Mathematical Models and Methods for Lithium-ion Batteries

In our modern age, where electronic devices are indispensable and electric vehicles are gaining momentum, the significance of efficient energy storage systems cannot be overstated. At the forefront of this energy revolution are lithium-ion batteries, which allow for an efficient, lightweight, and safe storage of electrical energy. Their development was awarded the 2019 Nobel Prize in Chemistry and their further improvement to enhance energy storage capacity, safety, durability, and cost-effectiveness, while also reducing reliance on rare materials and minimizing error rates, is a huge aspect in both research and industry.

Beneath the surface of these seemingly straightforward batteries lies a complex world of chemical reactions, electrochemical processes, and intricate material interactions. This is where mathematical modeling steps in as a vital tool to deepen our understanding of lithium-ion batteries. Based on physical principles it is possible to understand their complex, non-linear behavior and to predict their performance under a range of conditions. Mathematical modeling allows for virtual material design and testing and paves the way for their continuous improvement and optimization. This predictive power not only expedites the design and development of new battery technologies but also aids in the creation of strategies to enhance their efficiency, lifespan, and overall safety.

Lithium-ion battery - scales

Lithium-ion batteries span various spatial and temporal scales. The device which is commonly used in electric vehicles is termed battery module, which itself consists of multiple battery cells and a battery management system. The latter monitors, controls, and manages various aspects of the individual battery cells, for example the balancing of voltage and current among the cells during operation. A single battery cell is either a spiral wound or a stack of electrochemical unit cells, which consits of an anode, a seperator and a cathode. Each of these three phases is itself a porous medium with an individual microstructure and electrochemical performance. The porous electrodes, which consist of so-called active material particles kept together by some binding material as well as conductive additives, are soaked with electrolytes through which the transport of lithium ions is established. At each interface between an an electrode particle and the electrolyte an electrochemical double layer forms, where ions in the electrolyte and electrons on the surface of the electrode particle balance their charges. Resolving these double layers even further yields the actual electrode surfaces, on which the intercalation reaction (Li^+ + e^+ leftrightharpoons Li) occurs.

Fig. 1: Overview of the different length scales in Lithium-ion batteries.

Lithium-ion battery - functional principle

The anode and cathode materials of modern lithium-ion batteries have the ability to host lithium within their crystal structure. The anode is typically made of graphite and when the battery is charged, lithium ions move from the cathode to the anode through the electrolyte. This process is called intercalation, and the intercalation reaction can be written as (Li^+ + e^+ + C_6 leftrightharpoons LiC_6). The cathode is usually made of a metal oxide, like lithium cobalt oxide ((LiCoO_2)), lithium iron phosphate ((LiFePO_4)) or Lithium-Nickel-Manganese-Cobalt-Oxide ((Li(Ni_x Mn_y Co_1-x-yO_2)). During charging, lithium ions are extracted from the cathode material and travel through the electrolyte to the anode. This is achieved by applying a voltage to the cell which moves an electron from the cathode to the anode via an external circuit. During discharge, this process is reversed, and the electrons flowing through an outer circuit power th e device of interest, while lithium ions travel from the anode to the cathode through the electrolyte. However, such batteries do not only deliver a current during discharge, but they also inherently generate a voltage difference between anode and cathode. This so-called cell voltage (E) is determined by the actual materials used in the anode and cathode and depends additionally on the status of charge (q in [0,1]) of the battery: A fully charged battery ((q=1)) delivers a higher voltage than a discharged battery ((q=0)). Additionally, the cell voltage (E) depends (parametrically) on the discharge current (C_h), where a larger (constant) current yields a steeper voltage decline (see Figure 2). Mathematical models for Lithium-ion batteries aim to predict this behavior on the basis of coupled non-equilibrium thermo-electrodynamics.

Fig. 2: Computed cell voltage (E) as function of the Capacity.

Mathematical Models

The mathematical modeling of Lithium-ion batteries is a multilayered process. As Fig. 1 shows, various length scales arise in modern battery systems, and depending on the actual application of the model, different degrees of resolution are required. Two mainly different approaches arise in this context: Top-down modeling, where the largest scale is described by some heuristic modeling approach, which is refined if some necessity arises, and bottom-up approaches, where physically sound models for the smallest scale are stated and systematically upscaled to the desired length scale. At WIAS we rely almost exclusively on bottom-up approaches and derive models up to the length scale of full battery cells.

Based on the framework of non-equilibrium thermodynamics, transport equations for the electrolyte phase and the intercalation particles are stated. This yields in general some non-linear partial differential equations which describe the migration of lithium ions through the electrolyte as well as the diffusion of intercalated lithium in the solid active phases, i.e.?the anode and the cathode. We derive so-called material models in terms of free energy densities, which capture the physicochemical nature of the material, for instance, the above-mentioned aspect that intercalation materials can host lithium on interstitial sites in their crystal lattice. These material models are then validated on experimental data, for example on the open-circuit voltage (OCP) of a specific battery material.

Fig. 3: Parametric representation of OCP for Lithium-Nickel-Manganese-Cobalt-Oxide.

The transport equations of the intercalation electrodes and the electrolyte phases are coupled through electrochemical reactions (see also XX->electrochemical systems for details) for the Lithium insertion, i.e.?the reaction (Li^+ + e^+ leftrightharpoons Li). The modeling of this reaction is based on non-equilibrium surface thermodynamics and yields essentially boundary conditions for the transport equations of the two adjacent phases.

Several other modeling aspects are further considered and continuously developed in the field of battery modeling at WIAS, for instance, mechanical effects upon intercalation, side reactions such as the solid-electrolyte-interphase, metal deposition or gas evaporation, phase transitions within the electrode materials as well as new battery types such as Lithium-sulfur or sodium-ion based batteries.

Mathematical Methods

Once all the transport equations and boundary conditions are stated on the scale of the porous electrode, as well as its geometry, mathematical techniques are used to bridge the scale to the electrochemical unit cell. A very important technique is homogenization via multi-scale expansions, which allows us, under the assumption of some geometric periodicity of the porous medium, to deduce a coupled, non-linear partial differential equation system on the next length scale. The great benefit of this method is, that the geometry of the porous electrodes arises only as effective parameters in the transport equations in the electrochemical unit cell. These effective parameters can be computed numerically for a given periodic structure, for instance, the geometry of Fig. 4.

Fig. 4: Realistic, periodic microstructure of a porous battery electrode

Despite homogenization methods, many other mathematical methods, such as Asymptotic expansions, Gradient-Flow- and Entropy-methods as well as the functional analysis to prove the existence and uniqueness of the resulting equation system are employed in the field of battery modeling at WIAS and are continuously further developed by various research groups in the institute.

Numerical Simulations

One the spatial cell of the electrochemical unit cell, one obtains in the simplest case a mathematical model for four variables: (i) the concentration (y_E(x,t)) of lithium ions in the electrolyte, the electrostatic potential (ii) in the electrolyte (phi_E(x,t)) and (iii) in the solid electrode, and (iv) the concentration of intercalated lithium (y_A(x,r,t)). While (y_E,phi_E) and (phi_S) depend only on the macro-scale (x), that is the position within the homogenized porous electrode pointing from anode to cathode, the concentration of lithium (y_A) in the active particle remains dependent on (x) and (r), where (r) is the radial position within each particle. This is a consequence of the rather small solid-state diffusivity of lithium in the lattice host material. Figure 5 shows the numerically computed profiles of these variables at some time (t).

Fig. 5: (Left) Profiles of the variables during discharge at some time (t) across the anode ((x in [0,0.bar 3])), the separator ((x in [0.bar3,0.bar 6])) and the cathode ((xin [0.bar 6,1])). (Right) Profile of intercalated lithium at some time (t) as a function of the macro-scale variable (x) and the micro-scale variable (r).

An important application of such a mathematical model is that it can predict how the cell voltage (E) changes as a function of the status of charge (q), i.e.?the total amount of intercalated lithium in the anode, and simultaneously with respect to the discharge current (C_h). This is of special importance because in experimental conditions this requires a rather large amount of cells to be discharged.

Several further numerical techniques, such as Finite-Element- and Finite-Volume-Methods, contribute to the simulation of Lithium-ion batteries at WIAS and are continuously further developed by various research groups in the institute.


Publications

  Articles in Refereed Journals

  • W. Dreyer, P. Friz, P. Gajewski, C. Guhlke, M. Maurelli, Stochastic many-particle model for LFP electrodes, Continuum Mechanics and Thermodynamics, 30 (2018), pp. 593--628, DOI 10.1007/s00161-018-0629-7 .
    Abstract
    In the framework of non-equilibrium thermodynamics we derive a new model for porous electrodes. The model is applied to LiFePO4 (LFP) electrodes consisting of many LFP particles of nanometer size. The phase transition from a lithium-poor to a lithium-rich phase within LFP electrodes is controlled by surface fluctuations leading to a system of stochastic differential equations. The model is capable to derive an explicit relation between battery voltage and current that is controlled by thermodynamic state variables. This voltage-current relation reveals that in thin LFP electrodes lithium intercalation from the particle surfaces into the LFP particles is the principal rate limiting process. There are only two constant kinetic parameters in the model describing the intercalation rate and the fluctuation strength, respectively. The model correctly predicts several features of LFP electrodes, viz. the phase transition, the observed voltage plateaus, hysteresis and the rate limiting capacity. Moreover we study the impact of both the particle size distribution and the active surface area on the voltagecharge characteristics of the electrode. Finally we carefully discuss the phase transition for varying charging/discharging rates.

  • W. Dreyer, C. Guhlke, R. Müller, A new perspective on the electron transfer: Recovering the Butler--Volmer equation in non-equilibrium thermodynamics, Physical Chemistry Chemical Physics, 18 (2016), pp. 24966--24983, DOI 10.1039/C6CP04142F .
    Abstract
    Understanding and correct mathematical description of electron transfer reaction is a central question in electrochemistry. Typically the electron transfer reactions are described by the Butler-Volmer equation which has its origin in kinetic theories. The Butler-Volmer equation relates interfacial reaction rates to bulk quantities like the electrostatic potential and electrolyte concentrations. Since in the classical form, the validity of the Butler-Volmer equation is limited to some simple electrochemical systems, many attempts have been made to generalize the Butler-Volmer equation. Based on non-equilibrium thermodynamics we have recently derived a reduced model for the electrode-electrolyte interface. This reduced model includes surface reactions but does not resolve the charge layer at the interface. Instead it is locally electroneutral and consistently incorporates all features of the double layer into a set of interface conditions. In the context of this reduced model we are able to derive a general Butler-Volmer equation. We discuss the application of the new Butler-Volmer equations to different scenarios like electron transfer reactions at metal electrodes, the intercalation process in lithium-iron-phosphate electrodes and adsorption processes. We illustrate the theory by an example of electroplating.

  • W. Dreyer, R. Huth, A. Mielke, J. Rehberg, M. Winkler, Global existence for a nonlocal and nonlinear Fokker--Planck equation, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 66 (2015), pp. 293--315.
    Abstract
    We consider a Fokker-Planck equation on a compact interval where, as a constraint, the first moment is a prescribed function of time. Eliminating the associated Lagrange multiplier one obtains nonlinear and nonlocal terms. After establishing suitable local existence results, we use the relative entropy as an energy functional. However, the time-dependent constraint leads to a source term such that a delicate analysis is needed to show that the dissipation terms are strong enough to control the work done by the constraint. We obtain global existence of solutions as long as the prescribed first moment stays in the interior of an interval. If the prescribed moment converges to a constant value inside the interior of the interval, then the solution stabilises to the unique steady state.

  • W. Dreyer, C. Guhlke, R. Müller, Overcoming the shortcomings of the Nernst--Planck model, Physical Chemistry Chemical Physics, 15 (2013), pp. 7075--7086, DOI 10.1039/C3CP44390F .
    Abstract
    This is a study on electrolytes that takes a thermodynamically consistent coupling between mechanics and diffusion into account. It removes some inherent deficiencies of the popular Nernst-Planck model. A boundary problem for equilibrium processes is used to illustrate the new features of our model.

  • W. Dreyer, C. Guhlke, R. Huth, The behavior of a many-particle cathode in a lithium-ion battery, Physica D. Nonlinear Phenomena, 240 (2011), pp. 1008--1019.

  • W. Dreyer, M. Gaberšček, C. Guhlke, R. Huth, J. Jamnik, Phase transition and hysteresis in a rechargeable lithium battery, European Journal of Applied Mathematics, 22 (2011), pp. 267--290.

  • W. Dreyer, C. Guhlke, M. Herrmann, Hysteresis and phase transition in many-particle storage systems, Continuum Mechanics and Thermodynamics, 23 (2011), pp. 211--231.
    Abstract
    We study the behavior of systems consisting of ensembles of interconnected storage particles. Our examples concern the storage of lithium in many-particle electrodes of rechargeable lithium-ion batteries and the storage of air in a system of interconnected rubber balloons. We are particularly interested in those storage systems whose constituents exhibit non-monotone material behavior leading to transitions between two coexisting phases and to hysteresis. In the current study we consider the case that the time to approach equilibrium of a single storage particle is much smaller than the time for full charging of the ensemble. In this regime the evolution of the probability to find a particle of the ensemble in a certain state, may be described by a nonlocal conservation law of Fokker-Planck type. Two constant parameter control whether the ensemble transits the 2-phase region along a Maxwell line or along a hysteresis path or if the ensemble shows the same non-monotone behavior as its constituents.

  • W. Dreyer, J. Jamnik, C. Guhlke, R. Huth, J. Moškon, M. Gaberšček, The thermodynamic origin of hysteresis in insertion batteries, Nature Materials, 9 (2010), pp. 448--453.

  Talks, Poster

  • M. Maurelli , A McKean--Vlasov SDE with reflecting boundaries, CASA Colloquium, Eindhoven University of Technology, Department of Mathematics and Computer Science, Netherlands, January 10, 2018.

  • W. Dreyer, J. Fuhrmann, P. Gajewski, C. Guhlke, M. Landstorfer, M. Maurelli, R. Müller, Stochastic model for LiFePO4-electrodes, ModVal14 -- 14th Symposium on Fuel Cell and Battery Modeling and Experimental Validation, Karlsruhe, March 2 - 3, 2017.

  • P. Gajewski, M. Maurelli, Stochastic methods for the analysis of lithium-ion batteries, Matheon Center Days, April 20 - 21, 2015, Technische Universität Berlin, April 21, 2015.

  • C. Guhlke, Hysteresis due to non-monotone material behaviour inside many-particle systems, SIAM Conference on Mathematical Aspects of Materials Science (MS10), May 23 - 26, 2010, Philadelphia, USA, May 23, 2010.

  • C. Guhlke, Hysteresis due to non-monotone material behaviour inside many-particle systems, DPG Spring Meeting 2010, March 21 - 26, 2010, Regensburg, March 25, 2010.

  • W. Dreyer, Hysteresis and phase transition in many-particle storage systems, 13th International Conference on Hyperbolic Problems: Theory, Numerics, Applications (HYP 2010), June 14 - 19, 2010, Beijing, China, June 17, 2010.

  • W. Dreyer, On a paradox within the phase field modeling of storage systems and its resolution, 8th AIMS International Conference on Dynamical Systems, Differential Equations and Applications, May 25 - 28, 2010, Technische Universität Dresden, May 26, 2010.

  • W. Dreyer, On a paradox within the phase field modeling of storage systems and its resolution, PF09 -- 2nd Symposium on Phase-Field Modelling in Materials Science, August 30 - September 2, 2009, Universität Aachen, Kerkrade, Netherlands, August 31, 2009.

  • W. Dreyer, Phase transitions and kinetic relations, Séminaire Fluides Compressibles, Université Pierre et Marie Curie, Laboratoire Jacques-Louis Lions, Paris, France, September 30, 2009.

  • W. Dreyer, Phase transitions during hydrogen storage and in lithium-ion batteries, EUROTHERM Seminar no. 84: Thermodynamics of Phase Changes, May 25 - 27, 2009, Université Catholique de Louvain, Namur, Belgium, May 27, 2009.