Lithium-ion (Li-ion) batteries have become the most common rechargeable batteries for consumer electronics and automotive applications due to their high-energy densities, decent power density, relatively high cell voltages, and low weight-to-volume ratios.
The term Li-ion battery refers to an entire family of battery chemistries. The common properties of these chemistries are that the negative and the positive electrode materials serve as hosts for lithium ions and that the battery contains a nonaqueous electrolyte.
The increased demand and the pressure for improving battery performance have intensified the need for mathematical modeling. Modeling and simulations allow for the analysis of an almost unlimited number of design parameters and operating conditions at a relatively small cost. Experimental tests are used to provide the necessary validation of the models.
The Newman Model
Mathematical models can describe and predict cell voltage and current density during discharge, recharge, transient studies, as well as include mechanisms for aging and failure. The influence of different material properties and design parameters can be studied under these conditions.
The workhorse for high-fidelity modeling of Li-ion batteries is the so-called Newman model. This model has been validated by many scientists over the years. It has also been further developed and extended by others, for example, to account for designs with multiple electrode materials, the formation of a solid electrolyte interface, and alternative electrode kinetics. The original 1D model has later also been formulated by COMSOL for 2D, 2D axisymmetric, and 3D models.
Performance Models
A typical experiment that can be accurately described by a physics-based battery model is a discharge-recharge cycle as shown in Figure 2, where a high-energy battery for mobile applications is simulated.
In Figure 2, the green line represents the current density. The current density is defined as positive during the initial period of discharge of 2000 seconds, which is then followed by a period of rest (0 current) for 300 seconds. The battery is then recharged (negative current) for 2,000 seconds before it is allowed to rest again.
The response of the battery voltage to this cycle is shown by the blue curve and is very accurately predicted by the model. The voltage decays with the discharge time due to losses caused by mass transport resistance, concentration and activation overpotential, and due to thermodynamics. The battery voltage increases as the battery is recharged, again due to the same losses, but now with the opposite sign. When the battery is allowed to rest, the voltage slowly reaches a steady open cell voltage.
The advantage of the performance models is that they can be used to find out and analyze the processes that are responsible for the limitations in the performance of the battery and the losses that are responsible for these limitations. The models can also be used to evaluate how the energy and power density are changed when the design of the electrode is varied and how the electrode materials are utilized in the cell design.
Thermal Management and Safety
Most of the losses in a battery, for example ohmic losses and activation overpotentials, generate heat. In addition, in cold weather and during startup, the battery system may require heating in order to work. The cooling and heating of the battery system require thermal management.
Using a physics-based model, the different sources of heat are directly available from the model. The advantage of using a thermal model is that the temperature inside the cell can be estimated from the measurement at the surface. This allows for studying unwanted effects such as internal short circuits, where hot spots may be the cause of thermal runaway.
Temperature variations are predominant within large cells, since uneven current distributions cause uneven heat productions. Heating and cooling design for normal operation and for regular startup is focused on minimizing weight and power consumption.
The design of the thermal management system in a battery system is substantially complicated by the fact that it has to be able to cope with malfunctioning cells. Malfunction is usually caused by short circuiting of the electrodes due to metal deposits at the cathode that grow across the electrolyte and make electronic contact with the anode.
Mechanical damage is another cause of short circuits in batteries. If a foreign metallic object penetrates the battery pack or if the battery pack is damaged by being squashed, it can provide an internal conduction path that creates a short circuit. One standard safety test for Li-ion batteries is the “nail test” in which a nail is driven into the battery to create a short circuit. The nail conducts the current as an external circuit with a very small load, while the area around the nail behaves as during a discharge.
Characterization and State-of-Health
Li-ion batteries lose capacity and the internal resistance increases over time. After a while, the battery is unable to deliver the energy or power that is demanded. The reactions that are responsible for this aging can be included in a performance model.
There are many factors that influence the performance, and it is often difficult to separate the effects of different design and operation parameters on performance. A key to separating the influence of the different involved phenomena is the fact that they often have different time constants. For example, electrochemical reactions are usually fast compared to molecular diffusion.
A method that is becoming more common for analyzing the state-of-health of batteries is electrochemical impedance spectroscopy (EIS). This method is based on measuring impedance at different frequencies, thus separating processes with different time constants.
Physics-based performance models of EIS may be combined with experimental measurements to study effects of aging and decay of the battery material at the cell level.
Beyond the Newman Model
The latest development for understanding the electrodes in batteries is to use heterogeneous models that treat the geometry of the material in detail, in contrast to the homogeneous model. This is achieved by constructing the geometry from micrographs.
The example above shows a hypothetical heterogeneous structure with the graphite particles described as ellipsoids and the pore electrolyte filling the void between the skeleton formed by the ellipsoids. A structural analysis coupled to the detailed electrochemistry, with volume expansion caused by lithium intercalation, reveals that the necks of the skeleton structure are subjected to the highest stresses and strains. Cracks may therefore form there over repeated cycles and increase the ohmic losses, which contribute to the deterioration of the battery performance.
Multiphysics Models and Partial Differential Equations
The most accurate way of describing the Li-ion battery is through physics-based models formulated with partial differential equations. Further development of these batteries requires new models and new formulations, such as the heterogeneous model exemplified above. Models must be able to describe the fundamental processes that determine battery performance in order to give the deeper understanding required for developing new materials and new designs. There is no way around this: models and simulations are the shortcuts.
