A Reaction Model for Li Deposition at the Positive Electrode of the Braga-Goodenough Li-S Battery

The Braga-Goodenough all-solid-state Li-S battery demonstrates Li deposition at the positive electrode during discharge. This Li deposition behavior is explained herein in terms of a newly proposed multi-stage mechanism that can be summarized by the shorthand formula E[(ECC)c]n in which E stands for an electrochemical step, C stands for a chemical step, c indicates steps that are catalytic and n indicates a part of the process that is repeated n times. The catalytic part of the reaction, (ECC)c, cycles the deposition of Li. The n in the E[(ECC)c]n formula represents the number of cycles of the Li deposition step, (ECC)c. An intermediate radical S 8- formed during the first E step, the one-electron reduction of S 8 , plays an essential catalytic role in the process. The thermodynamics of the second E step were examined by taking account of the electrochemistry involving two consecutive one-electron steps and of the theory of generalized charge neutrality levels in respect of Schottky barriers. The thermodynamics relating to the Li deposition in the second E step was considered to result in a free energy change of at least ΔG< -2.34eV. The thermodynamics relating to the overall steps of the mechanism were also examined.

. Two other batteries, which were equipped with Li and Na negative electrodes respectively, were reported at the same time 1) . All three batteries exhibited metal (Li or Na) depositions in the positive active mass during their discharge reactions. The most detailed data provided for these  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60 A c c e p t e d M a n u s c r i p t F o r R e v i e w O n l y 2 systems were those revealed for the Li-S type battery with the newly developed Li-glass solid electrolyte, Li2.99Ba0.005O1+xCl1-2x 1) . This battery was demonstrated to provide a capacity of more than ten times that of conventional Li-ion batteries, while depositing Li metal in the positive active mass during discharge. In this Li-S type, the battery reactions and their characteristics were thought to be quite different from the conventional view of battery function. The paper describing these batteries 1) raises a number of questions. One relates to the thermodynamics of the Li deposition at the positive electrode. Despite the remarkable performance characteristics of this battery, hitherto, the chemistry of the system has never been disclosed, apart from consideration of electrostatic behavior, capacitor equivalent circuits and so on [1][2][3][4] .
S8 positive electrodes exhibit higher specific energy than those of Limetal oxides for the conventional Li-ion batteries. The enhancement of the positive electrode capacity of a battery equipped with a metal negative electrode is of great importance for energy storage devices to achieve high specific power and energy.
However, this Braga-Goodenough battery behavior had never been observed before.
In this paper, an E[(ECC)c]n mechanism to elucidate the Braga-Goodenough battery characteristics is proposed. In this formulation of the mechanism E stands for an electrochemical step, C stands for a chemical step, c indicates steps that are catalytic and n indicates a part of the process that  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60 A c c e p t e d M a n u s c r i p t Concerning Li deposition cycles, the number of its cycles corresponding to the Braga-Goodenough battery capacity of ten with respect to S8 was determined to be 18 by means of this mechanism. Li deposition cycles on the basis of this mechanism indicate a capacity of greater than the ten times advantage previously reported 1) .

Experimental
In order to evaluate the electronic states and the critical radius corresponding to the intermediate radical S8and the molecule S8, quantum chemistry calculations using Gaussian 09, Revision E.01, were performed with the accuracy of APFD/6-311+G(2d,p). S8-is a radical species, so that its calculations are to be expressed as SOMO calculations hereinafter.
Concerning experimental data relating to the Braga-Goodenough battery, all data in this paper are referred to those from reference 1 1) . The mechanism proposed herein is examined on the basis of the data in 1) .
An E[(ECC)c]n Model of Li Deposition at the Positive Electrode Figure 1 shows the Braga-Goodenough battery diagrams. Li deposition during discharge takes place at interfaces between the discharge reaction intermediate S8and Li + glas described below.
This cycle number n is thought to depend on the formation of S8in step [1]. This n relates to the number of contacts between S8and Li + (sf) surface states. This number of contacts is considered to be large since the ratio of the critical radius of S8to that of Li + is ca. 10 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60 A c c e p t e d M a n u s c r i p t signifies a change in the reaction site of Li + (sf) contacting the same S8 -. Then the one-electron reduction of each Li + (sf) contacting S8takes place n times until the number of contacts between the two is completely utilized thus demonstrating the role of S8as a catalyst. Figure 2 shows a schematic representation of the above role of S8as a catalyst and the cycle number n corresponding to the number of depositions of Li. Three species in the figure are illustrated with circles. The radii of the three circles as shown represent the radii of the three spherical species. The critical radius of 483pm S8is based on the SOMO calculations and the radii of 59pm Li + and 152pm Li can be referred to elsewhere. In this figure, a total of ten Li and Li + (sf) sites are illustrated. Five of the ten are already discharged to form Li, indicating the end of five cycles (n=5). In this case in Fig.2, the remaining 5 Li + (sf) are still contacting S8 -, so that there is a possibility of a total Li deposition cycle number of ten. Another significant factor of the S8role as a catalyst is its characteristics as a good conductor (0.64eV Eg as indicated by the SOMO calculations). If S8was a poor conductor, then all the continued one-electron reduction of Li + (sf) would be difficult, as shown in Fig.2.
On the other hand, the Eg of S8 is an insulator of 5.21eV by the calculation.
S8 + 2Li + (sf) + 2e → S8 -Li + (sf) (ad) + Li [6] In this E[(ECC)c]n mechanism the first step [1] corresponding to the first one-electron charge transfer step controls the overall kinetics from the viewpoint of the formation of S8 -. This dependence on step [1] is based on the kinetics of the two consecutive one-electron charge transfer reactions in the E[(ECC)c]n mechanism. An overall two-electron electrochemical reaction has been considered elsewhere [15][16][17][18][19][20][21][22][23][24] . In this study of S8, there is also an overall two-electron reduction. In this respect, the EE mechanism for S8 In the all-solid-state Li-S battery, the coordinates of the reduction site of S8 and that of S8-are to be equivalent to each other from a viewpoint of kinetics because S8-is formed at S8 coordinates. As described above and in Fig.2, on forming S8-, many contacts with Li + (sf) of Li + glare formed because the critical ion radius of S8-is nearly one order of magnitude greater than that of Li + . In the Braga-Goodenough battery, once S8-forms, it is considered that S8-adsorbs Li + (sf) of Li + gldue to their coulombic affinity and thus the adsorbent S8 -Li + (sf) (ad) is formed. Then the consecutive one-electron reduction will occur in step [2]. This reaction step [2] corresponds to the essential one to explain the Braga-Goodenough battery behavior because Li deposition occurs in this step.
Here, there are two possibilities for further one-electron reduction schemes. One is the consecutive reduction of the S8-side in S8-Li + (sf) (ad) to form S8 2 -. The other is the reduction of the Li + (sf) side in S8 -Li + (sf) (ad) to form metal Li. In this study, E[(ECC)c]n, the latter reaction was chosen. When the former scheme is chosen, the Braga-Goodenough battery behavior is not observed, and the reactions are those of conventional Li-S type batteries. This diagnostic criterion is based on the behavior of the Braga-Goodenough battery forming metal Li on discharge. Accordingly, the second charge transfer step must rely on the thermodynamics of ΔG<0, i.e., eV2 -eV0<0, where eV0 is the electron energy level for Li + /Li of the negative electrode and eV2 is that for S8 -Li + (sf) (ad)/ S8 -Li. Otherwise, as a spontaneous reaction of the battery discharge, no energy flow and no Li deposition would occur in this mechanism. According to the discharge behavior of the Braga-Goodenough battery 1) , the battery voltage of 2.34V was the threshold for the observation of Li deposition in the positive active mass under discharge conditions. That is, as long as the battery voltage is higher than 2.34V on discharge, the cycling of reactions [2] to [4] continues to deposit Li in the positive active mass.
The negative electrode reaction corresponding to the overall positive electrode reaction [5] of this E[(ECC)c]n mechanism is to be expressed as follows: (n+1)Li + ←→ (n+1)Li + (sf) [8] Therefore, the overall reaction of the negative electrode is as follows: The overall reaction of this battery under battery voltages higher than 2.34V conditions can be expressed by [10] taking account of the summation of [5] and [9].
S8 + Li → S8 -Li + (sf) (ad) [10] This overall reaction product of S8 -Li + (sf) (ad) is the same as the product of the 1 st one-electron charge transfer step [1]. As long as Li deposits by the reaction cycle from [2] to [4] in the positive active mass on discharge, the reduction of S8 is limited to a one-electron reduction. Then the product of S8of the one-electron reduction of S8 plays an essential role as a catalyst in the reaction cycle steps from [2] to [4]. The product of S8 -Li + (sf) (ad) in [10] is still an intermediate from the viewpoint of the S8 two-electron reduction. When the battery voltage is less than 2.34V, this E[(ECC)c]n mechanism is over, i.e., Li deposition stops during discharge 1) . The Braga-Goodenough battery reactions after the termination of E[(ECC)c]n mechanism are shown in the Appendix.
Concerning the charge reactions of the E[(ECC)c]n mechanism, those of the positive and the negative electrodes correspond to the reverse reactions of both [5] and [9]. Thus, the overall charge reaction [11] corresponds to the reverse reaction of [10] as follows:  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60 A c c e p t e d M a n u s c r i p t calculations, S8behaves as a good conductor, and this property is to be of importance to play a role of a catalyst in this mechanism. Incidentally, the conductivity of Li + of Li + glis greater than 10 -2 Scm -1 at 25 ˚C 1) . Thus, the resistance of Li + transfer in the Li + glis considered to be sufficiently small compared with the battery kinetics, that it does not affect the steps in the E[(ECC)c]n mechanism. In respect of the practical discharge rate of the Braga-Goodenough battery, it was estimated to be 0.018C by taking account of the data in 1) .

Thermodynamics of Li deposition in the E[(ECC)c]n mechanism
The thermodynamics of the Li deposition is a fundamental issue to elucidate the positive electrode reactions of the Braga-Goodenough battery.
In the potential regions of the positive electrode, the well-known electrode reaction of Li + /Li redox couple never takes place. In the E[(ECC)c]n mechanism, it is considered that the thermodynamics issue is integrated into the one-electron reduction of S8 -Li + (sf) (ad) in step [2].
In order to elucidate the electron energy states of the adsorbent S8 -Li + (sf) (ad) formed in the 1 st charge transfer step [1], two procedures from different viewpoints are employed herein. One is the thermodynamics in electrochemistry with regard to a two consecutive one-electron charge transfer reaction, the so-called EE mechanism [15][16][17][18][19][20][21][22][23][24] . The other is the theory of the generalized charge neutrality level φ G CNL [25][26][27][28][29] . This theory of φ G CNL newly emerged from the recent consideration of Schottky barriers, especially, for the understanding of effective work functions with electronic structures at interfaces between metals and high-k dielectric insulators [25][26][27][28][29] . Elucidating electron energy levels of the adsorbent of S8 -Li + (sf) (ad) by taking account of both electrochemistry and heterojunction physics is considered to be of great importance to clarifying the behavior of the Li deposition thermodynamics in terms of this E[(ECC)c]n mechanism. The thermodynamics ΔG<0 for the Li deposition in step [2] are equivalent to an indication that the electron energy levels of S8 -Li + (sf) (ad) are lower than those of Li + /Li of the negative electrode.
In practice, on discharge, electrons relating to this E[(ECC)c]n mechanism are all transferred from the negative electrode of Li + /Li, so that the energy levels of the Li deposition in step [2] must be lower than those of Li + /Li.
When this mechanism is appropriate for Li deposition behavior of the Braga-Goodenough battery, it is considered that these two types of examinations from different viewpoints will be consistent with each other.
From the first point of view of the redox potential with regard to S8/S8 2-, the electron energy levels of an intermediate radical of S8are to be examined [15][16][17][18][19][20][21][22][23][24] . However, it is already known that the electron energy levels of S8should be intermediate between those of S8 and S8 2-. In particular, S8has two levels in the orbitals occupied with unpaired electrons 15) . One is corresponding to the level of redox potential of S8/S8and the other is corresponding to that of S8 -/S8 2-. Thus, the electron energy level of S8is lower than S8 and higher than S8 2-. S8 2is the lowest of the electron energy levels and the highest in potential among the three species. In the E[(ECC)c]n mechanism, the chemical forms of S8Li + (sf), S8 -Li + (sf), and S8 -Li in steps [1] and [2] are different from those of S8, S8 -, and S8 2-, so that the absolute values of these levels, e.g., between S8 -Li + (sf), and S8-, cannot be identical to each other.
However, the potential relationship with regard to these species in the E[(ECC)c]n mechanism is considered to be of the same order of S8Li + (sf)< S8 -Li + (sf)< S8 -Li as that of S8<S8 -<S8 2-. That is, the energy level of S8 -Li + (sf) is explicitly lower than that of the negative electrode Li + /Li. Figure 4 shows the images of electron energy levels versus density of states for each reaction species of steps [1] and [2] in the E[(ECC)c]n mechanism. As described above, in particular, the electron energy levels of S8 -Li + (sf) (ad) are separated into two Gaussians. Here, E1 and E2 are equilibrium electrode potentials in the EE processes corresponding to the steps [1] and [2], respectively. ε1 and ε2 are the Fermi levels corresponding to E1 and E2, respectively. The relationship between E1 and E2 is E1<E2 15,[16][17][18][19][20][21][22][23][24] . As long as the battery voltage is higher than 2.34V, this E[(ECC)c]n mechanism can continue. Thus, the thermodynamics relating to the Li deposition in step [2] is considered to be ΔG< -2.34eV at least.
However, this φCNL could not clarify the Schottky barriers at interfaces between metals and high-k materials like HfO2. In the case of the interfaces of metals/HfO2, the penetration of MIGS into HfO2 was observed at most one to two atomic layers [25][26][27][28][29] , so that the full contact of electronic states could not be realized at metals/HfO2 interfaces. One of the causes of this shallow penetration of MIGS into HfO2 was ascribed to the high ionicity of HfO2 [25][26][27][28][29] .
HfO2 is a typical high-k insulator with a band gap of greater than 5eV. The Li-glass electrolyte used in the Li-S type battery also demonstrates a band gap of greater than 8eV with typical ionic bond glass structure 1) . Then Braga et al. described that high dielectric property of this glass electrolyte is the significant factor in this battery performance 1) .
The theory φ G CNL is built on an orbital hybridization between unoccupied and occupied states of two materials at an interface [25][26][27][28][29] . In this theory of φ G CNL, the charge transfer between S8and Li + glis realized only by such orbital hybridization between unoccupied and occupied states. This electron transfer is absolutely confined to just interfaces between S8-and Li + (sf) because the insulator of the ionic bond structure Li + gldemonstrates greater than 8eV band gap. S8 -, however, demonstrates metal like characteristics with a 0.64eV band gap as indicated by the SOMO calculations. Since Li + glis an ionic bond insulator material, its conduction band contains the frontier orbitals of Li + cation and the valence band mainly comprises the frontier orbitals of O 2-, Cl -30) in Li + gl -. Therefore, the heterojunction generated by the contact between S8and Li + (sf) is an analogy to the case of heterojunctions between metals and HfO2. It is considered that the algorithm of the analysis of the Schottky barriers between metals and HfO2 can also be applied to the analysis of electron levels of heterojunctions appearing in the E[(ECC)c]n mechanism.
When the second term increases, an electron level of φ G CNL goes up, demonstrating that the Fermi level of the metal side of S8also goes up with the contact.
Δρ means transfer charges from metals to insulators and is proportional to the right hand side of the equation. At Δρ=0 in Eq. [13], the Ef reveals the Fermi level of S8after contacting Li + (sf) of Li + gland it is identical to φ G CNL of Eq. [12]. The expression of Ef in Eq. [13] corresponds to the S8side as a metallike electronic structure and that of φ G CNL in Eq. [12] corresponds to the insulator side.
φCNL =EVB + Eg DVB/ (DVB+DCB) [14]  Under these conditions, from Eq. [12], the second term of the right-hand side decreases compared with that of eq. [14], so that φ G CNL and Ef  That is, the Fermi level of S8after contacting Li + (sf) and the φ G CNL of S8 -Li + (sf) formed after the contact decreases, indicating that the potential of S8 -Li + (sf) is higher than that of S8 -. The relationships of S8 -Li + (sf) >S8 Li + (sf) in potential were already considered above from the viewpoint of the EE process in the E[(ECC)c]n mechanism. From the different viewpoint of φ G CNL analysis of S8 -Li + (sf), the same finding was obtained. That is, electron levels of the absorbent S8 -Li + (sf) in steps [1] and [2] involving the intermediate radical S8indicate higher levels than those of S8before the formation of the adsorbent S8 -Li + (sf). The mobility gap corresponds to the energy gap between the mobility edge level of electrons and the mobility edge of holes and consists of three terms which are diffuse band tail states from mobility edges for electrons, diffuse band tail states from mobility edges for holes, and the gap states. Although more than 8eV band gap of this Li + gl-1) is a typical insulator and the above band model is based on semiconductor amorphous materials, this band model is considered to be fundamentally applicable to Li + gl-.
The detailed relations of E1, E2 in Fig.4, the equilibrium constants of K1 and K2, and an overall positive electrode potential of E (+) will be discussed in the section on thermodynamics cycle.

Li Deposition Cycle of the Braga-Goodenough Battery
On the basis of this E[(ECC)c]n mechanism from equations [1] to [4], when the discharge reaction initiates, at first, the one-electron reduction of S8 proceeds and then the Li deposition cycle is repeated. As shown in Appendix, after the termination of this mechanism, the final product of S8 -Li + (sf) (ad) is reduced by further one-electron reduction to form Li2Sx and to the final form of Li2S. Thus, the Braga-Goodenough Battery consists of two kinds of discharge reactions. One is the E[(ECC)c]n mechanism to form the discharge product of S8 -Li + (sf) (ad) and the total discharges within this mechanism corresponds to (n+1)e from Eq. [5]. The other discharge reaction is to form the discharge product of Li2S by the one-electron reduction of S8 -Li + (sf) (ad). In other words, the total amount of discharge can be expressed in terms of the number of electrons, as (n+2)e for this Li-S type battery where ne corresponds to nF discharge per nLi deposition in terms of the cycle number n in the E[(ECC)c]n mechanism and 2e corresponds to 2F discharge per S8. F is the Faraday constant. Therefore, the ratio of the total amount of discharge to the theoretical amount of capacity of S8 in the positive active mass can be expressed in percentage as follows: 100(n+2)/2 (%) [15] Figure 6 shows the relationship between total battery discharge capacity/S8 capacity versus the cycle number n by Eq. [15]. As shown in Fig.6, as long as the battery reaction obeys this E[(ECC)c]n mechanism, tenfold capacity against the theoretical capacity of S8 can be observed at cycle 18.

Thermodynamics Cycle
In general, on the basis of the energy cycle, the Gibbs free energy ΔG of the overall reaction [5] is consistent with each step ΔG summation in the E[(ECC)c]n mechanism. That is, ΔG=ΣΔGi (i=1-4) where ΔG i corresponds to each step [i] from the step [1] to [4]. Here, from the steps from [1] to [5], ΔG1= -FE1, ΔG2= -nFE2, ΔG3= -RTln(K1) n , ΔG4= -RTln(K2) n , and ΔG= -(n+1)FE (+) can be expressed, where E (+) corresponds to the positive electrode potential in the overall reaction step [5]. That is, Eq. [16] is obtained as follows: [16] where n is the cycle number of the E[(ECC)c]n mechanism; R, T, and F are their usual significance. At 25 ˚C, Eq. [14] is rewritten in the form in Eq. [17] as follows: E (+) = E1/(1+n) + E2/(1+(1/n)) + 0.059/(1+(1/n)) Log(K1K2) [17] where E (+) , E1, and E2 are all corresponding to equilibrium potentials not  [20] According to the E[(ECC)c]n mechanism, cycle number n corresponds to n number of Li deposition. Then the deposited Li is all transported from the negative electrode of the battery. In Eq. [20], E1 and n are eliminated from the derived general form of E (+) in Eq. [16]. Thus, in order to obtain the practical relationship between the cycle number n and K1K2, the rest of the parameters of E (+) , E1, and E2 must be fixed. In order to examine the cycle number n versus K1K2, E (+) , E1, and E2 have to be estimated by taking account of the battery practical data 1) . The positive and negative electrode potentials are -0.40 VSHE 1) and -3.05 VSHE 1) , respectively. Incidentally, the Fermi level of SHE corresponds to -4.44eV versus the absolute potential. On the basis of these data, E1 is considered to correspond to -0.40 VSHE because no S8 -Li + (sf) (ad) is formed before the battery discharge. Before the formation of S8 -Li + (sf) (ad), the potential of the positive electrode is thought to be established by S8 alone.
After discharge, the maximum voltage of the Braga-Goodenough battery was 2.68 V 1) . If this value did not involve overpotential components, the positive electrode potential after discharge would be (-3.05 VSHE+2.68 V) = -0.37 VSHE.
In practice, the value of 2.68 V explicitly includes overpotential components during discharge. On the basis of the discharge performance data 1) , the negative electrode dominates the overall overpotential components of this battery discharge behavior. That is, the estimated positive electrode potential  is formed by the discharge reaction, E2 can be defined as shown in Fig.4. Then, α of Eq. [21] is introduced here on account of the relationship E2>E1.
Here, when Li + cations are arranged on the sphere surface of S8 -, the maximum number of Li + per S8 -, (Li + /S8 -)max, under closest packing conditions is to be approximated by Eq. [23].
As shown in Fig.1, the positive active mass consists of mainly three kinds of components involving 47 S8, 43 Li + gl -, and 10 carbon black in weight % 1) . Thus, the ratio of S8 volume to the total volume and the porosity of these materials are thought to be the practical factors to affect (Li + /S8 -)max.
These numbers correspond to the probability of 1/3 and the porosity of 50%, respectively, which are considered to be reasonable values. Since the cycle number corresponding to the battery capacity of ten with respect to S8 was 18, the n lim = 44 value is more than twice this cycle number.
According to Braga et al., this battery was not optimized 1) , and the rate determining electrode of the battery discharge was apparently its negative electrode 1) . On optimizing the design of this battery in accordance with the capacity determining factor of n lim and n lim = 44, a capacity could be 23 times S8 by Eq. [15].
In this mechanism, the morphology of the deposited Li is thought to be a dispersion of deposited Li nuclei unless explicit susceptibility to coalescence of deposited Li nuclei occurs. It is thought that this is consistent with  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60 A c c e p t e d M a n u s c r i p t The thermodynamics of Li deposition at the second E step was examined by taking account of electrochemistry with respect to two consecutive oneelectron steps and of the theory of generalized charge neutrality levels in respect of Schottky barriers. Consequently, the thermodynamics relating to the Li deposition in the second E step was considered to bring forth at least ΔG< -2.34eV.
The thermodynamics cycle of the overall mechanism was also examined.