ECMath MATHEON

ECMath Project SE4:
Mathematical modeling, analysis and novel numerical concepts for anisotropic nanostructured materials

Project heads Christiane Kraus, Gitta Kutyniok and Barbara Wagner
Staff Esteban Meca Álvarez and Arne Roggensack
Internal cooperation OT1 (Hintermüller, Mielke, Surowiec, Thomas), SE2 (Glitzky, Mielke), SE5 (Hintermüller), SE8 (Dreyer, Friz) and SE13 (Eigel, Hömberg, Henrion, Schneider)
External cooperation Helmholtz-VI "Microstructure control for thin film solar cells"
This research is carried out in the framework of Matheon supported by Einstein Foundation Berlin.

Project background

Nanostructuring is fundamental in order to functionalize and optimize modern materials. Developing structuring techniques enables, in particular, the design of next generation thin-film solar cells as well as batteries. While the focus here is towards applications to battery design, the underlying model equations for intercalations processes in Li-ion batteries are very similar to those describing grain growth in poly-Si and CIGSe for application in thin-film solar cell design and synergies are expected. They are phase-field models of Allen-Cahn or Cahn-Hilliard type coupling elasticity, damage and chemical reactions at phase boundaries. Their inherent anisotropic nature is of particular importance (see Figure 2).

Challenges and Results

In order to understand the nanostructured behavior of lithium-ion batteries we have to take into account different physical and chemical effects. The first one is the propagation of the phase boundary between the lithium-rich and the lithium-poor phase (see Figure 3). This comes along with high stresses and high strains in the electrode material and suggests the usage of a phase-field approach of Cahn-Larché type. The intensity of the stresses leads to cracks in the delithiated material (see Figure 3) which have to be incorporated in the model. But also the anisotropy of the material as well as the choice of correct boundary conditions at the interface with the electrolyte has to be considered.

We have developed a novel model that couples the highly nonlinear Cahn-Hilliard reaction (CHR) model with a doubly nonlinear differential inclusion for the damage variable, extending the work of Singh et al. [SCB08] and Zeng and Bazant [ZB14]. One of the main differences of the CHR model to the Cahn-Larché system lies in the chemical active boundary condition which models the lithium intercalation at the surface of the battery's electrode. Mathematically, this is realized by a nonlinear Newton boundary condition for the chemical potential which already leads to challenging analytical and numerical tasks. In [31], we have shown the existence of weak solutions of this non-linear coupled PDE model.

Our numerical methods are based on an adaptive nonlinear multigrid algorithm for the finite-volume discretization. Our results show the formation of a sharp phase boundary between the lithiated and the amorphous silicon that continues to move as a front through the thin layer (see Figure 5). Interestingly, our model captures the non-monotone stress loading curve and rate dependence, as observed in recent experiments and connects the characteristic features of the curve with the structure formation within the layer [29] (see Figure 6).

In [30], we carried out a matched asymptotic analysis to derive the corresponding sharp-interface model that also takes into account the dynamics of triple junctions, i.e. the points where the sharp interface in the bulk of the thin layer intersects the free boundary with the electrolyte. We numerically compare the interface motion predicted by the sharp-interface model with the long-time dynamics of the phase-field model.

In addition to the derivation of appropriate models we have also developed innovative analytical and numerical tool to efficiently capture the various anisotropic properties [21, 22, 27].

Events

B. Wagner and G. Kutyniok are co-organizers, together with A. Münch and J. Tanner (Oxford), of the European Summer School in Modelling, Analysis and Simulation: Crime and Image Processing to be held July 4th - 8th, 2016 at the University of Oxford.

B. Wagner organized one of the BIMoS-Days (Berlin International Graduate School in Model and Simulation based Research), giving several lectures on methods of Multiscale Modeling on Feb. 8, 2016.

G. Kutyniok co-organized the Oberwolfach-Workshop Applied Harmonic Analysis and Sparse Approximation, Oberwolfach, August 16th - 22th, 2015 together with I. Daubechies, H. Rauhut und T. Strohmer.

In coopertion with ECMath project D-SE2, B. Wagner co-organized the ECMath-funded workshop Nanostructures for Photovoltaics and Energy Storage, an interdisciplinary and international workshop, which was co-financed by PVcomB and a grant from EPSRC(UK), took place at the TU Berlin (December 8th - 9th, 2014). The workshop was also the kick-off meeting of the newly established ECMI Special Interest Group “Sustainable Energy”.

Submitted Articles

  1. A. Roggensack and C. Kraus, "Existence of weak solutions for the Cahn-Hilliard reaction model including elastic effects and damage"
  2. E. Meca, A. Münch and B. Wagner, "Sharp-Interface Formation during Lithium Intercalation into Silicon"
  3. E. Meca, A. Münch and B. Wagner, "Thin-film electrodes for high-capacity lithium-ion batteries: Influence of phase transformations on stress"
  4. W. Dahmen, G. Kutyniok, W.-Q. Lim, C. Schwab and G. Welper, "Adaptive Anisotropic Petrov-Galerkin Methods for First Order Transport Equations"
  5. P. Grohs, G. Kutyniok, J. Ma and P. Petersen, "Anisotropic Multiscale Systems on Bounded Domains"
  6. C. Heinemann, C. Kraus, E. Rocca and R. Rossi, "A temperature-dependent phase-field model for phase separation and damage"
  7. G. Kutyniok, V. Mehrmann and P. Petersen, "Regularization and Numerical Solution of the Inverse Scattering Problem Using Shearlet Frames"
  8. W. Dreyer, J. Giesselmann and C. Kraus, "Modeling compressible electrolytes with phase transition"

Refereed Publications

  1. M. Dziwnik, A. Münch and B. Wagner (2016), "An anisotropic phase-field model for solid-state dewetting and its sharp interface limit", to appear in: Nonlinearity.
  2. M.D. Korzec, A. Münch, E. Süli and B. Wagner (2016), "Anisotropy in wavelet based phase-field model", to appear in: Discrete and Continuous Dynamical Systems - Series B.
  3. G. Kutyniok, W.-Q. Lim and R. Reisenhofer (2016), "ShearLab 3D: Faithful Digital Shearlet Transforms based on Compactly Supported Shearlets", ACM Transactions on Mathematical Software. Vol. 42 (5)
  4. B. Bodmann, G. Kutyniok and X. Zhuang (2015), "Gabor Shearlets", Applied and Computational Harmonic Analysis. Vol. 38, pp. 87-114.
  5. C. Heinemann and C. Kraus (2015), "Existence of weak solutions for a hyperbolic-parabolic phase field system with mixed boundary conditions on non-smooth domains", SIAM Journal on Mathematical Analysis. Vol. 47(3), pp. 2044-2073.
  6. C. Heinemann and C. Kraus (2015), "Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects", Discrete and Continuous Dynamical Systems - Series A. Vol. 35 , pp. 2565-2590.
  7. C. Heinemann and C. Kraus (2015), "A degenerating Cahn-Hilliard system coupled with complete damage processes", Nonlinear Analysis: Real World Applications. Vol. 22, pp. 388-403.
  8. C. Heinemann and C. Kraus (2015), "Complete damage in linear elastic materials - modeling, weak formulation and existence results", Calculus of Variations and Partial Differential Equations. Vol. 54, pp. 217-250.
  9. M. Hennessy, V. Burlakov, B.Wagner, A. Goriely and A. Münch (2015), "Controlled topological transitions in thin film phase separation", IAM Journal for Applied Mathematics. Vol. 75, pp. 38-60.
  10. C. Kraus, E. Bonetti, C. Heinemann and A. Segatti (2015), "Modeling and analysis of a phase field system for damage and phase separation processes in solids", Journal of Differential Equations. Vol. 258, pp. 3928-3959.
  11. W. Dreyer, J. Giesselmann and C. Kraus (2014), "A compressible mixture model with phase transition", Physica D: Nonlinear Phenomena. Vol. 273-274, pp. 1-13 .
  12. M. Dziwnik, M. Korzec, A. Münch and B. Wagner (2014), "Stability Analysis of Unsteady, Nonuniform Base States in Thin Film Equations", SIAM Multiscale Model. Simul.. Vol. 12(2), pp. 755-780.
  13. C. Heinemann and C. Kraus (2014), "Degenerating Cahn-Hilliard systems coupled with mechanical effects and complete damage processes", Mathematica Bohemica. Vol. 139(2), pp. 315-331.
  14. M. Hennessy, V. Burlakov, A. Münch, B. Wagner and A. Goriely (2014), "Propagating topological transformations in thin immiscible bilayer films", EPL. Vol. 105, pp. 66001-p1 - 66001-p6.
  15. M. Korzec, M. Roczen, M. Schade, B. Wagner and B. Rech (2014), "Equilibrium shapes of polycrystalline silicon nanodots", Journal of Applied Physics. Vol. 115, pp. 074304-01 - 074304-12.
  16. G. Kutyniok and P. Grohs (2014), "Parabolic Molecules", Foundations of Computational Mathematics. Vol. 14, pp. 299-337.
structure of a li-ion battery enlarge
Fig. 1: Schematic view of a Li-ion battery
TEM image of a Li0.5FePO4 crystal
Fig. 2: a) TEM image showing the domains in Li0.5FePO4 crystal, aligned along the c-axis, b) TEM image of thin Li0.5FePO4 crystal, showing crack in bc plane (Reproduced by permission of The Electrochemical Society from G. Chen et al., Electron microscopy study of the LiFePO4 to FePO4 phase transition. Electrochem. Solid State Lett. (2006) 9 (6): A295-A298)
Formation of a FePO₄ layer during delithiation of a LiFePO₄ crystal
Formation of a FePO₄ layer during delithiation of a LiFePO₄ crystal
Cracks in a partially delithiated LiFePO₄ single crystal
Cracks in a partially delithiated LiFePO₄ single crystal
enlarge
Fig. 3: Formation of a FePO₄ layer during delithiation of a LiFePO₄ crystal (Reprinted with permission from K. Weichert et al., Phase Boundary Propagation in Large LiFePO4 Single Crystals on Delithiation. J. Am. Chem. Soc. (2012) 134 (6): 2988-2992, Copyright 2012 American Chemical Society)
Simplified model of an electrode
Fig. 4: Simplified model of an electrode
Numerical results enlarge
Fig. 5: Concentration and x component of stress before and after phase separation. The concentration is higher near a defect where the highly lithiated phase nucleates. Observe how stress is localized in the
lithiated phase, with very high values near the triple junctions [30].
structure of a li-ion batteryStress loading curve enlarge
Fig. 6: Stress loading curve. The average stress reaches a minimum when phase separation begins (point a). Then the stress becomes less compressive as the lithium moves to the "softer" phase, until phase separation is complete (point b) [29].

Proceedings

  1. D. Knees, R. Kornhuber, C. Kraus, A. Mielke and J. Sprekels (2014), "Phase transformation and separation in solids", In Matheon - Mathematics for Key Technologies. pp. 189-203. EMS Publishing House, Zürich.
  2. G. Kutyniok, W. Dahmen, C. Huang, W.-Q. Lim and u.G.W. C. Schwab (2014), "Efficient Resolution of Anisotropic Structures", In Extraction of Quantifiable Information from Complex Systems. pp. 25-51. Springer.
  3. B. Wagner (2014), "The mathematics of nanostructuring free surfaces", In Matheon - Mathematics for Key Technologies. EMS Publishing House, Zürich.
  4. B. Wagner and W. Dreyer (2014), "Mathematical modeling of multisclae problems", In Matheon - Mathematics for Key Technologies. EMS Publishing House, Zürich.

Books

  1. C. Heinemann and C. Kraus (2014), "Phase Separation Coupled with Damage Processes" Springer-Verlag.

Further Publications

  1. G. Kutyniok (2014), "Geometric Multiscale Analysis: From Wavelets to Parabolic Molecules", Internationale Mathematische Nachrichten. Vol. 225, pp. 1-16.
  2. G. Kutyniok, W.-Q. Lim and G. Steidl (2014), "Shearlets: Theory and Applications", GAMM-Mitteilungen. Vol. 37, pp. 259-280.

References

[SCB08]
G. K. Singh, G. Ceder, and M. Z. Bazant. Intercalation dynamics in rechargeable battery materials: General theory and phase-transformation waves in LiFePO4. Electrochimica Acta, 53:7599-7613, 2008.
[ZB14]
Y. Zeng and M. Z. Bazant. Phase Separation Dynamics in Isotropic Ion-Intercalation Particles. SIAM J. Appl. Math., 74(4):980-1004, 2014.