Thermodynamic Modeling and Analysis of Phase Transitions

Publications

Articles in Refereed Journals

• L. Schmeller, D. Peschka, Sharp-interface limits of Cahn--Hilliard models and mechanics with moving contact lines, , 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.

• 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 .

Preprints, Reports, Technical Reports

• K. Remini, L. Schmeller, D. Peschka, B. Wagner, R. Seemann, Shape of polystyrene droplets on soft PDMS: Exploring the gap between theory and experiment at the three-phase contact line, Preprint no. 3160, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3160 .
  Abstract, PDF (6173 kByte)
  The shapes of liquid polystyrene (PS) droplets on viscoelastic polydimethylsiloxane (PDMS) substrates are investigated experimentally using atomic force microscopy for a range of droplet sizes and substrate elasticities. These shapes, which comprise the PS-air, PS-PDMS, and PDMS-air interfaces as well as the three-phase contact line, are compared to theoretical predictions using axisymmetric sharp-interface models derived through energy minimization. We find that the polystyrene droplets are cloaked by a thin layer of uncrosslinked molecules migrating from the PDMS substrate. By incorporating the effects of cloaking into the surface energies in our theoretical model, we show that the global features of the experimental droplet shapes are in excellent quantitative agreement for all droplet sizes and substrate elasticities. However, our comparisons also reveal systematic discrepancies between the experimental results and the theoretical predictions in the vicinity of the three-phase contact line. Moreover, the relative importance of these discrepancies systematically increases for softer substrates and smaller droplets. We demonstrate that global variations in system parameters, such as surface tension and elastic shear moduli, cannot explain these differences but instead point to a locally larger elastocapillary length, whose possible origin is discussed thoroughly.

• 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.

• A. Erhardt, Cardiac dynamics of a human ventricular tissue model with focus on early afterdepolarizations, Preprint no. 3147, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3147 .
  Abstract, PDF (7946 kByte)
  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.

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.

• 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.

• O. Klein, Uncertainty quantification for a model for a magnetostrictive material involving a hysteresis operator, Berliner Oberseminar Nichtlineare partielle Differentialgleichungen (Langenbach-Seminar), WIAS, Berlin, May 21, 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.

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.