Factors Contributing to Path Hysteresis of Displacement and

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Factors Contributing to Path Hysteresis of Displacement and Conversion Reactions in Li Ion Batteries Donghee Chang, Min-Hua Chen, and Anton Van der Ven* Materials Department, University of California Santa Barbara, 1361A Engineering II, Santa Barbara, California 93106-5050, United States S Supporting Information *

ABSTRACT: We investigate the thermodynamic and kinetic attributes of electrode materials that are necessary to suppress path hysteresis during displacement and conversion reactions in Li ion batteries. We focus on compounds in the Li−Cu−Sb ternary composition space, as the displacement reaction between Li1+ϵCu1+δSb and Li3Sb can be cycled reversibly. A first-principles analysis of migration barriers indicates that Cu, while not as mobile as Li in the discharged phase (Li3Sb), nevertheless should exhibit mobilities similar to that of Li in common intercalation compounds. A calculation of phase stability in the ternary Li−Cu−Sb system predicts that the intermediate phases along the reversible charge/discharge path are stable in a large Cu chemical potential window. This ensures that intermediate phases are not bypassed upon Li extraction even when large thermodynamic driving forces are needed to reinsert Cu into the discharged electrode. Our study suggests that the suppression of path hysteresis during displacement reactions requires (i) a high mobility of the displaced metal and (ii) the thermodynamic stability of intermediate phases along the reversible path in a wide metal chemical potential window. Even in the absence of path hysteresis, displacement and conversion reactions suffer from polarization needed to set up thermodynamic driving forces for metal extrusion and reinsertion. This polarization can be estimated with a Clausius−Clapeyron analysis.

1. INTRODUCTION Intercalation compounds are in many ways ideal for electrochemical energy storage applications. Nevertheless, their capacity, as measured by the amount of Li or Na that can be removed and reinserted into their crystal structure, is limited. Crystallographic and thermodynamic constraints often restrict the redox reaction to at most a one-electron valence shift per transition metal within the intercalation compound. There is therefore considerable interest in electrochemically active materials that undergo reaction mechanisms with alkali metals that overcome the capacity limitations of intercalation compounds. Displacement and conversion reactions,1−3 in which the insertion of the shuttled ion results in the displacement or extrusion of a transition metal from the electrochemically active compound, is often coupled with much larger valence shifts of the transition metal than is achievable in intercalation compounds. Displacement and conversion reactions utilize the full charge state of the metal cation M+z within the electrode material, reducing it to the M0 charge state as the metal precipitates out upon discharge.2,4,5 Chemistries that undergo conversion and displacement reactions, such MFx (M = Fe, Mn, Co, Ni, Cr, or Cu),6−8 MxOy (M = Ru, Co, Ni, Cu, or Fe),2,9−11 MgH2,12 MPx (M = Cu, Co, Mn, or Fe),13−16 MxSy (M = Cu, Co, Ni)17,18 and Cu2.33V4O1119 have much higher capacities compared to the best intercalation compounds. The kinetic mechanisms of displacement and conversion reactions are substantially more complex than intercalation © XXXX American Chemical Society

processes and remain poorly understood. Compounds that undergo displacement and conversion reactions suffer from a variety of limitations that need to be overcome.5,6,10 These include a poor reversibility and large capacity losses during charge and discharge. Furthermore, almost all electrode compounds that undergo a displacement or conversion reaction exhibit an unacceptably large hysteresis in the voltage profile between charge and discharge. One exception to this trend is lithiated Cu2Sb, a candidate anode material for Li-ion batteries that exhibits only a limited degree of polarization in its voltage profile when cycled between Li1+ϵCu1+δSb and Li3Sb.3,20−22 The electrochemical reaction of Cu2Sb with Li results in a sequence of intermediate compounds with varying concentrations of Li and Cu, but with strong structural similarities to the original compound.20 Although Li insertion causes large variations in lattice parameters, the fcc Sb sublattice of Cu2Sb is preserved in the lithiated compounds. Here we perform a comprehensive first-principles study of the thermodynamic properties associated with the Cu2Sb+Li displacement reaction as well as a variety of kinetic properties related to Li and Cu diffusion. Our aim is to elucidate the intrinsic properties of the various reaction products that facilitate reversibility and minimize hysteresis between disReceived: June 22, 2015 Revised: September 15, 2015

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DOI: 10.1021/acs.chemmater.5b02356 Chem. Mater. XXXX, XXX, XXX−XXX

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Chemistry of Materials

structure. Ionic positions and lattice parameters of each structure were fully relaxed. We also explored diffusion within the fully lithiated phase, Li3Sb, by calculating migration barriers of various likely hop mechanisms. The nudged elastic band (NEB) method implemented in VASP was used to calculate energy barriers associated with Li and Cu hops. The calculations of migration barriers were performed in super cells having 81 interstitial sites (27 Sb atoms).

charge and charge. A previous theoretical study of a model displacement reaction, which relied on first-principles thermodynamic and kinetic data and a phase field model to describe kinetics at the electrode particle level, identified specific thermodynamic and kinetic properties intrinsic to the electrode chemistry and crystal structure that cause hysteresis in displacement and conversion reactions.23 The study demonstrated that a primary source of hysteresis arises from a difference in reaction path between charge and discharge. Phase field simulations of the displacement reaction identified the triggers that lead to the selection of a different reaction path between charge and discharge. These included23 (i) a large difference in the diffusion coefficients between Li and the displaced ion and (ii) a lack of a thermodynamic driving force for the reinsertion of the displaced ion upon removal of Li during charge when retracing the discharge reaction path. In view of the minimal polarization exhibited during the Li1+ϵCu1+δSb to Li3Sb displacement reaction, our focus here is to ascertain whether the thermodynamic and kinetic properties of the reaction products of this chemistry differ substantially from other electrode chemistries that exhibit path hysteresis.

3. RESULTS 3.1. Thermodynamic Properties. The shape of the free energy surface as a function of concentration plays an important role in determining whether path hysteresis will be triggered during a displacement or conversion reaction.23 We investigated the thermodynamic properties of the various phases that form during the lithiation of Cu2Sb, first at 0 K using DFT-PBE as implemented in VASP and subsequently at finite temperature by applying Monte Carlo simulations to a ternary cluster expansion parametrized with the DFT-PBE energies. The Li−Cu−Sb ternary contains several stable compounds. Cu2Sb has an fcc Sb sublattice and belongs to space group P/ 4nmm (no. 129).33 Its crystal structure is shown in Figure 1(a).

2. METHOD The evolution of the displacement reaction of Li with Cu2Sb can be tracked in a ternary composition space spanned by Li, Cu, and Sb. Several intermediate phases are stoichiometric compounds, while others form solid solutions over a limited concentration interval. All the phases that form during the displacement reaction maintain an fcc Sb sublattice, differing only in the arrangement and composition of Li and Cu over the tetrahedral and octahedral interstitial sites of fcc Sb.20,21 While electronic and vibrational excitations contribute to the free energy of the various phases, the most important excitations in this system arise from configurational rearrangements of Li and Cu over the interstitial sites of Sb. Contributions to thermodynamic properties arising from configurational degrees of freedom in the Li−Cu−Sb system were calculated by applying grand canonical Monte Carlo simulations to a ternary cluster expansion parametrized with first-principles density functional theory (DFT) energies. A cluster expansion describes the dependence of the energy of a multicomponent crystal on the degree of ordering among its constituents.24,25 We found it convenient to use Li3Sb, in which Li fills all octahedral and tetrahedral sites of an fcc Sb sublattice, as a reference configuration for the cluster expansion. Other compounds in the Li−Cu−Sb ternary having an fcc Sb sublattice can be obtained from Li3Sb by replacing a subset of Li with vacancies and Cu. The construction of a ternary cluster expansion and the Monte Carlo simulations were performed with the CASM program.26,27 More details about the cluster expansion and its parametrization using DFT energies can be found in Supporting Information. Vibrational free energies were calculated for the stable phases along the Cu−Sb binary within the quasi-harmonic approximation. These were calculated because Cu2Sb is not predicted to be stable at 0 K with different approximations to DFT. Details of these calculations can also be found in Supporting Information. First-principles total energies to parametrize the ternary cluster expansion and the harmonic phonon Hamiltonians were calculated within the generalized gradient approximation (GGA) to DFT as parametrized by Perdew−Burke−Ernzerhof (PBE).28 We used the Vienna Ab initio simulation package (VASP) plane wave pseudopotential code29,30 with the projector augmented wave method (PAW) to describe the interactions between valence and core electrons.31,32 The valence states of the various PAW pseudopotentials were Li 1s, 2s, and 2p, Cu 3d10 and 4p1, and Sb 5s2 and 5p3. The plane-wave basis set cutoff energy was set to 400 eV, and a gamma point-centered k-point mesh with a density consistent with a 12 × 12 × 12 k-point mesh density in the primitive Li3Sb cell reciprocal lattice was used for each

Figure 1. Crystal structures of (a) Cu2Sb (P4/nmm)33 and (b) Li2CuSb (F4̅3m)34

One Cu per Cu2Sb formula unit fills the octahedral interstitial sites while the other Cu orders over half of the tetrahedral interstitial sites of the fcc Sb sublattice. The Cu occupying tetrahedral sites segregate between alternating (001) planes of the fcc Sb sublattice, thereby giving the crystal a tetragonal symmetry. Electrochemical lithiation of Cu2Sb results in the extrusion of Cu coupled with the formation of a ternary compound having nominal stoichiometry Li2CuSb with space group F4̅3m (no. 216) shown in Figure 1(b). However, the composition of Li2CuSb is not well established, with evidence of an off-stoichiometric solid solution.20,34,35 Li2CuSb also has an fcc sublattice. The Cu orders over half the tetrahedral sites; however, the Cu ordering differs from that in Cu2Sb, forming a zinc-blende ordering when considering only the Cu and Sb sublattices. The Li ions of Li2CuSb fill all the octahedral sites and remaining tetrahedral sites. Experimental and firstprinciples evidence suggests that Li2CuSb has a tendency to tolerate Li vacancies over the Li-tetrahedral sites.20,34,35 Further lithiation leads to additional Cu extrusion and the formation of Li3Sb, which again has an fcc Sb sublattice with Li ions filling all interstitial tetrahedral and octahedral sites. The compound belongs to the Fm3̅m (no. 225) space group.36,37 In addition to Cu2Sb, Li2CuSb, and Li3Sb, a hexagonal Li2Sb is also stable in B


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Figure 2. (a) The Li−Cu−Sb ternary phase diagram at 300 K constructed using free energies calculated with grand canonical Monte Carlo simulations applied to a ternary cluster expansion. (b) Phase stability as a function of the Li and Cu chemical potentials, μLi and μCu at 300 K. The reference states for the chemical potentials are fcc Cu (μoCu = 0) and bcc Li (μoLi = 0).

removal from Li3Sb to form Li2.96Sb and Li2.98Sb occurs most easily by placing Li vacancies on the octahedral sites. Phase stability can also be plotted as a function of the Li and Cu chemical potentials, μLi and μCu, as shown in Figure 2(b). This phase diagram was calculated by minimizing the grand canonical free energies of each phase at fixed chemical potentials μLi and μCu. The reference states for the Li and Cu chemical potentials in Figure 2(b) are bcc Li and fcc Cu, respectively. Hence, pure Cu is stable for positive Cu chemical potentials, independent of the Li chemical potential, while pure Li is stable for positive Li chemical potentials. Pure Sb is stable for very negative Li and Cu chemical potentials. The lithiumrich Li2Sb and Li3Sb phases become stable at increasing μLi and sufficiently negative μCu. The solid solution Li1+εCu1+δSb phase is stable over a sizable Li and Cu chemical potential range. The voltage of an electrochemical cell is related to the Li chemical potentials of the cathode and anode according to

the Li−Cu−Sb ternary. Li2Sb, however, does not form during lithiation of Cu2Sb or during subsequent Li removal from the Li3Sb+Cu two-phase mixture. Li2Sb is the only compound that does not have an fcc Sb sublattice. Instead the Sb sublattice is isomorphic to the omega phase of Zr.38,39 Figure 2(a) shows the calculated ternary phase diagram at 300 K. The phase diagram was determined by applying the common tangent construction to calculated free energies. The free energies for the Li-containing phases were those calculated with grand canonical Monte Carlo simulations applied to the ternary cluster expansion. For Cu2Sb, we used the vibrational formation free energy to ensure that it appears as a stable phase because DFT does not predict it to be stable at 0 K relative to a two-phase mixture of Cu and Sb (see Supporting Information). We did not include vibrational free energy contributions for any of the Li-containing phases. The implicit assumption of this approximation is that the inclusion of vibrational contributions will not qualitatively change the topology of the phase diagram away from the Cu−Sb binary. The calculated phase diagram shows a stable ternary solid solution, having composition Li1+ϵCu1+δSb with ϵ ranging between 0.3 and 0.64 and with δ ranging between 0 and 0.03. The octahedral sites of the fcc Sb sublattice in the solid solution are exclusively filled by Li, while the Cu fill half the tetrahedral sites forming an fcc sublattice. The remaining tetrahedral sites contain disordered Li and vacancies, with a very dilute concentration of Cu at the more Cu-rich concentrations. These sublattice concentrations, predicted with a ternary cluster expansion describing disorder over all interstitial sites, are qualitatively consistent with a study by Matsuno35,40 who used a binary cluster expansion describing Li-vacancy disorder over the tetrahedral sites occupied by Li and the octahedral sites. The Monte Carlo simulations applied to the ternary cluster expansion suggest that there is very little Cu disorder and a minimal concentration of Cu over the tetrahedral sites occupied primarily by Li and vacancies. Several phases are predicted to be stable along the Li−Sb binary. These include the cubic Li3Sb phase and the hexagonal Li2Sb phase. We also found that the introduction of dilute vacancies to cubic Li3Sb in large unit cells at compositions Li2.96Sb and Li2.98Sb leads to stable phases that reside on the convex hull.41 In contrast to the Li1+ϵCu1+δSb compounds, Li

V = −(μLi − μLi0 )/e

(1)

where μLi is the Li chemical potential in the positive electrode and μ0Li is the Li chemical potential of the reference anode, which we choose to be metallic Li. Figure 3 shows the calculated voltage as a function of Li content x in LixCuySb along a path in ternary composition space denoted by the dashed arrow in Figure 2(a). Because the electrode passes through three-phase and two-phase regions containing pure Cu along this path, the Cu chemical potential in LixCuySb is equal to that of fcc Cu, which due to our choice of reference state is equal to zero (i.e., μCu = μoCu = 0). The first plateau in the voltage profile (0 < x < 1.3) corresponds to a passage through the Cu + Cu2Sb + Li1.3Cu1.03Sb three-phase region where the Li chemical potential remains constant as the relative phase fractions of the three coexisting phases changes upon Li insertion or removal. The three-phase region is a point in the chemical potential phase diagram (point a in Figure 2(b)). The voltage profile between 1.3 < x < 1.64 decreases continuously because the electrode then passes through the two-phase region between a solid solution of Li1+εCu1+δSb and Cu. The second plateau emerges as the electrode passes through the three-phase region between Li1.64CuSb + Li3Sb + Cu. This three-phase region corresponds to point b in the chemical potential phase C


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Figure 3. Calculated voltage as a function of Li content x in LixCuySb along a path in ternary composition space denoted by the dashed arrow in Figure 2(a). Because pure Cu is in phase equilibrium with the other phases along this path, the Cu chemical potential is equal to that of metallic Cu. The calculated equilibrium voltage curve in the Li−Sb system as a function of Li concentration41 is also shown (dashed red line).

diagram Figure 2(b). Also shown in Figure 3 (dashed red line) is the voltage when Li reacts with Sb in the absence of Cu. This voltage profile exhibits two plateaus separated by a small step due to the stability of the hexagonal Li2Sb phase.41 It is the voltage profile that emerges when the Cu chemical potential is very negative relative to the chemical potential of metallic Cu (i.e., μCu < −0.15 eV, based on the phase diagram of Figure 2(b)). 3.2. Kinetic Properties. An important trigger of path hysteresis in conversion and displacement reactions was found to be a large difference in the diffusion coefficients of Li compared to that of the displaced cation in the fully lithiated phase.23 To elucidate why the hysteresis in the displacement reaction of Li1+ϵCu1+δSb to Li3Sb is small compared to other chemistries, we investigated the migration barriers for Li and Cu diffusion in Li3Sb, the fully lithiated end product of the displacement reaction. Past first-principles studies of the Li−Sb alloying reaction have predicted a very high Li mobility in Li3Sb.41,42 Dilute vacancies in Li3Sb prefer to occupy the octahedral sites. The migration barrier for a Li hop from a tetrahedral site into a vacant octahedral site in Li3Sb is predicted to be ∼100 meV, a barrier that is substantially lower than those of most intercalation compounds. The Li mobility in Li3Sb is therefore very high. Here we also calculated the migration barrier for a Li hop from a tetrahedral site to an octahedral site with a neighboring Cu in an adjacent tetrahedral site. Figure 4(a) shows the migration barrier of this Li hop, with the calculated migration barrier of 80 meV in the presence of a neighboring Cu being even lower than that in the absence of Cu. Also, as evident in Figure 4, the presence of Cu changes the site preference for the vacancy, with the vacancy now preferring the tetrahedral site over the octahedral site by 25 meV. This site preference for the vacancy is similar to that exhibited by the ternary Li1+ϵCu1+δSb solid solution. We also considered likely migration paths of Cu in Li3Sb. We found that an isolated Cu in Li3Sb is unstable in an octahedral site next to a vacant nearest neighbor tetrahedral site. The energy of the crystal increases monotonically as Cu migrates from a tetrahedral site to an adjacent vacant octahedral site. Cu hops from a tetrahedral site to a nearest neighbor vacant octahedral site therefore cannot occur. As a result, Cu migration in the presence of an isolated vacancy is only possible between

Figure 4. Migration barriers for Li hops and Cu hops in Li3Sb. (a) A Li hops from a tetrahedral site to an adjacent vacant octahedral site with a neighboring Cu in an adjacent tetrahedral site. (b) Cu hops from a tetrahedral site to an adjacent vacant tetrahedral site (an isolated vacancy). (c) Cu hops from a tetrahedral site to an adjacent vacant tetrahedral site in the presence of a neighboring vacant octahedral site (the Cu hops into a pair of vacancies).

nearest neighbor tetrahedral sites as shown in Figure 4(b). The migration barrier for this hop is predicted to be 1.2 eV, a value that is substantially higher than that of Li in Li3Sb and of Li in other common intercalation compounds. This hop mechanism is therefore unlikely to contribute much to macroscopic Cu diffusion. Li3Sb is likely to have a high concentration of vacancies upon Li removal from the electrode during charging due to its exceptionally high Li mobility and the thermodynamic stability (both energetically and entropically driven) of a Li-deficient solid solution of Li3−ζSb.41 We therefore also explored Cu migration mechanisms and barriers in the presence of a pair of Li vacancies. Figure 4(c) shows the calculated migration barrier as Cu migrates from a tetrahedral site to an adjacent vacant tetrahedral site in the presence of a neighboring vacant octahedral site. The hop path is curved, passing through the D


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Li3Sb will only occur if μCu is less than μoCu, or equivalently if ΔμCu = μCu − μoCu < 0. A mechanism to modify μCu within Li3Sb is by the electrochemical extraction of Li from Li3Sb. Figure 5 compares

octahedral site as illustrated in Figure 4(c). The migration barrier for this hop is 440 meV, which, while higher than that of Li diffusion in Li3Sb, is of the same order as Li migration barriers in common intercalation compounds.43−45 This result suggests that vacancy clusters mediate Cu diffusion in Li3Sb, in a similar way that vacancy clusters mediate Li diffusion in lithiated layered and spinel intercalation compounds.46 The very fast Li mobility should ensure that the reorganization of vacancy clusters after each Cu hop is not rate limiting to Cu diffusion. The low-barrier divacancy Cu hop mechanism in a “sea” of very mobile Li ions implies that Cu should have a high diffusion coefficient, similar to that of Li in many fully lithiated intercalation compounds.

4. DISCUSSION Conversion and displacement reactions in the electrodes of Liion batteries promise substantially higher capacities than can be achieved with intercalation compounds. However, the requirement to spatially redistribute a second species in addition to Li ions often results in an unacceptably large hysteresis between charge and discharge. In many cases this hysteresis can be attributed to a difference in the reaction path between charge and discharge.23,47−49 A theoretical study of hysteresis accompanying the displacement reaction of Li with CuTi2S4 identified two triggers in the fully discharged state that cause path hysteresis:23 (i) a large difference in mobility between Li and the displaced cationic species (i.e., Cu), with the displaced cation being substantially less mobile than Li and (ii) a lack of a thermodynamic driving force for the reinsertion of the displaced cation upon retracing the discharge path during charge. Cu2Sb is unusual in that its voltage polarization is relatively small compared to other electrode chemistries that undergo a displacement or conversion reaction.20 While the first voltage plateau due to the conversion of Cu2Sb to a two-phase mixture of the ternary Li1+εCu1+δSb phase and metallic Cu during Li insertion is usually not reversible, the subsequent reactions are reversible.21 The reaction path followed during the second voltage plateau in which Li1+εCu1+δSb reacts further with Li to form Li3Sb and metallic Cu is readily retraced experimentally upon Li extraction to reform Li1+εCu1+δSb. Hence, any difference between the discharge voltage and the charge voltage is due to dissipative polarization and not to a qualitative difference in reaction pathways. The crystallographic similarity between Li1+εCu1+δSb and Li3Sb, both sharing an fcc Sb sublattice, suggests that Li removal from the fully discharged electrode should be accompanied by the simultaneous reinsertion of Cu into Li3Sb. Otherwise Li extraction will lead to the formation of either Li2Sb or, as occurs during the Li−Sb alloying reactions, the formation of pure Sb.41,50,51 As can be seen in Figure 3, the qualitative shapes of the equilibrium voltage profiles for the two reaction paths (one with reversible Cu reinsertion and the other without any Cu reinsertion) differ significantly. The reinsertion of Cu into Li3Sb during Li extraction requires a thermodynamic driving force. The discharged state consists of a two-phase coexistence between Li3Sb and Cu precipitates. In equilibrium, the Cu chemical potential within Li3Sb, μCu, equals that of metallic Cu, μoCu (which when using metallic Cu as a reference state is equal to zero). Our calculated free energies predict that Li3Sb contains a very dilute concentration of dissolved Cu when it coexists with metallic Cu. A spontaneous increase in this Cu concentration within

Figure 5. Several trajectories in chemical potential space as Li is extracted from Li3Sb. Path I corresponds to pure Li extraction from Li3Sb without Cu reinsertion while path II corresponds to Li extraction from Li3Sb with Cu reinsertion.

several trajectories in chemical potential space as Li is extracted from Li3Sb. Equilibrium between Li3Sb and Cu reside along the line between points b and c in Figure 2(b). Path I, in Figure 5, corresponds to the removal of Li in the absence of Cu reinsertion. As is clear from Figure 5, Li extraction without Cu insertion (path I) will generate a negative Cu chemical potential within Li3Sb. The electrochemical removal of Li from Li3Sb realized by increasing the voltage (and thereby decreasing the Li chemical potential), will, therefore, set up chemical driving forces for Cu in the surrounding metallic precipitates to enter Li3Sb. The trajectory of path I, however, never intersects the stability domain of the Li1+εCu1+δSb solid solution. Hence, to avoid transforming directly to Li2Sb or pure Sb, Cu insertion and diffusion must be sufficiently rapid compared to the rate of Li extraction to deflect the trajectory in chemical potential space along a path similar to path II, which does pass through the Li1+εCu1+δSb domain. As can be seen in Figure 5, path II corresponds to a scenario where Li removal is accompanied by simultaneous insertion of Cu. Figure 5 clearly shows that the Cu mobility within Li3Sb must be sufficiently high to minimize the driving force ΔμCu needed to reinsert Cu. While our calculated migration barrier for Cu diffusion in the presence of divacancies is higher than that of Li in Li3Sb, it is nevertheless not much higher than typical Li barriers predicted for high rate capable intercalation compounds such as layered LixCoO2, spinel LixTiO2, and LixTiS2.43−45 Furthermore, diffusion in these intercalation compounds is also mediated by divacancies or triple vacancies.46 They rarely require more than 0.1 V over/under E


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Chemistry of Materials potentials (equivalent to ±ΔμLi) to achieve a reasonable charge or discharge rate. This suggests that driving forces of ΔμCu = μCu − μoCu ∼ −0.1 eV should be sufficient to reinsert Cu into Li3Sb on the time scale of typical charge rates. These are within the range of the Cu chemical potential stability window of Li1+ϵCu1+δSb (Figure 2(b)).52 Our results, therefore, suggest that the experimental reversibility of the Li1+εCu1+δSb to Li3Sb reaction can be attributed to a high Cu mobility within Li3Sb and a sufficiently wide Cu chemical potential stability window of Li1+ϵCu1+δSb such that it is not bypassed when a Cu driving force for Cu reinsertion, ΔμCu, emerges as Li is extracted. Despite the reversibility of the Li1+εCu1+δSb to Li3Sb displacement reaction, the need to create driving forces for Cu extrusion (ΔμCu = μCu − μoCu > 0) and opposite driving forces for Cu reinsertion (ΔμCu = μCu−μoCu < 0) will result in a polarization of the voltage profile between charge and discharge. On discharge (Li insertion), ΔμCu must be postive to set up a driving force for Cu extrusion. As an example, Figure 6 compares the calculated voltage curve for ΔμCu = 0.1 eV

dV * 1 dμLi* 1 ΔNCu =− = * * e dμCu e ΔNLi dμCu

(2)

In eq 2, μCu * and μLi * refer to chemical potentials along the equilibrium two-phase boundary in Figure 5 while e is the charge of an electron. The ΔNLi and ΔNCu correspond to the changes in Li and Cu concentrations when transforming from one phase to the other. Equation 2 can be used to estimate the polarization of a voltage plateau, ΔV*, due to the need of a nonzero ΔμCu according to ΔV * ≈

⎛ 1 ΔNCu ⎞ dV * ΔμCu = ⎜ ⎟Δμ * * dμCu ⎝ e ΔNLi ⎠ Cu

(3)

In general, a shallow slope in the coexistence line in a chemical potential phase diagram (as measured by dμCu * /dμLi *) results in a large shift in the plateau voltage for a fixed Cu driving force, ΔμCu. Equation 3 shows that the larger the change in Li concentration per displaced metal cation, the smaller the polarization ΔV* for a fixed driving force ΔμCu. We emphasize that these estimates of polarization are in addition to other sources of polarization that are also present in intercalation compounds. These include gradients in Li concentration and the dissipation of driving forces during the phase transformation between Li1+ϵCu1+δSb and Li3Sb. The crystallographic similarity between Li1+ϵCu1+δSb and Li3Sb, with both phases having a common fcc Sb sublattice, undoubtedly plays a favorable role in fostering reversibility of the Li1+ϵCu1+δSb to Li3Sb displacement reaction. This along with the fact that Li1+ϵCu1+δSb already contains both Li and Cu suggests that the reaction occurs topotactically.21 Most conversion reactions involving transition metal oxides10,54,55 and fluorides,6,8,56 in contrast, occur reconstructively. Reconstructive conversion reactions offer more flexibility as to the mechanisms of ion redistribution with a possibility that the anions may spatially redistribute in addition to the metal cations. For these systems, the thermodynamic and kinetic triggers of path hysteresis should be more numerous than in displacement reactions. The above Clausius−Clapeyron relation, eq 2, will give an estimate of the polarization needed to set up driving forces for metal extrusion and reinsertion. We expect these driving forces in oxides and fluorides to be well in excess of that needed for Cu extrusion from Li1+ϵCu1+δSb and Cu reinsertion into Li3Sb. Furthermore, in contrast to Li3Sb, the discharge products of transition metal oxides and fluorides are the highly ionic and electronically insulating Li2O and LiF salts. The Li mobility within Li2O and LiF are likely significantly lower than the remarkably high mobility in Li3Sb.41,42 Hence, substantial polarization can be expected in the oxides and fluorides simply to electrochemically remove Li from Li2O and LiF. The current analysis has focused exclusively on reaction pathways within the electrode and has neglected the role of electrode−electrolyte interactions in causing voltage hysteresis during conversion and displacement reactions. Reactions with components of the electrolyte can differ during discharge and charge, thereby also resulting in path hysteresis.9,49 These interactions and asymmetric reactions also remain to be understood and controlled in order to minimize voltage hysteresis.

Figure 6. Polarization in the voltage profile between discharge and charge due to the requirement of thermodynamic driving forces for Cu extrusion and reinsertion. Blue line represents the calculated voltage curve for a discharge reaction when ΔμCu = 0.1 eV. Orange line represents the calculated voltage curve for a charge reaction when ΔμCu = −0.1 eV. The equilibrium voltage curve (ΔμCu = 0 eV) is shown with the black line.

relative to the equilibrium voltage curve (for ΔμCu = 0), showing a polarization of the plateau by almost −0.08 V. During charging (Li removal), a negative ΔμCu is needed to set up a driving force for Cu reinsertion. The voltage curve with ΔμCu = −0.1 eV is also shown in Figure 6 and exhibits a polarization in the opposite direction of almost 0.07 V relative to the equilibrium plateau. This results in an overall polarization between charge and discharge of approximately 0.15 V when forces of ΔμCu = ±0.1 eV are needed to drive Cu extrusion and insertion. The degree of polarization due to the requirement of Cu chemical potential driving forces during a three-phase displacement reaction is determined by the slope of the phase boundary separating Li3Sb and Li1+ϵCu1+δSb in the chemical potential phase diagram of Figure 5. The slope of two-phase coexistence lines in chemical potential space can be estimated with a Clausius−Clapeyron analysis,53 which when related to the plateau voltage, V*, using eq 1, yields F



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5. CONCLUSION We have investigated thermodynamic and kinetic properties of the reaction products of the Li1+ϵCu1+δSb to Li3Sb displacement reaction with the aim of identifying the attributes that facilitate reversibility and prevent path hysteresis. Path hysteresis during displacement and conversion reactions is triggered during charging of the electrode as Li removal and simultaneous metal reinsertion into the fully discharged phase can occur along a multitude of paths. The results of the current study indicate that two properties of the electrode material are crucial to suppress path hysteresis during a displacement reaction: (i) The mobility of the displaced metal cation must be sufficiently high such that metal reinsertion can occur at the imposed charge rate and (ii) the intermediate phases along the reversible reaction path are stable in a sufficiently wide metal chemical potential window. Our first-principles analysis predicts that these requirements are satisified in Li3Sb, consistent with the experimental reversibility of the Li1+ϵCu1+δSb to Li3Sb displacement reaction. Even in the absence of path hysteresis, displacement and conversion reactions will suffer from polarization due to the requirement of thermodynamic driving forces for metal extrusion and reinsertion. The extent of polarization in a voltage plateau can be estimated with a Clausius−Clapeyron relation of the slope of equilibrium phase boundaries in chemical potential space. This polarization is in addition to that dissipated by Li diffusion and phase transformation kinetics. While the above two criteria are crucial to avoid path hysteresis, other thermodynamic and kinetic factors are also likely important to minimize polarization in more complex conversion reactions or to suppress path hysteresis due to reactions with the electrolyte.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.chemmater.5b02356. Computational details of the ternary cluster expansion, Monte Carlo simulations, quasi-harmonic phonon calculations, and HSE calculations (PDF)

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AUTHOR INFORMATION

Corresponding Author

*Phone: +1 805 893 7920. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the NorthEast Center for Chemical Energy Storage, an Energy Frontier Research Center funded by the U.S Department of Energy, and the Office of Basic Energy Science under award no. DE-SC0012583. The first-principles calculations were performed using computational resources provided by the National Energy Research Scientific Computing Center (NERSC), supported by the Office of Science and U.S. Department of Energy, under contract number DE-AC02-05CH11231, in addition to support from the Center for Scientific Computing at the CNSI and MRL: an NSF MRSEC (DMR-1121053) and NSF CNS0960316. Images of crystal structures were produced with VESTA.57 G


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