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  HelmholtzVI "Microstructure control for thin film solar cells" 
Project background
Nanostructuring is fundamental in order to functionalize and optimize modern materials. Developing structuring techniques enables, in particular, the design of next generation thinfilm solar cells as well as batteries. While the focus here is towards applications to battery design, the underlying model equations for intercalations processes in Liion batteries are very similar to those describing grain growth in polySi and CIGSe for application in thinfilm solar cell design and synergies are expected. They are phasefield models of AllenCahn or CahnHilliard 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 lithiumion batteries we have to take into account different physical and chemical effects. The first one is the propagation of the phase boundary between the lithiumrich and the lithiumpoor phase (see Figure 3). This comes along with high stresses and high strains in the electrode material and suggests the usage of a phasefield approach of CahnLarché 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 CahnHilliard 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 CahnLarché 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 nonlinear coupled PDE model.
Our numerical methods are based on an adaptive nonlinear multigrid algorithm for the finitevolume 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 nonmonotone 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 sharpinterface 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 sharpinterface model with the longtime dynamics of the phasefield 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 coorganizers, 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 BIMoSDays (Berlin International Graduate School in Model and Simulation based Research), giving several lectures on methods of Multiscale Modeling on Feb. 8, 2016.
G. Kutyniok coorganized the OberwolfachWorkshop 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 DSE2, B. Wagner coorganized the ECMathfunded workshop Nanostructures for Photovoltaics and Energy Storage, an interdisciplinary and international workshop, which was cofinanced by PVcomB and a grant from EPSRC(UK), took place at the TU Berlin (December 8th  9th, 2014). The workshop was also the kickoff meeting of the newly established ECMI Special Interest Group “Sustainable Energy”.
Submitted Articles

A. Roggensack and C. Kraus, "Existence of weak solutions for the CahnHilliard reaction model including elastic effects and damage"Preprint: WIASPreprint 2231, 2016

E. Meca, A. Münch and B. Wagner, "SharpInterface Formation during Lithium Intercalation into Silicon"Preprint: Preprint92016, TU Berlin

E. Meca, A. Münch and B. Wagner, "Thinfilm electrodes for highcapacity lithiumion batteries: Influence of phase transformations on stress"Preprint: Preprint52016, TU Berlin

W. Dahmen, G. Kutyniok, W.Q. Lim, C. Schwab and G. Welper, "Adaptive Anisotropic PetrovGalerkin Methods for First Order Transport Equations"Preprint: arXiv:1601.00193

P. Grohs, G. Kutyniok, J. Ma and P. Petersen, "Anisotropic Multiscale Systems on Bounded Domains"Preprint: arXiv:1510.04538

C. Heinemann, C. Kraus, E. Rocca and R. Rossi, "A temperaturedependent phasefield model for phase separation and damage"Preprint: WIASPreprint 2164, 2015

G. Kutyniok, V. Mehrmann and P. Petersen, "Regularization and Numerical Solution of the Inverse Scattering Problem Using Shearlet Frames"Preprint: Preprint262014, TU Berlin

W. Dreyer, J. Giesselmann and C. Kraus, "Modeling compressible electrolytes with phase transition"Preprint: WIASPreprint 1955, 2014
Refereed Publications

M. Dziwnik, A. Münch and B. Wagner (2016), "An anisotropic phasefield model for solidstate dewetting and its sharp interface limit", to appear in: Nonlinearity.

M.D. Korzec, A. Münch, E. Süli and B. Wagner (2016), "Anisotropy in wavelet based phasefield model", to appear in: Discrete and Continuous Dynamical Systems  Series B.Preprint: Preprint412014, TU Berlin

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)DOI: 10.1145/2740960

B. Bodmann, G. Kutyniok and X. Zhuang (2015), "Gabor Shearlets", Applied and Computational Harmonic Analysis. Vol. 38, pp. 87114.

C. Heinemann and C. Kraus (2015), "Existence of weak solutions for a hyperbolicparabolic phase field system with mixed boundary conditions on nonsmooth domains", SIAM Journal on Mathematical Analysis. Vol. 47(3), pp. 20442073.DOI: 10.1137/130949099

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. 25652590.

C. Heinemann and C. Kraus (2015), "A degenerating CahnHilliard system coupled with complete damage processes", Nonlinear Analysis: Real World Applications. Vol. 22, pp. 388403.

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. 217250.

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. 3860.DOI: 10.1137/130950227

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. 39283959.

W. Dreyer, J. Giesselmann and C. Kraus (2014), "A compressible mixture model with phase transition", Physica D: Nonlinear Phenomena. Vol. 273274, pp. 113 .

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. 755780.DOI: 10.1137/130943352

M. Hennessy, V. Burlakov, A. Münch, B. Wagner and A. Goriely (2014), "Propagating topological transformations in thin immiscible bilayer films", EPL. Vol. 105, pp. 66001p1  66001p6.

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. 07430401  07430412.DOI: 10.1063/1.4863467

G. Kutyniok and P. Grohs (2014), "Parabolic Molecules", Foundations of Computational Mathematics. Vol. 14, pp. 299337.
lithiated phase, with very high values near the triple junctions [30].
Proceedings

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. 189203. EMS Publishing House, Zürich.

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. 2551. Springer.

B. Wagner (2014), "The mathematics of nanostructuring free surfaces", In Matheon  Mathematics for Key Technologies. EMS Publishing House, Zürich.

B. Wagner and W. Dreyer (2014), "Mathematical modeling of multisclae problems", In Matheon  Mathematics for Key Technologies. EMS Publishing House, Zürich.
Books

C. Heinemann and C. Kraus (2014), "Phase Separation Coupled with Damage Processes" SpringerVerlag.
Further Publications

G. Kutyniok (2014), "Geometric Multiscale Analysis: From Wavelets to Parabolic Molecules", Internationale Mathematische Nachrichten. Vol. 225, pp. 116.

G. Kutyniok, W.Q. Lim and G. Steidl (2014), "Shearlets: Theory and Applications", GAMMMitteilungen. Vol. 37, pp. 259280.
References
 [SCB08]
 G. K. Singh, G. Ceder, and M. Z. Bazant. Intercalation dynamics in rechargeable battery materials: General theory and phasetransformation waves in LiFePO4. Electrochimica Acta, 53:75997613, 2008.
 [ZB14]
 Y. Zeng and M. Z. Bazant. Phase Separation Dynamics in Isotropic IonIntercalation Particles. SIAM J. Appl. Math., 74(4):9801004, 2014.