## 2 Materials and Methods  
The N1114-FTSI and N122(2O1)-FTFSI ionic liquids were synthesized and purified according to a procedure which uses water as the only processing solvent [32; 33]. The precursors _N_-trimethyl-_N_-butyl-ammonium bromide (N1114 Br, Solvionic, 99 wt.%) and \(N\),_N_-diethyl-_N_-methyl-_N_(2-methoxyethyl)-ammonium bromide (N122(2O1)Br, Solvionic, 99 wt.%) were previously treated trough activated carbon (Sigma-Aldrich, Darco-G60) [33] in deionized water (Millipore ion-exchange resin deionizer). Then, the purified aqueous precursors were reacted with LiFTFSI (Provisco, \(>\)98 wt.%) to obtain (anion exchange reaction) the IL materials, which were rinsed with deionized H\({}_{2}\)O to remove LiBr, LiFTFSI excess and water-soluble impurities. Finally, the ILs were vacuum dried (oil-free pump) at room temperature for 2 h and, successively, at 80 \({}^{\circ}\)C for 48 h. The chemical structure of the (N1114)\({}^{+}\) and N122(2O1)\({}^{+}\) cations (Figure 1) was validated by \({}^{1}\)H NMR measurements (Figure S1 as Supplementary Data), this confirming the feasibility of the synthesis route. The moisture and bromide content of the ionic liquids, determined by Karl-Fisher titration and X-ray fluorescence according to a route previously reported [33], was found to be lower than 2 and 5 ppm, respectively. The overall purity level of the N1114-FTSI and N122(2O1)-FTFSI ionic liquids, checked by UV-Vis spectrophotometry measurements as reported in previous work [33], was found to be above 99 wt.%. The thermal stability of the synthesized ILs was investigated by thermogravimetry. For this purpose, a Setsys Evolution 1200 TGA system by Setaram (KEP Technologies Gropu, Mougins--Sophia Antipolis, France) was used. Measurements were conducted in an argon atmosphere (60 mL/min) with a temperature rate of 10 \({}^{\circ}\)C/min and an initial mass of about 25 mg. The possible presence of low temperature phase transitions was investigated by means of differential scanning calorimetry measurements, using a Mettler Toledo DSC 3 system (Mettler Toledo, Columbus, OH, USA) equipped with a Huber TC100 Cryocooler (Peter Huber Kaltemaschinenbau AG, Offenburg, Germany). An argon flux of 50 mL min\({}^{-1}\) and a temperature rate of 5 \({}^{\circ}\)C min\({}^{-1}\) were applied. The mass of the samples was in the range of about 20 mg. The ionic conductivity of the ILs and their mixtures with LiTFSI (Solvionic, \(>\)99.9 wt.%, battery grade) was investigated as a function of the temperature. The molar fraction of LiTFSI in the LiTFSI-IL mixtures was 0.2, corresponding to a molar concentration around 0.8 M (i.e., close to that of commercial, organic solvent-based LIB electrolytes). The binary LiTFSI-IL electrolytes were prepared in a controlled argon-atmosphere glove-box (Jacomex SAS, Dagneux, France, H\({}_{2}\)O and O\({}_{2}\) content \(<\) 1 ppm) by dissolving the proper amount of lithium salt in the IL. The electrolytes (handled in the glove-box) were housed in sealed, glass conductivity cells (AMEL 192/K1, Amel Electrochemistry, Milano, Italy) equipped with two porous Pt electrodes (i.e., the cell constant was determined through a 0.1 N KCl aqueous solution). In order to reach a stable low temperature state, the cells were dipped in liquid nitrogen for 30-60 s and then quickly transferred into the climatic chamber (Binder GmbH, Tuttlingen, Germany, MK53, used for the temperature control) at \(-\)40 \({}^{\circ}\)C [34]. Then,  
Figure 1: Chemical structure of the FTFSI\({}^{-}\) anion, and N1114\({}^{+}\) and N122(2O1)\({}^{+}\) cations. the cells were kept at \(-40\)\({}^{\circ}\)C for at least 18 h prior to starting the conductivity tests, run from \(-40\) to 80 \({}^{\circ}\)C at low scan rate (1 \({}^{\circ}\)C h\({}^{-1}\)) for better evidencing the phase transitions [35], through an AMEL 160 conductivity-meter. The anodic stability of the 0.2 LiTFSI-0.8 IL electrolytes was investigated using carbon working electrodes (in order to better simulate the situation in battery cells [30]).