Project C3: Spinodal decomposition of polymer-solvent systems

We consider the phase separation of dynamically asymmetric mixtures, in particular polymer solutions, after a sudden quench. Crucial aspects are (i) hydrodynamic momentum transport and (ii) the lack of time-scale separation between molecular relaxation and coarsening. This gives rise to complex dynamical processes such as the transient formation of network-like structures of the slow-component-rich phase, its volume shrinking, and lack of dynamic self-similarity, which are frequently summarized under the term viscoelastic phase separation.

The relevant length and time scales of the physical phenomena are too large for microscopic (all atom) simulations. Alternative mesoscopic models based on a bead-spring description of polymer chains coupled to a hydrodynamic background, i.e., the Navier-Stokes equations for the solvent, allow to capture the basic physical principles but they are still computationally demanding. Therefore, macroscopic (two-fluid) models have been proposed in the literature which involve only averaged field quantities and therefore allow to simulate complex fluids on large length and time scales. These macroscopic models typically involve phenomenological nonlinear constitutive equations, whose parameters have to be carefully adjusted in order to match experimental data or data obtained from simulation of reliable micro- or mesoscopic models. In addition, the stability, thermodynamic consistency, and physical soundness of the macroscopic descriptions have to be carefully evaluated and appropriate numerical methods need to be developed for their simulation.

The main goal of the project is to obtain stable, consistent, and physically interpretable descriptions of complex flow dynamics in phase-separating polymer systems on multiple length and time scales. Systematic coarsegraining is applied to establish a solid theoretical foundation of the models on the meso- and macroscale, and to explain their interdependence. Consistency of the meso- and macroscopic models is achieved by careful calibration of macroscopic models to data obtained in experiments or independent mesoscopic simulations. In addition, the mathematical well-posedness of the macroscopic models is investigated and their systematic numerical approximation is studied.

In the second funding period, the focus of the project was on the systematic derivation, theoretical justification, analysis and numerical simulation of macroscopic (two-fluid) viscoelastic phase separation models. On the one hand, we mathematically analysed the model proposed by Zhou, Zhang and Gwiazda et al. and its phenomenologically modified variants, in particular “model I”, described in Brunk et al.. On the other hand, we have derived a new model (“model II”) in Spiller et al. via coarse-graining of a simple mesoscopic model. Both models are compatible with the GENERIC approach or they were directly derived using GENERIC. They may be viewed as complementary: The strength of model II is its well-defined molecular interpretation. To some extent this is missing in the model I, while we presently understand the mathematical properties of model I much better. For this reason, we intend to continue studying both models in parallel, at least in the immediate and mid-term future. In addition to these investigations, the physical properties of model I were qualitatively compared with mesoscopic simulations via dynamic structure factors. Similarly, model parameters were estimated by regularization methods.

In the third funding period, we plan a closer inspection of the mutual relations between mesoscopic and macroscopic descriptions. To this end, we will introduce an additional level of modeling based on kinetic equations. Although such models have already been discussed intensively in the literature, their systematic derivation from micro- and mesoscopic descriptions, their numerical approximation, as well as the further coarse-graining leading to purely macroscopic models still deserves further investigation in particular with respect to the viscoelastic phase separation. The following research topics will therefore be addressed in the third funding period of the project:

derivation of kinetic models (Navier-Stokes-Fokker-Planck systems) for viscoelastic phase separation starting from micro- or mesoscopic descriptions;
analysis and numerical simulation of kinetic models by both deterministic and stochastic approaches;
derivation of macroscopic models by further coarse-graining and their analysis and simulation

Validation of the kinetic models will be obtained by extensive mesoscopic simulation of viscoelastic phase separation by extension of a well-established coupled Lattice Boltzmann–Molecular Dynamics (LB-MD) method. In addition, the calibration of model parameters in the kinetic models will be studied by means of parameter estimation techniques.

The problem under investigation is a typical example of dynamic coarse-graining of a complex polymer system in a non-trivial non-equilibrium situation and therefore belongs to the research area C. The project is an inter-disciplinary research effort that requires the joint expertise of physics and mathematics. It will further benefit from collaboration with the external cooperation partner, Herbert Egger (Linz), who was a PI in the second funding period and therefore is familiar with the project goals. Further collaboration within the TRR is planned with projects A3, A8, A9, B1, B6, C1, C5, C7, and C8.

Publications

A simple, fully-discrete, unconditionally energy-stable method for the two-phase Navier-Stokes Cahn-Hilliard model with arbitrary density ratios
A. Brunk, M.F.P. ten Eikelder
J. Comp. Phys. 548, 114558 (2026)
Link: https://doi.org/10.1016/j.jcp.2025.114558

Structure-preserving approximation of the non-isothermal Cahn-Hilliard system based on the entropy equation
A. Brunk, D. Höhn, M. Lukáčová-Medvid'ová
Applied Mathematics and Computation, 522, 129987 (2026)
Link: https://doi.org/10.1016/j.amc.2026.129987

Analysis and discretization of the Ohta-Kawasaki equation with forcing and degenerate mobility
Aaron Brunk, Marvin Fritz
Partial differential equations and applications 6, 54 (2025)
Link: https://doi.org/10.1007/s42985-025-00362-x

Analysis and structure-preserving approximation of a Cahn-Hilliard-Forchheimer system with solution-dependent mass and volume source
Aaron Brunk, Marvin Fritz
ESAIM 59, 2991-3020 (2025)
Link: https://doi.org/10.1051/m2an/2025081

Error analysis for a second order approximation of a viscoelastic phase separation model
Brunk, A., Egger, H., Habrich, O., Lukáčová-Medvid'ová, M.
Numer. Math. 157, 1449–1489 (2025)
Link: https://doi.org/10.1007/s00211-025-01488-1

Error analysis of the Monte Carlo method for compressible magnetohydrodynamics
E. Feireisl, M. Lukáčová-Medvid'ová, B. She, Y. Yuan
Mathetmatical Models Methods Applied Science 35 (2025), no. 9, 2047–2097 (2025)
Link: https://doi.org/10.1142/S0218202525500332

A posteriori error control for the Allan-Cahn equation with variable mobility
A. Brunk, J. Giesselmann, M. Lukáčová-Medvid'ová
SIAM J. Num. Anal. 63, 1540-1560 (2025)
Link: https://doi.org/10.1137/24M1646406

A second-order structure-preserving discretization for the Cahn-Hilliard/Allen-Cahn system with cross-kinetic couplin
Aaron Brunk, Herbert Egger, Oliver Habrich
Applied Numerical Mathematics 206, 12-28 (2024)
Link: https://doi.org/10.1016/j.apnum.2024.07.016

On existence, uniqueness and stability of solutions to Cahn-Hilliard/Allen-Cahn systems with cross-kinetic coupling
A. Brunk, H. Egger, T.D. Oyedeji, Y.Yan, B.-X. Yu
Nonlinear Analysis 77, 104051 (2024)
Link: https://doi.org/10.1016/j.nonrwa.2023.104051

Structure-Preserving Approximation of the Cahn-Hilliard-Biot System
Aaron Brunk, Marvin Fritz
Numerical methods for partial differential equations 41, 323159 (2024)
Link: https://doi.org/10.1002/num.23159

On uniqueness and stable estimation of multiple parameters in the Cahn-Hilliard equation
Aaron Brunk, Herbert Egger, Oliver Habrich
Inverse Problems 39, 065002 (2023)
Link: https://doi.org/0.1088/1361-6420/acca44

Existence and weak-strong uniqueness for global weak solutions for the viscoelastic phase separation model in three space dimensions
Aaron Brunk
Discrete and continuous dynamics systems 43, 2120-2136 (2023)
Link: https://doi.org/10.3934/dcds.2023004

A second-order fully-balanced structure-preserving variational discretization scheme for the Cahn–Hilliard–Navier–Stokes system
Brunk, A., Egger, H., Habrich, O., and Lukáčová-Medviďová, M.
Mathematical Models and Methods in Applied Sciences, 33(12), 2587-2627, (2023)
Link: https://doi.org/10.1142/S0218202523500562

Stability and discretization error analysis for the Cahn–Hilliard system via relative energy estimates
Brunk, A., Egger, H., Habrich, O., and Lukáčová-Medviďová, M.
ESAIM: Mathematical Modelling and Numerical Analysis, 57(3), (2023)
Link: https://doi.org/10.1051/m2an/2023017

Relative energy and weak–strong uniqueness of a two-phase viscoelastic phase separation model
Aaron Brunk and Mária Lukáčová-Medviďová
ZAMM Z. Angew. Math. Mech. 103(7), Paper No. e202100240, 25 pp, (2023)
Link: https://doi.org/10.1002/zamm.202100240

Global existence of weak solutions to viscoelastic phase separation: Part I Regular Case
Aaron Brunk, Mária Lukáčová-Medvid'ová
Nonlinearity 35, 3417-3458, (2022)
Link: https://doi.org/10.1088/1361-6544/ac5920

Global existence of weak solutions to viscoelastic phase separation: Part II Degenerate Case
A. Brunk, M. Lukáčová-Medvid'ová
Nonlinearity 35, 3459-3486, (2022)
Link: https://doi.org/10.1088/1361-6544/ac591e

Existence, Regularity and Weak-Strong Uniqueness for Three-Dimensional Peterlin Viscoelastic Model
Brunk, A., Lu, Y., and Lukáčová-Medvid'ová, M.
Commun. Math. Sci. 20, 2022, 201-230 (2022)
Link: https://doi.org/10.4310/CMS.2022.v20.n1.a6

A model reduction approach for inverse problems with operator valued data
Jürgen Dölz, Herbert Egger, Matthias Schlottbohm
Numerische Mathematik 148, 889-917 (2021)
Link: https://doi.org/10.1007/s00211-021-01224-5

On the Stability of Harmonic Coupling Methods with Application to Electric Machines
H. Egger, M. Haruyunyan, M. Merkel, S. Schops
Scientific Computing in Electrical Engineering 36, 117-125 (2021)
Link: https://doi.org/10.1007/978-3-030-84238-3\_12

On the structure preserving high-order approximation of quasistatic poroelasticity
H. Egger, M. Sabouri
Mathematics and Computers in Simulation 189, 237-252 (2021)
Link: https://doi.org/10.1016/j.matcom.2020.12.029

Multistep and Runge-Kutta convolution quadrature methods for coupled dynamical systems
H. Egger, K. Schmidt, V. Shashkov
J. Comp. Appl. Math. 387, 112618 (2021)
Link: https://doi.org/10.1016/j.cam.2019.112618

A fast and oblivious matrix compression algorithm for Volterra integral operators
J. Dölz, H. Egger, V. Shashkov
Adv. Comp. Math. 47, 81 (2021)
Link: https://doi.org/10.1007/s10444-021-09902-6

A Convolution Quadrature Method for Maxwell's Equations in Dispersive Media
Jürgen Dolz, Herbert Egger, Vsevolod Shashkov
Sci. Computing in Electrical Engineering (SCEE 2020) 36, 107-115 (2021)
Link: https://doi.org/10.1007/978-3-030-84238-3\_11

Systematic derivation of hydrodynamic equations for viscoelastic phase separation
Dominic Spiller, Aaron Brunk, Oliver Habrich, Herbert Egger, Mária Lukáčová-Medvid'ová and Burkhard Dünweg
Journal of Physics: Condensed Matter 33 (36), 364001 (2021)
Link: https://doi.org/10.1088/1361-648X/ac0d17

Analysis of a viscoelastic phase separation model
Aaron Brunk, Burkhard Dünweg, Herbert Egger, Oliver Habrich, Mária Lukáčová-Medvid'ová, Dominic Spiller
Journal of Physics: Condensed Matter 33 (23), 234002 (2021)
Link: https://doi.org/10.1088/1361-648X/abeb13

A Second-Order Finite Element Method with Mass Lumping for Maxwell's Equations on Tetrahedra
Herbert Egger, Bogdan Radu
SIAM Journal on Numerical Analysis 59 (2), 864-885 (2021)
Link: https://doi.org/10.1137/20m1318912

On the Energy Stable Approximation of Hamiltonian and Gradient Systems
Herbert Egger, Oliver Habrich, Vsevolod Shashkov
Computational Methods in Applied Mathematics 21 (2), 335-349 (2020)
Link: https://doi.org/10.1515/cmam-2020-0025/html

On a Second-Order Multipoint Flux Mixed Finite Element Methods on Hybrid Meshes
Herbert Egger, Bogdan Radu
SIAM Journal on Numerical Analysis 58 (3), 1822-1844 (2020)
Link: https://doi.org/10.1137/19M1236862

Chemotaxis on networks: Analysis and numerical approximation
Herbert Egger, Lukas Schöbel-Kröhn
ESAIM: Mathematical Modelling and Numerical Analysis 54 (4), 1339-1372 (2020)
Link: https://doi.org/10.1051/m2an/2019069

Structure Preserving Discretization of Allen–Cahn Type Problems Modeling the Motion of Phase Boundaries
Anke Böttcher, Herbert Egger
Vietnam Journal of Mathematics 48 (4), 847-863 (2020)
Link: https://doi.org/10.1007/s10013-020-00428-w

A mass-lumped mixed finite element method for acoustic wave propagation
Herbert Egger, Bogdan Radu
Numerische Mathematik 145 (2), 239-269 (2020)
Link: https://doi.org/10.1007/978-3-030-44101-2\_2

On the transport limit of singularly perturbed convection–diffusion problems on networks
Herbert Egger, Nora Philippi
Mathematical Methods in the Applied Sciences 44 (6), 5005-5020 (2020)
Link: https://doi.org/10.1002/mma.7084

Semiautomatic construction of lattice Boltzmann models
Dominic Spiller, Burkhard Dünweg
Physical Review E101 (4), (2020)
Link: https://doi.org/10.1103/PhysRevE.101.043310

Structure preserving approximation of dissipative evolution problems
H. Egger
Numerische Mathematik 143 (1), 85-106 (2019)
Link: https://doi.org/10.1007/s00211-019-01050-w

A hybrid mass transport finite element method for Keller–Segel type systems
J.A. Carrillo, N. Kolbe, M. Lukáčová-Medvid'ová
Journal of Scientific Computing 80, 1777-1804 (2019)
Link: https://doi.org/10.1007/s10915-019-00997-0

Energy-stable linear schemes for polymer-solvent phase field models
P. Strasser, G. Tierra, B. Dünweg, M. Lukáčová-Medvid'ová
Computers and Mathematics with Applications 77 (1), 125-143 (2019)
Link: https://doi.org/10.1016/j.camwa.2018.09.018

Existence of global weak solutions to the kinetic Peterlin model
P. Gwiazda, M. Lukáčová-Medvid'ová, H. Mizerova, A. Szwierczewska-Gwiazda
Nonlinear Analysis: Real World Applications 44, 465-478 (2018)
Link: https://doi.org/10.1016/j.nonrwa.2018.05.016

Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange-Galerkin method. Part I: a nonlinear scheme
Lukáčová-Medviďová, M.; Mizerová, H.; Notsu, H.; Tabata, M.
ESAIM Mathematical Modelling and Numerical Analysis 51 (5), 1637–1661. (2017)
Link: https://doi.org/10.1051/m2an/2016078

Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange-Galerkin method, Part II: A linear scheme
M. Lukáčová-Medvid'ová, H. Mizerova, H. Notsu, M. Tabata
Mathematical Modelling and Numerical Analysis, (2017)
Link: https://doi.org/10.1051/m2an/2017032

An improved dissipative coupling scheme for a system of Molecular Dynamics particles interacting with a Lattice Boltzmann fluid
Nikita Tretyakov, Burkhard Dünweg
Computer Physics Communications 216, 102-108 (2017)
Link: https://doi.org/10.1016/j.cpc.2017.03.009

Global existence result for the generalized Peterlin viscoelastic model
Mária Lukáčová-Medvid'ová, Hana Mizerova, Sarka Necasova, Michael Renardy
SIAM Journal on Mathematical Analysis, 2950-2964 (2017)
Link: https://doi.org/10.1137/16M1068505

Energy-stable numerical schemes for multiscale simulations of polymer-solvent mixtures
M. Lukáčová-Medvid'ová, B. Duenweg, P. Strasser, N. Tretyakov
Mathematical Analysis of Contimuum Mechanics and Industrial Applications II ,Editor:Patrick van Meurs, Masato Kimura, Hirofumi Notsu,ChapterChap5: Interface Dynamics ,Pages1-12,Springer International Publishing AG/ Eds. Patrick van Meurs, Masato Kimura, Hirofumi Notsu (2017)
Link: https://doi.org/10.1007/978-981-10-6283-4_13

The Cassie-Wenzel transition of fluids on nanostructured substrates: Macroscopic force balance versus microscopic density-functional theory
Nikita Tretyakov, Periklis Papadopoulos, Doris Vollmer, Hans-Jürgen Butt, Burkhard Dünweg, Kostas Ch. Daoulas
The Journal of Chemical Physics 145 (13), 134703 (2016)
Link: https://doi.org/10.1063/1.4963792

Energy dissipative characteristic schemes for the diffusive Oldroyd-B viscoelastic fluid
Mária Lukáčová-Medvid'ová, Hirofumi Notsu, Bangwei She
International Journal for Numerical Methods in Fluids, 523-557 (2016)
Link: https://doi.org/10.1002/fld.4195