Thermodynamic Modeling and Analysis of Phase Transitions

Publications

Articles in Refereed Journals

• 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.

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.

Talks, 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.

External Preprints

• 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.