Thermodynamische Modellierung und Analyse von Phasenübergängen

Publikationen

Artikel in Referierten Journalen

• CH. Keller, J. Fuhrmann, M. Landstorfer, B. Wagner, A model framework for ion channels with selectivity filters based on continuum non-equilibrium thermodynamics, Entropy. An International and Interdisciplinary Journal of Entropy and Information Studies, 27 (2025), pp. 981--1013, DOI 10.3390/e27090981 .
  Abstract
  A mathematical model framework to describe ion transport in nanopores is presented. The model is based on non-equilibrium thermodynamics and considers finite size effects, solvation phenomena as well as the electrical charges of membrane surfaces and channel proteins. Par- ticular emphasis is placed on the consistent modelling of the selectivity filter in the pore. It is treated as an embedded domain in which the constituents can change their chemical properties. The diffusion process through the filter is governed by an independent diffusion coefficient and at the interfaces, de- and resolvation reactions are introduced as Neumann interface conditions. The evolution of the molar densities is described by drift-diffusion equations, where the fluxes depend on the gradient of the chemical potentials and the electric force. The chemical potentials depend on the molar fractions and on the pressure in the electrolyte and accounts for solvation effects. The framework allows the calculation of current-voltage relations for a variety of chan- nel properties and ion concentrations. We compare our model framework to experimental results for calcium-selective ion channels and show the general validity of our approach. Our parameter studies show that calcium and sodium currents are proportional to the surface charge in the se- lectivity filter and to the diffusion coefficients of the ions. Moreover, they show that the negative charges inside the pore have a decisive influence on the selectivity of divalent over monovalent ions.

• L. Schmeller, D. Peschka, Sharp-interface limits of Cahn--Hilliard models and mechanics with moving contact lines, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 22 (2024), pp. 869--890, DOI 10.1137/23M1546592 .
  Abstract
  We construct gradient structures for free boundary problems with moving capillary interfaces with nonlinear (hyper)elasticity and study the impact of moving contact lines. In this context, we numerically analyze how phase-field models converge to certain sharp-interface models when the interface thickness tends to zero. In particular, we study the scaling of the Cahn--Hilliard mobility with certain powers of the interfacial thickness. In the presence of interfaces, it is known that the intended sharp-interface limit holds only for a particular range of powers However, in the presence of moving contact lines we show that some scalings that are valid for interfaces produce significant errors and the effective range of valid powers of the interfacial thickness in the mobility reduces.

• A. Erhardt, D. Peschka, Ch. Dazzi, L. Schmeller, A. Petersen, S. Checa, A. Münch, B. Wagner, Modeling cellular self-organization in strain-stiffening hydrogels, Computational Mechanics, 75 (2025), pp. 875--896 (published online on 31.08.2024), DOI 10.1007/s00466-024-02536-7 .
  Abstract
  We develop a three-dimensional mathematical model framework for the collective evolution of cell populations by an agent-based model (ABM) that mechanically interacts with the surrounding extracellular matrix (ECM) modeled as a hydrogel. We derive effective two-dimensional models for the geometrical set-up of a thin hydrogel sheet to study cell-cell and cell-hydrogel mechanical interactions for a range of external conditions and intrinsic material properties. We show that without any stretching of the hydrogel sheets, cells show the well-known tendency to form long chains with varying orientations. Our results further show that external stretching of the sheet produces the expected nonlinear strain-softening or stiffening response, with, however, little qualitative variation of the overall cell dynamics for all the materials considered. The behavior is remarkably different when solvent is entering or leaving from strain softening or stiffening hydrogels, respectively.

• A. Erhardt, Cardiac dynamics of a human ventricular tissue model with focus on early afterdepolarizations, Frontiers in Physics, section Biophysics, 13 (2025), pp. 1569121/1--1569121/15, DOI 10.3389/fphy.2025.1569121 .
  Abstract
  The paper is aimed to investigate computationally complex cardiac dynamics of the famous human ventricular model of ten Tusscher and Panfilov from 2006. The corresponding physical system is modeled by a set of nonlinear differential equations containing various of system parameters. In case a specific physical parameter crosses a certain threshold, the system is forced to change dynamics, which might result in dangerous cardiac dynamics and can be precursors to cardiac death. For the performance of an efficient numerical analysis the original model is remodeled and simplified in such a way that the modified models perfectly matches the trajectory of the original model. Moreover, it is demonstrated that the simplified models have the same dynamics. Furthermore, using the lowest dimensional model it is systematically shown by means of bifurcation analysis that combinations of reduced slow and rapid potassium channels and enhanced sodium channel may lead to early afterdepolarizations. Finally, synchronization and the effect of EADs on larger scale (macro scale) is investigated numerically by studying the corresponding monodomain model. To this end we study the pattern formation of an one dimensional network of epi-, mid-myo- and endocardial cells and a two dimensional epicardial monodomain equation.

• M. Heida, M. Landstorfer, M. Liero, Homogenization of a porous intercalation electrode with phase separation, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 22 (2024), pp. 1068--1096, DOI 10.1137/21M1466189 .
  Abstract
  In this work, we derive a new model framework for a porous intercalation electrode with a phase separating active material upon lithium intercalation. We start from a microscopic model consisting of transport equations for lithium ions in an electrolyte phase and intercalated lithium in a solid active phase. Both are coupled through a Neumann--boundary condition modeling the lithium intercalation reaction. The active material phase is considered to be phase separating upon lithium intercalation. We assume that the porous material is a given periodic microstructure and perform analytical homogenization. Effectively, the microscopic model consists of a diffusion and a Cahn--Hilliard equation, whereas the limit model consists of a diffusion and an Allen--Cahn equation. Thus we observe a Cahn--Hilliard to Allen--Cahn transition during the upscaling process. In the sense of gradient flows, the transition goes in hand with a change in the underlying metric structure of the PDE system.

• M. Schlutow, T. Stacke, T. Dörffel, P.K. Smolarkiewicz, M. Göckede, Large eddy simulations of the interaction between the atmospheric boundary layer and degrading arctic permafrost, Journal of Geophysical Research: Atmospheres, 129 (2024), pp. e2024JD040794/1--e2024JD040794/17, DOI 10.1029/2024JD040794 .

• M. Brokate, M. Ulbrich, Corrigendum and Addendum: Newton differentiability of convex functions in normed spaces and of a class of operators, SIAM Journal on Optimization, 34 (2024), pp. 3163--3166, DOI 10.1137/24M1669542 .

• A. Erhardt, Cardiac dynamics of a human ventricular tissue model with focus on early afterdepolarizations, , 13 (2025), pp. 1569121/1--1569121/15, DOI 10.3389/fphy.2025.1569121 .
  Abstract
  The paper is aimed to investigate computationally complex cardiac dynamics of the famous human ventricular model of ten Tusscher and Panfilov from 2006. The corresponding physical system is modeled by a set of nonlinear differential equations containing various of system parameters. In case a specific physical parameter crosses a certain threshold, the system is forced to change dynamics, which might result in dangerous cardiac dynamics and can be precursors to cardiac death. For the performance of an efficient numerical analysis the original model is remodeled and simplified in such a way that the modified models perfectly matches the trajectory of the original model. Moreover, it is demonstrated that the simplified models have the same dynamics. Furthermore, using the lowest dimensional model it is systematically shown by means of bifurcation analysis that combinations of reduced slow and rapid potassium channels and enhanced sodium channel may lead to early afterdepolarizations. Finally, synchronization and the effect of EADs on larger scale (macro scale) is investigated numerically by studying the corresponding monodomain model. To this end we study the pattern formation of an one dimensional network of epi-, mid-myo- and endocardial cells and a two dimensional epicardial monodomain equation.

Preprints, Reports, Technical Reports

• M. Heida, M. Landstorfer, Modeling of porous battery Electrodes with multiple phase transitions -- Part I: Modeling and homogenization, Preprint no. 3251, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3251 .
  Abstract, PDF (1108 kByte)
  We derive a thermodynamically consistent multiscale model for a porous intercalation battery in a half-cell configuration. Starting from microscopically resolved balance equations, the model rigorously couples cation and anion transport in the electrolyte with electron transport and solid- state diffusion in the active material through intercalation reactions. The derivation is based on non-equilibrium thermodynamics and periodic homogenization. The central novelty of this work lies in the systematic incorporation of multi-well free energy functions for intercalated cations into a homogenized DFN-type porous-electrode framework. This modeling choice leads to non-monotonic chemical potentials and enables a macroscopic descrip- tion of phase separation and multiple phase transitions within the electrode. While multi-well free energies are well established at the particle scale, their integration into homogenized porous- electrode models has so far been lacking. By extending the homogenization framework to include Cahn--Hilliard-type regularizations, phase-transition effects are retained at the electrode level. The resulting model exhibits an intrinsically coupled 3D+3D structure, in which macroscopic transport in the electrolyte is coupled to fully resolved microscopic diffusion within active parti- cles. This coupling naturally induces memory effects and time lags in the macroscopic voltage response, which cannot be captured by reduced single-scale models. Although the microscopic dynamics possess an underlying gradient-flow structure, we adopt a formal asymptotic approach to obtain a tractable DFN-type model suitable for practical simulations. This paper constitutes Part I of a three-part series and is devoted to the systematic derivation and mathematical formulation of the model. Numerical analysis, discretization strategies, simula- tion studies of transient cycling behavior, and experimental validation are deferred to Parts II and III. Part II focuses on finite C-rates, while Part III addresses open-circuit voltage conditions, where the predictive capabilities of the framework are investigated in detail.

• M. Landstorfer, Ch. Pohl, F. Brosa Planella, K. Manmi, A model for SEI-growth based on non-equilibrium thermodynamics, Preprint no. 3250, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3250 .
  Abstract, PDF (1092 kByte)
  The growth of the solid electrolyte interphase (SEI) is a dominant degradation mechanism in lithium-ion batteries, governing capacity fade, coulombic efficiency, and long-term performance. Despite extensive experimental investigation, quantitative understanding of SEI formation and evolution remains limited by its nanoscale thickness, complex chemistry, and strong sensitivity to operating conditions. Existing zero-dimensional models capture individual rate-limiting mechanisms but typically treat the SEI as an idealized interface layer, neglecting spatially resolved transport, solvent consumption, and dynamic interface motion. In this work, we present a continuum-level model for SEI growth grounded in non-equilibrium thermodynamics. The SEI is treated as a distinct thermodynamic domain and modeled as a mixed ion - electron conductor, while the SEI - electrolyte interface is described as a moving boundary. The framework systematically derives transport laws and reaction kinetics from electrochemical poten- tials and interfacial free energies, ensuring thermodynamic consistency. A finite electrolyte reservoir is explicitly included, allowing solvent depletion to emerge naturally as a limiting mechanism for SEI growth. The general formulation consists of coupled partial differential equations for all domains and interfaces. Under open-circuit voltage conditions, the system reduces to a tractable set of ordinary differential equations describing lithium concentration in the active material, solvent concentration, and SEI thickness. Numerical simulations under charging, rest, and cycling conditions reproduce experimentally observed features such as linear and square root of time growth regimes, voltage shifts due to parasitic current consumption, capacity contributions from lithium stored in the SEI, and self- discharge during rest. Two distinct termination mechanisms - active lithium depletion and solvent exhaustion - are identified. Overall, the proposed framework unifies multiple SEI growth mechanisms within a single thermodynamically consistent model and provides a mechanistic basis for improved lifetime prediction and optimization of battery formation and operating protocols.

• W. Kenmoe Nzali, Ch. Bayer, D. Kreher, M. Landstorfer, Volatile electricity markets and battery storage: A model-based approach for optimal control, Preprint no. 3248, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3248 .
  Abstract, PDF (1340 kByte)
  Grid connected energy storage systems provide a strategic advantage by exploiting electricity market price fluctuations, thereby significantly reducing energy consumption costs. This paper presents a general framework for minimizing electricity consumption costs by formulating the problem as a stochastic optimal control problem for a stationary battery storage device (SBSD). We propose a realistic model for electricity spot prices calibrated with real data, alongside a detailed model of battery dynamics with practical parameters. The control problem is solved in a discrete time setting by combining dynamic programming with the least squares Monte Carlo method, allowing us to approximate the value function and the optimal policy under both state of charge and voltage constraints. Using the derived optimal policy, we estimate the lower bound of electricity consumption costs across multiple price trajectories. The results demonstrate that the SBSD can substantially reduce consumption costs, with savings increasing with battery duration. After one year, a battery with 12 hours duration achieves approximately 11% cost reduction, while 24 hours battery achieves 21%, compared to a scenario without storage. Finally, we estimate the amortization time (the period required for cumulative savings to offset the initial investment). After 6.7 years for the 12 hours battery and 9.9 years for the 24 hours battery, the amortization time is reached.

• CH. Keller, B. Wagner, A. Münch, An asymptotic model of the Poisson--Nernst--Planck--Stokes system for ion transport in narrow channels, Preprint no. 3243, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3243 .
  Abstract, PDF (1462 kByte)
  Ion transport through narrow channels is determined by the interaction between electrochemical and hydrodynamic effects, which are influenced by the channel geometry, ion concentrations, pressure and potential gradients, and surface charges. Understanding the mechanisms that control electrokinetic phenomena such as ion selectivity and flow transitions is crucial for elucidating biological functions and for further developing the design of artificial nanofluidic systems. On the continuum scale, these processes are described by the coupled Poisson-Nernst-Planck-Stokes equations (PNPS). However, direct numerical simulations in two or three dimensions are computationally intensive and provide only limited insights into the underlying physical and mathematical structure. Taking advantage of the small aspect ratio characteristic of nanopores, we derive a systematic asymptotic reduction of the PNPS boundary value problem. In contrast to existing one-dimensional reductions, which assume a Debye length much smaller than the channel radius, our analysis identifies a distinct asymptotic regime in which the Debye length is comparable to the channel width. This framework extends the applicability of reduced PNPS models and recovers previous approximations as limiting cases. The resulting model provides clarity and predictability for a wide range of settings. We demonstrate the influence of geometry and flow on ion transport in trumpet-shaped nanopores, flow transitions that occur due to electrostatic and hydrodynamic forces, and the conductivity properties of a protein-based channel.

• Z. Amer, A. Avdzhieva, M. Bongarti, P. Dvurechensky, P. Farrell, U. Gotzes, F.M. Hante, A. Karsai, S. Kater, M. Landstorfer, M. Liero, D. Peschka, L. Plato, K. Spreckelsen, J. Taraz, B. Wagner, Modeling hydrogen embrittlement for pricing degradation in gas pipelines, Preprint no. 3201, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3201 .
  Abstract, PDF (12 MByte)
  This paper addresses aspects of the critical challenge of hydrogen embrittlement in the context of Germany's transition to a sustainable, hydrogen-inclusive energy system. As hydrogen infrastructure expands, estimating and pricing embrittlement become paramount due to safety, operational, and economic concerns. We present a twofold contribution: We discuss hydrogen embrittlement modeling using both continuum models and simplified approximations. Based on these models, we propose optimization-based pricing schemes for market makers, considering simplified cyclic loading and more complex digital twin models. Our approaches leverage widely-used subcritical crack growth models in steel pipelines, with parameters derived from experiments. The study highlights the challenges and potential solutions for incorporating hydrogen embrittlement into gas transportation planning and pricing, ultimately aiming to enhance the safety and economic viability of Germany's future energy infrastructure.

• K. Hopf, J. King, A. Münch, B. Wagner, Interface dynamics in a degenerate Cahn--Hilliard model for viscoelastic phase separation, Preprint no. 3149, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3149 .
  Abstract, PDF (474 kByte)
  The formal sharp-interface asymptotics in a degenerate Cahn--Hilliard model for viscoelastic phase separation with cross-diffusive coupling to a bulk stress variable are shown to lead to non-local lower-order counterparts of the classical surface diffusion flow. The diffuse-interface model is a variant of the Zhou--Zhang--E model and has an Onsager gradient-flow structure with a rank-deficient mobility matrix reflecting the ODE character of stress relaxation. In the case of constant coupling, we find that the evolution of the zero level set of the order parameter approximates the so-called intermediate surface diffusion flow. For non-constant coupling functions monotonically connecting the two phases, our asymptotic analysis leads to a family of third order whose propagation operator behaves like the square root of the minus Laplace--Beltrami operator at leading order. In this case, the normal velocity of the moving sharp interface arises as the Lagrange multiplier in a constrained elliptic equation, which is at the core of our derivation. The constrained elliptic problem can be solved rigorously by a variational argument, and is shown to encode the gradient structure of the effective geometric evolution law. The asymptotics are presented for deep quench, an intermediate free boundary problem based on the double-obstacle potential.

Vorträge, Poster

• CH. Keller, Die Mechanik des Lebens: Physik und Mathematik von Ionenkanälen, Mathematisch-Physikalisches Kolloquium, Technische Hochschule Nürnberg Georg-Simon-Ohm, March 25, 2025.

• A. Erhardt, Mathematical modeling and analysis of cardiac dynamics, Kolloquium des SFB 1114, Freie Unversität Berlin, April 24, 2025.

• CH. Pohl, Modeling of solid-electrolyte interphase growth with non-equilibrium thermodynamics, 248th ECS Meeting, Session 'Electrolytes & Interfaces in Li-ion Batteries and Beyond', Chicago, USA, October 12 - 16, 2025.

• CH. Keller, Die Mechanik des Lebens: Physik und Mathematik von Ionenkanälen, Mathematisch-Physikalisches Kolloquium, Technische Hochschule Nürnberg Georg-Simon-Ohm, March 25, 2025.

• B. Wagner, Shape of polystyrene droplets on soft PDMS: Exploring the gap between theory and experiment at the three-phase contact line, SPP 2171 Spring Conference: Dynamic Wetting of Flexible, Adaptive, and Switchable Substrates, Max Planck Institute for Polymer Research, Mainz, February 20, 2025.

• M. Landstorfer, M. Heida, Ch. Pohl, Modeling lithium-ion batteries with phase separation using non-equilibrium thermodynamics and homogenization theory, Oxford Battery Modelling Symposium (OBMS), Oxford, UK, July 24 - 25, 2025.

• M. Landstorfer, Aspects of battery modeling with non-equilibrium thermodynamics and homogenization theory, Group Seminar: Transfer Group, May 21 - 23, 2025, Johann Radon Institute for Computational and Applied Mathematics (RICAM) of the Austrian Academy of Sciences, Linz, Austria, May 22, 2025.

• M. Landstorfer, The double layer capacitance of aqueous and aprotic electrode-electrolyte interfaces: Thermodynamic modeling and experimental data, 76th Annual Meeting of the International Society of Electrochemistry (ISE), Electrochemistry: From Basic Insights to Sustainable Technologies, September 7 - 12, 2025, International Society of Electrochemistry, Lausanne, Mainz, September 8, 2025.

• CH. Keller, A drift-diffusion model to describe ion channel dynamics, Applied Mathematics and Simulation for Semiconductor Devices (AMaSiS 2024), Berlin, September 10 - 13, 2024.

• CH. Keller, A model framework for calcium ion channels: Consistent modeling of selectivity filters, The European Conference on Mathematical and Theoretical Biology (ECMTB 2024), July 22 - 26, 2024, University of Castilla La Mancha, Toledo, Spain, July 25, 2024.

• T. Dörffel, M. Landstorfer, M. Liero, Modeling battery electrodes with mechanical interactions and multiple phase transistions upon ion insertion, MATH+ Day, Urania Berlin, October 18, 2024.

• T. Dörffel, Modeling a hurricane boundary layer through matched asymptotics, 36th Conference on Hurricanes and Tropical Meteorology, Session 16C Tropical Cyclones - Observing and Simulating the Boundary Layer, May 6 - 10, 2024, Long Beach, USA, May 9, 2024.

• T. Dörffel, Modeling a hurricane boundary layer through matched asymptotics, Workshop ``Model Hierarchies in Atmosphere, Ocean, and Climate Sciences'', June 30 - July 5, 2024, Mathematisches Forschungsinstitut Oberwolfach, July 2, 2024.

• A. Erhardt, Mathematical modeling of cell-hydrogel interactions, 23rd Symposium on Trends in Applications of Mathematics to Mechanics (STAMM 2024), April 3 - 5, 2024, Julius-Maximilians-Universität Würzburg, April 3, 2024.

• B. Wagner, Dynamics of cellular patterns on hydrogel sheets, University of Oxford, Mathematical Institute, UK, November 22, 2024.

• CH. Bayer, W. Kenmoe Nzali, D. Kreher, M. Landstorfer, Volatile electricity markets and battery storage: A model-based approach for optimal control, MATH+ Day, Urania Berlin, October 18, 2024.

• J. Fuhrmann, Ch. Keller, M. Landstorfer, B. Wagner, Development of an ion-channel model-framework for in-vitro assisted interpretation of current voltage relations, MATH+ Day, Urania Berlin, October 18, 2024.

• O. Klein, A model for a magneto mechanical device: Forward and inverse uncertainty quantization, Leibniz MMS Days 2024, Kaiserslautern, April 10 - 12, 2024.

• O. Klein, On a model for a magneto mechanical device: Forward and inverse uncertainty quantification, 2nd Workshop of the MATH+ Thematic Einstein Semester ``Mathematics of Small Data Analysis'', Berlin, January 17 - 19, 2024.

• M. Landstorfer, T. Dörffel, Mathematical modeling of intercalation batteries with non-equilibrium thermodynamics and homogenization theory, Oxford Battery Modelling Symposium 2024, UK, April 15 - 17, 2024.

• M. Landstorfer, Mathematical modeling of lithium-ion intercalation cells based on non-equilibrium thermodynamics, Seminar, Università degli Studi di Brescia, Dipartimento di Ingegneria Meccanica e Industriale, Italy, September 11, 2024.

• M. Landstorfer, Thermodynamic modeling of aqueous and aprotic electrode-electrolyte interfaces and their and double layer capacitance, 75th Annual Meeting of the International Society of Electrochemistry, Symposium S13 - Double layer reloaded: Theory meets experiments, August 18 - 23, 2024, Montréal, Canada, August 23, 2024.

Preprints im Fremdverlag

• G.L. Celora, R. Blossey, A. Münch, B. Wagner, The diffusive dynamics and electrochemical regulation of weak polyelectrolytes across liquid interfaces, Preprint no. arXiv:2502.14555, Cornell University, 2025, DOI 10.48550/arXiv.2502.14555 .
  Abstract
  We propose a framework to study the spatio-temporal evolution of liquid-liquid phase separation of weak polyelectrolytes in ionic solutions. Unlike strong polyelectrolytes, which carry a fixed charge, the charge state of weak polyelectrolytes is modulated by the electrochemical environment through protonation and deprotonation processes. Leveraging numerical simulations and analysis, our work reveals how solution acidity (pH) influences the formation, interactions, and structural properties of phase-separated coacervates. We find that pH gradients can be maintained across coacervate interfaces resulting in a clear distinction in the electro-chemical properties within and outside the coacervate. By regulating the charge state of weak polyelectrolytes, pH gradients interact and modulate the electric double layer forming at coacervate interfaces eventually determining how they interact. Further linear and nonlinear analyses of stationary localised solutions reveal a rich spectrum of behaviours that significantly distinguish weak from strong polyelectrolytes. Overall, our results demonstrate the importance of charge regulation on phase-separating solutions of charge-bearing molecules and the possibility of harnessing charge-regulated mechanisms to control coacervates and shape their stability and spatial organisation.