Energy variational approach to study charge inversion (layering) near charged walls

Abstract. We introduce a mathematical model, which describes the charge inversion phenomena in systems with a charged wall or boundary. This model may prove helpful in understanding semiconductor devices, ion channels, and electrochemical systems like batteries that depend on complex distributions of charge for their function. The mathematical model is derived using the energy variational approach that takes into account ion diffusion, electrostatics, finite size effects, and specific boundary behavior. In ion dynamic theory, a wellknown system of equations is the Poisson-Nernst-Planck (PNP) equation that includes entropic and electrostatic energy. The PNP type of equation can also be derived by the energy variational approach. However, the PNP equations have not produced the charge inversion/layering in charged wall situations presumably because the conventional PNP does not include the finite size of ions and other physical features needed to create the charge inversion. In this paper, we investigate the key features needed to produce the charge inversion phenomena using a mathematical model, the energy variational approach. One of the key features is a finite size (finite volume) effect, which is an unavoidable property of ions important for their dynamics on small scales. The other is an interfacial constraint to capture the spatial variation of electroneutrality in systems with charged walls. The interfacial constraint is established by the diffusive interface approach that approximately describes the boundary effect produced by the charged wall. The energy variational approach gives us a mathematically self-consistent way to introduce the interfacial constraint. We mainly discuss those two key features in this paper. Employing the energy variational approach, we derive a non-local partial differential equation with a total energy consisting of the entropic energy, electrostatic energy, repulsion energy representing the excluded volume effect, and the contribution of an interfacial constraint related to overall electroneutrality between bulk/bath and wall. The resulting mathematical model produces the charge inversion phenomena near charged walls. We compare the computational results of the mathematical model to those of Monte-Carlo computations. 1. Introduction. Ionic solutions are in many ways the most important mixtures people, and thus mathematicians, encounter. All of life occurs in ionic solutions; much of chemistry is done in ionic solutions, all our thoughts are nerve cells working in ionic solutions. Understanding ionic solutions has been entwined with understanding life since ions were discovered and their fluxes investigated by Fick, a biologist, actually a physiologist Complex distributions of charge are important in many applications. The switching and amplifying characteristics of transistors, and thus integrated circuits, arise in depletion layers of charge Charge inversion (layering) is an alternating accumulation of matter and charge near a charged wall, surface, or boundary. When the charged wall is exposed to electrolyte solution, the electric force (and sometimes potential) reverses polarity in some regions due to the complex spatial dependence of the accumulation of counterions near the wal

Evolution of the Stethoscope: Advances with the Adoption of Machine Learning and Development of Wearable Devices

The stethoscope has long been used for the examination of patients, but the importance of auscultation has declined due to its several limitations and the development of other diagnostic tools. However, auscultation is still recognized as a primary diagnostic device because it is non-invasive and provides valuable information in real-time. To supplement the limitations of existing stethoscopes, digital stethoscopes with machine learning (ML) algorithms have been developed. Thus, now we can record and share respiratory sounds and artificial intelligence (AI)-assisted auscultation using ML algorithms distinguishes the type of sounds. Recently, the demands for remote care and non-face-to-face treatment diseases requiring isolation such as coronavirus disease 2019 (COVID-19) infection increased. To address these problems, wireless and wearable stethoscopes are being developed with the advances in battery technology and integrated sensors. This review provides the history of the stethoscope and classification of respiratory sounds, describes ML algorithms, and introduces new auscultation methods based on AI-assisted analysis and wireless or wearable stethoscopes

A MATHEMATICAL MODEL FOR THE HARD SPHERE REPULSION IN IONIC SOLUTIONS

We introduce a mathematical model for finite size (repulsive) effects in ionic solutions. We first introduce an appropriate energy term into the total energy that represents the hard sphere repulsion of ions. The total energy then consists of the entropic energy, electrostatic potential energy, and the repulsive potential energy. The energetic variational approach is then used to derive a boundary value problem (the ‘Euler-Lagrange’ equations) that includes contributions from the repulsive term. The resulting system of partial differential equations is a modification of the Poisson-Nernst-Planck (PNP) equations widely if not universally used to describe the drift-diffusion of electrons and holes in semiconductors, and the movement of ions in solutions and protein channels. The modified PNP equations include the effects of the finite size of ions that are so important in the concentrated solutions near electrodes, active sites of enzymes, and selectivity filters of proteins. Finally, we do some numerical experiments using finite element methods, and present their results as a verification of the utility of the modified system

Energy Variational Analysis EnVarA of Ions in Water and Channels: Field Theory for Primitive Models of Complex Ionic Fluids

1 online resource (PDF, 60 pages, includes illustrations