Analysis of crossover-induced capacity fade in redox flow batteries with non-selective separators

1Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA 2Computational Science and Engineering Program, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA 3Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA 4Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA

In 2017 about 60% of electricity in the USA was generated from fossil fuels whereas only ∼10% was generated using renewable energy sources such as wind and solar. 1 However, the use of renewables (excluding hydropower) has grown by 15% in the USA over the past 10 years, 2 and it is projected that by 2050 22% of total electricity generation in the USA will be produced by wind and solar power plants. 3 Therefore, it is imperative to develop technologies to enable electrical energy storage (EES) from such renewable sources of energy that are intermittent by nature. Redox flow batteries (RFBs) are attractive EES solutions because of their ability to independently scale energy capacity and power density by way of scaling electrolyte volume and reactor area, respectively. Such a capability could enable the integration of renewable energy resources, as well as load leveling and peak shaving on the electric grid. [4][5][6][7][8] The commercialization of RFB technology at grid-scale is currently challenged by high system prices. 9,10 Accordingly, the United States Department of Energy has set an aggressive target price of $120 kW/h for the commercialization of RFB EES. 11 A typical RFB is comprised of a reactor (electrode compartments separated by an electron insulating separator) and two tanks (electrolyte storehouse) connected through a network of pipes and pumps. The electrolyte is made by dissolving redox-active molecules in aqueous (AqRFB) or non-aqueous (NAqRFB) solvents along with inert salt. We note that charged redox-active molecules can be synthesized as salt wherein the particular redox-active molecule is crystallized together with inert ions of opposite charge. Hereafter, we use "salt" to refer to the inert ions dissolved in solution, apart from those originally crystallized with as-synthesized redox-active molecules. Existing AqRFB chemistries include Fe/Cr, 12 all vanadium, 13 polysulfide/bromine, 14 zinc/bromine, 15,16 metallocenes, 17,18 and organic redox molecules such as TEMPO, 19 quinones, 20 and redoxactive polymers 21 (RAPs). The development of NAqRFBs with nonaqueous electrolytes 9 or even ionic liquids 22 is motivated by the fact that these electrolytes provide a wider electrochemical stability window allowing higher energy density storage devices. 23,24 Separators electrically isolate the two electrolytes in RFBs and allow conduction of inert salt ions to maintain charge balance across the two electrolytes. Regardless of the types of RFB, ion-selective membranes are most commonly used as separators to prevent crossover of redox active species with non-zero oxidation state. 25,26 However, these ion-exchange membranes have limited lifetime and are expensive, accounting for almost 20% of total battery cost. 10,27 Alternatively, a non-selective separator can conceivably be used, albeit with inherent capacity loss due to crossover. Membrane-free RFBs such as hybrid RFBs, 28,29 laminar flow RFBs, [30][31][32] and immiscible electrolyte RFBs 33 have been proposed. The scalability of these systems is challenging as they either use a solid electrode, flow in micro-channels or viscous ionic liquids respectively. In addition to membrane costs, the costs associated with supporting electrolyte are high particularly for NAqRFBs (∼$5/kg 9 ) which use fluorinated derivatives as inert salt. The primary purpose of inert salt ions is to facilitate long-range ion conduction between electrodes while retaining redox-active species within their respective electrodes. Therefore controlling the usage of solvent and inert salt is vital for commercialization of RFBs. 34 In this regard, a proof-of-concept NAqRFB omitting the use of inert salt was demonstrated by using redox species in their respective electrodes paired with a common inert counterion to facilitate long-range ion conduction. 35 The flowing electrochemistry in RFB reactors complicates their design, operation, and electrolyte development, and mathematical modeling of the transport phenomena therein can aid in linking RFB development to technical targets in a cost-, resource-, and time-effective manner. While ab initio atomistic simulations can be used to predict the equilibrium properties of RFB electrolytes, 36 continuum simulations that predict the spatial and temporal distribution of each species and their reaction rates are needed to understand the influence of electrolyte properties on RFB performance. The first two-dimensional, transient model of a vanadium RFB was developed by Shah and co-workers 37 following which several other models were developed [38][39][40][41] particularly accounting for crossover through ion-exchange membranes. [42][43][44] Other redox chemistries such as hydrogen/bromine, 45,46 polysulfides, 47 and organic/inorganic molecules 48 have been modeled as well. Recently, 49 we simulated a RAP-based NAqRFB using a non-selective separator and established the importance of maintaining a high flow rate to minimize the capacity losses inherent to the mixing processes in RFB tanks, the results of which were later used to assess the impact of electrolyte viscosity and ionic conductivity of redox-active polymer electrolytes on RFB performance. 50 However, most previous models simulated specific redox chemistries with certain diffusion coefficients at certain concentrations of both inert ions and redox-active species, one notable exception is zero-dimensional, time-independent modeling that employed ad hoc flux equations with species conversion rules that were used to specify materials selection criteria for RFB separators. 51 In the current work, we use physics-based modeling to predict the capacity utilization and fade of RFB electrolytes with a range of diffusion coefficient values and concentrations in order to extract guidelines for the design of electrolytes and the operation of RFBs that minimize crossover rates. To predict crossover rates accurately, we model the transient evolution of species concentration using a twodimensional version of porous electrode theory with Nernst-Planck fluxes, wherein electrolyte is transported between a reactor and wellmixed tanks. In contrast with previous RFB models we use Marcus-Hush-Chidsey theory 52 to predict redox kinetic rates, instead of the commonly used Butler-Volmer (BV) equation. BV is an empirical model where the rate constants exponentially depend on the overpotential η as k BV ox/red = k 0 e ± 0.5Fη RT , where k 0 is the standard rate constant. Although BV is accurate for low overpotentials, at high overpotentials BV predicts extremely large and unrealistic values of rate constants. This effect is particularly important in modeling the capacity fade due to the crossover of redox species where large overpotentials (>0.5 V for both AqRFBs and NAqRFBs) are experienced. On the other hand, the Marcus-Hush-Chidsey (MHC) kinetic theory 52,53 accounts for reorganization energy of solvent λ and the finite density of states in the electrode. Further, the MHC theory (both in its original integral form 52 and simplified form 53 ) predicts a saturating reaction constant with increasing overpotential as shown in Fig. 1.
In the present work, we present a two-dimensional porous electrode model broadening the scope to any single-electron redox species, and we account for the crossover of all redox species involved in the system. Crossover occurs whenever there exists a gradient in pres- sure (advection), concentration (diffusion), or electrolyte potential (migration) across the porous separator. Using this model, we study the crossover of redox species by mapping the variation of capacity utilization and capacity fade with two variables: (1) molar-salt ratio 34 (γ = c salt /c redox ) and (2) salt-diffusion ratio (ζ = D salt /D redox ). Further, non-dimensional scaling analysis is conducted based on the Damköhler numbers for redox active and salt species in the electrolyte. Operating regimes for RFBs with non-selective separators are defined using these non-dimensional parameters. In particular, we find that operating RFBs with sufficient supporting electrolyte (γ > 0.6) and with high redox Damköhler number ensures both high capacity utilization and long cycle life. These findings establish a rational paradigm for the selection and engineering of low-cost, long-life RFB electrolytes.

Modeling Approach
Reactor layout and electrochemistry.-The RFB simulated here consists of a reactor and two well mixed tanks storing its electrolytes   (Fig. 2). The two porous carbon electrodes in the reactor host the redox reactions and are electrically separated by a non-selective porous separator. This assembly is encased by two current collectors. The reactor uses interdigitated flow fields (IDFFs) that have alternating inlet and outlet channels, a detailed schematic of which is shown in our previous publication. 49 The IDFF offers optimal performance with respect to the wetting of electrodes (for conducting electrochemical reactions) and reducing pressure drop. 54 Monitoring pressure drops is important in the context of NAqRFBs as electrolyte viscosity increases rapidly with redox concentration. 21,50 Moreover, operating RFBs at higher magnitudes of pressure could force bulk movement of electrolytes through the separator if the respective electrolytes are pumped at different rates or have different viscosities. Such effects could induce electrolyte imbalance and, thus, capacity loss, while potentially affecting membrane life. Also, pumping costs increase, reducing the net effective energy stored/delivered by the battery system. We note that crossover due to bulk electrolyte flow is negligible here since viscosity is assumed to be constant in both electrolytes.
A complete IDFF reactor can be constructed by periodically repeating a 2D 'unit' cell as shown in Fig. 2. We therefore conduct our analysis on a representative unit cell of length L e (along the direction of the separator), electrode thickness H e , and porosity ε e with flow inlet and outlet channel width of L ent . The non-selective separator is modeled as an electrolyte-imbibed, electron-insulating porous medium with thickness H sep , porosity ε sep , and tortuosity τ sep . The geometrical and transport properties of the reactor and separator are listed in Table I. The two electrolytes in their pristine condition contain high potential and low potential redox couples, respectively. The primary redox reactions in the high potential electrode and low potential electrode (green dotted boxes in Fig. 2) along with equilibrium potentials at 50% state-of-charge (SOC) are: The equilibrium potentials are set to produce a 3 V working voltage RFB. Although a 3 V RFB corresponds to a non-aqueous RFB, the mechanisms explained in this work are applicable in designing both aqueous and non-aqueous RFBs as long as (1) a non-selective separator is used and (2) cell voltage exceeds the overpotential level where Marcus-Hush-Chidsey kinetics become overpotential independent. As the RFB is cycled, crossover of all the four different redox species (R1 z R1 , O1 z O1 , R2 z R2 , O2 z O2 ) from their primary electrode to the counter electrode forces a second set of redox reactions in the counter electrode (pink dotted boxes in Fig. 2). Depending on the charge numbers zi of the redox species, the reactions can be classified either as salt-splitting (SS) or rocking-chair (RC) configuration. In SS configuration one species in each redox couple is neutral requiring the presence of supporting electrolyte for ion transfer across the separator. In RC configuration all redox species have similarly signed non-zero oxidation state, 34 and these ions, along with their associated counter ions, can transfer across the separator to maintain electroneutrality. Theoretically RC requires less salt concentration than SS and could therefore be a potential solution toward economical low-salt RFBs. 34 In this study, we compare the capacity utilization and capacity fade of RC and SS configurations as a function of salt concentration and the diffusion coefficient of redox species.
Electrochemical kinetics and governing equations.-A transient, two-dimensional numerical model based on porous-electrode theory 55 is developed to model the competitive transport processes that govern RFB performance, particularly transport mechanisms related to crossover. The porous electrode model developed here is a homogenized one, employing a local, volume-averaged formulation for transport processes including electron conduction in electrodes, ion diffusion and migration in electrolytes, and reaction kinetics on the surface of carbon fibers.
The electrochemical kinetics of the redox couples are assumed to follow Marcus-Hush-Chidsey (MHC) kinetics. The MHC model is based on microscopic theory of electron transfer and reorganization energy, denoted as λ, that relates the interaction between the electrode surface, redox species, and the solvent. 52,56-58 For a general redox reaction given as R ↔ O + e − , the oxidation k ox and reduction k red reaction rate constants for MHC theory are given as: . [1] where η is overpotential, A is a pre-exponential factor, T is temperature and x is the electron energy relative to the Fermi level within the electrode. We use an analytical approximation for k ox/red (developed previously 53 ) expressed in terms of dimensionless overpotential (η * = η k B T /e ) and re-organization energy (λ * = λ k B T ): The net reaction current density i n for the redox reaction R ↔ O + e − can therefore be expressed as [3] where c R and c O are the concentrations of the reduced and oxidized species respectively. A positive overpotential drives the oxidation reaction and a negative overpotential drives the reduction reaction. The overpotential at the electrode/electrolyte interface, given as η = φ s − φ e − φ eq , is a measure of how high or low the potential of the electrode φ s is with respect to the solution potential φ e and the equilibrium potential φ eq of the active compounds. The equilibrium potential at any state of charge / discharge is determined here by the Nernst equation as: where E 0 is the equilibrium potential at 50% state of charge and n e is the number of electrons transferred in the redox reaction (here n e = 1).
The RFB system modeled here consists of 6 different species: the four redox active species (R1 z R1 , O1 z O1 , R2 z R2 , O2 z O2 ) and two supporting electrolyte ions (A + , B − ). The local concentrations of these six species are affected by redox reactions and transport processes. The mass conservation equation for each of these species takes the form: where c i is the local concentration in solution and − → N i is the superficial flux. The source term S i is related to the reaction current density and the available surface area per unit electrode volume a as S i = ±ai n /F for the reduced and oxidized species, while S i = 0 for the supporting electrolyte ions. The counter ions of each of the redox species are assumed to have the same properties as supporting electrolyte ions. Also, in the porous separator, no reactions occur (S i = 0). Equation 5 is solved together with the electroneutrality condition throughout the entire domain: The superficial flux − → N i arises due to gradient in pressure (bulk fluid flow), concentration (diffusion process), and solution potential (migration). The Nernst-Planck equation from dilute-solution theory accounts for these three effects with the superficial flux given as: The superficial velocity − → u of the electrolyte with viscosity μ, flowing through the porous media with permeability k h under the pressure field p is determined by Darcy's law − → u = −(k h /μ)∇ p and total mass conservation ∇ · − → u = 0. Unlike bulk solutions, the morphology of the porous medium (with porosity ε and tortuosity τ) affects the diffusion of species. The effective diffusion coefficient D i ef f is given as D i ef f = D 0 ε/τ, where D 0 is the corresponding bulk value. Note that the effective diffusion coefficient of all species in the electrode region is different from that of the separator region. The local pore-scale mass transfer coefficients scale with fluid velocity, 59 and at high Péclet number conditions (Pe = u d f /D, where d f is fiber diameter and u is the mean superficial velocity at the midplane between inlet and outlet) such mass transfer resistance can be neglected. In the present work Pe∼120, and we therefore assume facile pore-scale mass transfer. The electrode is assumed to be comprised of woven cylindrical carbon fibers. For such an electrode morphology transverse hydrodynamic dispersion has minimal impact on species transport, 60 and we therefore neglect the effects of dispersion in this work. The bulk ionic conductivity of the electrolyte (κ 0 ) can be derived by combining the ion transport and electroneutrality equations: RT . Therefore, this model naturally accounts for local changes in conductivity of the electrolyte with the state of charge/discharge and with supporting electrolyte composition. Lastly, electronic current conservation is implemented to couple the gradient in solid phase potential (φ s ) to the reaction current density.
The electrochemical and transport properties of all redox species and supporting electrolytes simulated are listed in Table II. The sep-arator is assumed to be electronically insulating. The RFB is subjected to galvanostatic cycling by applying a constant current density i app = 10m A/cm 2 on the high potential electrode current collector that theoretically corresponds to one hour charge and discharge cycle. The counter electrode current collector is grounded. The left and right sides of the reactor domain are modeled by imposing symmetric (i.e., mirrored) boundary conditions on both potential and concentration fields, so as to mimic a periodically repeating series of IDFF channels. Our previous study 49 shows that the flow rate in the reactor should be maintained above twenty times the stoichiometric flow rate in order to minimize mixing losses in the tanks and to achieve capacity utilization of about 85% of the theoretical maximum. Therefore, in this study we adopt the flow rate entering the reactor to be thirty times the stoichiometric flow rate with a tank-to-electrode volume ratio of twenty. This serves as the boundary condition for Darcy's equation to establish the 2D superficial velocity field in the electrodes.
The governing equations presented are non-linear and tightly coupled. Therefore, we use numerical methods to discretize the governing equations and to iteratively solve them. In this study, we use the finitevolume technique to discretize the equations on a 2D domain. The discretized equations are solved in MATLAB using a coupled Newton-Raphson iteration algorithm. [61][62][63] Convergence is achieved when the difference in concentration fields is less than 10 −8 mol/L and that of potential fields is less than 10 −5 V during consecutive iterations.
Verification is an important step in numerical modeling to check the correctness of implementation as computer programming language. We verify the present model by choosing appropriate conditions under which analytical solutions exist and by comparing the predictions of the 2D simulation with those analytical solutions. We take inspiration from Wolfrum and co-workers 64 who studied the redox active shuttling of the catechol-quinone redox couple in a nano gap electrode. The shuttling between the two electrodes is limited by diffusion processes and, under no-flow conditions, the diffusion limited current density in a channel of height h with n redox active molecules is given as i di f f = n Dze h 2 . 64 Under similar conditions of operation, the diffusion limited current density predicted by our simulation is in 98% agreement with the analytical expressions over three orders of magnitude variation in redox species concentrations. This comparison verifies our implementation of the species transport equation. Other verification steps related to implementation of mass balance, advection transport, and reaction kinetics (as described in our previous work 49 ) have been performed, and the results of these tests were well within tolerance limits.

Results and Discussion
The primary aim of this study is to explain the mechanisms governing the crossover rate of redox active species and their effect on capacity fade. To ensure charge neutrality during the cycling process, redox active species can transport alongside with supporting electrolyte ions by crossing the separator, leading to material loss. The crossover rate is affected by the concentration, diffusion coefficient, and charge number of redox species relative to the supporting electrolyte. Such processes can lead to interesting phenomenon such as redox active shuttling between the two electrodes. Although zerodimensional models have been proposed to understand capacity fade due to crossover, 51 an in depth understanding of the local processes that produce capacity fade is missing. Here, a 2D model which tracks all dissolved species at all instants in time allows us to quantify the degree of crossover and the spatially resolved reaction currents. With this model we map the performance and capacity fade of RFBs against variations in initial salt concentrations and redox active species diffusion coefficients for both RC and SS configurations. Such analysis can be used to engineer redox active molecules and separator materials.
As there are multiple species and reactions involved, we refer to the redox reactions which are intended for either the high potential electrode (HPE) or the low potential electrode (LPE) as primary reactions. The redox reactions the species experience upon crossing over the separator are referred to as "crossed" reactions (see Figure 2).
We introduce two dimensionless numbers of practical importance to elucidate our results. Molar salt ratio γ is defined as the ratio of the concentration of salt to the concentration of the redox species in the electrolyte γ = c salt /c redox . Throughout our analysis, the concentration of redox active species is fixed at 0. Cycling behavior and reaction current distribution.-We first study the cycling performance of an RFB in terms of capacity utilization and coulombic efficiency for ten charge/discharge cycles. For this analysis, a representative RFB with 0.5 M salt concentration (γ = 1) and with redox species diffusion coefficient ζ = 63 times smaller than that of salt ions, a ratio that is similar magnitude to the "rejection ratio" measured previously 21 for RAPs relative to inert salt ions. For the chosen values of D salt , current density, and salt concentration this simulation produces a salt Damköhler Da salt equal to 8, which we show later is small enough to facilitate "sufficient supporting electrolyte" behavior where capacity losses are dominated by the diffusive crossover of redox species (rather than by ohmic polarization). Here, we focus on this particular case to illustrate the influence of crossover on reaction current distribution and charge capacity. The cell voltage curve for this case is shown in Figure 3 along with the variation of the capacity utilization during discharge at every cycle. Here, the coulombic efficiency, which was previously correlated to the degree of crossover, 51 is 97% indicating that with every cycle there is certain loss of active species due to crossover. Because of this effect the discharge capacity utilization fades by 20% within the first ten cycles.
Reaction current distribution and crossover fluxes.-The dynamics of crossover and redox active species shuttling processes vary significantly within a single charge/discharge cycle step. With variation in state-of-charge, the processes of diffusion and migration can either favor or oppose each other in transferring redox active species across the separator, depending on the direction of current. When the primary redox couple of the HPE (R1 z R1 /O1 z O1 ) crosses over into the LPE, the large negative overpotential environment in the LPE drives the reduction reaction where almost all O1 z O1 reduces to R1 z R1 . This overpotential is negative throughout the charge/discharge cycle. Similarly, when the primary redox couple of the LPE (R2 z R2 /O2 z O2 ) crosses over to the HPE, the large positive overpotential environment drives the oxidation reaction converting all R2 z R2 to O2 z O2 .
The total pore-scale reaction current density i n (x, y, t) is a result of electrode-electrolyte interfacial redox reactions and is different from the galvanostatic applied current density at the current collectors. This reaction current density is calculated from the MHC kinetic model using Equations 2 and 3. Figure 4 shows the development of cell voltage with state-of-charge during the second charge/discharge cycle. Snapshots of the primary and crossed reaction current densities are also shown at two points: near the end of the charge step (A) and early in the discharge step (B). The corresponding cell-voltage is shown as a loop varying with state-of-charge to delineate the capacity utilization during both charge and discharge steps (80% in this case) and the polarization measured by the averaged difference of the voltages during charge and discharge. The cell voltage plot in Fig. 4 also shows that the state-of-charge consumed during the discharge process falls short to complete the loop, indicating coulombic inefficiency due to crossover.
The pore-scale reaction current density is positive during the oxidation process and negative during reduction. While charging the RFB, the redox couple in the HPE undergoes oxidation generating O1 z O1 while the redox couple in the LPE undergoes reduction generating R2 z R2 . The oxidation reaction in the HPE is reflected by positive values of i n, primar y (x, y, t) in the HPE and reduction reaction in the LPE are associated with negative values of i n, primar y (x, y, t) in the LPE, as shown in panel A1 of Fig. 4. Since a large flow rate is chosen, the reaction currents are well distributed within the electrode volume, effectively using the available surface area for conducting reactions. This observation of uniform reaction distribution within the electrodes at high flow rates is consistent with the our previous findings 49 and operating the RFB at lower flow rates will bias reactions near the inlets increasing polarization losses. Similarly, during the discharge process (Fig. 4, B1), the primary reaction currents are negative (reduction) in the HPE and positive (oxidation) in the LPE. The primary reaction current distribution i n, primar y (x, y, t) shown at points A and B is representative of the entire charge/discharge step.
The crossed reaction current density i n,cr oss (x, y, t) is affected by local overpotential and the availability of redox active species in the counter electrode (degree of crossover). As explained above, the redox couple R1 z R1 /O1 z O1 , which is primary to the HPE, undergoes reduction upon crossing over to the LPE. Therefore when R1 z R1 crosses over, it accumulates in the LPE without undergoing any redox re-action. On the contrary when O1 z O1 crosses over into the LPE, it undergoes fast reduction to R1 z R1 due to large negative overpotentials irrespective of the RFB's state-of-charge. The degree of crossover of each species, characterized by its diffusion and migration rates, varies significantly with state-of-charge. The HPE is rich in the oxidized species O1 z O1 during the end of charging or early in the discharge process, favoring diffusion of O1 z O1 from the HPE to the LPE immediately reducing to R1 z R1 in the LPE. Therefore, as shown in Fig. 4 panels A2 and B2, i n,cr oss (x, y, t) of the crossed couple R1 z R1 /O1 z O1 in the LPE becomes negative (reduction) and dominant near the separator region. At the same instant in the charge/discharge cycle, the LPE electrode is plentiful in R2 z R2 , which, upon crossing over to the HPE, undergoes oxidation to O2 z O2 . Thus, i n,cr oss (x, y, t) in both electrodes is significant in magnitude only during the second half of the charging process and during the first half of discharging process. Crossover magnitudes are relatively small otherwise, a mechanistic explanation of which is given later by comparing diffusion and migration contributions. Also, note that the values of i n,cr oss (x, y, t) are at least an order of magnitude lower than i n, primar y (x, y, t) and the sign of i n,cr oss (x, y, t) is maintained consistently in the HPE and in the LPE (i.e., oxidation and reduction respectively) regardless of the charge/discharge cycling phase of the RFB.
To qualitatively estimate the distribution of the crossed reaction current density i n,cr oss (x, y, t), the kinetically limited current I lim kinetic in the electrode is compared to the species ionic crossover current passing through the separator I crossover . At high overpotentials (|η| >> k B T /e) MHC theory predicts a limiting kinetic rate constant: For the present simulations we estimate I lim kinetic /I crossover ∼10 4 , implying that active species should react instaneously once they have crossed through the separtor. This expectation is confirmed by our simulations where reaction hotspots are observed adjacent to the separator, as shown in panels A2 and B2 of Fig. 4.
Diffusion and migration mechanisms govern crossover processes. The relative contribution of these two processes toward crossover varies significantly with active-species charge number, its diffusion coefficient, and the transient concentration fields of all the redox active and salt species involved. Although zero-dimensional, timeindependent heuristic models haven been proposed to quantify the degree and effects of crossover, 51,68 a comprehensive theory resolving local, time-dependent fields is needed to design RFBs and to engineer their electrolytes. Using the two-dimensional, transient model presented here, we track the time varying diffusion and migration rates of selected redox species, and we further elucidate the findings about crossed reaction current density in Fig. 4. The variation of the concentration of O1 z O1 in the HPE electrode is shown for a single charge/discharge cycle in Fig. 5a along with cell voltage. The diffusion and migration crossover fluxes are shown in Fig. 5b in terms of crossover current fraction along with thumbnails indicating the direction of crossover during charge/discharge. The crossover current fraction is defined as the ratio of the ionic current due to a particular species passing through the separator and the total applied galvanostatic current at the current collector. The crossover current fraction is positive when species crossover from HPE to LPE and is negative otherwise. The ionic current of O1 z O1 is calculated by multiplying each component of the molar flux (Eq. 7) through the porous separator with the oxidation state of O1 z O1 (here z O1 = +3). During the charging process the concentration of O1 z O1 increases in the HPE along with solution potential and cell voltage. Therefore, during the charging process, the gradient of O1 z O1 concentration and solution potential across the separator continuously increases, leading to increased diffusion and migration fluxes with time. The migration fluxes have a squared dependence on the oxidation state of the species involved. As can be seen from Fig. 5b, migration dominates diffusive crossover fluxes for a high oxidation state redox molecule (z O1 = +3) during the charging process. During the discharge process, the polarity of electrolyte potential difference across the separator reverses, and the electric field induced migration is thereafter directed from the LPE to the HPE, opposing the direction of diffusive fluxes. Therefore, the net crossover of O1 z O1 is insignificant and is reflected by the decreasing magnitude of the crossed reaction current density during discharge in the LPE (compare LPE in panels A2 and B2 of Fig. 4). On the contrary, the crossover of the redox molecule R2 z R2 (z R2 = +1) from the LPE to the HPE is sizable during the discharge process. Figure 5c shows the variation of R2 z R2 concentration in the LPE, and Fig. 5d shows the transient variation of the crossover flux components. During the charge process, the diffusive and migration fluxes oppose each other, resulting in net crossover that is practically insignificant. With the change in polarization of electrolyte potential during discharge, the diffusion and migration mechanisms ensure a net crossover of R2 z R2 from the LPE to the HPE. Therefore, we observe small crossed reaction current density in the HPE during charge (Fig. 4, panel A2), and it is more significant during discharge (Fig. 4, panel B2). In summary, the redox species with the largest charge number in the HPE experiences the maximum degree of crossover, and this process is dominated by migration. Therefore, it is crucial to engineer such high oxidation state redox active molecules to have low diffusion coefficients.  capacity fade for ten charge/discharge cycles are the metrics used here to characterize the performance and cycle life of RFBs. The SS configuration requires a threshold salt concentration for conduction through the separator to maintain charge neutrality in either electrode. In SS configuration, every mole of active species undergoing redox reactions requires at least half a mole of salt (which splits into two ions) to maintain charge neutrality. The RC configuration has no such restrictions and therefore could be a potential solution in designing salt free RFBs. 34 Here we show that, although the RC configuration allows operating RFBs at low salt concentration levels, the capacity utilization at such extremes is very low and is impractical. Both RC and SS configurations require a certain minimum amount of salt to minimize electric field and thereby obtain sufficient discharge capacity utilization. In both these configurations the performance of RFBs is mapped using two non-dimensional variables: (1) the molar-salt ratio defined as the ratio of salt concentration to redox-active species concentration in electrolyte (γ = c salt /c redox ) and (2) the salt-diffusion ratio which is the ratio of the diffusion coefficient of salt ions to redox active species (ζ = D salt /D redox ). The concentration of redox active molecules is fixed to c redox = 0.5M. Larger values of γ means that more salt is dissolved in the electrolyte. In addition, the diffusion coefficient of salt ions is set to be representative of fluoride salt derivatives D salt = 5 × 10 −10 m 2 /s and D redox is varied to parameterize redox active molecule design. A fast diffusing redox active molecule (small value of ζ) has a diffusion coefficient comparable to that of salt ions and therefore has more crossover. The variation of ionic conductivity in this electrolyte design space of concentration (γ) and diffusion coefficient (ζ) is given as κ 0 = z i 2 c i D i F 2 /RT with contributions coming from the four redox-active species and the two salt ions. Electrolytes with large γ and low ζ are most conductive and possess lowest ohmic polarization. Despite this advantage such electrolytes necessarily experience high crossover rates. On the other hand designing redox molecules with high ζ can reduce crossover, but ohmic polarization increases due to decreased ionic conductivity.

Effects of supporting electrolyte and diffusion coefficient.-In
A total of forty-two and forty-eight cases spanning the design space of γ (0 → 2) and ζ (10 → 1000) were simulated for both RC and SS configurations, respectively. A contour map of discharge utilization and capacity fade for RC configuration is shown in Fig. 6. The dashed lines correspond to the difference in discharge capacity for the first and tenth cycles expressed as a percentage of theoretical capacity. We observe that, although salt-free RFBs (γ → 0) are attractive for the development of economical RFBs, their capacity utilization is as low as 20%. This low discharge capacity utilization at small salt concentrations is not due to a lack of redox active supply to the reactor but is a result of the deficiency of positive salt ions in the HPE during charging. The concentration and potential gradients across the separator drive the movement of salt ions from one electrode to another. To maintain the balance of charge at finite applied current density, the transport of both positive and negative salt ions across the separator is needed. When there is deficiency of one of the ions, the solution phase potential difference across the separator increases to force the migration transport of the available salt ions to ensure electroneutrality in both electrodes. In other words, the cell voltage increases to cause enough solution potential gradient across the separator for the negative salt ions to migrate from the LPE to the HPE and subsequently to balance out the applied current in the absence of positive salt ions. Therefore, cutoff voltage limits are prematurely reached. Note that the limiting mechanism in such low salt operating regimes is the availability of both positive and negative ions for conduction across the separator and is not necessarily a result of the bulk ionic conductivity of either electrolyte. The predictions of discharge utilization capacity for low salt RFBs with the RC configuration is about 20-30% from Fig. 6. This is consistent with experimental findings of proof-of-concept saltfree RFB in Ref. 35. Figure 6 shows that the salt concentration should be more than 60% of the redox active concentration to obtain at least 80% discharge utilization. Although high diffusion coefficients for redox active molecules may be attractive from the perspective of ionic conductivity, the capacity fade due to crossover will be enormous. The diffusion coefficient must be three orders of magnitude smaller than the salt diffusion coefficient to obtain a capacity retention of 95% for 10 cycles. For values of γ > 1, capacity fade becomes independent of salt concentration. We later identify this region as the excess supporting electrolyte domain commonly used in dilute solution theories. 67 The SS configuration theoretically requires salt concentration to be greater than half of the redox active concentration 34 (γ ≥ 0.5). However, such theoretical analysis assumes that capacity utilization is 100%, i.e., for every mole of redox actives undergoing reaction, half a mole of salt is needed to preserve charge neutrality by splitting into two ions. If the capacity utilization is less than 100%, the SS configured RFB can be operated with γ < 0.5. Also, the presence of salt is essential to ensure sufficient ionic conductivity, as some of the redox species in the SS RFB are neutral. Figure 7 shows the performance of SS configured RFBs as a function of molar salt ratio γ and salt diffusion ratio ζ. An appreciable discharge utilization of 80% can be achieved when γ > 0.5. Also, capacity fade rate becomes independent of concentration for γ > 0.6, consistent with excess supporting electrolyte behavior. Compared with the RC configuration (Fig. 6), the SS configuration (Fig. 7) has better capacity retention. The use of smaller charge number redox materials in the SS configuration decreases migration crossover fluxes, benefitting capacity retention. Contrary to the hypothesis that the RC configuration could be the potential pathway for developing low salt RFBs, Figs. 6 and 7 show that the SS configuration has similar performance to the RC configuration. Both configurations require minimum salt concentration levels (γ > 0.5) for profitable capacity utilization and the SS configuration has an advantage over RC in capacity retention by smaller migration crossover fluxes. Also, the SS configuration achieves the excess supporting electrolyte regime sooner than the RC configuration, reducing its sensitivity of performance to changes in initial salt concentration. That said, the results and conclusions presented thus far are for RFBs using a porous non-selective separator. Crossover and transport mechanisms will change with the use of selective membranes.  Figure 8a maps the variation of discharge capacity utilization in the first cycle and of capacity fade after the first ten cycles using z Da salt /Da redox and Da redox . Here, z is the arithmetic average charge number among all redox species involved with z = 2 for the present RC RFB simulations. On this space of z Da salt /Da redox and Da redox three regimes are labeled: (1) redox-shuttle limited, (2) ohmic polarization dominated, and (3) sufficient supporting electrolyte. In the redox shuttle limit regime, the transport rate of redox molecules is comparable to that of salt and therefore the redox molecules shuttle between the two electrodes constantly, leading to poor capacity retention. In the ohmic polarized regime (Da salt > 12) polarization limits capacity utilization due to its low concentration of supporting electrolyte. The most favorable regime for the efficient and long-term operation of RFBs with non-selective separators is the sufficient supporting electrolyte region, where discharge capacity utilization exceeds 80%. To achieve such conditions requires that z Da salt /Da redox < 0.3 and Da salt < 12, such that neither redox shuttling nor ohmic polarization limits cycling. From Figure 8a, operating RFBs in the sufficient supporting electrolyte region and at large values of Da redox will yield greatest capacity retention. Da redox can be increased either by increasing applied current density or by engineering redox species to have low diffusion coefficients.

Non-dimensional analysis using
In the sufficient supporting electrolyte regime, the ratio z Da salt /Da redox translates to a monotonic function of the transference number of redox active species t i , given as z Da salt /Da redox = t i /(1 − t i ). In the context of the molecular engineering of RFB elec- trolytes low redox-species transference number is cruicial to maintain high capacity utilization over many cycles. In the sufficient supporting electrolyte region capacity fade is weakly dependent on Da salt and thus depends primarily on Da redox . We use the six simulated data points in the sufficient supporting electrolyte regime that have smallest redox . This expression can be used in the engineering of redox active molecules and to determine operating conditions for a given RFB electrolyte that are necessary to achieve a target capacity retention level. For example, Da redox must be greater than 1.7×10 5 to obtain a capacity retention of 99.9% after ten cycles. Comparing Figs. 6 and 7, the capacity utilization and capacity fade for RC and SS configurations are qualitatively and quantitatively similar. Therefore, the non-dimensional analysis presented through Fig. 8 holds true for both RC and SS configurations.

Conclusions
In the context of engineering cost effective RFBs we develop a mechanistic understanding of the crossover process and the role of supporting electrolyte in RFBs using porous, non-selective separators. To do this we present an RFB model based on porous electrode theory with Marcus-Hush-Chidsey (MHC) kinetics and Nernst-Planck fluxes. The model predicts the spatial and temporal variation of two sets of redox couples (R1 z R1 /O1 z O1 &R2 z R2 /O2 z O2 ), two salt ions (A + , B − ), solid phase potential, and electrolyte potential. The redox couples could have a rocking-chair (RC) or salt splitting (SS) configuration, the former involving similarly charged ions and neutral species in the latter.
The pore-scale reaction current density distributions simulated here reveal that the reaction current densities resulting from crossover processes are at least an order of magnitude lower than the primary redox reactions and, thus, vary in time significantly during charge/discharge cycling. The diffusion and migration mechanisms that govern crossover (according to Nernst-Planck equation) either enhance or oppose each other in the crossover process depending upon (1) state-of-charge, (2) charge/discharge cycling, and (3) the properties of the redox species. We find that the capacity loss due to crossover is dictated by the highest oxidation state redox species in the high potential electrode and that rate is dominated by migration especially in the second half of the charging process where the cell potential is large. We further extend the study by mapping the discharge utilization and capacity fade of RFBs by varying the amounts of supporting electrolyte and the diffusion coefficient of the redox active molecules. Contrary to expectations of the RC configuration being a possible solution for cost-effective low salt RFBs, we find that both configurations have very low discharge utilization in low salt regimes. The deficiency of positive salt ions in the high potential electrode forces the negative salt ions alone to maintain the electroneutrality by transferring across the separator. Maintaining electroneutrality with negative salt ions alone can only be done when the potential gradient across the membrane is increased, resulting in prematurely reaching cell voltage limits of cycling without actually utilizing active molecules for energy storage. A threshold amount of salt (at least 50% concentration of redox actives in either electrolyte) is needed to obtain 80% of discharge utilization capacity. From the perspective of engineering redox active molecules, the diffusion coefficient of these molecules should be three orders of magnitude lower to secure more than 95% capacity retention for ten cycles at 10m A/cm 2 . In the scope of designing and operating RFBs, the variation of the discharge capacity utilization and the capacity fade of RFBs is presented on a 2D space containing the non-dimensional Damköhler numbers of the salt and the redox active species. Three different regimes dominated by ohmic-polarization, redox active shuttling and sufficient supporting electrolyte are identified in this 2D space. A correlation is established between capacity fade and redox Damköhler number in the sufficient supporting electrolyte regime where capacity fade becomes independent of salt Damköhler number. Designing and operating RFBs in the sufficient supporting electrolyte regime with large redox Damköhler numbers ensures high capacity utilization and capacity retention.
The transient porous electrode model with Marcus-Hush-Chidsey kinetics and the simultaneous tracking of six different ions is the first of its kind in non-aqueous redox flow battery literature. The redox active species that undergo crossover in 3V RFBs are subjected to high overpotentials. The Butler Volmer kinetic model exaggerates the reaction rates at such high overpotentials as opposed to MHC kinetic theory which is based on microscopic properties of the redox active molecules and their interaction with the electrolyte. Representative properties of the redox species and supporting electrolyte ions are chosen to delineate mechanistic understanding of crossover processes and the role of supporting electrolyte. Although specific properties of the species involved may vary from a practical standpoint, the physics that govern crossover presented here are still applicable in designing RFBs. However, the model in its current state neglects hydrodynamic dispersion and pore-scale mass transport resistances and assumes that the counter ions of each redox active species have transport properties similar to the supporting electrolyte ions. The present study motivates the optimization of salt concentration levels, non-selective separator properties, and the engineering of redox active molecules for cost-effective RFBs with long operational life.

Acknowledgments
This work was partially supported by the Joint Center for Energy Storage Research, an Energy Innovation Hub funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences. Additional support was provided by the Department of Mechanical Science and Engineering at the University of Illinois at Urbana-Champaign.