling between the Nernst–Planck–Poisson system and the incompressible Navier–Stokes equations. For the generally smooth doping profile, the quasineutral limit (zero-Debye-length limit) is justified rigorously in Sobolev norm uniformly in time. The proof is based on the elaborate energy analysis and the key point is to establish the uniform estimates with respect to the scaled Debye length.

Quasineutral limit of a time-dependent drift–diffusion–Poisson model for p-n junction semiconductor devices

In this paper the vanishing Debye length limit of the bipolar time-dependent drift–diffusion–Poisson equations modelling insulated semiconductor devices with p-n junctions (i.e., with a fixed bipolar background charge) is studied. For sign-changing and smooth doping profile with ‘good’ boundary conditions the quasineutral limit (zero-Debye-length limit) is performed rigorously by using the multiple scaling asymptotic expansions of a singular perturbation analysis and the carefully performed classical energy methods. The key point in the proof is to introduce a ‘density’ transform and two λ-weighted Liapunov-type functionals first, and then to establish the entropy production integration inequality, which yields the uniform estimate with respect to the scaled Debye length. The basic point of the idea involved here is to control strong nonlinear oscillation by the interaction between the entropy and the entropy dissipation.

QUASINEUTRAL LIMIT OF THE MULTI-DIMENSIONAL DRIFT-DIFFUSION-POISSON MODELS FOR SEMICONDUCTORS WITH PN-JUNCTIONS

In this paper the vanishing Debye length limit (space charge neutral limit) of bipolar time-dependent drift-diffusion models for semiconductors with p-n-junctions (i.e. with a fixed bipolar background charge) is studied in the multi-dimensional case. For generally smooth sign-changing doping profiles with good boundary conditions, the quasineutral limit (zero-Debye-length limit) is justified rigorously in the Sobolev's norm uniformly in time. The proof is based on the elaborate energy method and the relative entropy functional method which yields the uniform estimates with respect to the scaled Debye length.

Quasi-neutral Limit of the Drift Diffusion Models for Semiconductors: The Case of General Sign-Changing Doping Profile

In this paper the vanishing Debye length limit (space charge neutral limit) of bipolar time-dependent drift-diffusion models for semiconductors with p-n junctions (i.e., with a fixed bipolar background charge) is studied in one space dimension. For general sign-changing doping profiles, the quasi-neutral limit (zero-Debye-length limit) is justified rigorously in the spatial mean square norm uniformly in time. One main ingredient of our analysis is the construction of a more accurate approximate solution, which takes into account the effects of initial and boundary layers, by using multiple scaling matched asymptotic analysis. Another key point of the proof is the establishment of the structural stability of this approximate solution by an elaborate energy method which yields the uniform estimates with respect to the scaled Debye length.

Screened occurrence scattering time for electron–dust grain collisions in dusty plasmas

Plasma screening effects on occurrence time for electron–dust grain scattering in dusty plasmas are investigated using the eikonal method. The semiclassical path and the semiclassical wave function are applied to obtain the eikonal phase and amplitude. The interaction potential is obtained by using the image-charge method. The scaled semiclassical occurrence time is obtained as a function of the impact parameter, dust radius, scattering angle, and scattering energy. It has been found that the occurrence time delay is slightly increased as the scaled Debye length is increased for small scattering angles. However, the occurrence scattering time is increased sinusoidally with the increasing scattering angle.

Asymptotic Waves in the Hydrodynamical Model of Semiconductors Based on Extended Thermodynamics

An analysis of asymptotic wave solutions for the hydrodynamical model of semiconductors based on Extended Thermo-dynamics [1, 2, 3, 4] is presented. The evolution equations form a quasilinear hyperbolic system, coupled to the Poisson equation for the electric potential. The aim of this article is to describe the far field for such a model by a suitable adaptation of the asymptotic expansion proposed in [5, 6, 7]. The stationary solution for uniformly doped semiconductor has been taken as unperturbed state. We distinguish two cases: perturbations with wavelength smaller than the scaled Debye length and perturbations of the same order of the Debye length. In the first case the perturbation of the electric field is negligible and the resulting model equation exhibits nonlinear and relaxation effects. In the second case a drift term due to the presence of a non negligible perturbation of the electric field arises as well.

On Maxwellian equilibria of insulated semiconductors

A semi-linear elliptic integro-differential equation subject to homogeneous Neumann boundary conditions for the equilibrium potential in an insulated semiconductor device is considered. A variational formulation gives existence and uniqueness. The limit as the scaled Debye length tends to zero is analysed. Two different cases occur. If the number of free electrons and holes is sufficiently high, local charge neutrality prevails throughout the device. Otherwise, depletion regions occur, and the limiting potential is the solution of a free boundary problem.

A quantum regularization of the one-dimensional hydrodynamic model for semiconductors

The steady-state hydrodynamic equations for isothermal states including the quantum Bohm potential are analyzed. The one-dimensional equations for the electron current density and the particle density are coupled self-consistently to the Poisson equation for the electric potential. The quantum correction can be interpreted as a dispersive regularization of the classical hydrodynamic equations. Physically motivated Dirichlet and Neumann boundary conditions for the electron density are prescribed. The existence and uniqueness of strong solutions for sufficiently small current densities are proven. Furthermore, the classical limit (vanishing scaled Planck constant) and the zero-space-charge limit (vanishing scaled Debye length) are performed. The proofs are based on a transformation of variable for the electron density, yielding a fourth-order, elliptic equation for the new variable. As a by-product of the classical limit, the existence of subsonic solutions to the hydrodynamic system is obtained. Finally, numerical examples are presented showing that for ``large'' current densities, fast oscillations in the particle density occur as the scaled Planck constant tends to zero.

Applicability of the High Field Model: An Analytical Study Via Asymptotic Parameters Defining Domain Decomposition

In this paper, we present a mesoscopic-macroscopic model of self-consistent charge transport. It is based upon an asymptotic expansion of solutions of the Boltzmann Transport Equation (BTE). We identify three dimensionless parameters from the BTE. These parameters are, respectively, the quotient of reference scales for drift and thermal velocities, the scaled mean free path, and the scaled Debye length. Such parameters induce domain dependent macroscopic approximations. Particular focus is placed upon the so-called high field model, defined by the regime where drift velocity dominates thermal velocity. This model incorporates kinetic transition layers, linking mesoscopic to macroscopic states. Reference scalings are defined by the background doping levels and distinct, experimentally measured mobility expressions, as well as locally determined ranges for the electric fields. The mobilities reflect a coarse substitute for reference scales of scattering mechanisms. See [9] for elaboration.The high field approximation is a formally derived modification of the augmented drift-diffusion model originally introduced by Thornber some fifteen years ago [25]. We are able to compare our approach with the earlier kinetic approach of Baranger and Wilkins [5] and the macroscopic approach of Kan, Ravaioli and Kerkhoven [20].