Composition-dependent thermodynamic and mass-transport characterization of lithium hexa ﬂ uorophosphate in propylene carbonate

A complete set of thermodynamic and mass-transport properties is determined for solutions of LiPF 6 in propylene carbonate at 25 (cid:1) C, elucidating the composition dependences of six independent primary material parameters. Mass density is correlated with concentration to parameterize the partial molar volumes of electrolyte and solvent. Conductometry and Hittorf experiments yield correlations for equivalent conductance and cation transference number, respectively. A theoretical analysis connects the measured transference number to transport numbers measured by other standard approaches. Concentration-cell measurements yield the thermodynamic Darken factor; voltammetric restricted diffusion quanti ﬁ es diffusivity. A Bruggeman factor is measured to relate true transport properties to the effective properties in polarization cells with glass- ﬁ ber separators. Taken together, the data reveal how the three Stefan e Maxwell diffusivities vary with composition, in terms of functions that interpolate between 0.2 M and 2M. The property set is validated by simulations, which match voltage-relaxation experiments excluded from the parameterization.


Introduction
Lithium-ion cells dominate the commercial market for rechargeable batteries. As applications have broadened, the need for more advanced cell technology has also grown. The electrolyte is an essential part of any electrochemical storage system: ion mobility can limit high-power performance and concentration overpotentials can be important during dynamic operation. Accurate prediction of lithium-ion-battery behavior requires careful study of electrolyte thermodynamics and transport.
Concentrated-solution theory, first developed in detail by Newman, Bennion, and Tobias [1], has been widely adopted to simulate mass transport in electrolytic solutions. Within the theory, an isothermal, isobaric solution comprising n chemically distinct species incorporates nðn À1Þ=2 transport properties to describe all the possible pairwise diffusional drag interactions. For a binary electrolyte, comprising a simple salt dissolved in a neutral solvent, three macroscopic transport coefficients d ionic conductivity, cation transference number, and thermodynamic diffusivity d characterize the species/species interactions. Concentratedsolution theory also includes thermodynamic parameters, which rigorously determine how concentration variation within an electrolyte affects the voltage drop across it. A binary electrolyte requires three equilibrium properties d the Darken thermodynamic factor, as well as partial molar volumes for salt and solvent d to specify how its thermodynamic state depends on composition.
Work on transport and thermodynamic characterization abounds in the literature, but most papers focus on measuring one or two properties in isolation [2e10]. Full characterization of mass transport in nonaqueous or air-sensitive electrolytes is uncommon, presumably due to the lack of accurate experimental methods and the complexity of air-free experimental design.
Ma et al. [11] studied mass transport in a sodium-ion conductive polymer electrolyte by purely electrochemical methods d to our knowledge, the first attempt at complete mass-transfer characterization for a nonaqueous electrolyte. Subsequent efforts have set out to characterize various liquid lithium-ion-conductive electrolytes, with the most thorough recent efforts being made by the groups of Lindbergh and Gasteiger [12e17].
One of the better known data sets for lithium-ion battery applications was created by Valøen and Reimers [18], who characterized lithium hexafluorophosphate (LiPF 6 ) solutions in a mixed solvent comprising propylene carbonate (PC), ethylene carbonate, and dimethyl carbonate, using a suite of experiments similar but not identical to Ma et al. Diffusivities were measured by the galvanostatic polarization method, transference numbers by the Hittorf method, and salt activity coefficients by the concentration-cell method. Generally, the accuracies of activity measurements by concentration cells and diffusivity measurements by galvanostatic polarization depend strongly on the accuracies of transferencenumber measurements. The intrinsic error of Valøen's Hittorf apparatus was high, bringing the precision of the other reported properties into question.
Lindbergh and colleagues have made substantial efforts to characterize electrolyte properties based in concentrated-solution theory, taking a more general approach by directly fitting numerical solutions of transport simulations to the current/voltage relationships measured during dynamical electrochemical experiments [12e15]. The paper by Nyman et al. is particularly detailed, adding rigor to the concentrated-solution transport model by including local solute-volume effects explicitly [13]. Lindbergh's direct application of numerical modelling to polarization data dramatically simplifies experimental property measurement protocols, but nevertheless brings about conceptual questions. Because the observabilities of the various material properties that describe an electrolyte vary widely, large inaccuracies can arise during the process of multi-parameter inverse modelling. Overfitting can also be a problem: all of the material parameters are convoluted within the voltage data being fit, and some properties covary as composition changes.
This paper presents a suite of experiments that can be performed to produce a complete and precise transport and thermodynamic characterization of isothermal binary electrolytic solutions. The approach is applied to solutions of LiPF 6 in PC, over a concentration range useful for practical lithium-ion batteries (0.2 Me2 M at 25 C), providing accurate composition correlations for six independent primary material properties. Both LiPF 6 and PC commonly appear in commercial batteries, but we are aware of no work on this binary system that provides a data set sufficiently broad that one can simultaneously extract all the thermodynamic and transport parameters.
Data presented below establish the composition dependences of total mass density, equivalent conductance, Hittorf transference number, steady concentration-cell liquid junction potential, and thermodynamic diffusivity. Density and ionic conductivity are measured via densitometry and AC conductometry, respectively; the Hittorf method is used to quantify the transference number; concentration-cell data are analysed in combination with Hittorf results to determine the Darken factor; and voltammetric restricted diffusion is used to measure Fickian diffusivity. This internally consistent property set is ultimately used to quantify the composition dependences of the three StefaneMaxwell diffusivities, yielding insight about microscopic interactions in LiPF 6 /PC solutions.

Experimental methods
Solutions were formulated in an argon-filled glovebox (Inert Technologies) with low H 2 O (0.1 ppm) and O 2 (0.1 ppm) content, whose temperature was controlled at 25.0 ± 0.5 C by a thermostat. Solutions were made by dissolving LiPF 6 salt (99.99%, battery grade, Sigma Aldrich) in PC (99.9%, anhydrous, Sigma Aldrich); LiPF 6 salts were used as received. The PC solvent was dried over 3 Å molecular sieves for more than a week before use. The ultimate water content of the solutions, determined by Karl Fischer titration, was below 10 ppm. LiPF 6 /PC sample solutions were stored in aluminium bottles, which were vacuum dried at 100 C for 24 h before being filled. Plastic consumables were vacuum dried at 80 C overnight.
Densitometry. For density measurements, solutions were prepared with LiPF 6 concentrations that ranged from approximately 50 mM to 2 M. Gravimetric measurements are more precise than volumetric ones, however, because component masses within a solution are independent of temperature. Salt mass fraction u was therefore used as a primary composition measure. Mass measurements of both salt and solvent were performed with an analytical balance, providing far more precision than the traditional method of solution preparation with a volumetric flask. For each solution, the mass fraction was established to as many as six significant digits.
A high-precision oscillation density meter (DMA4100, Anton Paar) was brought into the glovebox and used to measure the density r of LiPF 6 /PC solutions. The density meter was rinsed with isopropanol (!99.9%, HPLC grade, Fisher Chemical) at least three times and dried in ambient argon between measurements. Every density reported here is an average of at least three measurements.
Given a solution with salt mass fraction u, measurements of density r with five-digit precision were performed to calculate each LiPF 6 solution's molarity c at 25:00±0:02 C, through where M e ¼ 151:905 gmol À1 is the molar mass of LiPF 6 . Thus the molarity of each sample solution was measured very precisely, at the expense of accuracy. Densitometry was further leveraged to account for variability in ambient temperature. For each density measurement, equation (1) was employed to compute the molarity; these data were used to establish a density correlation rðcÞ. Fits of data from sets of samples at 20.00 ± 0.02, 25.00 ± 0.02, and 30.00 ± 0.02 C produced parameterizations valid at each temperature. For experiments at temperatures different from 25 C, molarities can be corrected by linear interpolation of the parameters in rðcÞ.
Once known, density's composition dependence determines the partial molar volumes of solvent and salt in the solution, V 0 and V e , respectively, which generally vary with composition [19]. Newman shows in particular how partial molar volumes can be derived from a density/molarity correlation [20]. In a binary electrolytic solution, Here M 0 ¼ 102:089 gmol À1 is the molar mass of PC.
Conductivity measurement. Ionic conductivity k was measured with an AC conductivity meter (Orion A212, Thermo Scientific), using a custom airtight and watertight test cell that was immersed in a heating and cooling water bath (ARCTIC A40, Thermo Scientific). The meter was calibrated with commercial conductivity standard solutions at 25:00±0:02 C. Temperature calibration between the interior of the test cell and the bath was also performed, using baths of ultrapure water at 20:00±0:02, 25:00±0:02, and 30:00±0:02 C. Samples were loaded into the test cell in the glovebox. After being sealed under argon, the test cell was brought outside the glovebox and immersed in the water bath until it reached thermal equilibrium. Ionic conductivity was measured three times for each sample and is reported as the average. Hittorf experiment. Hittorf measurements were conducted to measure the cation transference number relative to the solvent velocity, t 0 þ . The Hittorf cell, shown in Fig. 1, was machined from a polyether ether ketone (PEEK) rod.
The cell has a 20 cm long cylindrical cavity, which is capped by planar electrodes at either end. Two PEEK stopcocks can be turned to divide the cavity into three parts: a cathodic chamber (top), an anodic chamber (bottom), and a neutral chamber (middle). Threaded access ports sealed by PEEK screws allow fluid to be added to or withdrawn from the three chambers. The 20 cm cavity length was chosen to be long enough that sustained galvanostatic polarization could be performed without affecting the concentration distribution in the neutral chamber. By volumetric pipetting ultrapure water through the access ports outside the glovebox at room temperature (23.8 C), the cathodic and anodic chamber volumes were both established to be V chamber ¼ 4:0± 0.1 mL.
Before an experiment, density of the test solution was measured at 25:00±0:02 C, and the correlation for rðcÞ established by densitometry was used to determine equilibrium salt molarity c and salt partial molar volume V e before preparing the Hittorf cell for loading. Two discs of lithium were cut from the ribbon and placed on either end of the cell before the endcaps were screwed on to seal it. Both stopcocks were turned to the open position, leaving a single, unsubdivided central cavity in the cell. The electrolytic solution was then injected into the cavity through one of the access ports, with care to fill the central cavity completely and avoid introducing bubbles, after which the ports were screwed closed.
To perform the Hittorf measurement, the sealed Hittorf cell was oriented vertically. A constant current I pulse was applied such that positive charge flowed from the bottom of the cell to the top, inducing a concentration difference in the electrolyte as lithium was stripped from (bottom) and plated onto (top) the electrodes. This directionality of current flow ensured that the mass density of the cell decreased monotonically as height within it increased, suppressing the effects of free convection. After being applied for duration T pulse , the applied current was shut off, the two stopcocks were closed, and the solutions were extracted via the access ports. Extracted solutions from all three chambers were stirred for 2 h to guarantee concentration uniformity, after which their densities were measured at 25:00±0:02 C. The final LiPF 6 molarity c f was then calculated by inversion of the density correlation, whereupon t 0 þ was calculated through The absolute value of the concentration difference in the numerator ensures that the same formula applies to both the anodic and cathodic chambers. It is important to be aware that transference-number measurement with a Hittorf cell yields an 'integral' property value d i.e., the measurement induces a nonuniform salt distribution in the cell, and therefore the t 0 þ value obtained from it generally represents an average over the concentration nonuniformity. It is desirable to have measurements that produce 'differential' properties, whose values depend only on the bulk concentration of the solution on which the measurement is performed.
A theoretical treatment of the Hittorf measurement and a detailed error analysis is provided in Appendix A. The analysis shows that Equation (3) does produce a differential transference number in cases where the density r of the solution being tested varies linearly with molar salt concentration c. For LiPF 6 below 2 M in PC, the higher-order molarity dependence of density is weak, so the intrinisic error associated with taking the integral value of t 0 þ yielded by Equation (3) as a differential measure of t 0 þ at the bulk electrolyte concentration is far smaller than the experimental error in V chamber .
In all cases the pulse duration was T pulse ¼ 20 h; the applied current was either 0.1 mA or 0.2 mA. Hittorf measurements were performed using solutions with seven different LiPF 6 mass fractions.
During a Hittorf pulse, the amount of charge passed needs to be large enough to introduce a statistically meaningful density difference between chambers. The pulse duration must be short enough, however, to keep the diffusion boundary layers from reaching the neutral chamber. Further, if the chosen current is larger than the cell's limiting current d which is usually the case if the cell is long d the pulse duration must be short enough to keep the salt concentration from being driven to zero at one of the electrodes. The choice of 20 h guaranteed a good signal-to-noise ratio in the density measurements and was shorter than Sand's time, while also keeping experimental protocols consistent across compositions.
Concentration cell. The concentration-cell method exploits the fact that a steady open-circuit voltage difference DU, which relates to the Darken factor c [8,13,20], develops when electrolytic solutions with identical constituents but different compositions are placed in chemical contact. The cell contains two chambers, separated by a porous material that allows chemical contact between the solutions but significantly slows interdiffusion. One chamber contains a 'test' solution, whose composition is varied between iterations of the experiment; the other holds a 'reference' solution, whose composition is fixed across experiments. Systematic uncertainty in the composition basis due to temperature fluctuations can be reduced by employing a temperatureindependent composition descriptor. The salt fraction y, which is similar to a mole fraction, is computed via where n is the number of ions in a salt formula unit (n ¼ 2 for LiPF 6 ).
The first of these equations is useful to compute y for a solution of unknown composition via inversion of the density correlation rðcÞ; The open-circuit potential (OCP) across a concentration cell relates to material properties through where R is the gas constant, F is Faraday's constant, and T is the absolute temperature. Another justification for introducing the composition variable y is that it makes equation (5) take a particularly simple form, which does not require additional excludedvolume factors [21]. Fig. 2 shows an exemplary concentration cell from the set used to gather the data reported here. Each cell was made of glass tubing in the shape of an 'H', with a grade D frit (pore size: 10e16 mm) embedded at the center to divide the interior into two chambers with 15 mL volumes. Measurements were performed in the glovebox. Commercial 1.0 M LiPF 6 /PC solutions (99.99%, battery grade, Sigma Aldrich) were used in the reference chamber; the salt composition of this standard was y ref ¼ 0:07648. 1 Both chambers of the cell were loaded with electrolyte volumes of 5.0 mL to minimize the difference in liquid heights, which can disturb results by driving Darcy flow through the frit. Lithium strips were trimmed from commercial lithium metal ribbon (99.9%, 0.75 mm thick Â 19 mm wide, Alfa Aesar) and submerged in the solutions to measure the steady-state OCP. All voltammetric data were gathered by an Autolab PGSTAT302 N potentiostat/galvanostat (Metrohm), with a module for electrochemical impedance spectroscopy (EIS) and a low-current amplifier. OCP was tracked by the potentiostat for at least 2 h, a period sufficient to ensure a steady state.
Voltammetric restricted diffusion. Electrolyte diffusivity was measured by tracking the relaxation of nonuniform concentration profiles within the planar parallel-electrode cell depicted on Fig. 3. An annular PEEK spacer (inner diameter: 4.98 mm, outer diameter: 12.69 mm, thickness: 4.99 mm) was fabricated and placed in a Swagelok cell. Two lithium discs were punched out of the metal ribbon and placed at either end of the spacer to serve as cationreversible electrodes.
Harned and French observed that concentration polarization decays exponentially at long times during open-circuit relaxations of planar diffusion media, and related the decay rate to the Fickian diffusivity [22]. Through a perturbation analysis, Newman and Chapman showed that this behavior is also expected for binaryelectrolyte slabs whose material properties depend on local composition [23]. Ma et al. further noted that when two reference electrodes are immersed in a locally nonuniform solution at open circuit, the OCP between them will vary in direct proportion to the difference between the concentrations of salt at their surfaces, so long as the difference is sufficiently small [11,24]. Thus, if one makes a semilog plot of OCP against time as a restricted-diffusion cell with nonuniform composition relaxes toward equilibrium, the curve eventually becomes linear, with a slope related to the diffusivity. The reproducibility of diffusivity measurements by this technique was recently confirmed for battery electrolytes by Ehrl et al. [16].
The interelectrode distance L in a restricted-diffusion cell must be short enough for the exponential diffusional relaxation to be observed within a reasonable amount of time. Exponential behavior reliably occurs when the relaxation timescale is much greater than the time for a diffusion boundary layer emanating from one electrode to reach the other, which can be estimated using Fick's second law. On the other hand, L cannot be too short in a cell with lithium foil electrodes, because metallic Li is very malleable and has a low yield stress. Deformation of the lithium when sealing the cell can easily induce error in L. The choice of L ¼ 4:99 mm ensured that a diffusivity measurement could be made in one or two days, while consistently avoiding short circuiting during cell fabrication.
Lithium's ductility limited the experimental accuracy of the 4.99 mm restricted-diffusion cell. The uncertainty in L due to the malleability of the two electrodes is of the order of the foil thickness; since the diffusional relaxation time scales as L 2 , this implies an intrinsic error around 20%. Following Lindbergh and colleagues [12,13], this error was mitigated by filling the spacer gap with an inert porous separator, which kept the exposed lithium surfaces from deforming plastically as the cell was screwed shut. In all experiments the spacer gap was filled with a glass fiber filter where D eff is the effective Fickian diffusivity of the salt in a solution-permeated glass-fiber separator. Practically, the linear regions of experimental plots of ÀlnðOCPÞ vs. time were fit by firstorder polynomials, and the slopes were used to compute D eff .
To produce restricted-diffusion relaxation plots, 20 h pulses of constant current (ranging between 1:02Am À2 and 2:55Am À2 across experiments) were applied from the bottom of the cell to its top, introducing arbitrary concentration polarization across the electrolyte. After the current was shut off, the OCP was tracked for 10 h as the cell relaxed. Experiments were all performed in the glovebox at its ambient temperature of 25:0±0:5 C.
Bulk impedance. EIS was performed on each restricted-diffusion cell to establish the effective ionic conductivity of its electrolytepermeated separator, k eff . All measurements were performed at the ambient glovebox temperature, 25:0±0:5 C. A potentiostat applied voltage excitations with 10 mV amplitude over frequencies from 100 mHz to 70 kHz, sampling the cell impedance at 10 points per decade. Nova 2.1 software was used to extrapolate the data above ca. 600 Hz down to the real intercept on a Nyquist plot by fitting with a ReRjjCPE circuit; this was identified as the bulk cell resistance U bulk . Finally, in which A is the cross-sectional area of the spacer gap, was used to compute k eff .

Property correlations
Volume and composition. Fig. 4 presents densitometry results; Table S1 of the supplementary information provides the same data in tabular form. DebyeeHückel theory suggests that density varies with molarity as r ¼ r 0 þ r 1 c þ r 2 c 3=2 ; (8) in which the parameters r 0 , r 1 , and r 2 depend on temperature and pressure only. Table 1 presents the parameters that resulted from data fitting.
The solvent density r 0 was established by separate densitometry experiments with pure PC. For solutions, nonlinear least squares fits of the data sets at each temperature, constrained such that rð0Þ ¼ r 0 , were used to establish parameters r 1 and r 2 . Each parameter shows a mild temperature dependence, which is essentially linear across the range studied. OnsagereFuoss theory [25] suggests higher-order terms that could be added to correlation (8), but their inclusion caused the confidence intervals of r 1 and r 2 to widen dramatically without improving R 2 for the fits.
Given the parameterization of density in equation (8), Newman's formulas (equation (2)) show that the partial molar volumes of salt and solvent are respectively, plotted for each temperature studied on Fig. 5.
The partial molar volume of LiPF 6 increases by approximately 5 mLmol À1 (9%) from 0:5 M to 2:1 M, while the partial molar volume of PC decreases by less than 0:4 mLmol À1 (0.35%) over the same range, showing that the solvent molar volume is relatively constant with composition. The temperature dependence of V e is weak, while V 0 varies by about 1% between 20 and 30 C; the temperature effect appears to be similar across the whole composition range. The increase of V e may suggest that the salt occupies an apparently higher volume fraction in concentrated solutions, perhaps because of greater solvation by PC.
Ionic conductivity. Conductivity measurements are tabulated in Table S2 and depicted on Fig. 6a. Conductivity passes through a maximum at a LiPF 6 concentration of $ 0:8 M. At low molarity, conductivity increases as the number of charge-carrying ions rises. At higher concentrations conductivity falls, as solution viscosity rises and salt dissociation weakens [4].
Division of ionic conductivity by the cation equivalent charge z þ and its formula-unit stoichiometry n þ produces the equivalent conductance, which was chosen for correlation using the form Here the first two terms were suggested by the analysis of Debye and Hückel [26]. The third term improved the data fit and yielded a parameter L 2 whose confidence interval was narrower than the coefficients of higher-order terms suggested by other theories, including terms involving other powers of ffiffiffi y p identified by Pitts [27], or a term proportional to ylny from OnsagereFuoss theory [25]. The parameters obtained by fitting data sets at different Fig. 4. Density of LiPF 6 in PC (marks) and correlations from equation (8) (curves) at 20, 25, and 30 C.

Table 1
Parameters of equation (8) for density of LiPF 6 in PC at 20, 25, and 30 C. temperatures are summarized in Table 2.
Each conductance parameter shows a relatively linear temperature dependence. This feature arose naturally with the y dependence in correlation (11), but did not do so for other attempted functional forms. Equivalent conductance data are shown graphically on Fig. 6b. Whereas conductivity goes through a maximum, the equivalent conductance decreases monotonically up to 2 M.
Transference number. Fig. 7 presents results for the cation transference number relative to the solvent velocity, t 0 þ , yielded by the Hittorf experiments. A linear fit of the measurements was made via the method of least squares, The data show the composition dependence of t 0 þ to be quite strong; cation transference decreases steadily, reducing by about half from infinite dilution up to 2 M.
In principle, both the anodic and cathodic chambers of a Hittorf cell can be analyzed to confirm measurements of t 0 þ . The derivation of equation (3) (see Appendix A) assumes, however, that the electrode half-reaction is Li þ þ e À #Li; without any competitive processes involving PC or PF À 6 . During experiments run at 0.2 mA, discoloration and trace amounts of a black precipitate were apparent in the cathodic solution, suggesting a side reaction. Although cathodic solutions at 0.1 mA did not appear discolored to the naked eye, the transference numbers produced by equation (3) exhibited scatter larger than the intrinsic error in V chamber ; inclusion of cathodic data at 0.1 mA did not significantly change the slope or intercept of correlation (12), while decreasing the t-scores of both parameters substantially. The evidence of a cathodic side reaction at 0.2 mA was deemed sufficient to call into question the validity of equation (3) at 0.1 mA, so the cathodic data were rejected. Complete Hittorf-cell data, including the cathodic solution densities for 0.1 mA pulses, are presented in Table S3. Further discussion of t 0 þ is left to section 5 below.
Darken factor. Measurements of DU (cf. equation (5)) were performed with test solutions of given mass fraction, with molarities ranging from approximately 0.1 Me2.0 M. Raw results for the concentration-cell measurements are plotted on Fig. 8a and tabulated in Table S4.
The slope of DU becomes much steeper as dilution rises. Nyman et al. observed that because both c and ð1 Àt 0 þ Þ approach nonzero constants as concentration tends to zero, the integral in equation (5) should vary with lny in this limit [13]. When the composition dependence of the transference number is already known d as is  the case here d a rescaled concentration-cell potential can be formed to correct for singular behavior near infinite dilution. The transformed varable Du, defined as is well behaved as y/0, and makes it easier to assess goodness-offit when developing property correlations for c. Experimental data that have been transformed to Du with correlation 12 are shown on Stewart and Newman successfully fit concentration-cell data [8] by using a formula for c derived from Guggenheim's extended DebyeeHückel activity equation [28], under the assumption that t 0 þ is constant. The Darken factor must go to 1 at infinite dilution, and DebyeeHückel theory suggests that it can otherwise be expressed as a power series in the square root of salt concentration [28,29]. A number of power series in ffiffiffi y p were tried, and the function was found to yield the best fit of the data. The correlation presented by equation (15) is plotted on Fig. 8c, 2 and has been combined with the transference-number correlation to produce the theoretical fits on Figs. 8a and b.
Debye and Hückel's limiting law [29] can be used to estimate c 1 .
Based on the analysis provided by Newman and Thomas-Alyea [20], one would expect that Table 2 Parameters of the equivalent-conductance correlation (equation (11)) for LiPF 6 in PC at 20, 25, and 30 C.    (14). (c) Thermodynamic factor from the correlation in equation (15). 2 The transformed voltages Du were fit by assuming the form obtained by substituting equations (12), (14) and (15) into equation (5).
T. Hou, C.W. Monroe / Electrochimica Acta 332 (2020) 135085 where ε 0 is the dielectric permittivity of vacuum, N A is Avogadro's number, and k ¼ 64:96 is the relative permittivity of PC [30]. This formula suggests that c 1 ¼ À 2:7. The value of À 0:3243 obtained by data fitting has the appropriate sign, but is about a factor of 8 lower than the theoretical prediction. Note that this represents the slope at ffiffiffi y p ¼ 0 of the curve on Fig. 8b; to query consistency with the DebyeeHückel model further, additional data would probably be needed at low ffiffiffi y p .

Diffusion coefficient.
Restricted-diffusion cells were tested to measure the effective Fickian diffusivity D eff , which was corrected to obtain the Fickian diffusivity of the pure solution, D, as follows. Before a given relaxation experiment, EIS was used to establish the effective bulk conductivity k eff . Then the cell was exposed to a current pulse, and the voltage relaxation was tracked after the pulse. Voltage relaxation data from all the restricted diffusion measurements are presented on Fig. S1. Every relaxation curve eventually became linear on a semilog plot with respect to time; slopes of lines fitted to the data in the linear regions (also shown on Fig. S1) were used to compute D eff through equation (6). Finally, the formula D D eff ¼ k k eff (18) was used to obtain D. Although the spread in D eff was relatively wide, the values of D extracted by this method were very reproducible, as seen from the data, which are reported on Fig. 9. All the raw experimental data, including D eff and k eff values, are provided in Table S5.
Although it has been used successfully in the past [13], one must apply the principle that underpins the conversion of D eff into D with care. The derivation of equation (18) assumes that cation flux and charge flux follow similar constitutive laws. In a restricteddiffusion relaxation, one has that i in the high-frequency portion of the EIS spectrum, concentration gradients vanish [31] and i ! fÀV ! F. Since these flux laws are similar, formal homogenization techniques can be used to prove that in a given porous medium, the same geometric factor scales both effective transport properties [32]. In short, the MacMullin number a depends only on pore geometry. 3 The data on Fig. 9 show that the Fickian diffusivity of LiPF 6 decreases steadily from around 250 mm 2 s À1 to 50 mm 2 s À1 as the concentration goes from 0:2 M to 2 M at 25:0 C. These measurements are somewhat lower than the results of Nishida et al. [7], who measured the diffusivity of LiPF 6 in PC with a Moir e technique, determining that Dz400 mm 2 s À1 . They also observed that diffusivity remained relatively constant between 0.5 M and 1.0 M, while the present results indicate a decrease of about 20% over the same range. It is worth noting that decreasing diffusivity with rising concentration has been observed in several prior studies of nonaqueous LiPF 6 solutions [13e15]. The thermodynamic diffusion coefficient D appears in transport laws that use a driving force based on gradients of salt chemical potential, rather than concentration. It relates to the Fickian diffusivity and Darken factor through D ¼ D c: (20) Correlation (15) was combined with equation (20) to transform the experimental data into measurements of D , whose composition dependence is also shown on Fig. 9. This thermodynamic diffusivity data was fit with the function D 1 mm 2 s À1 ¼ Multiple power series in ffiffiffi y p were attempted to perform the fitting; this version of the correlation produced the three-parameter function with the narrowest confidence intervals on its parameters. Bruggeman exponent. MacMullin-number measurements can be used to establish geometric factors for the porous separators used in the restricted-diffusion experiments. Tortuosity t is a key separator property, which relates to the MacMullin number through Various experimental programmes have investigated the tortuosity of porous media, particularly in the chemical engineering literature. A comprehensive study by Comiti and Renauld showed that the tortuosities of beds randomly packed with numerous materials are modelled well by the correlation expðt À 1Þ ¼ ε 1Àb ; where b is a fixed parameter characteristic of a given packing material [33]. For ε close to 1, t is also near 1, and one can approximate the ComitieRenaud correlation with the simpler form tzε 1Àb .
Insertion of this approximate form yields the so-called Bruggeman formula for the MacMullin number [12,13,34], in which b is commonly referred to as the Bruggeman exponent. 4 A very thorough paramaterization of equation (24) for polymeric lithium-ion battery separator materials was recently provided by Landesfeind et al. [36]. That work did not investigate glass-fiber separators, however. Figure 10 shows experimental values of the Bruggeman exponent obtained from the MacMullin number of every restricteddiffusion cell investigated. As would be expected, the Bruggeman exponent does not appear to correlate with electrolyte concentration. The average value is 1.42, showing good agreement with the commonly assumed value of 3=2, which was used with success by Geor en and colleagues [12]. Note that Nyman et al. found much higher Bruggeman exponents for the glass fiber they used [13].
Glass fiber filters are relatively incompressible, so it was not expected that their porosity changed when being inserted into the cavity of the restricted-diffusion-cell spacer. Microglass fiber has a random woven structure, however, so tortuosity t can differ substantially from sample to sample. The standard deviation observed for b was very large (0.49), suggesting this variability was wide.

Polarization-cell model
Concentrated-solution theory generally shows how the state of a binary electrolyte evolves. It is convenient here to employ a modified version of the concentrated-solution model put forward by Newman and Chapman [23], which is readily parameterized in terms of the properties correlated with composition in section 3. Fig. 11 summarizes the suite of characterization experiments, and also shows which transport and thermodynamic properties are involved in each measurement.
The polarization-cell model takes a simpler form if local molar salt concentration c is used as the primary thermodynamic state variable. Assume local electroneutrality, that the solvent carries no equivalent charge, and that the salt obeys the Guggenheim condition z þ n þ þ z À n À ¼ 0; (25) where z À is the anion's equivalent charge and n À , its stoichiometry in a formula unit. (Note that n ¼ n þ þ n À .) The local electrochemical state of the solution relates to current and composition gradients through a MacInnes equation, Here the secondary thermodynamic state variable F represents the voltage measured by a lithium-metal reference electrode reversible only to lithium cations, as discussed by Bizeray et al. [37]. Volume-averaged velocity v ! , , total molar cation flux N ! þ , and current density i ! serve as the basis of dynamical state variables.
Cation flux can be driven by composition gradients, migration, or convection, This constitutive relationship provides the key coupling between the local thermodynamic state (determined by c) and the dynamical basis variables. An effective diffusivity appears, in line with an assumption that the electrolytic solution permeates an inert porous medium. When writing boundary conditions, it sometimes helps to have explicit formulas for the anion flux N ! À and the solvent flux N ! 0 ; should these be needed they can be specified in terms of the dynamical basis which defines the volume-averaged solution velocity, and which states Faraday's law. The evolution of nonuniform composition distributions over time follows three coupled balance equations, which derive from individual material balances for the solvent, anion, and cation the salt distribution follows a material balance, which again contains a correction for the constant porosity of the diffusion medium; and current density is solenoidal, This formulation makes clear that only one fundamental balance needs to contain a time derivative, so relaxation is expected to occur with a single time constant. Transient simulations of the restricted-diffusion experiments described in section 2 can be executed by solving equation (28) through (32) over a one-dimensional Cartesian geometry, subject to boundary conditions that describe galvanic control with applied current IðtÞ. Assuming that only lithium ions react at the electrodes, the cation flux is proportional to the current and both the cation and anion fluxes vanish at where e ! x is a unit vector in the x direction. At the beginning of the experiment, cð0; xÞ ¼ c eq ; i.e., the electrolyte is initially at a uniform concentration, corresponding to an equilibrium state at open circuit. Finally, the voltage drop across the electrolyte can be computed using the integral Surface overpotentials also contribute to the observed cell voltage when IðtÞ is nonzero, in which case this equation does not provide complete information about the cell response. Alternatively, when IðtÞ ¼ 0, as it does for a cell held at open circuit, equation (35) should account entirely for the observed voltage.

Discussion
Parameter validation. Diffusion-coefficient measurements by the restricted-diffusion technique should only use long-time relaxation data in the regime of linear voltage response. At short times, boundary-layer growth dominates OCP relaxation behavior. Also, just after a voltage pulse, composition variation within the cell can be significant, so the local variation of transport properties is also convoluted into the OCP measurement. Ehrl et al. have substantiated these points experimentally, showing that relaxation experiments produce diffusivities that are more reproducible [16]. Precisely because of these issues, short-time relaxation data provide a good test to validate the parameter correlations.
The measured property correlations (equations (9), (11), (12), (15), (19), (21) and (24) were incorporated into the model in section 4 to simulate restricted diffusion with COMSOL multiphysics software. OCP relaxations were simulated for several restricteddiffusion measurements with different LiPF 6 concentrations and various applied constant-current pulses. Data from the cells that were simulated were excluded from the parameterization process. Fig. 12 compares simulation results to experimental relaxation data. Generally the simulations match experiments very well, with a good fit during the period just after the pulse.
Transport number. Some ambiguity about how transference numbers are defined persists in the literature. Many of the most detailed recent transport characterization efforts rely on the NernstePlanck dilute solution theory [38,39] or on models that use concentrated-solution theory, but neglect the Faradaic convection associated with volume-balance equation (30) [14,40,41]. Lindbergh and colleagues [14] based their transport analysis on the cation flux law which has no convective term and includes a different transference number t þ , which they call the 'transference number relative to the room'. As we shall see shortly, the formulation of a relationship between t 0 þ and t þ requires more than a simple change of reference velocity. We therefore instead choose to call t þ the 'cation transport number'. Note that in systems in the limit of infinite dilution wherein solvent does not react at the electrodes, t þ and t 0 þ equate. Thus the BruceeVincent transference number, which is derived from dilute-solution theory [42], may be more broadly understood as a cation transport number.
Clearly one can use constitutive equation (36) in place of equation (27) without a loss of information if and only if the current density and volume-average velocity vectors both point in the same direction. This condition holds true for one-dimensional rectilinear systems in which Faradaic processes are the only source of convection d i.e., most battery separators or polymer-electrolyte slabs d but can fail in two-or three-dimensional systems where reactions drive electrolyte stratification, or in systems where an externally driven liquid flow passes through porous electrodes in the same direction as the ionic current [21,43].
Generally, the cation transference number relative to the solvent velocity relates to the transport number in a one-dimensional system through When considered at an electrode surface where cations reduce completely and no other species react, boundary conditions (33) can be inserted to produce, after substantial algebraic manipulation using thermodynamic state relations, the equation which matches the relationship between t 0 þ and t þ put forward by Lindbergh et al. [14]. Use of the transport number can simplify models and make them easier to solve, but the property is problematic for a number of reasons. Most importantly, the convective term in equation (37) varies locally within a cell in a way that depends on the applied current or flow conditions, and the derivation of equation (38) further requires choosing a particular electrode half-reaction stoichiometry (equation (13)). Whereas t 0 þ is an isolable bulk material property, t þ is not.
Nevertheless, reports of transport numbers are prevalent in the literature, and the correlation for t þ that derives from equation (12) is consequently also presented on Fig. 7. Both t 0 þ and t þ decrease with salt content, but t 0 þ generally has a lower value, and the difference between the two numbers increases with LiPF 6 concentration. These trends agree with observations from literature [14]. Zhao et al. measured the transport number of LiPF 6 in PC with both the pfg-NMR method and the BruceeVincent method [9]. The maximum LiPF 6 concentration they used was 1.5 M. They found that t þ decreased from 0.557 at 0.25 M to 0.370 at 1.5 M. Riley et al. also measured the cation transport number for LiPF 6 in PC using the BruceeVincent method [5]. Their t þ values varied from 0.3 at 0.1 M to 0.1 at 1 M. Zhao's work showed much higher transport numbers than expected based on the present Hittorf measurements. It may be important that they chose a 50 mV voltage amplitude for EIS, much higher than the upper limit of 10 mV suggested by Bruce and Vincent [42]. It has also been suggested that pfg-NMR tends to overestimate transport numbers [10]. The results presented here agree with Riley's at low concentrations but disagree at higher concentrations, which could arise from the neglect of solutevolume effects in the BruceeVincent model [42].
Other groups have measured the cation transference or transport numbers for LiPF 6 in other solvents, which are summarized in Table 3

OnsagereStefaneMaxwell laws
Concentrated-solution theory is fundamentally rooted in the OnsagereStefaneMaxwell transport formalism. In an isothermal, isobaric binary electrolytic solution, principles of irreversible thermodynamics suggest that transport can be described by two StefaneMaxwell equations, in which c þ ¼ n þ c and c À ¼ n À c are the ion molarities, c 0 is the solvent molarity, c T ¼ c 0 þ c À þ c þ is the total molarity, and m 0 and m þ are the electrochemical potentials of solvent and cations, respectively; D ij is the StefaneMaxwell diffusivity of species i in species j. (The Onsager reciprocal relation among these coefficients is assumed to be a symmetry, D ij ¼ D ji .) This formulation of the constitutive laws may be preferable for engineering applications because it readily extends to multicomponent situations. Furthermore, the clear basis on thermodynamic driving forces allows straightforward extensions of the model to include multiphysical phenomena such as thermal diffusion [45] and pressure diffusion   [46]. Newman, Bennion, and Tobias began the development of concentrated-solution theory by proving that StefaneMaxwell equations (39) and (40) are isomorphic to constitutive laws (26) and (27) [1]: the StefaneMaxwell coefficients relate to macroscopic properties through [20] the gradient of solvent electrochemical potential relates to molarity and thermodynamic parameters as [37] À and voltage relates to the cation electrochemical potential. One can use the approach of Latz and Zausch [47] or Newman [20], both of which essentially show that the voltage F measured by a lithiummetal reference electrode at which lithium cations completely reduce relates to the cation electrochemical potential, Thus the OnsagereStefaneMaxwell model is fully parameterized in terms of the solution's volume-explicit equation of state, and the property correlations from equations (9), (11), (12), (15) and (21). (Note that the salt fraction is fundamentally defined as y ¼ c=c T [21].) Fig. 13 shows the StefaneMaxwell coefficients for LiPF 6 in PC, calculated using the measured transport-property correlations provided above. Both D 0À and D 0þ decrease by about an order of magnitude as the LiPF 6 concentration goes from 0.2 to 2 M, commensurate with the decrease in thermodynamic diffusivity. The difference between D 0À and D 0þ increases, consistent with the falling transference number. The former observation suggests that drag by the solvent becomes stronger on both ions as the salt concentration increases, whereas the latter indicates that Li þ carries a smaller portion of the current. This supports the notion that Li þ ions are coordinated with solvent molecules, and that these interactions increase more than they do for anions as concentration rises. Unlike the expectation from OnsagereFuoss theory that D þÀ rises with the square root of concentration, confirmed for aqueous KCl by Newman et al. [1,20], the cation-anion diffusivity D þÀ passes through a maximum at 0.8 M, corresponding to the maximum of ionic conductivity. This trend has been seen and discussed in other studies [13,14].

Conclusion
A suite of experiments was designed and implemented to produce composition-dependent measurements of component partial molar volumes, Darken thermodynamic factor, equivalent conductance, cation transference number relative to the solvent velocity, and thermodynamic diffusivity for the binary Li-ionbattery electrolyte LiPF 6 in PC, as well as the MacMullin number for the inert glass-fiber separator material used for restricteddiffusion measurements. These seven properties allow simulation of one-dimensional electrolytes through the system of model equations presented in section 4. The property correlations were validated by comparing simulations of diffusional relaxations after restricted diffusion cells were galvanostatically pulsed to experiments run under various conditions. Transference numbers from concentrated-solution theory with Faradaic convection were related to transport numbers from NernstePlanck theory and concentrated-solution theory without Faradaic convection. Ultimately, the measured properties were used to develop a complete parameterization of the OnsagereStefaneMaxwell transport model for isothermal, isobaric solutions of LiPF 6 in PC.
in which n þ represents the total number of moles of Li þ . The divergence theorem implies that where vV stands for the surface area bounding control volume V Here IðtÞ represents the applied current (anodic currents are positive). Since the fluid in the chamber maintains local electroneutrality, current density is divergence-free; in this onedimensional geometry, the current at x ¼ 0 must equal the current at x ¼ L. The experiment is run with a constraint that the current pulse must be short enough so the concentration boundary layer, which develops near the anode surface in response to IðtÞ, does not penetrate into the neutral chamber. Bearing this restriction in mind, there is no composition gradient at x ¼ L; equation (27) then implies that It remains to find the flow velocity at the junction between chambers, v , j L . Vector identities can be applied to rewrite volume continuity equation (30) as simplified by the fact that current density is divergence free. Liu and Monroe [21] considered Faradaic convection in a symmetric plating/stripping cell, of which the Hittorf cell here is an example. They showed for a metallic anode reversible only to cations that the surface velocity is v , j t;0 ¼ (A. 9) in which the identity 10) has been applied to make the expression more compact. Insertion of equation (A.9) into equation (A.6) yields , and the superscript eq indicates a value at the composition of the neutral chamber. Here EðtÞ stands for If the density of an electrolyte is a linear function of molar concentration, then V e is constant with respect to composition, equation (A.12) shows EðtÞ ¼ 0, and the Hittorf experiment yields a differential value of the transference number at c eq . When precision of the data mandates a more complicated density correlation, such as equation (8), then V e varies with composition, and one must consider the impact of EðtÞ. One can use equation (27) During a galvanostatic pulse, vc=vx achieves its largest magnitude at the anode surface and decreases monotonically to zero as x increases, so the square-bracketed term in the integrand of equation (A.12) has an upper bound of 1. Assuming that this value is uniform, which should greatly overestimate the error, and further assuming t 0 À =ðt 0 À Þ eq is of order unity, the formula EðtÞ ¼ O À V e j t;0 À V eq e Á c eq 1 À V eq e c eq ! (A.14) establishes the scale of the systematic error induced by neglecting local material-property variation when processing Hittorf-cell data.
Using the largest changes in electrolyte partial molar volume possible from Fig. 5 and the highest salt content studied, c eq ¼ 2:1 M, the cumulative value of EðtÞ (accrued by integration over the duration of a current pulse) cannot exceed 0.012, leading to 1:2% error when t 0 þ is computed with equation (3). This very conservative estimate is still smaller than the ±2:5% error in the chamber volume, justifying setting Ez0 in equation (A.11).
Therefore, if a constant current I pulse is applied to an initially equilibrated Hittorf cell for a time interval T pulse sufficiently short that the concentration boundary layer induced by the current does not penetrate into the neutral chamber, then the change in the cation molar content of the anodic chamber, Dn þ ¼ n þ ðt pulse Þ À n þ ð0Þ, relates to the cation transference number at the equilibrium salt molarity c eq through t 0 þ eq ¼ 1 À This formula is exact if the density of the electrolyte studied is linear with respect to molarity over the range of concentrations accessed during an experiment, and has intrinsic error far below 1:2% for the solutions studied here. Analysis of the cathodic chamber produces an expression identical to equation (A.15), except I is replaced by À I. Bruggeman exponent, see equation (24). ½unitless c Molar salt concentration; related to u through equation (1) and y by equation (4). ½M ¼ molL À1 c À Anion molarity; c À ¼ n À c. ½M ¼ molL À1 c þ Cation molarity; c þ ¼ n þ c. ½M ¼ molL À1 c 0 Solvent molarity; c 0 V 0 ¼ 1 À V e c. ½M ¼ molL À1 c T Total particle molarity, c T ¼ c 0 þ c À þ c þ ; see equation (44). ½M ¼ molL À1 c eq Equilibrium (globally averaged) salt concentration. Cation transport number, see equation (37).