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the film, ξ2 is the distance electrons can travel through the film, e.g. by tunneling, and l is the thickness of the Li2S film on the carbon substrate, calculated according to l= εcarbon −εLi2S1/3 ε carbon −1 ·rcarbon. (4.27) The radius of the carbon black particles, rcarbon as well as the constants ξ1 and ξ2 are additional input parameters. As long as l is smaller than ξ2, there is virtually no overpotential. Once enough Li2S is precipitated to cover the entire surface, the resistivity increases quickly, triggering the end of discharge. 4.2.7 Degradation A first extensive review of degradation mechanisms in Li-ion batteries was presented in Ref. [224] already in 1998. Many refinements were added over time, see e.g. Ref. [225]. However, with a few notable exceptions [226–231], not a lot of model- ing work has been done concerning these issues. For Li/S cells, the situation is even more desperate. We presented the first – and so far only – modeling study of Li/S cell degradation in Ref. [P3]. To enable degradation in the model, two additional ef- fects are added: First, the reduction of polysulfides may also happen at the lithium electrode. This is implemented by enabling the same set of charge transfer reactions (CTRs) that is assumed at the positive electrode. Since the reaction rate constants and activation energies strongly depend on the species’ interaction with the surface, they must be chosen differently, though. For simplicity, the forward reaction rate constants kfwd of all CTRs are scaled by the same factor ξCTR. Second, solid Li2S is allowed to precipitate (and accumulate) at the negative electrode according to S2−+2Li+2e− Li2S(s). Again, since the precipitation happens at the lithium surface, the reaction rate constant may be different than that used for the same reaction in the positive electrode. It is scaled by a factor of ξprecip. Additionally, the active surface area of the lithium electrode is modeled to depend on the amount of precipitated Li2S according to the heuristic expression V ε0,elyte −εLi2Sξ0 ALi|elyte = A0 · ε0, elyte , where the expression in brackets is the fraction of the porosity remaining after Li2S deposition and the factor ξ0 = 3.5 phenomenologically quantifies the impact on the lithium’s active surface area [192]. 83PDF Image | Lithium-Sulfur Battery: Design, Characterization, and Physically-based Modeling
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