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The state of the art on electrochemical modelling for Lithium-ion batteries
Framework and objectives
Electric vehicles (EV) are periodically promoted as a pollution-free and economic alternative to gasoline vehicles. In reality, this technology is still affected by low autonomy and the high costs . For these reasons, in the past their diffusion was limited. Despites these technological challenges, the EVs are the most promising solution for: mitigating the greenhouse effect, improving air quality for the citizen and ensuring the energetic stability from oil producing regions . The continuous drop in battery prices combined with the consciousness against the emissions of internal combustion engines, could boost the EV sales up to hundreds of thousands in 2020-2030 , . The electrical vehicles program is the core of the Renault’s strategical framework to achieve zero emissions in automobile transports. The objective is to guarantee, to everyone, the access to silent and emission-free vehicles with no-compromises between performances and safety. The Li-ion batteries, are one of the most promising technology able to achieve these goals. These batteries are largely diffused in consumer electronics because of their performances but, major improvements are still required to reduces costs, improving the safety and extend the lifetime for EV applications. Moreover, these batteries require expensive and time-consuming tests to ensure their safety, evaluate their performance and assess their degradation during the years , . Thus, better methods are required to predict these information during the cell design phase . Improvements for the cell design (e.g. better performances, faster charging protocols or reduced ageing) or the understanding of physicochemical phenomena, could be achieved with the support of modeling . In fact, an electrochemical model could simulate the behavior of lithium ion cells by using the chemical characteristics of the compounds and the design parameters.
In this field, the Newman’s model and its variants represent a reference. These models are nowadays an intense object of research and largely promoted in commercials software –.
The growth of articles on EVs, batteries and modelling are illustrated in Figure 1. Before 1989 the numbers of articles on EVs can be neglected while a constant number of articles on non-lithium ion is observed in Figure 1(A). Then, after the introduction of the first commercial lithium ion battery by Sony in 1991, the 9 publications on EVs increase exponentially by reaching 2000 articles only for in the last year. Thus, the articles on lithium-ion batteries follows the same exponential growth. As consequence, the general interest for new battery technologies is rising to supply the demand of electric vehicles. However, it is also evident how the development of electric vehicles is strictly connected to the introduction of lithium ion batteries. The number of papers on battery modelling is also rising but near 40 % of them, in 2016, are focused on lithium ion batteries as illustrated in Figure 1(B). Among all the publications on modeling, few hundreds concern physical based models. In conclusion, the development of modeling tools, from both industry (e.g. automotive producers and battery manufacturer) and academy (e.g. research centers) aiming to push forward the battery performances, is rapidly increasing.
Figure 1 – The histograms in (A-B) reports the number of publications in the periods coming from 1989 to 2016. In picture (A) the publications are for keywords: “Electric Vehicles”, “Lithium-Ion”, and “batteries” that are not lithium-ion. In picture (B) are reported the number of lithium ions model and the models of batteries that are not lithium ion. (Source: Web of Science®)
During a previous Ph.D. project in Renault, a simplified electrochemical model, developed by M. Safari in 2011, was able to simulate the battery (graphite/LFP) voltage up to 1C load , . Thus, this work, can be considered as the further step to model a lithium ion battery using the Newman’s theory to increase the comprehension on batteries and improving the performances of modelling. In fact, after the testing of some commercial software, Renault decided to develop its own tools to improve the know-how on lithium-ion battery modelling.
This work aims to develop the simplest possible electrochemical model, based on the Newman’s theory, to find a compromise between the predictability and the number of parameters. For this reason, the parameters available in literature are analysed to detect a range of values for each parameter and then investigate their contribution in the model. To generalise these results, the equations systems and the parameters are set in a dimensionless form, following an approach commonly used in fluid mechanics. Thus, the effects of each parameter are isolated in a limit case. In fact, for a limit case the parameter object of study, is the only parameter responsible for the result observed. Another question we want to answer, is how to accurately characterize the battery performances without worrying about the load history. In fact, we believe that only effective and accurate tests can be compared with the numerical simulations, while at the same time it uses only few parameters that are could be easily measured.
Thus, at the first the literature state of the art is identified and then critically discussed. An innovative non-dimensional system of equations able to generalise kinetic laws from the simulations based on the Newman’s model is proposed. The values of the parameters from the literature are regrouped in a database also useful for further simulations. Commercial LG cells are electrically and physiochemically characterized to evaluate the performances and identify the parameters for the model. Thus, a new protocol aiming to accurately establish the electrical performances of the cells is proposed. Then, the electrode balancing and how the shape of the isotherms affects the estimation of the state of charge are deeply studied. Finally, in the last section the proposed non-dimensional model is solved in COMSOL for limiting cases and the kinetic limitations are generalised.
In the next section, the working principles of the lithium ion batteries are shortly exposed.
Lithium-ion working principles
The Lithium-ion battery is a complex system where mass transports and chemical reactions acts together –. In this section, the working principles are illustrated while the characteristics of a commercial cell are investigated in detail § 4.2. The LIB system is working because the electrodes have the ability to reversibly host lithium in their structure. The cell is a sandwich composed of three porous components, as reported in Figure 2: two electrodes and a separator placed within.
Figure 2 – The lithium ion cells is constituted of two electrodes backed on two current collectors and a separator. The working principle of a lithium ion cell is illustrated. The red circles with a plus sign represents the lithium ions, while the black circle with a minus sign represent the electrons and the blue particles is the solvent. The dashed lines in blue and red represents the path of the electron and the lithium ions, respectively.
The void spaces, in these porous structures, are filled with an electrolyte composed of a mixture of solvents, additives and a lithium salt (e.g. LiPF6 in a mixture of ethylene carbonate and ethyl methyl carbonate). The purpose of the separator is to avoid the direct contact between the electrodes (in fact, this generate a short circuit) but to allow the flow of charged species. The electrodes are based on chemical compounds where the lithium can soak into them. The positive electrode is usually a lithium metal oxide, with a large choice of different chemical elements (e.g. lithium manganese oxide, lithium nickel cobalt aluminum oxide, lithium iron phosphate), while the negative electrode is usually based on carbon (or rarely other materials such as the lithium titanate). The potential of the electrodes depends on the chemical species and the amount of lithium in its structure. Since each species has a different potential, a voltage jump between the two electrodes is created. Thus, the positive electrode and the negative electrode are assigned to the compounds with the higher potential and the lower potential, respectively. Hence, the cell voltage is given by the difference between the potentials of these electrodes. The electrodes are backed on a metallic current collector, usually aluminum and copper for the positive electrode and the negative electrode, respectively. Then, an electric tab is soldered to each current collector. When these tabs are connected to a load via an externa wire, a flow of electrons circulates between the electrodes and thus the cell discharges, as illustrated in Figure 2 (A). Instead, when the cell is connected to an electric source the reaction is forced to reverse and consequently the cell is recharged, as illustrated in Figure 2 (B). The details of this process that produces electrons circulating in an external circuit are explained in the followings.
Either during the cell charge or discharge, lithium-ions intercalate in one electrode and deintercalated from the other. Thus, the ions shuttles between the electrodes creating a flow of charged species in the electrolyte. At the same time, this reaction of intercalation requires the participations of electrons: when an ion of lithium leaves the host structure of the electrode, an electron moves in the external circuit in direction of the other electrode. In parallel, an electron from the external circuit react with the lithium ion to intercalate in the host structure in the other electrode. Thus, a net flow of electric charges is moving: ions in the electrolyte and electrons in the external cables and electrodes.
The working principle of any electrochemical system is based on the possibility that a chemical species exists under two different forms. These two species are called oxidant and reductant, indicated by “Ox” and “Red” respectively. The transformation of the matter from “Ox” to “Red” goes via the electron shift at the atomic level, the so called redox reaction. When a chemical element (i.e. lithium) of this compound loses one electron this species undergoes to oxidation, instead, the reduction occurs, when the species gains an electron.
In this system, the electro-chemical reactions occur at the interface between the solid and the liquid phases. The porous structure of the electrodes, guarantees a higher active surface compared to a bulky electrode having the same dimensions. Consequently, the electrodes in lithium ion cells are porous, because higher is the active surface and higher is the power density.
During the charge, most of the lithium in the positive electrode leaves the host structure and goes into the host structure of the negative electrode. If the negative electrode is made of lithium metal, the lithium ions are simply deposed on its surface. To reduce the formation of dendrites (i.e. the deposition of solid lithium over the electrodes when their potential is close to zero), that could generate short circuits, the lithium metal is replaced with the less performant graphite as a negative electrode. In fact, the graphite has a lower specific capacity but because of its slightly higher potential than lithium metal, the lithium deposition is disfavored. However, another inconvenient on graphite, is that during the first lithiation the potential decreases and the electrolyte reacts with the carbon on the surface creating new compounds and releasing gas, such as CO2. This process is accompanied by a non-reversible consumption of lithium ions that remains trapped in these compounds. Thus, the surface of the active material is covered with the so-called solid electrolyte interface (SEI). This layer is composed of a mixture of lithium carbonates and many other complex compounds –. Fortunately, it constitutes a barrier between the active material and the electrolytes, preventing further reactions between the electrode and the electrolyte. However, the SEI (that is mostly formed during the first charge of the battery), could be broken up and it is subject to degradation due to both current and temperature cycling. Consequently, the SEI is re-formed at each time that the electrode’s surface is directly exposed to the electrolyte. One of the major degradation of the battery is attributed to the non-stability of SEI resulting in the direct contact between the graphite the electrolyte. Thus, the re-decomposition of the electrolyte creates a new layer of SEI to fill the cracks in the old SEI. This overview illustrates how the involved phenomena are complex and mutually coupled. Thus, an appropriate model is required to develop new cells and to simulate their behavior when they are integrated in a system (such as the EVs powertrain).
Lithium-ion battery modelling
The main purpose of modeling is to develop a mathematical representation able to simulate a system behavior. In lithium ion batteries, many complex phenomena are involved such as: the mass transport, migrations of ions, red-ox reactions (i.e. transformations of the chemical compounds when they react with electrodes) and side reactions. The current collectors can be neglected since their conductivity is orders of magnitude higher than the values of active materials or electrolytes as reported in § 3.5. Thus, the potential drop in the collectors may be reasonably neglected. For these reasons, any proposed model can partially describe its behavior. In recent years, many models are developed for different purposes , , such as the integration of a battery in a more complex electrical system (e.g. a EV powertrain) or to focus into the internal physics (e.g. mass transport and chemical reactions). Some niche models are based on stochastics, artificial neural networks or the fuzzy logic –. These approaches are not based on the physics of the system, but they still can reproduce its behavior. Thus, it is possible to divide all these models in two families based on: the phenomenology or the physics of the system. The phenomenological models can reproduce the battery behavior (i.e. the voltage drop under an external load) after a test campaign aiming to characterize electrically the cell performances in several operating conditions (e.g. temperature, state of charge, degradation, etc.). These models includes, as an example, empirical equations (Shepherd in 1965 ) and equivalent electric circuits. Instead, the electrochemical model describes the physics of the involved phenomena, such as (the list is not exhaustive): diffusion of ions, mechanical strain, charge transfer and migrations of ions.
However, some hybrid models containing elements of both families can also be found, as an example: the model developed by Rakhmatov & Vrudhula 2001  contains the diffusion of lithium in the solid phase and the empirical Peukert’s law, or the transmission line model that uses electric lumped elements to simulate the porous electrodes , .
Today, when real time computations are required (e.g. in the battery management systems, BMS), the simple approach with the equivalent electrical circuit is usually preferred , . In fact, in EVs is important to know, the battery state of charge, power fade, capacity fade, and instantaneous available power, that are used by the battery management systems (BMS) to estimate the present operating condition of the battery packs. This is achieved by adding the control theory to an equivalent electrical circuit. The equivalent electrical circuit is composed of several lumped circuital elements (e.g. voltage generators, resistors, capacitors). The values of this elements are estimated with experiments and electrical tests , . However, these components are usually functions of the state-of-charge (SOC), state of health (SOH) and temperature–. It should be mentioned that many researches are mplementing simplified electrochemical models (e.g. single particle or linearizing the charge transfer relationship, cf. § 1.4) with a control systems in the BMS –. The transmission line model (TLM), uses circuital elements disposed in the configuration illustrated in Figure 3 , –. In this schematics, the resistances are attributed to electrode and electrolyte conductivity, while the lithium diffusion in the electrodes and the charge transfer kinetics are represented by non-constant and frequency dependent impedances . Like in phenomenological models, these elements are influenced by the battery state of charge, the temperature and the state of degradation.
Table of contents :
1 The state of the art on electrochemical modelling for Lithium-ion batteries
1.1. Framework and objectives
1.2. Lithium-ion working principles
1.3. Lithium-ion battery modelling
1.4. Review on electrochemical models
1.5. Tests and simulations
2 Electrochemical equations system
2.1. Newman’s PDE equations system (diluted solutions)
2.2. Critical review of the literature
2.3. The system of equations proposed in this study
2.3.1 Comparison with Newman
2.3.2 Non-dimensional PDE equations system (positive and negative porous electrodes)
2.3.3 Non-dimensional PDE equations system for the Li-metal foil negative electrode
3 Analysis of the parameters from literature
3.1. Electrolyte conductivity
3.2. Electrolyte diffusivity
3.3. Transport number
3.4. Solid phase diffusivity
3.5. Solid phase conductivity
3.6. Kinetic reaction rate constant
3.7. Dimensional design parameters
3.8. Dimensionless parameters
3.9. The ageing effect on parameters
4 Electrical and physicochemical characterizations
4.1. Electrical characterization
4.1.1 Reproducibility analysis of a test protocol for galvanostatic discharges
4.1.2 Voltage dip during galvanostatic discharges
4.1.3 Galvanostatic discharge to 0.05 V
4.2. LGC INR18650MH1 chemical characterization
5 Electrode balancing
5.1. Introduction to electrode balancing
5.2. Introduction to isotherms and the states of lithiation in either complete cell and “half-cell” configurations
5.3. How the shape of the isotherms influences the accuracy on the initial states of lithiation
5.4. Identification of the state of lithiation in LGC INR18650MH1 half cell
5.5. The aging scenarios
6 Numerical simulations with COMSOL
6.1. Galvanostatic discharges
6.1.1 Kinetic redox limitation
6.1.2 Electrolyte mass transport
6.1.3 Electronic transport
6.1.4 Solid phase diffusivity
6.1.5 Mixed case: solid phase diffusivity and electronic transport
6.1.6 Mixed case: solid phase diffusivity and electrolyte diffusivity
6.1.7 Mixed case: electronic transport and electrolyte mass transport diffusivity
6.1.8 Conclusions and perspectives
6.2. Pulse-rest sequences
6.2.2 Time constants
6.2.4 First steps to an appropriate interpretative framework of GITT
7 Conclusions and perspectives
8.1. Mesoscopic 1D porous electrode model
8.2. Estimation of the state of charge at 4.3 V
8.3. Differential voltage
8.4. PDE equations system in COMSOL® (half-cell)
8.5. PDE equations system in COMSOL® (full-cell)
8.6. Simulation of the LGC INR18650MH1 with COMSOL®
8.7. C-rate profile used in galvanostatic discharges in § 4.1.2
8.8. C-rate profile used in galvanostatic discharges in § 4.1.3