Thermal metastability is an inescapable trait of lithium-ion batteries. However, canonically, only electrochemical signatures are studied as calorimetry imposes a controlled environment to isolate the self-heating signal. We propose an *in operando* approach for characterizing the thermal signatures. Using an inverse heat transfer formulation, we deconvolve the self-heating signature from other simultaneous heat transfer modes. Temporal variation of heat generation is subsequently estimated. This approach does not presuppose a particular electrochemical operation and is agnostic to materials used in the Li-ion cells. The generality and simplicity of this experimental approach rely on inverse thermal analysis and concurrent calibration of ambient natural convection response.

Lithium-ion (Li-ion) batteries have become the customary technology for high energy and high power applications, i.e., electric vehicles and grid storage.^{1–3} Thermal interactions are integral to such extreme functioning. Electrochemical complexations in porous battery electrodes are composed of charge transport and interfacial reactions at the microstructural scale.^{4,5} These pore-scale events manifest as battery internal resistance, $rin$, and lead to Joule heating. This self-heating is characterized as

where the second term, $T\Delta S\u0307$, specifies the entropic penalty to interconversion of chemical (intercalation of lithium) and electrical (voltage and current) energy modes. The self-heating behavior is intrinsic to electrochemical abuse scenario,^{6} e.g., external short and overcharge, as well. The self-heating, if not modulated appropriately, leads to unattenuated temperature rise and triggers thermally activated autocatalytic side reactions^{7} that burgeon to thermal runaway.^{8,9} Repeated operation prompts chemical degradation, altering the thermal response. Such temperature-dependent metastability of Li-ion cells makes an *in operando* characterization of heat generation vitally important.^{10,11} The temperature and heat generation rate jointly provide necessary insights into thermal metastability, thus defining the thermal signature for Li-ion batteries.

The thermal interactions have largely been overlooked since the calorimeter^{12–16} is the only tool at a researcher's disposal to evaluate heat generation for energetic systems. Calorimetry is prohibitive given the specific instrumentation needs, e.g., thermal isolation for adiabatic testing (not to mention the cost constraints). It is suitable for neither field testing nor continuous monitoring. These attributes are fundamentally tied into achieving a thermally isolated and noise-free test environment.

Here, we propose an elegant approach to probe the cell's thermal signature using an Inverse problem formulation.^{17} Essentially, the experimental difficulties are transformed into the more involved analysis of the measurand (i.e., temperature).

The electrochemical operation of the Li-ion cell (in ambient) is accompanied by temperature rise that follows the energy balance

There are two heat transfer mechanisms: conduction (internally) and convection (externally). Using Biot number argument,^{18} it can be shown that the ambient convection (i.e., natural convection in air) is the limiting mechanisms. Hence, the internal thermal gradients can be neglected, and the general energy balance simplifies to Eq. (2). Re-expressing Eq. (2), we get

where the terms are re-scaled using heat capacity, *mC*. Despite its apparent simplicity, the above expression cannot be directly used to analyze temperature measurements. Temperature is recorded as discrete data (usually at equal time intervals). Such measurements invariably contain noise, and the difference formula is ineffective to estimate the time dereivative,^{17} making the problem ill-posed. The calorimeter analysis does not suffer from a similar predicament as in the absence of convection term, and Eq. (2) can be recast as an integral problem. The heat generated in a finite time interval $\delta t$ becomes $\delta Q=\u222b\delta tQ\u0307dt=mC\delta T$ (without having to invoke derivatives).

Figure 1(a) presents the experimental setup with a cylindrical Li-ion cell operating in the ambient. Thermistors (calibrated to a 0.1 °C precision) measure surface temperatures as well as the ambient temperature. In the presence of considerable surface gradients, surface averaged temperature is to be used for analysis. The Li-ion cell (NCM cathode and graphite anode) is charged and discharged in an identical voltage range (2.8–4.2 V) using a potentiostat. Heat generation takes place during the electrochemical operations. An in-between rest period of two hours is used to ensure electrochemical and thermal equilibrium at the beginning of the next current operation (*i.e.,* heat generation sequence). Figures 1(b) and 1(c) show the evolution of cell voltage and temperature, respectively (only part of the rest phase data are shown here).

Since the measurements are carried out in the ambient whose convection characteristics can change in time, rest phase data are also analyzed to self-consistently calibrate the convection time constant, *τ*_{conv}. For a heated cell (both charge and discharge cause a temperature rise), temperature decays gradually during the rest phase, governed by the following expression:

whose analytical solution is

with cell temperature *T*_{0} at *t *=* *0 (the time axis is reset to zero at the start of each test phase). Let the temperature data (discrete) be denoted by $T\u0302$ and $T\u0302\u221e$ for cell and ambient, respectively. The experiments are carried out inside the laboratory where the ambient temperature drift is much slower than the duration of each test. Further, it can be shown that $T\u221e=mean(T\u0302\u221e)$, and the cell temperature data $T\u0302=T\u0302(tj)$ are analyzed to seek the functional trend expressed in Eq. (5) and subsequently identify the convention time constant. To quantify heat generation, the remaining difficulty is an accurate interpretation of time derivative in Eq. (3).

The principles of Inverse heat transfer^{17} suggest that in order to differentiate an experimental (discrete time) measurement, one should identify the underlying functional variation and subsequently employ analytical differentiation. Such an approach implicitly filters out noise (since noise is a high-frequency small amplitude signal and the interpreted analytical trends are insensitive to such components). Equation (5) is a homogeneous solution of the differential Eq. (3). Given that the heat generation rate *Q** is time-dependent, a suitable analytical temperature trend can be assumed as^{19}

where coefficients *a*_{i} and frequencies *f*_{i} (i.e., time constants 1/*f*_{i}) capture the essence of temporal variation. Note that the number of frequency components, *N*, is to be determined from the spectral width of the discrete signal. Equation (6) is agnostic to the details of battery operation. To determine the frequency information, Fast Fourier Transform (FFT) is carried out over the experimental time series $T\u0302(tj)$. This signal contains true frequencies spanning between 0 (stationary) and *f*_{s}/2 (Nyquist limit^{20}) where *f*_{s} = 1/Δ*t* is the sampling frequency. On the FFT spectrum, higher order frequency information is corrupted by noise. Figure 2(a) shows the time series, $T\u0302$, for a 2.25 Ah Li-ion cell charged at 4.5 A (2 C current). The corresponding FFT spectrum is shown in Fig. 2(b). The amplitudes are rescaled using the highest amplitude signal (i.e., the stationary component). Any frequencies representing less than 1% of the information are discarded. The rescaling allows for an automated data conditioning. The remaining frequencies are used to identify the coefficients in the expression Eq. (6). Figure 2(c) presents the derived temperature trend, *T* (the goodness of fit is better than 0.99), and visually justifies the accuracy of the procedure. Thus, a transformation from a discrete time signal $T\u0302$ to a continuous time signal *T* filters out the noise and provides distilled temporal variations. The analytical trend [Eq. (6)] is further differentiated to quantify the rate of temperature change (here “temperature rise”). Note that the temperature rise is higher at the beginning [Fig. 2(c)] and gradually plateaus as a higher temperature leads to a greater convective loss. The differentiated signal [Fig. 2(d)] captures such variations quite faithfully. Equation (3) is subsequently employed to obtain the heat generation rate in the continuous time domain. Figure 3 demonstrates the resultant time dependence of the heat generation rate. As a comparison, the heat generation trend of the subsequent discharge is also shown alongside. The two trends are qualitatively different and suggest the existence of thermal hysteresis. The direction of current flow is opposite for the two operations, and since the two electrodes are not identical in terms of material and microstructural aspects, an asymmetric thermal behavior originates.^{5} Moreover, the sign of the entropic heat changes when current switches the direction. Authors have recently predicted the existence of thermal hysteresis in Li-ion cells, and the present experiments confirm the peculiar response.

Internal resistance as used in Eq. (1) is separately measured at multiple intermediate locations during operation and their average values are reported in Table I. For higher currents, the Joule heating (i.e., *I*^{2}*r*_{in} term) is the leading contributor to heat generation and is tabulated as well (*Q*_{r}). The cells used weigh 45 g and specific heat is assumed to be 823 J/kg·°C.^{21} With these, the average rate of heat generation is computed as follows:

Measured quantity . | Charging . | Discharging . |
---|---|---|

$Q\xaf$ (W) | 0.9092 | 1.1049 |

r_{in} (mΩ) | 48.1 | 50.4 |

Q_{r} (W) | 0.9740 | 1.0206 |

Avg. $Q\xaf$ (W) | 0.9091 | 1.0959 |

Std. $Q\xaf$ (mW) | 7.82 | 28.3 |

Coeff. var. | 0.86% | 2.58% |

Measured quantity . | Charging . | Discharging . |
---|---|---|

$Q\xaf$ (W) | 0.9092 | 1.1049 |

r_{in} (mΩ) | 48.1 | 50.4 |

Q_{r} (W) | 0.9740 | 1.0206 |

Avg. $Q\xaf$ (W) | 0.9091 | 1.0959 |

Std. $Q\xaf$ (mW) | 7.82 | 28.3 |

Coeff. var. | 0.86% | 2.58% |

Both $Q\xaf$ and *Q*_{r} are of the same order (Table I). Note that the departure $(Q\xaf\u2212Qr)$ has opposite signs for charging and discharging operation. Since the difference between the two arises from entropic contributions and it changes sign, the estimations are logical. The reproducibility of the results is identified by carrying out ten charge-discharge operations. The corresponding statistics^{22} are reported in Table I and reveal that the measurements are reproducible (coefficient of variation = std/avg).

The proposed procedure is equally applicable to cells with different chemistries, shapes, and electrochemical history (e.g., fresh *vs*. aged cell). The analytical sophistication allows one to study thermal signatures for various electrochemical operations. Since the approach neither requires cell-level modifications (e.g., drilling a hole to place internal thermocouple^{23}) nor relies on the adiabatic environment, it is an elegant *in operando* non-invasive technique with the potential to be a commonplace measurement for a laboratory setting, batch testing, and continuous monitoring. Such an approach in principle can be extended to monitor cells in a battery pack after appropriately accounting for (i) geometrical arrangement of the cells and (ii) limiting mode of heat transfer. The proposed thermal tracking procedure could allow decision making, i.e., for a battery management system if reasonable estimates are available for frequencies and ambient convection from the preceding measurement set. The subjectivity of the experimentalist is also circumvented as no cell preparation is required. Essentially, the complexities of a controlled experiment are translated to the involved analysis of the measurements. We envision such a “*thermal signature probe*” to provide detailed insights into the thermal metastability of Li-ion cells.

Financial support in part from the Office of Naval Research DURA Program is gratefully acknowledged. P.P.M. would like to thank Dr. Michele Anderson from ONR for supporting the DURA project. A.N.M. developed the inverse analysis, designed the setup, and performed experiments. H.R.P. performed an initial set of experiments. P.P.M. contributed to the concept development and writing of the manuscript. The authors thank Daniel Juarez-Robles from ETSL for his assistance with the electrochemical test protocol.