Jan-Frederik Pietschmann

Nonlinear Cross-Diffusion with Size Exclusion

The aim of this paper is to investigate the mathematical properties of a continuum model for diffusion of multiple species incorporating size exclusion effects. The system for two species leads to nonlinear cross-diffusion terms with double degeneracy, which creates significant novel challenges in the analysis of the system. We prove global existence of weak solutions and well-posedness of strong solutions close to equilibrium. We further study some asymptotics of the model, and in particular we characterize the large-time behavior of solutions.

Rectification of Ion Current in Nanopores Depends on the Type of Monovalent Cations Experiments and Modeling

Rectifying nanopores feature ion currents that are higher for voltages of one polarity compared to the currents recorded for corresponding voltages of the opposite polarity. Rectification of nanopores has been found to depend on the pore opening diameter and distribution of surface charges on the pore walls as well as pore geometry. Very little is known, however, on the dependence of ionic rectification on the type of transported ions of the same charge. We performed experiments with single conically shaped nanopores in a polymer film and recorded current–voltage curves in three electrolytes: LiCl, NaCl, and KCl. Rectification degrees of the pores, quantified as the ratio of currents recorded for voltages of opposite polarities, were the highest for KCl and the lowest for LiCl. The experimental observations could not be explained by a continuum modeling based on the Poisson–Nernst–Planck equations. All-atom molecular dynamics simulations revealed differential binding between Li+, Na+, and K+ ions and carboxyl groups on the pore walls, resulting in changes to both the effective surface charge of the nanopore and cation mobility within the pore.

Rectification properties of conically shaped nanopores: consequences of miniaturization

Nanopores attracted a great deal of scientific interest as templates for biological sensors as well as model systems to understand transport phenomena at the nanoscale. The experimental and theoretical analysis of nanopores has been so far focused on understanding the effect of the pore opening diameter on ionic transport. In this article we present systematic studies on the dependence of ion transport properties on the pore length. Particular attention was given to the effect of ion current rectification exhibited in conically shaped nanopores with homogeneous surface charges. We found that reducing the length of conically shaped nanopores significantly lowered their ability to rectify ion current. However, rectification properties of short pores can be enhanced by tailoring the surface charge and the shape of the narrow opening. Furthermore we analyzed the relationship of the rectification behavior and ion selectivity for different pore lengths. All simulations were performed using MsSimPore, a software package for solving the Poisson-Nernst-Planck (PNP) equations. It is based on a novel finite element solver and allows for simulations up to surface charge densities of -2 e per nm(2). MsSimPore is based on 1D reduction of the PNP model, but allows for a direct treatment of the pore with bulk electrolyte reservoirs, a feature which was previously used in higher dimensional models only. MsSimPore includes these reservoirs in the calculations, a property especially important for short pores, where the ionic concentrations and the electric potential vary strongly inside the pore as well as in the regions next to the pore entrance.

ON A PARABOLIC FREE BOUNDARY EQUATION MODELING PRICE FORMATION

We discuss existence and uniqueness of solutions for a one-dimensional parabolic evolution equation with a free boundary. This problem was introduced by Lasry and Lions as description of the dynamical formation of the price of a trading good. Short time existence and uniqueness is established by a contraction argument. Then we discuss the issue of global-in-time-extension of the local solution which is closely related to the regularity of the free boundary. We also present numerical results.

Multiscale modeling of a rectifying bipolar nanopore: Comparing Poisson-Nernst-Planck to Monte Carlo

In the framework of a multiscale modeling approach, we present a systematic study of a bipolar rectifying nanopore using a continuum and a particle simulation method. The common ground in the two methods is the application of the Nernst-Planck (NP) equation to compute ion transport in the framework of the implicit-water electrolytemodel. The difference is that the Poisson-Boltzmann theory is used in the Poisson-Nernst-Planck (PNP) approach, while the Local Equilibrium Monte Carlo (LEMC) method is used in the particle simulation approach (NP+LEMC) to relate the concentration profile to the electrochemical potential profile. Since we consider a bipolar pore which is short and narrow, we perform simulations using two-dimensional PNP. In addition, results of a non-linear version of PNP that takes crowding of ions into account are shown. We observe that the mean field approximation applied in PNP is appropriate to reproduce the basic behavior of the bipolar nanopore (e.g., rectification) for varying parameters of the system (voltage, surface charge,electrolyte concentration, and pore radius). We present current data that characterize the nanopores behavior as a device, as well as concentration, electrical potential, and electrochemical potential profiles.

Identification of Chemotaxis Models with Volume-Filling

Chemotaxis refers to the directed movement of cells in response to a chemical signal called chemoattractant. A crucial point in the mathematical modeling of chemotactic processes is the correct description of the chemotactic sensitivity and of the production rate of the chemoattractant. In this paper, we investigate the identification of these nonlinear parameter functions in a chemotaxis model with volume-filling. We also discuss the numerical realization of Tikhonov regularization for the stable solution of the inverse problem. Our theoretical findings are supported by numerical tests.

Simultaneous identification of diffusion and absorption coefficients in a quasilinear elliptic problem

In this work we consider the identifiability of two coefficients

Numerical identification of a nonlinear diffusion law via regularization in Hilbert scales

a(u)

Lane formation in a microscopic model and the corresponding partial differential equation

and

Flow characteristics in a crowded transport model

c(x)