High C-Rate Differential Expansion and Voltage Model for Li-Ion Batteries
Authors: Mohtat Peyman , Siegel Jason B , Stefanopoulou Anna G 
Published: February 2020

The differential voltage analysis (DVA) is one of the conventional approaches for estimating capacity degradation. The phase transitions in the graphite electrode emerge as observable peaks in the differential voltage. The DVA method utilizes these peaks for estimating battery’s capacity. Unfortunately, at higher C-rates (above C/2) the peaks flatten, and their locations become unobservable. Hence the capacity estimation becomes highly uncertain. Li-ion batteries also expand (contract) during charging (discharging) in repeatable patterns. Thus, the derivative of cell expansion is also used for identifying phase transitions. The peaks in the voltage derivative curve and the second derivative of the expansion curve correspond to the same phase transitions in the material [1]. Unlike the differential voltage, the peaks are observable up to 1C rate in the differential expansion curve, which makes the differential expansion an excellent method for capacity estimation during fast charging scenarios (above C/2). To understand why that is the case, and at the same time develop a model-based estimator a multi-particle electrochemical model is developed. The electrochemical model is primarily based on the well-known Doyle Fuller Newman (DFN) model. The inclusion of electrodes with multiple materials is done in [2], and the approach to modeling multiple sized particles is analogous to the models for multi-material electrodes. The particles are in a parallel configuration, and similar to the single particle model (SPM) the molar flux and solid concentration are independent of location [3]. However, the electrolyte dynamics are considered in the model. The model also includes a particle expansion model and an energy balance model. Currently, the temperature is only coupled to the thermal expansion model.
The cell under consideration in this study is a 5 Ah graphite/NMC pouch cell fabricated at the University of Michigan Battery Lab. A fixture was designed to measure the mechanical response of the cell. The expansion was measured using a displacement sensor (Keyence, Japan) mounted on the top plate, and a battery cycler (Biologic, France) was used for measuring the voltage. The fixture was installed inside a climate chamber with the temperature set to 25˚C. The cell was charged with a constant current (CC) from the fully discharged state at different rates (C/20, C/10, C/5, C/2.5, 1C, and 2C) up to 4.2 V, followed by a constant voltage (CV) period until (I<C/50), followed by a pause of 3 h. The thermal tests were performed to characterize the thermal expansion and cooling behavior of the fixture. The expansion was also monitored during these tests and used in thermal expansion coefficient estimation. Furthermore, to measure the potential of graphite and NMC electrodes, coin cells were built. The Li/NMC (Li/graphite) coin cell was cycled at C/50 between 2.8 and 4.35 V (0.005 and 1.0 V). The lattice expansion data was taken from literature for graphite and NMC.
Figure 1(a) shows the overall model structure. Furthermore, Fig. 1(b) presents the voltage and expansion derivatives; from this figure, the aforementioned smoothing effect of the voltage derivative is seen. Note that the voltage peaks are unobservable at 1C rate. The voltage and expansion derivatives for the C/5 and 1C are selected to showcase the smoothing effect of the model, and they are depicted in Fig. 1(c) and (d). Finally, Fig. 1(e), (f), and (g) show the model prediction together with the data for voltage, expansion, and temperature rise, respectively. The multi-particle model with different particle sizes results in a distribution of concentration among the particles. This is simulating the non-uniform charging of particles that can happen at higher rates, which is causing the smoothing effect. Furthermore, the reason for the observability of the peaks at higher rates in expansion is that first, the changes between lattice parameters of two different phases are more significant compared to their voltage, and second, the voltage is a function of the surface concentration, while the expansion is a function of the concentration curve in the particle.
References:
[1] Z. J. Schiffer, et al., J. Electrochem. Soc., 163, A427-A433 (2016).
[2] P. Albertus, et al., J. Electrochem. Soc., 156 (7) A606–A618 (2009).
[3] B. S. Haran, et al., J. Power Sources, 75 (1) 56–63 (1998).

Figure 1