Supplementary Data for

Expanding the Operational Limits of the Single-Point Impedance Diagnostic for Internal Temperature Monitoring of Lithium-ion

Batteries

Neil S. Spinner1,2, Corey T. Love1, Susan L. Rose-Pehrsson1, and Steven G. Tuttle1,*

1U.S. Naval Research LaboratoryChemistry Division4555 Overlook Ave., SWWashington, DC 20375

2National Research CouncilThe National Academies of Science500 Fifth St., NWWashington, DC 20001

*Corresponding author, email: steven.tuttle@nrl.navy.mil; office: 202-767-0810; fax: 202-767-1716

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1. Dummy cell Nyquist plots – contribution from contact leads

Figure S1. Nyquist plots for 1000 -22 F resistor-capacitor in parallel dummy cell at temperatures 25°C to 70°C. Frequency range was 30 kHz and 10 mHz with a 5 mV perturbation.

2. Analytical Solution for Transient Heating of 18650 Lithium-ion Battery

An analytical approach was taken to the heating of 18650 lithium-ion batteries inside the microclimate benchtop temperature test chamber. This analysis, along with calculation of various lithium-ion battery thermophysical properties, was also shown in our recent work [1]. Assuming heat conduction took place solely in the radial direction, the time-dependent, radial heat equation can be applied, Equation S1:

ρC p

kr∂T∂t

=1r∂∂ r (r ∂T∂r ) (S1)

where r, Cp and kr are the battery density, heat capacity and radial thermal conductivity, respectively, T is temperature, t is time, and r is the radial position. The boundary and initial conditions can be written as:

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∂T∂r |r=0=0 (S2)

−k r∂T∂r |r=R=h ( T|r=R−T ∞ ) (S3)

T|t=0=T o (for all r) (S4)

where R is the battery radius (9 mm), h is the heat transfer coefficient, and T∞ is the ambient chamber temperature for each step change. For all experiments, the battery was situated horizontally underneath an air vent which provided some circulation and forced convection, and a value for h was calculated using various correlations to be approximately 35 W/m2-K [1-3].

An analytical solution to Eq. S1 with conditions shown in Eq. S2-S4 is

T (r , t )=T ∞+2 (T o−T ∞ )

R ∑n=1

∞

( 1λn )J 1 ( λnR )J 0 (λn r )J02 (λn R )+J1

2 (λnR )e−αλn

2 t (S5)

where J0 and J1 are Bessel functions of the first kind of orders 0 and 1, respectively, is the thermal diffusivity (equivalent to kr / Cp), and n are the eigenvalues of the transcendental equation

λn J1 (λn R )− hkrJ0 (λnR )=0 (S6)

Using thermophysical properties measured for the commercial LiCoO2 18650 lithium-ion battery found in our previous work [1], Eq. S5 was used to generate data at regular time intervals for an example step change. Table S1 lists the set of values for all thermophysical properties, and Figure S2 shows resulting temperature vs. radius curves for a sample step change from 0°C to 50°C at regular time intervals up to 1 hour.

Table S1. Thermophysical properties for commercial 18650 LiCoO2 lithium-ion battery used in this study, as measured in our previous work [1].

(kg/m3) kr (W/m-K) Cp (J/kg-K)

2721 0.3 896

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Due to the relatively low radial thermal conductivity of an 18650 lithium-ion battery, Fig. S2 shows there is initially a relatively large radial temperature distribution. However, after just 20 minutes the maximum difference between surface and internal temperatures is only 1°C, and the average battery temperature has reached about 95% of the total step change. After 1 hour, the radial distribution is essentially nonexistent and the battery has uniformly reached the set point temperature. Based on this analysis, it was reasonable to assume that the measured surface temperatures of the battery during both the temperature- and state of charge-varied impedance tests (3.5 hour dwell times, see Section 2.1 in the main document) and the single-point impedance diagnostic tests (2 hour dwell times, see Section 2.2) were uniform and equal to the internal battery temperature since the battery was given ample time to reach the set point value.

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Figure S2. Temperature vs. radius curves for time intervals up to 1 hour for 18650 LiCoO2

lithium-ion battery, for an example step change from 0°C to 50°C. Curves generated using analytical solution to the time-dependent, radial heat equation shown in Eq. S5.

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3. Additional Nyquist plots

All Nyquist plots shown collected between 30 kHz and 10 mHz with a 40 mV perturbation. Each data point is the average impedance value taken at every 10% SOC-increment from 0-100%, for selected temperatures between -10°C and 95°C. Error bars represent standard deviation for all SOCs.

Figure S3. Nyquist plots for temperatures -10°C to 15°C.

Figure S4. Nyquist plots for temperatures 20°C to 45°C

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Figure S5. Nyquist plots for temperatures 50°C to 75°C

Figure S6. Nyquist plots for temperatures 80°C to 95°C

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4. References

[1] N. S. Spinner, R. Mazurick, A. Brandon, S. L. Rose-Pehrsson and S. G. Tuttle, J. Power Sources, Submitted.

[2] I. Tosun, “Modeling in Transport Phenomena: A Conceptual Approach.” Elsevier (2007).

[3] D. W. Green, R. H. Perry, "Perry's Chemical Engineers' Handbook, 8th Edition." New York: McGraw-Hill (2008).

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