The viability of next generation lithium and beyond-lithium battery technologies hinges on the development of electrolytes with improved performance. Comparing electrolytes is not straightforward as multiple electrochemical parameters affect the performance of an electrolyte. Additional complications arise due to the formation of concentration gradients in response to dc potentials. We propose a modified version of Ohm’s law to analyze current through binary electrolytes driven by a small dc potential. We show that the proportionality constant in Ohm’s law is given by the product of the ionic conductivity, *κ*, and the ratio of currents in the presence (*i*_{ss}) and absence ($i\Omega $) of concentration gradients, *ρ*_{+}. The importance of *ρ*_{+} was recognized by Evans *et al.* [Polymer **28**, 2324 (1987)]. The product *κρ*_{+} is used to rank order a collection of electrolytes. Ideally, both *κ* and *ρ*_{+} should be maximized, but we observe a trade-off between these two parameters, resulting in an upper bound. This trade-off is analogous to the famous Robeson upper bound for permeability and selectivity in gas separation membranes. Designing polymer electrolytes that overcome this trade-off is an ambitious but worthwhile goal.

## I. INTRODUCTION

In a battery, the passage of ionic current between the cathode and anode is enabled by the electrolyte. The dependence of the current on the potential drop between the electrodes is at the core of battery design and engineering.^{1,2} The kind of device that can be powered by a battery is limited by the maximum current that can be passed safely through the electrolyte.

The starting point for understanding the relationship between the potential drop and current is Ohm’s law. For a simple conductor with one charge carrier, such as a copper wire [Fig. 1(a)], the current density, *i*, is proportional to the potential drop per unit length, Δ*V*/*L*, and Ohm’s law can be written as

where *σ* is the electronic conductivity of the material. All materials are electrically neutral and have at least two charge carriers; the one charge carrier approximation is valid because the compensating copper cations are essentially immobile. Current density vs Δ*V*/*L* for copper is presented in Fig. 1(b), where the slope, $m$, is given by 5.8 × 10^{5} S cm^{−1}.^{3} In this case, $m$ = *σ*. For a copper wire, carrier concentration gradients do not develop as the copper cations are stationary and charge neutrality is maintained.

An example of a rechargeable battery is shown schematically in Fig. 1(c). It consists of a lithium metal anode and a lithium iron phosphate, LiFePO_{4,} cathode separated by an EC:DEC/LiPF_{6} electrolyte in a porous separator. During discharge, the passage of ionic current through the electrolyte from the anode to the cathode is driven by an overpotential, *η*, which is the equilibrium potential of the cell minus the operating voltage, *U*^{0} − *V*.^{1} When an overpotential is present, concentration gradients develop across the electrolyte because both cations (in this case, Li^{+}) and anions ($PF6\u2212$) are mobile in the system. Under a constant overpotential, this would result in a time-dependent current density until the concentration gradient reaches a steady-state. Only Li^{+} ions are transported across electrode/electrolyte interfaces; this also affects the nature of the gradients. In Fig. 1(d), we plot the steady-state current density, *i*_{ss}, as a function of the overpotential per unit length, *η*/*L*, for the cell depicted in Fig. 1(c).^{4} It appears that the relationship between *i*_{ss} and *η*/*L* is approximately linear, similar to the copper wire. However, the slope, $m$ = 2.5 × 10^{−5} S cm^{−1}, is not equal to the conductivity of the electrolyte. It reflects numerous processes that include charge transfer between the electrodes and the electrolyte, diffusion of lithium in the cathode, and diffusion and migration of ions in the electrolyte. Thus, the relationship between *i*_{ss} and *η*/*L* in Fig. 1(d), although it appears linear, is not a manifestation of Ohm’s law.

In Fig. 1(e), a schematic for a symmetric cell consisting of an electrolyte sandwiched between two identical nonblocking electrodes is presented. In this perspective, we focus on symmetric cells comprising either lithium or sodium foil electrodes and electrolytes containing a lithium or sodium salt, respectively. This cell, popularized by the pioneering work of Evans, Vincent, and Bruce, and others,^{6–8} is similar to that shown in Fig. 1(c) with one crucial difference: *U*^{0} = 0 V. This cell enables a fair comparison of the ion transport properties of different electrolytes: the symmetry of the cell allows electrode effects to be deconvoluted from the properties of the electrolyte. In Fig. 1(f), we plot *i*_{ss} as a function of the potential drop across the electrolyte, ΔΦ/*L*, for a cell with lithium foil electrodes and an electrolyte comprising poly(ethylene oxide) (PEO) and lithium bis(trifluoromethanesulfonyl)imide (LiTFSI) salt.^{5} Here, the slope $m$ = 9.9 × 10^{−5} S cm^{−1} is not equal to the ionic conductivity of the electrolyte. However, unlike in a full battery, $m$ is related to the properties of the electrolyte alone. In our effort to design high performance electrolytes, it is the slope in Fig. 1(f) which we wish to maximize. Many publications, however, disregard this. It is fairly common, these days, to invent a new electrolyte, measure the ionic conductivity, and declare victory if it is greater than that of a baseline electrolyte.

The purpose of this perspective is to analyze symmetric cell data obtained from different electrolytes. Evans, Bruce, and Vincent^{6,9} and Watanabe *et al.*^{10} modeled symmetric cells containing dilute and ideal electrolytic solutions. In later studies, Newman and co-workers^{1,11} considered symmetric cells containing concentrated electrolytic solutions and developed the relationships between $m$ and intrinsic transport and thermodynamic properties of the electrolyte. This perspective is focused on small applied potentials wherein the concentration dependence of the relevant electrolyte properties can be neglected. Based on the work in Refs. 5–11, we develop a framework for measuring the Ohm’s law coefficient which allows us to produce a rank ordered list of electrolytes based on their ability to maximize the flux of lithium or sodium cations. We conclude by discussing the limitations of our approach as, ultimately, the rank ordering of electrolytes needs to be reassessed in the presence of significant concentration gradients for practical devices.

## II. THEORY

Electrolytes of interest comprise a salt $(Mz+)\nu +(Xz\u2212)\nu \u2212$ dissolved in a matrix. Characterization of ion transport typically begins with measurement of the ionic conductivity, *κ*, by ac impedance spectroscopy. A powerful feature of ac impedance spectroscopy is that *κ* is measured without introducing significant concentration gradients. When a dc potential, ΔΦ, is applied across an electrolyte of dimension *L* in a symmetric cell [Fig. 1(e)], there are, by definition, no concentration gradients at the first instant of polarization (*t* = 0^{+}). The initial current density, *i*_{0}, at *t* = 0^{+} is given by

With time, i.e., at *t* > 0, salt concentration gradients develop in the cell and eventually the gradient becomes time-invariant. The measured current density decreases with time as these concentration gradients develop and reaches a steady value at long times. We refer to the current obtained at long times as *i*_{ss}.

In the limit of small applied potentials, an expression for *i*_{ss} can be derived based on concentrated solution theory,^{11,12}

where Ne is a dimensionless parameter that we call the Newman number. Ne is given by

where *R* is the gas constant, *T* is the temperature, *F* is the Faraday constant, *D* is the restricted diffusion coefficient of the salt, *c* is the salt concentration, $t+0$ is the transference number of the cation with respect to the velocity of the solvent, *γ*_{±} is the mean molal activity coefficient of the electrolyte, and *m* is the salt molality. The parameter *a* is related to the stoichiometry of the salt,

where *ν* is the total number of cations and anions to which the salt dissociates, *ν*_{+} is the total number of cations to which the salt dissociates, and *z*_{+} is the charge number of the cation. (For a salt comprising univalent ions, *a* = 2.) Equations (3) and (4) are based on Newman’s concentrated solution theory wherein electrolytes are characterized by three transport parameters, *κ*, *D*, and $t+0$, and a thermodynamic factor, $Tf=1+d\u2009ln\u2009\gamma \xb1d\u2009ln\u2009m$. This theory builds on the work of Onsager^{13} who recognized that ion transport in binary electrolytes is governed by three Stefan-Maxwell diffusion coefficients, $D0\u2212$, $D0+$, and $D+\u2212$. Relationships between *κ*, *D*, and $t+0$ and the Stefan-Maxwell diffusion coefficients are given in Ref. 11.

While all four parameters (*κ*, *D*, $t+0$, and *T*_{f}) dictate the time-dependent current at a given applied potential, explicit knowledge of all these parameters is not required to determine *i*_{ss} or Ne. In fact, Ne can be determined in a single experiment by measuring *i*_{0} and *i*_{ss} at constant dc polarization, ΔΦ, over the electrolyte,

Bruce and Vincent pioneered the measurement of *i*_{ss}/*i*_{0}.^{6,9}

where

and *c*_{0} is the solvent concentration. Equations (7) and (8) were first derived by Balsara and Newman.^{11} Only in the limit *c* → 0, *β* → 0 does

a result presented by Bruce and Vincent.^{9} Determining the range of concentration over which *β* is small enough such that Eq. (9) is valid requires knowledge of the Stefan-Maxwell diffusion coefficients. For dilute 0.01M aqueous potassium chloride (Fig. 14.1 of Ref. 1), $D+\u2212$ = 1.1 × 10^{−7} cm^{2} s^{−1}, $D0+$ = 1.9 × 10^{−5} cm^{2} s^{−1}, *c*_{0} = 56 mol l^{−1}, *β* = 0.031, and Eq. (9) is a good approximation. However, most practical electrolytes are not dilute. For a 1M aqueous potassium chloride solution, $D+\u2212$ = 1.9 × 10^{−6} cm^{2} s^{−1}, $D0+$ = 2.0 × 10^{−5}, *c*_{0} = 53.6 mol l^{−1}, and *β* = 0.20. For 2.6M PEO/LiTFSI (Figs. 3 and 4 of Ref. 14), $D+\u2212$ = 4.0 × 10^{−9} cm^{2} s^{−1}, $D0+$ = 1.1 × 10^{−8} cm^{2} s^{−1}, *c*_{0} = 16 mol l^{−1}, and *β* = 0.44. Equation (9) is not a good approximation for either 1M KCl or 2.6M PEO/LiTFSI.

We thus define the current fraction, *ρ*_{+}, which can be rewritten on the basis of Eq. (6) as

The current fraction is an intrinsic property of an electrolyte, irrespective of whether it is dilute or concentrated. The transference number, $t+0$, is defined as the fraction of current carried by the cation in a solution of uniform salt concentration and is only approximated by *ρ*_{+} when *β* is small. For this reason, we prefer to use *ρ*_{+} to refer to the current fraction, $iss/i0$, rather than using $t+0$ or “the transference number” as is commonly done in the literature. This point was alluded to by Bruce and Gray in 1995, who referred to this current fraction as “the limiting current fraction.”^{15}

The discussion thus far ignores the resistance of the electrode/electrolyte interface. In practice, when a dc voltage, Δ*V*, is supplied to a symmetric cell, the potential drop across the electrolyte, ΔΦ, will be reduced by an amount equal to the product of the interfacial resistance and the current. Assuming other sources of ohmic loss are negligible,

where *R*_{i} is the interfacial impedance that is readily measured by ac impedance spectroscopy, *A* is the electrochemically active surface area of the electrode, and *i* is the current density through the symmetric cell. We can combine Eqs. (2), (3), (10), and (11) to obtain a useful expression,

where *i*_{ss} and *i*_{0} refer to the steady-state and initial current density through a symmetric cell as in Eq. (11). The importance of corrections for interfacial resistance was recognized by Evans, Bruce, and Vincent^{6} and Watanabe *et al.*^{10} We use the term *ρ*_{+,0} in Eq. (12) to clarify that this current fraction is based on a measured value of *i*_{0}, which we discuss next.

In order to apply Eqs. (10)–(12), the value of *i*_{0} must be measured. A practical approach is to take the first data point measured after the potential is applied. However, this method is inherently problematic because the current is a strong function of time in the first instant of polarization. An example of such a measurement is shown in Fig. 2. A small potential, Δ*V* = 8.9 mV, was applied across a lithium symmetric cell (*A* = 0.079 cm^{2} and *L* = 0.050 cm) containing a 35 kg mol^{−1} PEO/LiTFSI electrolyte with salt concentration *r =* 0.010, where *r* is defined as the molar ratio of lithium ions to ethylene oxide moieties. A sampling rate of 1 ms^{−1} was used for the first few seconds. Figure 2 presents the current response over the entire time window (400 min) required to reach a steady-state, and the inset highlights the first 10 ms. Over the first 10 ms, the current is approximately constant with time. Thus, we have confidence that the current density we measure, *i*_{0} = 0.051 mA cm^{−2}, truly captures the initial current.

An alternative that has been proposed^{6,16–23} is to calculate *i*_{0} by combining Eqs. (2) and (11). In this case,

We can rearrange Eq. (13) to solve for *i*_{0}. We refer to this calculated current density as $i\Omega $ because it is a statement of Ohm’s law [Eq. (1)],

For the electrolyte and cell used in Fig. 2, *κ* = 0.33 mS cm^{−1} and *R*_{i} = 495 Ω, yielding $i\Omega $ = 0.047 mA cm^{−2} (shown as a red dashed line in Fig. 2). We see reasonable agreement between *i*_{0} and $i\Omega $ from this experiment. The advantage of using $i\Omega $ instead of *i*_{0} is that it is based on parameters that are easily measured (Δ*V*, *L*, *R*_{i}, *κ*, and *A*). A further rationale for this is discussed in Sec. IV. For the purposes of this paper, we define *ρ*_{+} as

## III. DATA

To select the systems used in this perspective, we studied the 472 papers which cited Evans, Vincent, and Bruce’s 1987 paper titled “Electrochemical measurement of transference numbers in polymer electrolytes”^{6} since 2010. Only a small fraction of these papers reported all parameters necessary for our analysis. Those parameters are listed in Table I.

Parameter . | Symbol . | Description . |
---|---|---|

Ionic conductivity, blocking | κ_{b} | Ionic conductivity of the electrolyte measured by ac impedance using blocking electrodes (e.g., stainless steel) |

Ionic conductivity, nonblocking | κ_{nb} | Ionic conductivity measured by ac impedance using nonblocking electrodes (e.g., lithium metal) |

Applied voltage | ΔV | Constant voltage applied by the potentiostat in order to elicit a steady-state current density |

Current density, initial | i_{0} | Initial current density measured after polarization at ∆V |

Current density, steady-state | i_{ss} | Current density measured at steady-state in response to ∆V |

Interfacial resistance, initial | R_{i,0} | Interfacial resistance measured by ac impedance spectroscopy just before ∆V is applied |

Interfacial resistance, steady-state | R_{i,ss} | Interfacial resistance measured by ac impedance spectroscopy after the steady-state current is reached |

Bulk resistance | R_{b} | Bulk resistance measured in the cell during the steady-state current experiment |

Cell thickness | L | Distance between electrodes; electrolyte thickness |

Interfacial area | A | Nominal electrode area in contact with the electrolyte |

Parameter . | Symbol . | Description . |
---|---|---|

Ionic conductivity, blocking | κ_{b} | Ionic conductivity of the electrolyte measured by ac impedance using blocking electrodes (e.g., stainless steel) |

Ionic conductivity, nonblocking | κ_{nb} | Ionic conductivity measured by ac impedance using nonblocking electrodes (e.g., lithium metal) |

Applied voltage | ΔV | Constant voltage applied by the potentiostat in order to elicit a steady-state current density |

Current density, initial | i_{0} | Initial current density measured after polarization at ∆V |

Current density, steady-state | i_{ss} | Current density measured at steady-state in response to ∆V |

Interfacial resistance, initial | R_{i,0} | Interfacial resistance measured by ac impedance spectroscopy just before ∆V is applied |

Interfacial resistance, steady-state | R_{i,ss} | Interfacial resistance measured by ac impedance spectroscopy after the steady-state current is reached |

Bulk resistance | R_{b} | Bulk resistance measured in the cell during the steady-state current experiment |

Cell thickness | L | Distance between electrodes; electrolyte thickness |

Interfacial area | A | Nominal electrode area in contact with the electrolyte |

The four categories of electrolytes covered in this study are pictured in Fig. 3: homopolymer electrolytes (HPEs) containing a lithium salt and no solvent, gel polymer electrolytes (GPEs) containing a cross-linked polymer mixed with a solvent and a lithium salt, polymer electrolytes containing a sodium salt (NaPE), and multicomponent polymer electrolytes (MCPEs) containing a polymer mixed with a salt and at least one additional component. The additional component in the MCPEs may be another polymer (blended or covalently bonded), an ionic liquid, or a ceramic particle. All of the electrolytes were designed to transport lithium ions except for those placed in the sodium electrolyte category. A long-form description of each electrolyte, its category, and its reference is provided in Table II.

Electrolyte description . | Category . | Reference . |
---|---|---|

Polyethylene oxide with lithium bis(trifluoromethanesulfonyl)imide salt (PEO/LiTFSI) with 0.017 moles of LiTFSI per mole of ether oxygen (r = 0.017) | HPE | 21 |

PEO/LiTFSI with r = 0.08 | HPE | 17 |

Poly(diethylene oxid-alt-oxymethylene) with LiTFSI (P(2EO-MO)/LiTFSI) with 0.04 moles of LiTFSI per mole of oxygen (r = 0.04) | HPE | 18 |

P(2EO-MO)/LiTFSI with r = 0.08 | HPE | 18 |

P(2EO-MO)/LiTFSI with r = 0.14 | HPE | 18 |

Perfluoroether containing 8 carbon atoms with dimethyl carbonate end groups and bis(fluorosulfonyl)imide lithium salt (C8-DMC/LiFSI) with 5.84 wt. % LiFSI | HPE | 22 |

C8-DMC/LiFSI with 19.9 wt. % LiFSI | HPE | 22 |

Perfluoropolyether with hydroxyl end groups containing 10 fluoro-ether oxygens (PFPE_{D10}-Diol) and 9.1 wt. % LiTFSI | HPE | 23 |

Perfluoropolyether with dimethyl carbonate end groups containing 10 fluoro-ether oxygens (PFPE_{D10}-DMC) and 9.1 wt. % LiTFSI | HPE | 23 |

PEO/LiTFSI gel mixed with tetraethylene glycol dimethyl ether (TEGDME) | GPE | 24 |

80 wt. % methoxy-PEO-methacrylate and 20 wt. % hexadecal-PEO-methacrylate copolymerized into a matrix (PMH20) with lithium perchlorate (LiClO_{4}) salt | GPE | 25 |

Cross-linked PEO plasticized by TEGDME with LiTFSI | GPE | 26 |

PEO/LiTFSI blended with poly[(trifluoromethyl)sulfonyl acrylamide] (PA-LiTFSI) | MCPE | 27 |

Li_{7}La_{3}Zr_{2}O_{12} (LLZO) dispersed in poly(vinylidene fluoride-hexa-fluoropropylene) (PVDF-HFP) | MCPE | 28 |

Polyhedral oligomeric silsesquioxane (POSS) grafted with ionic liquid (IL) side chains doped with LiTFSI | MCPE | 29 |

Perfluoropolyether with 2 ethylene oxide units on each end terminated with dimethyl carbonate end groups containing 10 fluoro-ether oxygens (PFPE_{E10}-DMC) and 9.1 wt. % LiTFSI | MCPE | 23 |

Corn starch cross-linked with γ-(2,3-epoxypropoxy)propyltrimethoxy-silane with LiTFSI | MCPE | 30 |

PEO blended with sodium carboxyl methyl cellulose (Na-CMC) with sodium perchlorate (NaClO_{4}) | NaPE | 31 |

Organic ionic plastic crystals consisting of triisobutylmethylphosphonium bis(fluorosulfonyl)imide with added bis(fluorosulfonyl)imide sodium salt (NaFSI) | NaPE | 32 |

Electrolyte description . | Category . | Reference . |
---|---|---|

Polyethylene oxide with lithium bis(trifluoromethanesulfonyl)imide salt (PEO/LiTFSI) with 0.017 moles of LiTFSI per mole of ether oxygen (r = 0.017) | HPE | 21 |

PEO/LiTFSI with r = 0.08 | HPE | 17 |

Poly(diethylene oxid-alt-oxymethylene) with LiTFSI (P(2EO-MO)/LiTFSI) with 0.04 moles of LiTFSI per mole of oxygen (r = 0.04) | HPE | 18 |

P(2EO-MO)/LiTFSI with r = 0.08 | HPE | 18 |

P(2EO-MO)/LiTFSI with r = 0.14 | HPE | 18 |

Perfluoroether containing 8 carbon atoms with dimethyl carbonate end groups and bis(fluorosulfonyl)imide lithium salt (C8-DMC/LiFSI) with 5.84 wt. % LiFSI | HPE | 22 |

C8-DMC/LiFSI with 19.9 wt. % LiFSI | HPE | 22 |

Perfluoropolyether with hydroxyl end groups containing 10 fluoro-ether oxygens (PFPE_{D10}-Diol) and 9.1 wt. % LiTFSI | HPE | 23 |

Perfluoropolyether with dimethyl carbonate end groups containing 10 fluoro-ether oxygens (PFPE_{D10}-DMC) and 9.1 wt. % LiTFSI | HPE | 23 |

PEO/LiTFSI gel mixed with tetraethylene glycol dimethyl ether (TEGDME) | GPE | 24 |

80 wt. % methoxy-PEO-methacrylate and 20 wt. % hexadecal-PEO-methacrylate copolymerized into a matrix (PMH20) with lithium perchlorate (LiClO_{4}) salt | GPE | 25 |

Cross-linked PEO plasticized by TEGDME with LiTFSI | GPE | 26 |

PEO/LiTFSI blended with poly[(trifluoromethyl)sulfonyl acrylamide] (PA-LiTFSI) | MCPE | 27 |

Li_{7}La_{3}Zr_{2}O_{12} (LLZO) dispersed in poly(vinylidene fluoride-hexa-fluoropropylene) (PVDF-HFP) | MCPE | 28 |

Polyhedral oligomeric silsesquioxane (POSS) grafted with ionic liquid (IL) side chains doped with LiTFSI | MCPE | 29 |

Perfluoropolyether with 2 ethylene oxide units on each end terminated with dimethyl carbonate end groups containing 10 fluoro-ether oxygens (PFPE_{E10}-DMC) and 9.1 wt. % LiTFSI | MCPE | 23 |

Corn starch cross-linked with γ-(2,3-epoxypropoxy)propyltrimethoxy-silane with LiTFSI | MCPE | 30 |

PEO blended with sodium carboxyl methyl cellulose (Na-CMC) with sodium perchlorate (NaClO_{4}) | NaPE | 31 |

Organic ionic plastic crystals consisting of triisobutylmethylphosphonium bis(fluorosulfonyl)imide with added bis(fluorosulfonyl)imide sodium salt (NaFSI) | NaPE | 32 |

For each electrolyte in Table II, we calculated *ρ*_{+} using Eq. (15) and the values of the parameters we obtained from the publication. For some references, all parameters were listed explicitly. In others, we needed to estimate the parameters from raw data such as Nyquist impedance spectra or current vs time plots. In three cases, the parameters needed were supplied in a personal communication from the authors.^{17,18,32} Finally, if our calculated value for *ρ*_{+,0} differed substantially from the reported value (usually referred to by others as *t*_{+}), the reference was not included in this study. Only 13 out of the 472 papers satisfied all of the constraints. The most common reason a paper was excluded from our analysis was not reporting *L* and *A*. Unfortunately, we could not find any papers which characterized single ion conductors that met all our requirements.

## IV. CHARACTERIZATION OF ELECTROLYTE PERFORMANCE

In most papers, the reported current fraction is based on the measured value of *i*_{0}. One criterion for including papers in this study was that all parameters needed to calculate $i\Omega $ from Eq. (14) were reported. We were thus able to calculate *ρ*_{+} using Eq. (15) and compare it with the reported value, *ρ*_{+,0}, obtained using Eq. (12). Figure 4 is a plot of *ρ*_{+} vs *ρ*_{+,0} for the 19 electrolytes listed in Table II. For references that report only *ρ*_{+}, we plot *ρ*_{+} = *ρ*_{+,0}: these are represented by filled in symbols. Points which lie on the dotted-dashed line in Fig. 4 indicate that the measured value of *i*_{0} was consistent with the calculated value of $i\Omega $. A significant number of data points in Fig. 4 fall well below the dotted-dashed line. A likely reason for this is the use of a sampling rate that is too slow to capture *i*_{0} accurately. Because the current density falls rapidly at early times (see Fig. 2), use of a less frequent sampling rate will result in a lower value of *i*_{0} and thus an inflated value of *ρ*_{+,0}.

While using $i\Omega $ to calculate *ρ*_{+} has been proposed by some,^{6,16–23} the literature is dominated by reports of *ρ*_{+,0} based on measured values of *i*_{0}. Our analysis suggests that *ρ*_{+} is a more robust method for determining the current fraction of an electrolyte. For consistency, all calculations will utilize $i\Omega $ beyond this point.

In principle, the conductivity of an electrolyte measured by ac impedance spectroscopy is a material property that should not depend on the electrodes used in the experiment. Either nonblocking electrodes (lithium or sodium metal) or blocking electrodes (stainless steel, aluminum, etc.) can be used when conducting ac impedance spectroscopy. Conductivities measured using nonblocking or blocking electrodes are denoted *κ*_{nb} and *κ*_{b}, respectively. Figure 5 presents *κ*_{nb} vs *κ*_{b} for the electrolytes in Table II. For many electrolytes, *κ*_{nb} is significantly lower than *κ*_{b}. A few electrolytes show the opposite trend. It is not immediately clear whether *κ*_{nb} or *κ*_{b} should be used to quantify the performance of an electrolyte. To answer this question, we rearrange Eqs. (3) and (10) to obtain

This is a statement of Ohm’s law for an electrolyte at steady-state under small polarization, where *κρ*_{+} can be defined as the effective conductivity of the electrolyte at steady-state. In Fig. 6(a), we plot *κ*_{b}*ρ*_{+} vs $iss\Delta \Phi /L$, while in Fig. 6(b), we plot *κ*_{nb}*ρ*_{+} vs $iss\Delta \Phi /L$. The data in Fig. 6(b) are consistent with Eq. (16), while the data in Fig. 6(a) are not. Figure 6 shows that only *κ*_{nb} can be used to accurately describe the experimental steady-state current. This is because *ρ*_{+} and *κ*_{nb} are both measured in symmetric cells with nonblocking electrodes. For consistency, as we compare *ρ*_{+} of electrolytes, we must also use *κ*_{nb} when evaluating the performance of an electrolyte. Future studies aimed at characterizing new electrolytes should report both *κ*_{b} and *κ*_{nb}. For cases where *κ*_{b} and *κ*_{nb} differ substantially, attempts should be made to understand the root cause as it may be an indication of electrolyte degradation or inconsistencies in cell fabrication. For ether-based polymer electrolytes, it may be an indication of physical (i.e., nonelectrochemical) dissolution of lithium or sodium metal from the electrodes.^{33}

## V. TRADE-OFF BETWEEN CONDUCTIVITY AND SELECTIVE CATION TRANSPORT

In an electrolyte, both cations and anions are mobile, but our main interest is to maximize the flux of the working cation. This is similar to a gas separation process wherein a membrane is used to concentrate a desired species.^{34,35} In gas separation, a pressure gradient is used to drive transport through the membrane, which is designed such that one species is more permeable. Selective transport in this system is characterized by two parameters: (1) the permeability of species *i*, *P*_{i}, which relates the molar flux and driving force (Δ*P*/*L*), where Δ*P* is the pressure drop across a membrane of thickness *L*, and (2) the selectivity of species *i, α*_{ij}*,* which is defined as *P*_{i}/*P*_{j}, where *j* refers to the other species being transported. Ideally, one would like to maximize both *P*_{i} and *α*_{ij}. The difficulty in realizing this ideal was noted by Robeson, who showed that membranes with high permeability typically had low selectivity, while membranes with high selectivity had low permeability.^{36} When data from a large number of membranes were compiled on a plot of selectivity vs permeability, a clear upper bound was evident. Robeson presented a straight line on a log-log plot of selectivity vs permeability such that all compiled data lay below this line. This is referred to as the Robeson upper bound for gas separation.

We present a similar analysis for ion transport in polymer electrolytes under a small dc potential. Selective transport in this system is characterized by two parameters: (1) the conductivity, *κ*, relates the total current, with contributions from both ions, and driving force (ΔΦ/*L*), and (2) the current fraction, *ρ*_{+}, which is a measure of selectivity for cation transport. Ideally, one would like to maximize *κ* and *ρ*_{+}.^{37–39} In Fig. 7, we plot *ρ*_{+} vs *κ*_{nb} for the electrolytes in Table II. The line in Fig. 7 is analogous to the Robeson upper bound. The upper bound is defined empirically by *ρ*_{+} = −0.64 − 0.34 log *κ*_{nb}, where *κ*_{nb} is in S cm^{−1} and *ρ*_{+} is bounded between 0 and 1.

The best electrolyte would be one that supports the highest steady-state current density for a given applied potential, i.e., maximizing the slope in Fig. 1(f), $m$ *= κ*_{nb}*ρ*_{+}. Since both parameters have been calculated, we can rank order the electrolytes of interest. This is done in Table III, where the third column gives the product *κ*_{nb}*ρ*_{+}. For completeness, we also give values of *κ*_{b}, *κ*_{nb}, Ne, *ρ*_{+}, and $t+0$ (when known). The top six electrolytes are identified by their rank in Fig. 7. Interestingly, *ρ*_{+} is less than or equal to 0.2 for all six. In other words, the best electrolytes to date rely on high ionic conductivity rather than selective transport of cations, and efforts to achieve a value of *ρ*_{+} closer to 1 have come at the cost of a disproportionate reduction in ionic conductivity. Considerable research has focused on surpassing the Robeson upper bound because there is no physical reason that a membrane cannot surpass it. The same is true for polymer electrolytes: future research aimed at surpassing the upper bound presented in Fig. 7 seems warranted.

Rank . | Electrolyte . | κ_{nb}ρ_{+} (mS/cm)
. | κ_{b} (mS/cm)
. | κ_{nb} (mS/cm)
. | Ne . | ρ_{+}
. | $t+0$ . | Category . |
---|---|---|---|---|---|---|---|---|

1 | PEO/LiTFSI with r = 0.017^{21} | 0.28 | 0.34 | 1.8 | 5.4 | 0.16 | HPE | |

2 | PEO/LiTFSI gel mixed with TEGDME^{24} | 0.21 | 1.6 | 1.6 | 6.8 | 0.13 | GPE | |

3 | Cross-linked cornstarch with LiTFSI^{30} | 0.17 | 0.34 | 1.0 | 4.9 | 0.17 | MCPE | |

4 | PEO/LiTFSI with r = 0.08^{17} | 0.16 | 2.2 | 1.58 | 9.07 | 0.10 | 0.43 | HPE |

5 | Organic ionic plastic crystals with NaFSI^{32} | 0.14 | 2.1 | 6.6 | 45 | 2.2 × 10^{−2} | NaPE | |

6 | P(2EO-MO)/LiTFSI with r = 0.08^{18} | 0.10 | 1.1 | 0.54 | 4.3 | 0.19 | HPE | |

7 | P(2EO-MO)/LiTFSI with r = 0.04^{18} | 6.9 × 10^{−2} | 0.69 | 0.34 | 3.9 | 0.20 | HPE | |

8 | LLZO dispersed in PVDF-HFP^{28} | 4.9 × 10^{−2} | 0.11 | 0.16 | 2.3 | 0.31 | MCPE | |

9 | P(2EO-MO)/LiTFSI with r = 0.14^{18} | 3.9 × 10^{−2} | 0.33 | 0.24 | 5.2 | 0.16 | HPE | |

10 | C8-DMC with 19.9 wt. % LiFSI^{22,40} | 3.7 × 10^{−2} | 5.5 × 10^{−2} | 4.5 × 10^{−2} | 0.23 | 0.81 | −0.07 | HPE |

11 | PMH20/LiClO_{4}^{25} | 2.4 × 10^{−2} | 8.9 × 10^{−2} | 0.11 | 3.4 | 0.23 | GPE | |

12 | C8-DMC with 5.84 wt. % LiFSI^{22,40} | 1.2 × 10^{−2} | 8.5 × 10^{−3} | 1.3 × 10^{−2} | 9.0 × 10^{−2} | 0.92 | −0.97 | HPE |

13 | Cross-linked PEO/LiTFSI with TEGDME^{26} | 6.7 × 10^{−3} | 0.110 | 1.5 × 10^{−2} | 1.3 | 0.43 | GPE | |

14 | POSS with IL side chains and LiTFSI^{29} | 3.6 × 10^{−3} | 0.120 | 0.10 | 27 | 4.0 × 10^{−2} | MCPE | |

15 | PEO/LiTFSI blended with PA-LiTFSI^{27} | 2.2 × 10^{−3} | 0.141 | 3.3 × 10^{−3} | 0.64 | 0.61 | MCPE | |

16 | PFPE_{D10}-DMC with 9.1 wt. % LiTFSI^{23} | 1.4 × 10^{−3} | 4.8 × 10^{−2} | 1.7 × 10^{−3} | 0.14 | 0.88 | MCPE | |

17 | PFPE_{E10}-DMC with 9.1 wt. % LiTFSI^{23} | 9.1 × 10^{−4} | 2.2 × 10^{−2} | 2.7 × 10^{−3} | 1.8 | 0.36 | MCPE | |

18 | PEO/Na-CMC blend with NaClO_{4}^{31} | 3.0 × 10^{−4} | 0.10 | 6.5 × 10^{−2} | 210 | 4.8 × 10^{−3} | NaPE | |

19 | PFPE_{D10}-Diol with 9.1 wt. % LiTFSI^{23} | 7.4 × 10^{−5} | 3.70 × 10^{−2} | 7.9 × 10^{−5} | 5.0 × 10^{−2} | 0.95 | MCPE |

Rank . | Electrolyte . | κ_{nb}ρ_{+} (mS/cm)
. | κ_{b} (mS/cm)
. | κ_{nb} (mS/cm)
. | Ne . | ρ_{+}
. | $t+0$ . | Category . |
---|---|---|---|---|---|---|---|---|

1 | PEO/LiTFSI with r = 0.017^{21} | 0.28 | 0.34 | 1.8 | 5.4 | 0.16 | HPE | |

2 | PEO/LiTFSI gel mixed with TEGDME^{24} | 0.21 | 1.6 | 1.6 | 6.8 | 0.13 | GPE | |

3 | Cross-linked cornstarch with LiTFSI^{30} | 0.17 | 0.34 | 1.0 | 4.9 | 0.17 | MCPE | |

4 | PEO/LiTFSI with r = 0.08^{17} | 0.16 | 2.2 | 1.58 | 9.07 | 0.10 | 0.43 | HPE |

5 | Organic ionic plastic crystals with NaFSI^{32} | 0.14 | 2.1 | 6.6 | 45 | 2.2 × 10^{−2} | NaPE | |

6 | P(2EO-MO)/LiTFSI with r = 0.08^{18} | 0.10 | 1.1 | 0.54 | 4.3 | 0.19 | HPE | |

7 | P(2EO-MO)/LiTFSI with r = 0.04^{18} | 6.9 × 10^{−2} | 0.69 | 0.34 | 3.9 | 0.20 | HPE | |

8 | LLZO dispersed in PVDF-HFP^{28} | 4.9 × 10^{−2} | 0.11 | 0.16 | 2.3 | 0.31 | MCPE | |

9 | P(2EO-MO)/LiTFSI with r = 0.14^{18} | 3.9 × 10^{−2} | 0.33 | 0.24 | 5.2 | 0.16 | HPE | |

10 | C8-DMC with 19.9 wt. % LiFSI^{22,40} | 3.7 × 10^{−2} | 5.5 × 10^{−2} | 4.5 × 10^{−2} | 0.23 | 0.81 | −0.07 | HPE |

11 | PMH20/LiClO_{4}^{25} | 2.4 × 10^{−2} | 8.9 × 10^{−2} | 0.11 | 3.4 | 0.23 | GPE | |

12 | C8-DMC with 5.84 wt. % LiFSI^{22,40} | 1.2 × 10^{−2} | 8.5 × 10^{−3} | 1.3 × 10^{−2} | 9.0 × 10^{−2} | 0.92 | −0.97 | HPE |

13 | Cross-linked PEO/LiTFSI with TEGDME^{26} | 6.7 × 10^{−3} | 0.110 | 1.5 × 10^{−2} | 1.3 | 0.43 | GPE | |

14 | POSS with IL side chains and LiTFSI^{29} | 3.6 × 10^{−3} | 0.120 | 0.10 | 27 | 4.0 × 10^{−2} | MCPE | |

15 | PEO/LiTFSI blended with PA-LiTFSI^{27} | 2.2 × 10^{−3} | 0.141 | 3.3 × 10^{−3} | 0.64 | 0.61 | MCPE | |

16 | PFPE_{D10}-DMC with 9.1 wt. % LiTFSI^{23} | 1.4 × 10^{−3} | 4.8 × 10^{−2} | 1.7 × 10^{−3} | 0.14 | 0.88 | MCPE | |

17 | PFPE_{E10}-DMC with 9.1 wt. % LiTFSI^{23} | 9.1 × 10^{−4} | 2.2 × 10^{−2} | 2.7 × 10^{−3} | 1.8 | 0.36 | MCPE | |

18 | PEO/Na-CMC blend with NaClO_{4}^{31} | 3.0 × 10^{−4} | 0.10 | 6.5 × 10^{−2} | 210 | 4.8 × 10^{−3} | NaPE | |

19 | PFPE_{D10}-Diol with 9.1 wt. % LiTFSI^{23} | 7.4 × 10^{−5} | 3.70 × 10^{−2} | 7.9 × 10^{−5} | 5.0 × 10^{−2} | 0.95 | MCPE |

While our analysis focuses on the bulk properties of the electrolyte, we recognize the importance of the electrolyte/electrode interface. Both interfacial resistance and the stability of the electrolyte/electrode interface contribute to the efficacy of an electrolyte in a battery. Our approach accounts for interfacial resistance [Eq. (11)–(15)]. The rank ordering of electrolytes is, however, based on bulk properties alone.

## VI. DISCUSSION

The relationship between *ρ*_{+} and transport properties of concentrated electrolytes is quantified by Eqs. (4)–(7). In Table III, there are some electrolytes for which Ne is small (i.e., Ne ≤ 0.1) and others for which Ne is large (i.e., Ne ≥ 10). In the limit of small Ne, $11+Ne\u22481\u2212Ne$ and Eq. (3) reduces to

which implies that the effective conductivity of the electrolyte at steady-state is marginally reduced from that at *t* = 0^{+} by a factor equal to (1 − Ne). When Ne is large, 1 + Ne ≈ Ne and Eq. (3) can be combined with Eq. (4) and written as

The surprising conclusion from Eq. (18) is that there is a class of ion conductors for which the relationship between *i*_{ss} and ΔΦ/*L* is independent of conductivity.

Maximizing *ρ*_{+} is equivalent to minimizing Ne. It is clear from Eq. (4) that Ne may be reduced by either reducing *κ*, reducing (1 − $t+0$)^{2}, reducing *T*_{f}, or increasing *D*. Ultimately, we desire small values of Ne and large values of *κ*: thus, reducing Ne by reducing *κ* is not desirable. On the other hand, reducing (1 − $t+0$)^{2}, reducing *T*_{f}, and increasing *D* are desirable routes to increase *ρ*_{+}. There are very few publications where $t+0$, *T*_{f}, and *D* are measured.^{2,14,17,41,42} Table III presents values of $t+0$ in cases where it has been reported. Note that there is little correspondence between *ρ*_{+} and $t+0$.^{17}

Our discussion has been limited to electrolytes under small applied dc potentials. Whether polarizations are large or small, the salt concentration gradients in the cell affect the current-voltage relationship. At large potential gradients obtained in practical batteries [Figs. 1(c) and 1(d)], the concentration dependence of *κ*, *D*, $t+0$, and *T*_{f} can no longer be ignored, and rank ordering electrolytes would require numerical calculations described in Refs. 5 and 38.

## VII. CONCLUSION

Ion transport through a binary battery electrolyte is governed by four concentration dependent parameters: *κ*, *D*, $t+0$, and *T*_{f}. Under large applied potentials typical of many battery applications, explicit knowledge of these four parameters and their concentration dependence is required to predict the relationship between *i* and ΔΦ/*L*. The problem is simplified for small applied potentials wherein two parameters govern the relationship between *i* and ΔΦ/*L*: *κ* and *ρ*_{+}. Data obtained from symmetric cells with nonblocking electrodes can be used to determine *ρ*_{+} using Eqs. (14) and (15). In principle, *κ* can be determined using either blocking (*κ*_{b}) or nonblocking electrodes (*κ*_{nb}). Our study of the literature revealed a surprising discrepancy between these two measurements reported in a significant number of publications (see Table III). When a discrepancy was found, *κ*_{nb} was often significantly lower than *κ*_{b} although a few electrolytes show the opposite trend. While the analysis reported here is based on *κ*_{nb}, it is likely that practical electrolytes are those wherein the two conductivities are within experimental error, i.e., those that are unaffected by contact with the alkali metal of interest. Our analysis is restricted to publications wherein both *κ*_{nb} and *ρ*_{+} were rigorously measured. Ideally, both *κ*_{nb} and *ρ*_{+} should be maximized. However, there appears to be a trade-off between these two parameters, resulting in an upper bound (*ρ*_{+} = −0.64 − 0.34 log *κ*_{nb}, where *κ*_{nb} is in S cm^{−1}) that is analogous to the one exposed by Robeson for the relationship between permeability and selectivity in gas separation membranes. Designing polymer electrolytes to surpass this upper bound may enable next-generation lithium and sodium batteries. In the limit of small applied potentials, the proportionality factor between *i* and ΔΦ/*L* for binary electrolytes at steady-state is the product *κ*_{nb}*ρ*_{+}. This relationship is analogous to Ohm’s law for electronic conductors. When comparing electrolyte performance, the preferred electrolyte is the one for which *κ*_{nb}*ρ*_{+} is maximized. We use this principle to rank order electrolytes. We hope this perspective will serve as a guide for quantifying the efficacy of future electrolyte designs.

## NOMENCLATURE

## ACKNOWLEDGMENTS

This work was supported by the Joint Center for Energy Storage Research (JCESR), an Energy Innovation Hub funded by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under Contract No. DEAC02-06CH11357. J.A.M. was supported by a National Science Foundation Graduate Research Fellowship (Grant No. DGE 2752814). We are grateful to Dr. Danielle Pesko for supplying portions of the data used in this perspective.

## NOMENCLATURE

### Symbol

*A*electrode area (cm

^{2})*a*salt stoichiometric coefficient

*c*salt concentration (mol cm

^{−3})*c*_{0}solvent concentration (mol cm

^{−3})*D*restricted diffusion coefficient of the salt (cm

^{2}s^{−1})- $D0+$
Stefan-Maxwell diffusion coefficient describing the interactions between the solvent and cation (cm

^{2}s^{−1})- $D0\u2212$
Stefan-Maxwell diffusion coefficient describing the interactions between the solvent and anion (cm

^{2}s^{−1})- $D+\u2212$
Stefan-Maxwell diffusion coefficient describing the interactions between the cation and anion (cm

^{2}s^{−1})*F*Faraday constant (96 485 C mol

^{−1})*i*current density (mA cm

^{−2})*i*_{0}initial current density measured after polarization at ∆

*V*(mA cm^{−2})*i*_{ss}current density measured at steady-state in response to ∆

*V*(mA cm^{−2})- $i\Omega $
initial current density calculated using Ohm’s law at

*t*= 0^{+}, see Eq. (14) (mA cm^{−2})*L*electrolyte or membrane thickness (cm)

- $m$
slope

*m*salt molality (mol kg

^{−1})*M*general cation

- Ne
Newman number

*P*_{i}permeability of species

*i*(mol m^{−1}s^{−1}Pa^{−1})*R*universal gas constant (8.314 J mol

^{−1}K^{−1})*r*molar ratio of cations to oxygens in the electrolyte

*R*_{b}bulk resistance of the electrolyte measured by ac impedance spectroscopy (Ω)

*R*_{i}interfacial resistance between electrolyte and nonblocking electrode (Ω)

*R*_{i,0}interfacial resistance measured by ac impedance spectroscopy just before Δ

*V*is applied (Ω)*R*_{i,ss}interfacial resistance measured by ac impedance spectroscopy after the steady-state current is reached (Ω)

*T*temperature (K)

*T*_{f}thermodynamic factor

*t*time (s)

- $t+0$
transference number of the cation with respect to the velocity of the solvent

*U*^{0}battery open circuit potential (V)

*V*battery operating voltage (V)

*X*general anion

*z*_{+}charge number of cation

*z*_{−}charge number of anion

### Greek

*α*_{i,j}selectivity of species

*i*compared to species*j**β*dimensionless parameter defined by Eq. (8)

*γ*_{±}mean molal activity coefficient of the electrolyte

- ΔΦ
dc potential drop across an electrolyte, excluding ohmic drop across interfaces (V)

- Δ
*P* pressure drop across a membrane (Pa)

- Δ
*V* dc potential drop across a symmetric cell (V)

*η*overpotential (V)

*κ*ionic conductivity (S cm

^{−1})*κ*_{b}ionic conductivity measured using blocking electrodes (S cm

^{−1})*κ*_{nb}ionic conductivity measured using nonblocking electrodes (S cm

^{−1})*v*total number of ions to which the salt dissociates

*v*_{+}number of cations in the dissociated salt

*v*_{−}number of anions in the dissociated salt

*ρ*_{+}current fraction obtained using $i\Omega $

*ρ*_{+,0}current fraction obtained using

*i*_{0}*σ*electronic conductivity (S cm

^{−1})