Evaluation and Modeling of Long-Term Endurance Measurement on Electric Double Layer Capacitors to Increase Reliability of Lifetime Predictions

This section presents and discusses measurements made under the above-described conditions. The discussion includes the interpretation of the data, with deterioration mechanisms described in the literature, and the calculation of apparent acceleration factors by comparing measurements made at different temperatures. The average relative capacitance and ESR, measured over a range of about 9000 h at 65 °C, are plotted in Figure 1 and Figure 2, respectively, and show a substantial deterioration for EDLCs with applied voltage and a more moderate decline for the short-circuited sample. The semitransparent area along the graph indicates the 10th percentile and 90th percentile. The capacitance for the parts that were permanently kept at their rated voltage of 2.7 V decreases well below 40% relative to their initial value. The corresponding ESR of those parts has deteriorated to more than tenfold relative to their initial value. The short-circuited parts, however, are still well within the EOL criteria. The parts have a defined EOL at 30% capacitance loss and 100% ESR increase relative to their rated parameters. The datasheet endurance is guaranteed for 1000 h at 65 °C and with an applied voltage of 2.7 V. All parts are well within the EOL limits for the endurance time. The dispersion of the capacitance and ESR values increases with time. The mean average deviation for the capacitance and ESR measurements is about 40% and 30%, respectively. Especially above 3000 h, the dispersion is well above 10%, indicating a dispersion in aging rates and thus highlighting that any aging or degradation forecast can only be an estimate, even for highly standardized commercial parts. This increasing dispersion at large measurement times may be identified as the wearout phase, in which the failure rate rises rapidly [28]. Usually, wearout only occurs for electronic devices beyond the service life, i.e., above the EOL criteria. Unfortunately, based on this study, it is not possible to identify the physical or electrochemical mechanisms responsible for the wearout. Commonly, EDLCs have production tolerances around 40% and are typically delivered with capacitances slightly above the rated value. Therefore, the measured dispersion of <10% below the EOL criteria is within this limit and could be assessed as acceptable. This phase may be identified as the early-life or steady-state phase, in which failures occur at a decreasing or constant rate [28]. The reliability prediction assumes that within the early life, which is well below EOL criteria, the failure rate is constant and the lowest within the life cycle of electronic equipment. The general progression of the capacitance values vs. time, as shown in Figure 1, is comparable to measurements previously reported in the literature [7,23,26]. Slight discontinuities arise from variations associated with the measurement. Usually, the capacitance decreases in the first phase, lasting around 1000 h to 3000 h, with a power law or square root time dependence. Literature associates this degradation with electro-thermal degradation of electrolytes and or loss of electrode surface area [18,23]. The second phase, which starts above 4000 h to 5000 h or so, is not well-documented but is still discussed or hinted at in some publications. [26,43] It is speculated that the second phase, i.e., wearout phase, is associated with further loss of active electrode area and the decrease of potential barrier between electrolyte and carbon electrode interface [26,43]. Since the loss of electrolyte is challenging to detect, it might be that the lack of availability of conducting ions is an additional cause for the device performance [42]. The ESR in Figure 2 increases over several orders of magnitude. For the 3F and 50 F types, the rate of ESR increase seems to accelerate around 1000 h and to decelerate slightly around 8000 h. Below 6000 h, the relative change in ESR and capacitance for large capacitor types is lower than for smaller capacitor types, i.e., larger capacitor types degrade more slowly than smaller ones. The short-circuited 50 F capacitors degrade much more slowly than the capacitors charged at the rated voltage of 2.7 V, showing that without voltage, the degradation is reduced significantly but not entirely. This may allow speculation that the volume loss of electrolytes could contribute to the degradation [42]. Smaller capacitors may be subject to a larger loss of electrolyte relative to their housing size. Additionally, the type of sealing may also have an effect. The 350 F type is a snap-in design with a roll-fold sealing that secures the baseplate and the container, providing better sealing than the rubber stopper of the standard cylindrical design. However, at this point, this is speculation; the loss of electrolyte by evaporation needs to be determined in another gravimetrical experiment. Although this study does not include chemical or physical analysis and therefore does not allow for discussions concerning the deterioration mechanism, others have studied this area in detail. The decrease in capacitance and increase in ESR could be attributed to the agglomeration of conductive carbon particles and the reduced crystallinity of the carbon materials, but also to the erosion of the current collector itself and the delamination between the collector and the activated carbon [5]. The erosion is caused by electrochemical decomposition of the activated carbon, a more pronounced process in the anode than in the cathode. The performance degradation of the supercapacitor could thus be attributed to the reduced ion diffusion efficiency in the collapsed, blocked, or narrowed pores of the activated carbon layer and the separator, which in this study was a polypropylene film. Sedlakovas’ study on diffuse resistance corroborates these results [26]. Concerning the usability of the capacitors, the test results indicate that even after the datasheet endurance limit of 1000 h, the capacitors are still functioning. Applications requiring a current demand below the rated voltage could tolerate an increase of ESR above the EOL limit. The decision on what an acceptable degradation is can only be made if the power demand of the application is known. However, by moderate oversizing, knowing the degradation rate could enable the engineer to prolong the estimated lifetime of the application and thus save resources. Choosing a larger capacitor adds an extra cost but might be environmentally and financially less costly than replacing the entire device. To assess the extrapolation of the degradation to lower temperatures, 3 F and 50 F capacitor types have been tested at room temperature, i.e., about 24 °C. The median of the resulting capacitance and ESR measurements, taken over 4 years, is given in Figure 3 and Figure 4, respectively, and illustrates that the device parameters are within the EOL criteria, which are defined as 30 % capacitance loss and 100 % ESR increase. As for Figure 1 and Figure 2, the semitransparent area in Figure 3 and Figure 4 indicates the 10th percentile and 90th percentile value. Within the capacitance decrease of about 20% and for the ESR increase of about 40%, the long-term measurements at 65 °C and 24 °C allow the calculation of the temperature factor. Let $$C_{0} \left(t_{0} ; T_{0}\right)$$ and $$E S R_{0} \left(t_{0} ; T_{0}\right)$$ be the relative capacitance and ESR, respectively, at time $$t_{0}$$ for a given temperature $$T_{0}$$ and $$C_{1} \left( t_{1} ; T_{0} g \left(T\right) \right)$$ and $$E S R_{1} \left( t_{1} ; T_{0} g \left(T\right) \right)$$ the relative capacitance and ESR, respectively, at time $$t_{1}$$ for a given temperature $$T_{1}$$ then $$C_{0} \left(t_{0} ; T_{0}\right) = C_{1} \left( t_{1} g ; T_{1} \right)$$ and $$E S R_{0} \left(t_{0} ; T_{0}\right) = E S R_{1} \left( t_{1} g ; T_{1} \right)$$ with $$t_{0} = t_{1} g$$. This calculation can be done numerically by rescaling the time axis of an interpolated function of $$C_{0}$$ and $$E S R_{0}$$ until the $$C_{0} = C_{1}$$ and $$E S R_{0} = E S R_{1}$$. The thermal acceleration factor $$g$$ can be determined independently based on the capacitive and resistive changes. For the sake of simplicity, a subscript for $$g$$, indicating the type of measurement, is not introduced. A graphical example for the determination of $$g$$ is shown in Appendix B in Figure A3. The measurements for $$T_{0} = 65$$ °C, given in Figure 1 as well as Figure 2, and for $$T_{1} = 24$$ °C (room temperature), given in Figure 3 and Figure 4 are used to calculate the factor stepwise. Figure 5 and Figure 6 shows $$B = g^{\frac{10 K}{\Delta T}}$$ as well as the activation energy $$E_{a} = log ⁡ \left(g\right) k_{B} \frac{\Delta T}{T_{1} T_{0}}$$ versus measurement time and relative deterioration as calculated from the capacitance (Figure 1 and Figure 3) and ESR (Figure 2 and Figure 4) measurements, respectively. The capacitance, as well as the ESR measurement, yields about the same results in terms of base and activation energy, which is given in units of eV as plot marker annotations. The range of the activation energies, as well as the base for the corresponding acceleration factor, is comparable to literature values [23,30,31]. Apart from the unsteady behavior for the ESR-based measurements around 3000 h, it is generally noticeable that $$B$$ and $$E_{a}$$, increase with time, i.e., with relative deterioration of the device. It suggests that the effect of $$\Delta T$$ increases with time and an acceleration factor $$\propto 2^{\left(\frac{\Delta T}{10 K}\right)}$$ could be considered an estimate. In this measurement, the 50 F capacitor type has, on average, larger activation energies than the 3 F type. Hence, the temperature is reduced below the max. temperature (65 °C), as used in the endurance test, will tend to a stronger gain in lifetime for the 50 °F-type than for the 3 °F-type. An acceleration factor with deterioration-dependent or time-dependent activation energy would be required to get an exact description of the deterioration over time. The flattening of the slope of the function $$B$$ with the increase of measurement time in Figure 5 and Figure 6 may indicate a limit, and that for larger deteriorations and measurement times, $$B$$ indeed can be well approximated by a constant. This set of measurements suggests that the widely used base of $$B = 2$$ for the temperature factor is a lower estimate that best fits deterioration well below the EOL conditions of EDLCs. This is a conservative estimate for the product’s reliability, which might tend to underestimate the lifetime at lower temperatures. Whether or not a deterioration or time-dependent description is necessary is open for discussion; after all, lifetime calculations are estimates, and the application’s operational parameters are rarely precisely known. Also, the production tolerances of EDLCs are about 40%, so one could argue that the current approach is sufficiently precise. On the other hand, using an unsuitable acceleration factor may lead to overestimating the lifetime. The time periods during which these studies are conducted, and the number of samples, make it challenging to test at several temperatures. In this campaign, it was therefore only possible to test at 65 °C and room temperature. Indeed, the validity of the data can be enhanced when tested against other long-term experiments conducted at different temperatures. However, the collected data proves that the apparent activation energies are not constant (see Section 3). If a non-constant apparent activation energy is found for these two temperatures, then it is plausible to assume that it could also be true for other temperatures. Based on the reviewed literature and the above discussion, finding a model based on electrochemical or physical principles and mechanisms appears challenging. Presumably, the complexity of the processes makes it difficult to formulate an approach leading to a closed mathematical expression. There are certainly estimations possible, concerning the temperature dependence, based on the Arrhenius theory, utilizing apparent activation energies. However, those appear only valid for a certain time or deterioration regions. Thus, closed mathematical expressions, which have the advantage of being relatively easily implementable, may currently only be found phenomenologically. Although such models do not give any insights into the physical processes, they can express device and environmental characteristics by a set of parameters. The challenge is to find an accurate model that describes all features of the deterioration behavior with a small set of parameters. In the next section, a phenomenological description $$B$$ vs. time, as well as the exponents $$t^{∗} A \left(t^{∗}\right)$$ are introduced to give a mathematical approximation of the measurement, allowing for a more precise degradation forecast.