Asymptotic analysis of the Poisson-Boltzmann equation describing electrokinetics in porous media

We consider the Poisson-Boltzmann equation in a periodic cell, representative of a porous medium. It is a model for the electrostatic distribution of $N$ chemical species diluted in a liquid at rest, occupying the pore space with charged solid boundaries. We study the asymptotic behavior of its solution depending on a parameter $\beta$ which is the square of the ratio between a characteristic pore length and the Debye length. For small $\beta$ we identify the limit problem which is still a nonlinear Poisson equation involving only one species with maximal valence, opposite to the average of the given surface charge density. This result justifies the {\it Donnan effect}, observing that the ions for which the charge is the one of the solid phase are expelled from the pores. For large $\beta$ we prove that the solution behaves like a boundary layer near the pore walls and is constant far away in the bulk. Our analysis is valid for Neumann boundary conditions (namely for imposed surface charge densities) and establishes rigorously that solid interfaces are uncoupled from the bulk fluid, so that the simplified additive theories, such as the one of the popular Derjaguin, Landau, Verwey and Overbeek (DLVO) approach, can be used. We show that the asymptotic behavior is completely different in the case of Dirichlet boundary conditions (namely for imposed surface potential)

Role of non-ideality for the ion transport in porous media: derivation of the macroscopic equations using upscaling

This paper is devoted to the homogenization (or upscaling) of a system of partial differential equations describing the non-ideal transport of a N-component electrolyte in a dilute Newtonian solvent through a rigid porous medium. Realistic non-ideal effects are taken into account by an approach based on the mean spherical approximation (MSA) model which takes into account finite size ions and screening effects. We first consider equilibrium solutions in the absence of external forces. In such a case, the velocity and diffusive fluxes vanish and the equilibrium electrostatic potential is the solution of a variant of Poisson-Boltzmann equation coupled with algebraic equations. Contrary to the ideal case, this nonlinear equation has no monotone structure. However, based on invariant region estimates for Poisson-Boltzmann equation and for small characteristic value of the solute packing fraction, we prove existence of at least one solution. To our knowledge this existence result is new at this level of generality. When the motion is governed by a small static electric field and a small hydrodynamic force, we generalize O'Brien's argument to deduce a linearized model. Our second main result is the rigorous homogenization of these linearized equations and the proof that the effective tensor satisfies Onsager properties, namely is symmetric positive definite. We eventually make numerical comparisons with the ideal case. Our numerical results show that the MSA model confirms qualitatively the conclusions obtained using the ideal model but there are quantitative differences arising that can be important at high charge or high concentrations.Comment: 46 page

Unified Treatment of Even and Odd Anharmonic Oscillators of Arbitrary Degree

We present a unified treatment, including higher-order corrections, of anharmonic oscillators of arbitrary even and odd degree. Our approach is based on a dispersion relation which takes advantage of the PT-symmetry of odd potentials for imaginary coupling parameter, and of generalized quantization conditions which take into account instanton contributions. We find a number of explicit new results, including the general behaviour of large-order perturbation theory for arbitrary levels of odd anharmonic oscillators, and subleading corrections to the decay width of excited states for odd potentials, which are numerically significant.Comment: 5 pages, RevTe

Reconstructing the nucleon-nucleon potential by a new coupled-channel inversion method

A second-order supersymmetric transformation is presented, for the two-channel Schr\"odinger equation with equal thresholds. It adds a Breit-Wigner term to the mixing parameter, without modifying the eigenphase shifts, and modifies the potential matrix analytically. The iteration of a few such transformations allows a precise fit of realistic mixing parameters in terms of a Pade expansion of both the scattering matrix and the effective-range function. The method is applied to build an exactly-solvable potential for the neutron-proton $^3S_1$-$^3D_1$ case.Comment: 4 pages, 4 figure

Vernacular Knowledge and Water Management - Towards the Integration of Expert Science and Local Knowledge in Ontario, Canada

Complex environmental problems cannot be solved using expert science alone. Rather, these kinds of problems benefit from problem-solving processes that draw on ‘vernacular’ knowledge. Vernacular knowledge integrates expert science and local knowledge with community beliefs and values. Collaborative approaches to water problem-solving can provide forums for bringing together diverse, and often competing, interests to produce vernacular knowledge through deliberation and negotiation of solutions. Organised stakeholder groups are participating increasingly in such forums, often through involvement of networks, but it is unclear what roles these networks play in the creation and sharing of vernacular knowledge. A case-study approach was used to evaluate the involvement of a key stakeholder group, the agricultural community in Ontario, Canada, in creating vernacular knowledge during a prescribed multi-stakeholder problem-solving process for source water protection for municipal supplies. Data sources – including survey questionnaire responses, participant observation, and publicly available documents – illustrate how respondents supported and participated in the creation of vernacular knowledge. The results of the evaluation indicate that the respondents recognised and valued agricultural knowledge as an information source for resolving complex problems. The research also provided insight concerning the complementary roles and effectiveness of the agricultural community in sharing knowledge within a prescribed problem-solving process

On Finite-Time Stabilization of Evolution Equations: A Homogeneous Approach

International audienceGeneralized monotone dilation in a Banach space is introduced. Classical theorems on existence and uniqueness of solutions of nonlinear evolution equations are revised. A universal homogeneous feedback control for a finite-time stabilization of linear evolution equation in a Hilbert space is designed using homogeneity concept. The design scheme is demonstrated for distributed finite-time control of heat and wave equations

Generalized Nonanalytic Expansions, PT-Symmetry and Large-Order Formulas for Odd Anharmonic Oscillators

The concept of a generalized nonanalytic expansion which involves nonanalytic combinations of exponentials, logarithms and powers of a coupling is introduced and its use illustrated in various areas of physics. Dispersion relations for the resonance energies of odd anharmonic oscillators are discussed, and higher-order formulas are presented for cubic and quartic potentials