The effect of ionic association on the electrochemistry of redox mediators for Li–O2 batteries: developing a theoretical framework

A theoretical framework to explain how interactions between redox mediators (RMs) and electrolyte components impact electron transfer kinetics, thermodynamics, and catalytic efficiency is presented. Specifically focusing on ionic association, 2,5-di-tert-butyl-1,4-benzoquinone (DBBQ) is used as a case study to demonstrate these effects. Our analytical equations reveal how the observed redox couple's potential and electron transfer rate constants evolve with Li+ concentration, resulting from different redox activity mechanisms. Experimental validation by cyclic voltammetry measurements shows that DBBQ binds to three Li+ ions in its reduced state and one Li+ ion in its neutral form, leading to a maximum in the electron transfer kinetic constant at around 0.25 M. The framework is extended to account for other phenomena that can play an important role in the redox reaction mechanisms of RMs. The effect of Li+ ion solvation and its association with the supporting salt counteranion on the redox processes is considered, and the role of “free Li+” concentration in determining the electrochemical behaviour is emphasized. The impact of Li+ concentration on oxygen reduction reaction (ORR) catalysis was then explored, again using DBBQ and modelling the effects of the Li+ concentration on electron transfer and catalytic kinetics. We show that even though the observed catalytic rate constant increases with Li+ concentration, the overall catalysis can become more sluggish depending on the electron transfer pathway. Cyclic voltammograms are presented as illustrative examples. The strength of the proposed theoretical framework lies in its adaptability to a wider range of redox mediators and their interactions with the various electrolyte components and redox active molecules such as oxygen. By understanding these effects, we open up new avenues to tune electron transfer and catalytic kinetics and thus improve the energy efficiency and rate capability of Li–O2 batteries. Although exact results may not transfer to different solvents, the predictions of our model will provide a starting point for future studies of similar systems, and the model itself is easily extensible to new chemistries.

Which depicts that the total rate law can be expressed as a function of a new rate constant composed of the addition of the three pathway's rate constants.
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Effect for supporting salt association
From equation (25), the free lithium concentration can be introduced in the apparent standard potential expression, to obtain a new expression of the apparent rate constant in a system with parallel association equilibria with the supporting salt counteranion.Here is the derivation for pathway 1, analogous treatment can be done for pathways 2 and c to obtain the total rate changes observed in Figure 5.
Starting from equation (8) and considering the free [Li + ] given by equation ( 26), the apparent standard potential can be written as: ) And the evolution of k01,app, from equation ( 15) can be rewritten as: The transition concentrations are given by the condition of ( ) which the relations between associated and free quinone species are equal, i.e.

Effect of solvation energy
The complexation of Li + ions by solvent molecules can be modelled by the following reaction, characterized by a formation constant of the complex related to the free energy stabilization of Li + in solution Ka sv .
If we made the assumption that the activity of the solvent is not too different from the unity, under the assumption that the solvent molecules taking part on the interaction are negligible with respect to the total number of solvent molecules (which is not strictly valid at around 1 M, but it allows for a qualitative description), the expression for free lithium concentration simplifies to equation (S3) for the free lithium concentration, which produces a simple shift in the concentration axis Figure S3: Effect of solvation energy in potential, rate constant and pathways of the apparent simplified mechanism.Steady state equations for ORR catalysis A simplified catalytic mechanism can be described by (assuming n=1, see main text): Under the assumptions described above, and the premise of a steady state concentration of the intermediate species, we can express the rate law and apparent catalytic rate constant, kapp Cat , for each of the mechanisms described in the main text as for mechanism in equation (30), for mechanism in equation ( 31), and for mechanism in equations (32).k*cat, in all three cases corresponds to the rate constant under a vast excess of Li + concentration compared to catalysis products, Li2O2 and LiO2, i. e. at very fast scan rates or the beginning of the scan in cyclic voltammetry, and at low capacities or very slow discharge rates in battery configuration.
Since it has been shown that DBBQ monoanion needs to be bound to a metal ion to effectively react with O2, in this case bound to Li + , the apparent catalytic rate depends on the Li + concentration and the Ka R .It is important to note that the [Li + ] refers to the free lithium, which will change as the reaction evolves until a steady state is established.In all three catalytic pathways, the lithium concentration has qualitatively the same effect, however, accurate determination of the variation of kcat obs could help dilucidated which is the preferred pathway.In a similar manner, determining the rate variation when using different concentrations of quinone and oxygen can help determine the reaction order of these species and consequently the pathway.Even when this experiments are not straightforward, we can use these proposed mechanism to simulate qualitative changes in cyclic voltammetries, which are seldom analyzed in detail (see main text).

Discussion on the mechanism of ORR catalysis of DBBQ
The different proposed mechanisms of the reaction of DBBQ with oxygen leading to Li2O2 are stated in equations ( 3) to (6) from the main text, and are copied here for the reader's convenience It is reasonable to expect the rate of equations (4) and (5) to be slower than that of (3) given the higher activation energy required for electron tunnelling necessary for a redox process in comparison with a mere association reaction.Reactions (4) and (5) are likely also slower than (6.a) since the former require two quinone-containing particles to find each other in solutions that are usually quite diluted (around 10 mM).More evidence supporting the mechanism through reactions (6) is the negligible or very small reduction in the amount of degradation products found in Li-O2 batteries run with DBBQ-containing electrolytes in DME and TEGDME compared with batteries run without DBBQ. 1,2The only report showing a significant decrease of side reactions when using DBBQ is in the study using 0.5M LiClO4 / TEGDME electrolyte by Bawol et al. 3 It

Figure S1 :
Figure S1: Example cyclic voltammograms of solutions of 5mM DBBQ in varying LiTFSI concentrations in DMSO.The currents are normalized to the cathodic peak for an easier visual comparison, and the voltage referenced to Li1.5Mn2O4 in 1M LiTFSI/tetraglyme solution.

Figure S2 :
Figure S2: Enlargement of Figure 3 for the pathways that show the same functional behaviour with [Li + ], for n=3.The plots shown in the figures were computed using the parameters Ka1 R =1x10 5 , Ka2 R =1x10 3 , Ka1 O =1000, Ka2 O =10, and Ka3 R = Ka3 O =1.Pathways 1c and 2c are scaled to allow for an easier visual comparison. ]