Introduction

The battery outcome in terms of e.g., energy/power density is practically limited by Ohmic-(RΩ), charge transfer-(Rct) and mass transport (Rmt) resistances. They induce overvoltages and decrease the discharge voltage as well as the accessible capacity, especially at kinetically harsher conditions, i.e., high current rates and/or low temperature [1,2,3,4,5,6].Rating the practical relevance of RΩ, Rct, and Rmt, can be vital for systematic battery R&D and can be easily investigated during battery operation, i.e., during constant current (CC) cycling via analysis of overvoltages as illustrated in Fig. 1 [2, 7].

Fig. 1
figure 1

Schematic illustration of a voltage–time plot during a constant current application showing a chronological order of resistances, i.e., beginning with Ohmic-(RΩ), immediately followed by charge transfer- (Rct) and mass transport resistance (Rmt). At small currents the resistances behave linearly (= can be considered constant) and can be simply distinguished and calculated, while at high currents Rct and Rmt get dynamic and deviate from constant behavior

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At low current excitation, all resistance types behave linear and the respective overall resistance can be calculated according to Ohm’s law [5]. At high currents, the effective Rct becomes smaller as a result of an asymptotic voltage vs. current relation, obeying the Butler-Volmer equation, as shown in Fig. 1 (right-hand sight, red curve) [2, 5]. Though, the respective resistances are literature well-known and can be easily obtained via e.g., electrochemical impedance spectroscopy (EIS), their application-relevance is hardly discussed, in particular at application-relevant currents, as these are usually by orders of magnitude higher than the excitation currents applied in EIS. Besides the in-depth complexity of the Butler-Volmer equation which contains various system-specific constants, e.g., symmetry factor and exchange current density [8, 9], and which can be also expressed by the Tafel-plot [10] for the discussion of this work, the asymptotic voltage-current relation is predominantly relevant. In this context, the literature already points to possible overestimation of Rct due to its absence in Li metal batteries (LMB) at the Li interphases formed with ceramic-based solid electrolytes (SE) [2, 11,12,13,14].

Here, simple CC-technique-based experiments [2, 7] are highlighted to systematically evaluate the relevance of Rct at Li metal electrode|electrolyte interfaces for both, liquid electrolyte (LE), and SE at currents beyond the frequently used in EIS in order to get an application-oriented picture.

Results and discussion

At small constant current (CC), the respective cell resistances, i.e., Ohmic-(RΩ), charge transfer-(Rct) and mass transport resistance (Rmt) display a linear overvoltage/current behavior (Fig. 1) and can be calculated via Ohm’s law, as shown in Fig. 2 [2, 7, 15]. The initial voltage drop, i.e., the difference between open circuit voltage (OCV) and the first experimentally derived resistance data point, corresponds to RΩ and amounts to 45 Ω cm2 and 124 Ω cm2 for a liquid-(LE) and a solid electrolyte (SE)-based \(\mathrm{Li}\mid \,\, \mid\mathrm{Li}\) cell, respectively (Fig. 2a, b). The Rct and Rmt can be distinguished on the basis of their characteristic overvoltage vs. time profiles [2]. While the overvoltage due to Rct builds up logarithmically, thus is particularly pronounced after current onset, Rmt-induced overvoltage gets pronounced after a characteristic duration (Sand’s time). In LE-based cells Rmt is negligible due to fast ion transport within the standard electrolyte (here, 1 M LiPF6 in ethylene carbonate (EC): ethyl methyl carbonate (EMC); 3:7 by wt.), as depicted in Fig. 2a [3]. In contrast, a typical linear increase of overvoltage caused by Rmt can be seen in Fig. 2b for the lower-conducting poly(ethyleneoxide) (PEO)-based electrolyte with 1 M LiTFSI [16,17,18].

Fig. 2
figure 2

Voltage vs. time plots from CC measurements, comparing LE (a) and SE (b) in symmetric \(\mathrm{Li}\mid \,\, \mid\mathrm{Li}\) cells at 60 °C, measured after 24 h. Both cells show RΩ, while Rct and Rmt-induced characteristic overvoltages are solely observed in LE- and SE-based cells, respectively

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The Rct is only observed in the LE-based cell (Fig. 2a), in line with literature [2, 11,12,13,14] and can be related with different properties of the solid electrolyte interphase (SEI) on Li metal [19,20,21,22]. It might be attributed to a higher rate of surface exchange at the electrode|electrolyte interface, possibly driven by higher Brownian motion and diffusion, and finally resulting in enhanced electrolyte consumption and a thicker SEI.

The dependence of Rct from the the SEI nature is validated by varying storage time of \(\mathrm{Li}\mid \,\, \mid\mathrm{Li}\) cells at 60 °C, as shown in Fig. 3. Interestingly, also the LE-based cell has no Rct, initially (after 1 h), but gets more pronounced after 24 h (55 Ω cm2) and is even higher than RΩ (45 Ω cm2). In general, a longer storage time can be related with a thicker/more resistive SEI [19,20,21, 23,24,25] and increases both RΩ and Rct, i.e., transport resistance through SEI and activation barrier, respectively [19].

Fig. 3
figure 3

Voltage vs. time behavior of LE-based \(\mathrm{Li}\mid \,\, \mid\mathrm{Li}\) cells at 60 °C, after storage of a 1 h and b 24 h. The initial absence and progressive growth of Rct suggests a relation with a progressive SEI change

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For reasons of linearity between current and voltage, the applied current densities of 10 µA cm−2 are rather low compared to application currents (see Fig. 1). Practical areal capacities of battery cells range between 2 and 3 mAh cm−2 [26,27,28] and correspond to current densities of 4–6 mA cm−2, when aiming at fast charge rates of e.g., 2C. The asymptotic voltage-current relation of the Butler-Volmer equation theoretically suggests lower Rct at elevated currents, as mentioned in the “ Introduction” and, indeed, can be experimentally validated, as shown in Fig. 4.

Fig. 4
figure 4

Voltage vs. time plots of LE-based \(\mathrm{Li}\mid \,\, \mid\mathrm{Li}\) cells at 60 °C with varying current densities, a 10 µA cm−2, b 100 µA cm−2, and c 1000 µA cm−2. Rct decreases with increased current density, while RΩ remains almost constant, d which is in line with their characteristic voltage-current relations

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The increase in current density from 10 to 100 µA cm−2 and finally to 1000 µA cm−2 (Fig. 4a–c), leads to a decrease in Rct from 55 Ω to 38 Ω cm2 to finally 12 Ω cm2, respectively, while RΩ remains almost constant (45–46 Ω cm2), which is in line to linear (Ohmic) current-overvoltage relation, as schematically illustrated in Fig. 4d. From a practical point of view, higher applied currents than 1000 µA cm−2 are experimentally not reasonable in these cells as these relative harsh currents can likely affect Li surface morphology and additionally affect overvoltage [29]; which might be related with wavering voltage response in Fig. 4c. Nevertheless, the decrease in Rct can still be experimentally validated for application-relevant currents (< 1 mA cm−2), finally proving decreased relevance of this resistance contribution for faster charge [11], while enhanced contribution of RΩ; and in the case of SEs additionally Rmt can be observed [4].

Conclusion

Charge-transfer resistance (Rct) that can be described with the Butler-Volmer equation is a major resistance type in electrochemical storage devices alongside with ohmic (RΩ) and mass-transport (Rmt) resistances. Frequently observed via electrochemical impedance spectroscopy (low excitation currents), their practical relevance is discussed in this work for application-relevant higher and galvanostatic currents within \(\mathrm{Li}\mid \,\, \mid\mathrm{Li}\) cells [8, 19, 30]. Major differences in terms of the type of electrolyte (liquid vs. solid) are shown. Rct is basically absent for a polymer-based solid polymer electrolyte, while it progressively grows in contact with a conventional liquid-electrolyte (LE). Absence of Rct for solid electrolytes (SEs) is in line with previous literature, observed for ceramic-based SEs (Li7La3Zr2O12, LLZO) [11,12,13].

Given the asymptotic voltage vs. current relation of the Butler-Volmer Equation, Rct becomes less distinctive at elevated currents and is experimentally validated via increasing current densities; while the Ohmic resistance remains constant in the course of its linear voltage vs. current relationship [3]. Ohmic resistance can be concluded to make a major contribution to the total cell resistance at elevated currents (e.g., under fast charge conditions). In addition, for dual-ion conducting solids, the relevance of Rmt will increase with higher currents, as well, as shown in the literature [31].

Electrochemical impedance spectroscopy [32, 33] is, indeed, a sensible method to determine Rct, but in the course of the typically applied low excitation currents, it can lead to overestimation of Rct for real-life batteries.

Consequently, Rct can be interpreted as an activation hindrance of an electrochemical reaction at the electrode|electrolyte interface, and its relevance is affected by the nature of interphases and the type of electrolyte [19], and/or the magnitude of the applied currents [4] (Fig. 5).

Fig. 5
figure 5

Schematic overview of the overvoltage development, described by Butler-Volmer equation, and the accompanied charge transfer resistance as a function of current. Charge transfer resistance is relevant at low currents (almost linear overvoltage-current relation), while its relevance decreases with increased currents

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While \(\mathrm{Li}\mid \,\, \mid\mathrm{Li}\) cells are reasonable as a model system to investigate charge transfer for varied electrolyte systems, e.g., LEs and SEs, the impact of Rct in practical cell setup and conditions, e.g., composite electrodes and high applied currents, require further in-depth investigations.

Experimental

  1. (a)

    Materials

    Poly(ethyleneoxide) (PEO, MW 300.000 Da), 99.5%) was purchased from Sigma-Aldrich, Germany. Lithium bis(trifluoromethanesulfonyl)imide (LiTFSI, 99.9%) was purchased from Solvay, France. Lithium metal (Albemarle) was purchased from Albemarle, USA. Glass fiber paper (Whatman) was purchased from VWR, Germany. Lithium hexafluorophosphate (LiPF6) of 1 M in ethylene carbonate (EC):ethyl methyl carbonate (EMC) 3:7 (by weight) “LP57” was purchased by BASF, Germany. Material storage and sample preparations were performed in a dry-room with a dew point of −65 °C. PEO was dried under vacuum of 10−7 mbar at 45 °C and LiTFSI at 110 °C for 2 days before use. Whatman glass fiber paper was punched in discs (16-mm diameter) and dried at 100 °C under vacuum.

  2. (b)

    SPE membrane preparation

    PEO-based SPE polymer membranes were prepared by dry mixing PEO and LiTFSI with a molar ratio of 20:1 (EO:Li). The obtained mixture was stored in a pouch bag under vacuum overnight (60 °C). The resulting gum-like material was sandwiched between Mylar foil sheets and pressed at 100 °C with an applied pressure of 15 bar for 10 min.

  3. (c)

    Cell assembly

    Symmetric \(\mathrm{Li}\mid \,\, \mid\mathrm{Li}\) cells (two-electrode coin cell configuration) were assembled using polymer membranes and/or glass fiber separators (+ 200 µL liquid electrolyte) with 16- mm diameter, sandwiched between lithium metal electrodes (15-mm diameter).

  4. (d)

    Electrochemical measurements

    Constant current experiments were conducted on an Arbin Instruments battery cell test system at 60 °C in a climate chamber (Memmert). The used current densities are mentioned in the figure captions.