Hüsnü Dal
Middle East Technical University
Electro-Chemo-Mechanics and Fracture of Li-İon Battery Electrodes
Capacity fade in conventional Li-ion battery systems due to chemo-mechanical degradation during charge–discharge cycles is the bottleneck in high-performance battery design. Stresses generated by diffusion-mechanical coupling in Li-ion intercalation and deintercalation cycles, accompanied by swelling and shrinking at finite strains, cause micro-cracks, which finally disturb the electrical conductivity and isolate the electrode particles. This leads to battery capacity fade. As a first attempt towards a reliable description of this complex phenomenon, we propose a novel finite strain theory for chemo-elasticity coupled with phase-field modeling of fracture, which regularizes a sharp crack topology. We apply a rigorous geometric approach to the diffusive crack modeling based on the introduction of a global evolution equation of regularized crack surface, governed by the crack phase field. The irreversible evolution of the crack phase field is modeled through a novel critical stress-based growth function. A modular concept is outlined for linking of the diffusive crack modeling to the complex chemo-elastic material response of the bulk material. Here, we incorporate standard as well as gradient-extended Cahn–Hilliard-type diffusion for the Li-ions, where the latter accounts for a possible phase segregation. From the viewpoint of the methodology, the separation of modules for the crack evolution and the bulk response provides a highly attractive and transparent structure of the multi-physics problem. This structure is exploited on the numerical side by constructing a robust finite element method, based on an algorithmic decoupling of updates for the crack phase field and the state variables of the chemo-mechanical bulk response. The performance of the proposed coupled multi-field formulation will be demonstrated with representative initial boundary value problems.

Thomas Carraro
Heidelberg University
Multi-scale modelling and simulations of electrodes for fuel cells and batteries
The theory of porous electrodes has been widely developed and used in the last decades in electrochemistry. Therefore, reduced models derived by homogenization have been successfully applied in simulations of porous electrodes especially in the study of solid oxide fuel cells (SOFC). In addition, in the last ten years detailed simulations of three dimensional reconstructed electrodes have gained more and more importance for understanding the interplay of the electrochemical processes in dependence of geometrical properties. This approach has become achievable to study SOFC porous electrodes. However, three dimensional simulations of lithium ion battery (LIB) electrodes, due to their high complexity, have still a very limited application. In this talk we present several aspects of the numerical formulation, simulation and parameter estimation of homogenized and microscopic models of SOFC and LIB electrodes.

Nicolae Suciu
Mathematics Department, Friedrich-Alexander University of Erlangen-Nuremberg, and Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
A filtered density function approach for upscaling transport processes in groundwater
Evolution equations for probability density functions (PDFs) and filtered density functions (FDFs) of random concentrations weighted by conserved scalars are derived from Fokker-Planck equations describing stochastically equivalent processes in concentration-position spaces. This approach provides consistent numerical PDF/FDF solutions, given by the density in the concentration-position space of an ensemble of computational particles governed by the associated Ito equations. The solutions are obtained by a global random walk (GRW) algorithm, which is stable, free of numerical diffusion, and practically insensitive to the increase of the number of particles. The upscaled drift and diffusion coefficients describing the PDF transport in the physical space are estimated on single-trajectories of the advection-diffusion process governing the random concentration of the conserved scalar. The procedure uses the self-averaging property of the transport in velocity fields with short-range correlations. The drift and mixing coefficients describing the PDF transport in concentration spaces are parameterized by the trend and the noise inferred from analyses of simulated concentration time series as well as by classical mixing models. A Gaussian spatial filter, applied to a Kraichnan velocity field generator, is used to construct coarse-grained simulations (CGS) for FDF problems. The CGS approach helps to understand the significance of the FDF from a practical point of view and its relation to the PDF approach. The methodology to construct GRW-based PDF and FDF numerical solutions is illustrated for a problem of passive contaminant transport in heterogeneous groundwater systems.

Charles Monroe
University of Oxford
Electrolyte Transport Modelling
Transport models describing ion-conductive media are involved in simulations of electrochemical devices such as batteries and fuel cells. Physically describing the dynamics of an electrolytic medium requires simultaneous consideration of irreversible and equilibrium thermodynamics. This talk will focus on a number of theoretical problems associated with thermodynamically rigorous continuum-scale analysis of multicomponent materials that support simultaneous coupled transport of mass, charge, heat, and momentum. Issues associated with characterisation - and property measurement in particular - will be addressed, along with experimental techniques we have developed and implemented to establish valid multiphysics models. Practical examples including nonaqueous lithium-battery electrolytes and hydrated fuel-cell membranes will be discussed.

Maria Bruna
University of Oxford
Homogenisation of Spatially Varying Porous Media
We discuss the problem of diffusion in a porous medium with a spatially varying porosity. The particular microstructure we consider here comprises a collection of impenetrable spheres, though the methods developed are general. We present two different approaches for calculating the effective diffusion coefficient as a function of the microstructure. The first is a deterministic approach for quasi-periodic media based on the method of multiple scales; the second is a stochastic approach for random media based on matched asymptotic expansions.

Agnel Manuel Ramos
Universidad Complutense de Madrid
Mathematical analysis of a pseudo-two dimensional model for Lithium batteries

Ilenia Battiato
San Diego State University
Multi-scale dynamics of transport in Li ion batteries and limitations of macroscopic Models
Batteries are electrochemical energy storage devices that exhibit physico-chemical heterogeneity on a continuity of scales. As such, battery systems are amenable to mathematical descriptions on a multiplicity of scales that range from atomic to continuum. The need for predicting the system behavior under time dependent forcing over large spatial (system level) and ultra-long time scales (thousand of cycles) requires the adoption of spatially and temporally averaged (continuum) equations. Model accuracy is critical when predictions are needed to accurately estimate macroscopic battery response, including battery lifecycle, SoC and SoH. Macroscopic models treat the electrode as a continuum and are often employed to describe the mass and charge transfer of lithium since they are computational tractable and practical to model the system at the cell scale. Yet, they rely on a number of simplifications and assumptions that may be violated under given operating conditions. Based on perturbation methods of pore-scale equations, we derive the applicability conditions of macroscopic models in terms of relevant dimensionless numbers. Such conditions can be adaptively employed to establish when an algorithm refinement is necessary in order to preserve model predictivity and accuracy. Finally, we discuss how the proposed tool can be used to assess the validity of macroscopic models across different battery chemistry and conditions of operation.

Jochen Zausch
Fraunhofer-Institute for Industrial Mathematics
Coupled thermal-electrochemical simulation of Li-ion batteries on micro and cell scale
Lithium ion batteries are currently one of the most important electrochemical energy storage systems. This is especially true for electromobile applications where the demands on power and energy density, safety and reliability are much higher compared to consumer products like eg mobile phones. One important issue is the thermal management of the large battery systems used in electronic vehicles. A proper thermal management not only ensures safe operation conditions but also an even distribution of temperature throughout the battery pack. The latter is important to maintain a comparable state of health of the different cells.
Designing the thermal management system requires an understanding of the relevant thermal and electrochemical phenomena in a cell. In particular detailed knowledge of the amount and the distribution of heat that is produced inside the electrochemical cells is needed. Here, computer simulations can provide valuable insight into the complex and strongly coupled processes of ion, charge and heat transport.
We have developed a theory to describe these effects in a thermodynamically consistent way on the microscopic scale right above the scale of the electrical double layer. This model is implemented it in the simulation tool BEST (Battery and Electrochemistry Simulation Tool) and enables us to perform coupled thermal-electrochemical simulations in a complicated three-dimensional electrode microstructure including transport in electrodes and electrolyte and the interface kinetics. After a review of the model and details on the numerical method we will discuss and compare the different sources of heat and how they vary with space and state-of-charge.
Due to their computational costs microscale simulations, however, are limited in their applicability for the virtual battery design process. Inspired by the porous electrode theory of Newman and coworkers we show how our microscopic model can be upscaled to allow for a more efficient 3+1-dimensional macro-description on the cell scale. We show recent application examples and discuss differences between micro- and macro-model."

Scott Armstrong
Universite Paris- Dauphine
Quantitative theory of stochastic homogenization
Stochastic homogenization involves the study of solutions of partial differential equations with random coefficients, which are assumed to satisfy a "mixing" condition, for instance, an independence assumption of some sort. One typically wants information about the behavior of the solutions on very large scales, so that the ("microscopic") length scale of the correlations of the random field is comparatively small. In the asymptotic limit, one expects to see that the solutions behave like those of a constant-coefficient, deterministic equation. In this talk, we consider uniformly elliptic equations in divergence form, which has applications to the study of diffusions in random environments and effective properties of composite materials. Our interest is in obtaining quantitative results (e.g., error estimates in homogenization) and to understand the solutions on every length scale down to the microscopic scale. In joint work with Tuomo Kuusi and Jean-Christophe Mourrat, we introduce a new method for analyzing this problem, based on a higher-order regularity theory for equations with random coefficients, which, by a bootstrap argument, accelerates the exponent representing the scaling of the error the all the way to the optimal exponent given by the scaling of the central limit theorem.

Francesco Ciucci
HKUST
Data Mining Oxygen and Lithium Transport in Ionic Conductors.

Ionic conductors are key to many applications ranging from fuel cells to batteries. For example, oxygen conduction and catalysis has been a key research focus in the field of highly efficient solid oxide fuel cells and solid electrolytes are key for high safety lithium batteries. We developed a data-driven framework to assist the analysis of ionic transport. We elucidated the diffusion characteristics in reference to the local atomic arrangement of various conductors including oxygen conductors, such as PrBaCo2O5.5 and La-doped BaFeO3 [1,2], and lithium conductors, i.e., lithium-rich anti-perovskites (LiRAPs) [3] and lithium-stuffed garnets.
In particular, LiRAPs are a promising family of lithium electrolytes exhibiting ionic conductivities above 10−3 S cm−1 at room temperature, among the highest reported values to date. We investigated the defect chemistry and the associated lithium transport in Li3OCl, a prototypical LiRAP, using ab-initio density functional theory (DFT) calculations and classical molecular dynamics (MD) simulations. We found that the low energy pathways of Li between the Cl vacancies could explain the high conductivities and low activation energies of LiCl Schottky systems.

Nicolae Suciu
Romanian Academy, Tiberiu Popoviciu Institute of Numerical Analysis
Towards a filtered density function approach for reactive transport in groundwater
Evolution equations for probability density functions (PDFs) and filtered density functions (FDFs) of random species concentrations weighted by conserved scalars are formulated as Fokker–Planck equations describing stochastically equivalent processes in concentration-position spaces. This approach provides consistent numerical PDF/FDF solutions, given by the density in the concentration-position space of an ensemble of computational particles governed by the associated Itô equations. The solutions are obtained by a global random walk (GRW) algorithm, which is stable, free of numerical diffusion, and practically insensitive to the increase of the number of particles. The general FDF approach and the GRW numerical solution are illustrated for a reduced complexity problem consisting of the transport of a single scalar in groundwater. Randomness is induced by the stochastic parameterization of the hydraulic conductivity, characterized by short range correlations and small variance. The objective is to infer the statistics of the random concentration sampled at the plume center of mass, integrated over the transverse dimension of a two-dimensional spatial domain. The PDF/FDF problem can therefore be formulated in a two-dimensional domain as well, a spatial dimension and one in the concentration space. The upscaled drift and diffusion coefficients describing the PDF transport in the physical space are estimated on single- trajectories of diffusion in velocity fields with short-range correlations, owing to their self-averaging prop- erty. The mixing coefficients describing the PDF transport in concentration spaces are parameterized by the trend and the noise inferred from the statistical analysis of an ensemble of simulated concentration time series, as well as by classical mixing models. A Gaussian spatial filter applied to a Kraichnan velocity field generator is used to construct coarse-grained simulations (CGS) for FDF problems. The purposes of the CGS simulations are two-fold: first to understand the significance of the FDF approach from a practical point of view and its relation to the PDF approach; second to investigate the limits of the mixing models considered here and the desirable features of the mixing models for groundwater systems.

Jamie Foster
University of Portsmouth
Explaining the mechanisms for binder delamination in lithium-ion cells
We will present some recent results from a collaborative research effort which has been focussed on developing and analysing models that elucidate mechanisms causing delamination of the binder matrix from active material (AM) particle surfaces during cell assembly and operation. Two different mechanisms that have the potential to lead to polymer binder delamination have been identified and investigated, namely: (i) absorption of the electrolyte into the binder during cell assembly --- causing swelling of the polymer phase, and; (ii) expansion and contraction of the active materials during (de-)lithiation.
First, we develop a suitable poroviscoelastic model for a porous electrode that is comprised of: (i) a viscoelastic material (that may change its volume) model for the polymer binder; (ii) Darcy's equation for the flow of the electrolyte, and; (iii) solid inclusions (that may change their volume) to mimic the AM particles. Owing to the complex geometry of modern-day electrodes, solution of such a model on a realistic geometry is impractical, even with sophisticated numerical methods. Multiscale techniques (that exploit the disparity in length scales between that of a typical AM particle and the whole electrode) are therefore employed to coherently separate solution of the problem into two steps: the first for the microscopic scale, and the second for the macroscopic scale. Finally, these two problems are solved sequentially using numerical (finite-element) and analytical (asymptotic) techniques, respectively.
By doing so we are able to demonstrate that the two types of morphological damage caused during manufacture and operation are quite different. It turns out, that delamination caused by the binder absorbing the electrolyte, and subsequently swelling, causes detachment of the binder from the AM particles surfaces in planes parallel to the current collector (CC) and separator (S), whereas the volumetric changes of the AM associated with cycling tends to cause delamination in planes perpendicular to the CC and S. Thus, the damage caused during assembly has more important implications for decreasing the electronic conduction through the electrode.

Dhammika Widanalage
WMG, University of Warwick
A new signal design and model structure for Li-ion battery modelling.
The Pulse Power Current (PPC) is a widely used current signal to characterise and model Li-ion batteries in the form of an Equivalent Circuit model (ECM). An ECM has few model parameters and is the model used in a vehicle's battery management system (BMS). The model parameters however depend on the frequency bandwidth and power spectrum of the PPC signal which is very di erent to that of a drive cycle current. An ECM is also a linear model neglecting high current non-linear response e ects. Both factors therefore contribute to model prediction er- rors. The frequency discrepancy between the estimation and validation current pro le should therefore be analysed and the level of non-linearity in the voltage response should be characterised. Here we present a new signal design procedure, pulse-multisine, to generate a representative current signal and a new model structure, Non-linear ECM (NL-ECM), that accounts for non-linear voltage responses. The pulse-multisine signal and NL- ECM model when applied to a 18650 Li-ion battery resulted in a lower RMSE and pk-error by 13 % - 25 % and 52 % - 62 % respectively when compared to an ECM estimated with PPC data. Matching the bandwidth of the estimation current pro le to the validation pro le and modelling non-linearity is therefore important to extend the accuracy of a Li-ion battery model over a wide state-of-charge (SoC) and temperature range. Furthermore, the experimentation time, per SoC and temperature, for the pulse-multisine is several minutes compared to several hours for a PPC experiment.