Negative Stefan-Maxwell diffusion coefficients and complete electrochemical transport characterization of homopolymer and block copolymer electrolytes

Nanostructured block copolymers are of particular interest as electrolytes in batteries with lithium metal anodes. The performance of electrolytes in batteries can be predicted only if three transport coefﬁcients (ionic conductivity, κ , salt diffusion coefﬁcient, D , and cation transference number, t 0 + ) are known. We present complete electrochemical transport characterization of a microphase-separated SEO block copolymer electrolyte by reporting κ , D, and t 0 + as functions of salt concentration. We compare the properties of the block copolymer electrolyte with those of PEO homopolymer electrolytes. Negative values of t 0 + are observed in many cases. Recasting the transport parameters in terms of Stefan-Maxwell coefﬁcients provides insight into the nature of ion transport in these electrolytes. ©

There is continuing interest in developing polymer electrolytes for lithium batteries. [1][2][3] Block copolymer electrolytes with a rigid nonconducting block and a soft conducting block are of particular interest as they enable decoupling of electrochemical and mechanical properties. 4,5 The performance of electrolytes in batteries can be predicted using concentrated solution theory only if three transport coefficients are known. In many studies, [6][7][8][9][10][11] the transport coefficients used are ionic conductivity, κ, salt diffusion coefficient, D, and cation transference number, t 0 + . While the second law of thermodynamics requires κ and D to be positive, t 0 + can be positive or negative. We note in passing that most of the literature on polymer electrolytes is restricted to measurement and interpretation of ionic conductivity. In recent years, two approximate methods for measuring the transference number have gained in popularity: the steady-state current method 12,13 and the NMR method. [14][15][16][17] The relationship between the approximate transference numbers determined by these methods and t 0 + is complex. [18][19][20][21] In concentrated solution theory developed by Newman, 22 the transport of ionic species is governed by Stefan-Maxwell diffusion coefficients, D +0 , D −0 , and D +− . These coefficients quantify the relationship between the gradient of electrochemical potential (∇μ i ) and velocity of species (v i ) obtained under this gradient: where the subscripts refer to the cation, anion, and solvent, c i are the molar concentration of species i, and c T is the total molar concentration of the solution. In high-molecular-weight polymer electrolytes, the "solvent" is essentially immobile (v 0 = 0). We are thus mainly interested in the relationships between ∇μ + and ∇μ − and v + and v − . In concentrated solution of a salt (M z+ ) ν+ (X z− ) ν− , the transference number is related to Stefan-Maxwell diffusion coefficients, . [3] In the simplest case D +,0 and D −,0 are positive, and t 0 + is bounded between 0 and 1. If on the other hand t 0 + is negative, as reported in Refs. 18 and 20, then one or more Stefan-Maxwell diffusion coefficients must be negative. This implies that the salt does not dissociate into individual cation and anion; instead charged clusters are obtained. For example, a negative D −,0 implies that the net velocity of the anion due to electric migration alone is toward the negative electrode (i.e., the direction of v − is opposite to that of ∇μ − , see Equation 1). This is expected if the anion is clustered with two cations, and this cluster is fairly mobile. Direct evidence for the presence of such clusters can be obtained by electrophoretic nuclear magnetic resonance experiments (e-NMR). 15,23 Since the signs of Stefan-Maxwell diffusion coefficients are not constrained to be positive, many combinations of D +,0 and D −,0 can, in principle, lead to the observed negative transference numbers. For example, D +,0 = −1 (units are unimportant as we are mainly interested in ratios of D i j ) and D −,0 = + 2 gives t 0 + = −1. However, D +,0 = + 1 and D −,0 = −2 also gives t 0 + = −1. We show that one of these combinations violates thermodynamics. The advantage of determining Stefan-Maxwell diffusion coefficients (as opposed to the equivalent determination of κ, D, and t 0 + ) is that they provide insight into the magnitude and direction of ion migration.
In this paper, we present complete electrochemical characterization of a high-molecular-weight nanostructured polystyrene-bpoly(ethylene oxide) (SEO) copolymer electrolyte doped with a lithium salt. Measurements of κ, D, and t 0 + are used to determine Stefan-Maxwell diffusion coefficients. In several cases, we find that the diffusion coefficients are negative. Implications of these measurements on ion migration are discussed. For completeness, we compare results obtained from block copolymer electrolyte with those obtained from mixtures of poly(ethylene oxide) (PEO) homopolymer doped with lithium salt over the same concentration window.

Experimental
Materials.-Anhydrous tetrahydrofuran (THF) and benzene were purchased from Sigma-Aldrich, poly(ethylene oxide) (PEO, M w = 5 kg/mol) with -OH endgroups was purchased from Polymer Source, Electrolyte preparation.-PEO electrolytes were prepared according to the procedures outlined in Ref. 18. All electrolytes are homogeneous mixtures of PEO and LiTFSI. Electrolytes are prepared at varying salt concentrations, ranging from m = 0.23 to m = 6.82 mol/kg, where m is the molality of the conducting phase.
Block copolymer electrolytes (SEO) were prepared by mixing SEO block copolymer and LiTFSI in dry DMF using a hot plate at 110 • C for 12 h to obtain homogeneous solutions at different salt concentrations (from m = 0.45 to m = 6.82 mol/kg).
The characteristics of these electrolytes -density of the conducting phase, volume fraction of the conducting phase, φ c , salt concentration, c, conducting phase concentration, c 0 , and the molality of the conducting phase, m -are summarized in Table I for PEO electrolytes  and Table II for SEO electrolytes. The volume fraction of the conducting phase in SEO electrolytes, φ c , was calculated according to the equation used in Ref. 24. Salt concentration, c, was calculated according to where ρ is the density of the PEO conducting phase, r is the molar ratio of lithium atoms to ethylene oxide (EO) moieties, M EO and M LiTFSI are the molar masses of ethylene oxide unit (44.05 g mol −1 ), and LiTFSI (287.08 g mol −1 ). Conducting-phase concentration, c 0 , was calculated The molality of the electrolyte, m, is calculated according to Electrochemical characterization.-All sample preparation steps were performed inside an argon glove box (MBraun) in order to maintain water and oxygen levels below 1 and 5 ppm, respectively. PEO electrolytes for conductivity, steady-state current, restricted-diffusion measurements, and concentration-cell experiments were prepared according to the procedures outlined in Ref. 18. SEO electrolytes for conductivity measurements were prepared by heat-pressing the polymer at 90 • C into a 150 μm thick fiberglass−epoxy annular spacer (Garolite-10). The diameter of the electrolyte was taken to be the size of the hole in the annulus, 3.175 mm. High-purity aluminum foils, 17.5 μm thick, were pressed onto either side of the polymer as electrodes, and aluminum tabs (MTI corporation) were attached to the electrodes with polyimide tape. The sample assembly was vacuumsealed in an airtight aluminum-reinforced polypropylene pouch with tabs protruding out so the sample could be electrically probed. The thickness of the polymer sample was measured after conductivity measurements were performed using a precision micrometer. Impedance spectroscopy measurements were performed using a VMP3 potentiostat (Bio-Logic) with an ac amplitude of 20 mV in the frequency range 1 MHz to 1 Hz. The ionic conductivity of the conducting phase in polymer electrolytes, κ, is calculated from the measured sample thickness, L, the cross-sectional area of the spacer, S, and bulk resistance, R b,0 , which was determined by methods discussed in the literature. 25 The conductivity is given by: κ (T) = L/(S * R b,0 (T)).
Lithium symmetric cells were prepared for steady-state current and restricted-diffusion measurements of SEO electrolytes. Samples were made by pressing the polymer electrolyte into a Garolite-10 spacer at 90 • C for 10 s. After that, the electrolytes were sandwiched between two 150 μm lithium metal chips. Nickel tabs were secured to the lithium chips to serve as electrical contacts. The assembly was vacuum sealed in a laminated aluminum pouch material (Showa-Denko) before removal from the glove box. All samples were annealed at 90 • C for 2 h prior to electrochemical characterization.
Steady-state current and restricted-diffusion measurements of SEO electrolytes were performed using a Biologic VMP3 potentiostat. All measurements were performed at 90 • C. At the beginning of the experiment, cells were conditioned for 3 charge/discharge cycles at a low current density of 0.06 mA/cm 2 . Each conditioning cycle consisted of a 4 h charge followed by a 4 h rest and a 4 h discharge. ac impedance spectroscopy was performed prior to potentiostat polarization. Complex impedance measurements were acquired for a frequency range of 1 MHz to 1 Hz at an amplitude of 40 mV. The cell resistances were measured as a function of time by performing ac impedance spectroscopy every 10 min during polarization. Here, the center of the ac input signal was offset by V, and the amplitude was set to 20 mV to minimize disturbance of the polarization signal.
The transference number was determined according to where V is the applied potential, i SS is the current measured at steadystate, and R i,0 and R i,SS are the initial and steady-state resistances of the interface, respectively. This is the Bruce-Vincent-Watanabe approach for approximately measuring the transference number of electrolytes, and it is only valid for ideal solutions. 12,13 i is the initial current calculated according to this equation where R i,0 and R b,0 are interfacial and bulk resistances measured by ac impedance spectroscopy prior to polarization. Restricted-diffusion measurements of SEO electrolytes were performed using the polarization induced by the steady-state current experiment. The applied current was removed, and the cells were allowed to relax for up to 4 h while the open-circuit voltage, U, was measured at time intervals of 1 s. In the simple case where the (t) is a single exponential, the data are fit to the functional form where a and b are the fit parameters and k 0 is an empirically determined offset voltage. We posit that offset voltage, k 0 , arises from small differences in the polymer/lithium interfaces in the symmetric cells. The offset voltage is much smaller than U over most of the experimental window.
where b is from the fit of Equation 9 and L is the thickness of the electrolyte. The lower limits of the fits are such that Dt/L 2 >0.05. Concentration cells of SEO electrolytes were prepared using a similar cell configuration as that described in Ref. 20. SEO electrolytes were contained within Garolite-10 spacer. A channel approximately 2.5 cm long and 0.4 cm wide was cut in the Garolite-10 spacer. Half of the channel was filled with reference electrolyte (m ref = 1.93 mol/kg), and the other half was filled with electrolytes at various m. Lithium metal electrodes were placed on either end of the channel. Nickel tabs were secured to lithium metal electrodes, and the assembly was vacuum sealed in a laminated aluminum pouch material. Two or three concentration cells were prepared for each salt concentration. The open-circuit voltage, U, was measured for each cell at 90 • C using a Biologic VMP3 potentiostat.
The cells were annealed at 90 • C for a minimum of two hours prior to commencing the electrochemical measurements.

Results and Discussion
Ionic conductivity, κ, measured using ac impedance, is plotted as a function of salt concentration in Figure 1a  The approximate transference number, t +,SS, , based on the assumption of an ideal dilute electrolyte, is plotted as a function of salt concentration, m, in Figure 1c. Both PEO and SEO electrolytes exhibit similar trends. In PEO electrolytes, we find that t +,SS reaches a minimum of 0.06 at m = 3.64 mol/kg. In SEO electrolytes, we find that t +,SS reaches a minimum of 0.04 at m = 2.73 mol/kg. These trends are in good agreement with the literature. 26,18,20 The open-circuit potential across concentration cells wherein the salt concentration at one electrode was changed while that at the other was held fixed, U, was measured. The fixed salt concentrations were m = 1.36 mol/kg for PEO electrolytes and m = 1.93 mol/kg for SEO electrolytes. The results of these experiments are shown in Figure 1d. The curves in Figure 1d show the best-fit polynomial equations of the form U PEO = 47.478 − 70.320 (lnm) − 33.145 (lnm) 2 − 8.052 (lnm) 3 [11] U SEO = 56.30 − 82.69 (lnm) − 44.81(lnm) 2 + 1.77 (lnm) 3 [12] where m has units of mol/kg and U is in mV. The trends of U vs lnm in PEO and SEO are very similar; it is evident that attaching nonconducting polystyrene block to PEO has negligible effect on open-circuit potential. The cation transference number, t 0 + , is related to U by the following equation: 20 where z + is the charge on the cation and ν + is the number of cations in the dissociated salt (z − and ν − are defined similarly for the anion), φ c is the volume fraction of the conducting phase, and F is Faraday's constant. In our analysis, we approximate the block copolymer to be a three-component system; we ignore interactions between the salt and the nonconducting polystyrene block. We account for the presence of the nonconducting block by inserting the term φ c in Equation 13. Combining measurements of t +,SS , κ, D, and dlnm/dU shown in Figure 1, t 0 + was calculated as a function of salt concentration, m. Figure 2a shows the results of these calculations. Both PEO and SEO electrolytes exhibit maxima in the vicinity of m = 2 mol/kg. This is followed by a local minimum at m = 3.41 mol/kg for PEO and m = 2.73 mol/kg for SEO. The SEO minimum is sharper than the PEO minimum. In the case of PEO, t 0 + is negative in the vicinity of m = 3.18 mol/kg. In the case of SEO, negative t 0 + values are obtained at all concentrations below 3.41 mol/kg. At high concentrations (m greater than 4 mol/kg), t 0 + values of PEO and SEO are similar. Significant differences between t 0 + and t +,SS are evident at all concentrations, indicating that the assumption of ideality does not apply to PEO/LiTFSI and SEO/LiTFSI electrolytes.
The thermodynamic factor, 1+ dlnγ ± /dlnm is calculated using dU/dlnm and the anion transference number, t 0 − , according to [14] where t 0 − = 1 − t 0 + and ν = ν + + ν − . The thermodynamic factors are plotted in Figure 2b for PEO and SEO electrolytes. The thermodynamic factors for both electrolytes are similar, suggesting that the environments of the salt ions in both systems are similar. This is expected in strongly microphase separated block copolymer electrolytes compared to homopolymer electrolytes. It also provides justification for treating SEO/LiTFSI mixtures as a three-component system and for our assumption that the salt interacts only with the PEO domains. Over most of our concentration window, the thermodynamic factor is significantly greater than unity. This is another indication PEO/LiTFSI and SEO/LiTFSI electrolytes are nonideal. (The thermodynamic factor of ideal mixtures is unity).
The data for κ, D, t +,SS , and dlnm/dU were fit to polynomial expressions (shown as curves in Figures 1 and 2), and the coefficients obtained from fitting are given in Tables III and IV. The fits were broken up into two regions in PEO and SEO electrolytes for accuracy.
Following the Newman approach, 21 Stefan-Maxwell diffusion coefficients are calculated from the transport parameters given Figures 1 and 2 using the following equations: ) unless CC License in place (see abstract       where D is the Stefan-Maxwell diffusion coefficient of the electrolyte based on a thermodynamic driving force, c 0 is the concentration of solvent, c T = 2c + c 0 is the total electrolyte concentration, D +,0 is the Stefan-Maxwell diffusion coefficient describing the interactions between Li + and PEO, and D −,0 is the Stefan-Maxwell diffusion coefficient describing the interactions between TFSI and PEO, D +,− is the Stefan-Maxwell diffusion coefficient describing interactions between Li + and TFSI − . For the case of univalent salts, Equations 16-20 place constraints on the values of D i,j . In particular, if t 0 + is negative then D +,0 must be positive and D −,0 must be negative (see Equations 16 and 17). Returning to the example in the introduction, if t 0 + = −1, then the combination D +,0 = + 1 and D −,0 = −2 is consistent with the second law of thermodynamics (the combination D +,0 = −1 and D −,0 = + 2 is not). The overall Stefan-Maxwell diffusion coefficient D must be positive. This implies that D +0 must be smaller than the magnitude of D −,0 when D −,0 is negative (see Equation 20).
The Stefan-Maxwell diffusion coefficients, D +,0 , for PEO and SEO electrolytes, calculated using Equation 16, are plotted as a function of salt concentration, m, in Figure 3. The data points in Figure 3 present the salt concentrations at which transport coefficients were measured. The curves in Figure 3 were obtained by substituting the polynomial fits for D and the thermodynamic factor into Equation 15 to obtain an expression for D, and substituting this expression and the polynomial expression for t 0 + into Equation 16. We see in Figure 3, D +,0 of PEO and SEO electrolytes are positive and show similar trends. In general, D +,0 decreases with increasing m. Both sets of data exhibit local maxima at m = 3.64 mol/kg (PEO) and m = 2.73 mol/kg (SEO). The agreement between PEO and SEO data in Figure 3 indicates that the presence of polystyrene domains in SEO have little effect on the Stefan-Maxwell diffusion coefficients corresponding to cation motion through poly(ethylene oxide) domains.  The dependence of D +,− on m for PEO and SEO electrolytes is shown in Figures 5a and 5b. These data also contain singularities when as required by Equation 18. The poles in Figure 5 appear at salt concentrations where Equation 21 is satisfied. It is also important to note that in the vicinity of singularities when a particular D ij is large, it means that transport is governed by the other relevant Stefan-Maxwell diffusion coefficients; D ij appear in the denominator of the right side of Equation 1. For example, in the vicinity of m = 2.73 mol/kg where D −,0 approaches either +∞ or −∞, the flux of the anion is entirely determined by the magnitude and sign of D +,−.
In Figure 6a, we plot 1/D −,0 versus m for PEO and SEO electrolytes. In both cases we see two maxima separated by a minimum in the vicinity of m = 3 mol/kg. The low salt concentration maxima are shallow, relative to the high salt concentration maxima in both systems. Except for the highest salt concentration studied, there is reasonable agreement between the two electrolytes. In Figure 6b, we plot 1/D +,− versus m for PEO and SEO electrolytes. In both cases, we see a maximum in the middle of our salt concentration window. In  cases where poles are obtained in D ij versus salt concentration plots, it is preferable to change the abscissa to 1/D ij .
The physical implications of negative transference numbers and Stefan-Maxwell diffusion coefficients is that LiTFSI does not dissociate into Li + and TFSI − ions that migrate independently under the influence of the applied electric field. When t 0 + is negative, it implies that when a field is applied to an electrolyte with uniform composition, both the Li + and TFSI − are driven to the positive electrode (we assume that the motion of the polymer chains induced by the electric field can be ignored). This, in turn, may imply the presence of a multitude of charged clusters in the electrolyte. If, for example, there are three species in solution, Li + , TFSI − , and [Li(TFSI) 2 ] − , then a negative transference number would arise if the dominant mobile species were TFSI − , and [Li(TFSI) 2 ] − . The Stefan-Maxwell diffusion coefficients D ij that we have obtained represent the frictional interactions between all manifestations of a particular species. A powerful property of the Stefan-Maxwell formalism is that it applies regardless of the nature of the associations in solution. A limitation of the Stefan-Maxwell formalism is that it provides limited information on the nature of the associations that underlie the measured diffusion coefficients. Negative transference numbers in our electrolytes could arise due to the presence of [Li(TFSI) 2 ] − or [Li 2 (TFSI) 3 ] − or some other negatively charged cluster. Such information must be provided by other independent experiments (e.g. spectroscopic experiments such as Raman or NMR).

Conclusions
We present complete electrochemical transport characterization of a microphase-separated SEO block copolymer electrolyte by reporting κ, D, and t 0 + as well as the thermodynamic factor as functions of salt concentration. We compare the properties of the block copolymer electrolyte with those of PEO homopolymer electrolytes. Negative values of t 0 + are observed in many cases. The nature of frictional interactions between the components of the electrolytes (ions and polymer chains) is elucidated by determining the Stefan-Maxwell diffusion coefficients (D +,0 , D −,0, and D +,− ) from the electrochemical characterization data. It may be possible to examine these interactions independently using techniques such as electrophoretic NMR. [14][15][16][17]23 The complete electrochemical characterization data for SEO and PEO electrolytes presented here is necessary for modeling charge-discharge characteristics of cells that contain these electrolytes.