Method of the Four-Electrode Electrochemical Cell for the

Article pubs.acs.org/JPCC

Method of the Four-Electrode Electrochemical Cell for the Characterization of Concentrated Binary Electrolytes: Theory and Application M. Farkhondeh,† M. Pritzker,† C. Delacourt,‡ S. S.-W. Liu,† and M. Fowler*,† †

Department of Chemical Engineering, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada Laboratoire de Réactivité et de Chimie des Solides, CNRS UMR 7314, Université de Picardie Jules Verne, Amiens, France

‡

S Supporting Information *

ABSTRACT: A novel method based on a four-electrode cell to determine the transport properties of concentrated binary electrolytes is presented. The cell contains two potential sensors in addition to the working and counter electrodes. The sensors measure the closed-circuit as well as the open-circuit potential in response to an input galvanostatic pulse across the working and counter electrodes. An important advantage of this new method is that it requires only the application of a single pulse in addition to the appropriate concentration cell experiments. By fitting a suitable model to the data obtained from these experiments, the three independent transport properties of a concentrated binary electrolyte and thermodynamic factor can be determined. The proposed technique benefits considerably from the measurement of closed-circuit data for estimation of the transference number. A comprehensive 2D axisymmetric model based on concentrated-solution theory is developed to account for faradaic convection as well as the bipolar effect at the surface of the sensors. It is shown that the bipolar effect has negligible impact on the potential measurements under operating conditions relevant to these experiments. Consequently, a simpler 1D model can be used in place of the 2D model to estimate the transport properties without any loss in accuracy.

1. INTRODUCTION Electrochemical energy storage systems are playing a key role in the ever-growing renewable energy infrastructure. For this to be possible, the development of high power/energy batteries is essential for both mobile and stationary applications. Different systems including insertion- and alloying-type as well as sulfur and air electrodes which store various charge carriers such as Li+, Na+, Mg2+, Zn2+, etc. have been developed in recent years. In order for a battery to be effective, high potential (i.e., cathode) and low potential (i.e., anode) electrodes must be combined and separated by an electrolyte that enables ionic transport between the two electrodes. An ideal electrolyte should be chemically stable with respect to the anode and cathode potentials to guarantee long battery life. As a result, one of the constraints on the operating battery voltage is the stability of the electrolyte. Therefore, any improvement in the electrolyte chemistry (and additives) that leads to a more stable electrode/electrolyte interface will improve the energy and power density of a given anode/cathode pair.1 At the same time, the power is also dependent upon the electrolyte transport properties. Redox flow systems in which an anolyte and a catholyte are responsible for both charge storage and transport have also been receiving considerably more interest for large-scale stationary applications.2,3 Thus, it is not surprising that the search for ever more effective battery electrolytes has received a great deal of attention in recent years.1,4−15 Since the effectiveness of these electrolytes depends © 2017 American Chemical Society

strongly on their transport properties, the development of techniques that can measure these properties as straightforwardly and accurate as possible will lead to faster progress in developing new and useful batteries. In most cases, the electrolyte is a concentrated solution of one salt dissolved in one or more solvents.1 According to the Stefan−Maxwell equation, full characterization of a multicomponent concentrated solution requires estimation of N(N − 1)/2 parameters where N is the number of species in the solution.16 Despite a well-developed theory for concentrated solutions, direct estimation of its parameters is often impracticable for multicomponent systems with more than three species and limited, in practice, to binary electrolytes (i.e., N = 3).16 In a binary electrolyte where the solution of a binary salt in a single solvent produces three species (i.e., one cation, one anion, and one solvent), mass transport is fully characterized by three independent properties. These parameters are related to measurable properties, i.e., ionic conductivity of the solution, diffusion coefficient of the solute, and cation (or anion) transference number,16 which are concentration- and temperature-dependent and required for numerical simulation of electrochemical systems such as Li-ion batteries.17−21 Received: November 15, 2016 Revised: February 1, 2017 Published: February 1, 2017 4112

DOI: 10.1021/acs.jpcc.6b11501 J. Phys. Chem. C 2017, 121, 4112−4129

Article

The Journal of Physical Chemistry C

Figure 1. Exploded view of the four-electrode electrochemical cell.

very onset of the open-circuit step immediately after the current is interrupted, errors in the transference number obtained using this method can result if this effect is not taken into consideration.22,23,27,28,30,31,34,35,42,43 The thermodynamic factor reflects the solution nonideality and must be estimated in addition to the three transport properties. This is most commonly determined by standard concentration-cell experiments that enable the product of the thermodynamic factor and the anion transference number to be determined.16 Measurements from the semi-infinite diffusion analysis must be combined with those from the restricted diffusion and concentration-cell experiments, which are carried out using two separate setups. As a result, a large amount of experimentation conducted in three different cells is required to estimate a complete set of transport properties at a given concentration.22 Conventional methods assume that the transport properties do not change across the cell, and consequently the experimental conditions and the cell design must be strictly controlled to ensure that only a small concentration gradient exists across the cell. However, the assumption of uniform transport properties is made only to ensure that an analytical solution to the mass transport equations exists. If the charge and mass balance equations are solved numerically, the parameters can be allowed to depend on concentration (and hence spatial location),23,27,30,31,34,35 which relaxes the constraints on operating conditions and cell design. Restricted diffusion and semi-infinite diffusion experiments are commonly carried out in symmetric one-dimensional twoelectrode cells with the electrodes located at the two ends separated by the electrolyte. Current is applied to the electrodes for a certain period of time, and voltage is recorded across the electrode pair either immediately after the circuit is opened (semi-infinite diffusion) or while the concentration profile is relaxing (restricted diffusion). However, due to the complicated, not well-understood kinetics of the charge-transfer reaction at the surface of the electrodes, it is not possible to

Ionic conductivity is the most frequently measured property, usually by standard AC impedance methods. However, estimation of the other two parameters is rare and has been limited to a few of the most commonly used electrolytes.22−35 Restricted diffusion theory, as extended to concentrated solutions by Newman and Chapman,36 is a powerful means for estimating the salt diffusion coefficient at a given concentration. Restricted diffusion experiments track the temporal decay of an already established concentration gradient in a cell either electrochemically (i.e., via electric potential difference between two reversible electrodes)37 or nonelectrochemically (i.e., via changes of the local refractive index of the solution)36 using UV/vis absorption29 or via local conductometry.38 Various techniques to estimate the transference number are available: Hittorf,39 moving boundary,40 and emf41 methods directly estimate the transference number of concentrated solutions. However, each of these methods has drawbacks due to experimental complexities.36,41 Alternatively, the semiinfinite diffusion method has become a common practice by researchers since its development and first application to a solid polymer electrolyte by Ma et al.22 in 1995 and further validation for liquid electrolytes by Hafezi et al.42 in 2000. In this method, a symmetric two-electrode cell is first polarized by applying current for a short enough time that the concentration boundary layer developed at the electrode surfaces does not reach the center of the cell so that semi-infinite diffusion conditions can be assumed. The open-circuit voltage (OCV) of the cell immediately after the current is interrupted is related to the solute concentration build-up (or depletion) adjacent to the electrodes. Voltage measurements are conducted at numerous combinations of applied pulse current and duration while maintaining the semi-infinite diffusion assumption. A leastsquares fit of a plot of cell voltage versus current × time1/2 is required to obtain the transference number at a given bulk concentration. Moreover, since double-layer discharging/ charging dominates the response of each electrode at the 4113


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intermediate, and bottom plates) are made of PTFE. In order to be used outside of the glovebox, the cell is made airtight by means of two EPDM rubber O-rings positioned in the top and bottom plates and held together using six bolts and nuts passing through the cell body.

accurately relate the evolution of the closed-circuit cell voltage to the variations of the electrolyte concentration and potential across the cell. As a result, the two-electrode characterization methods are unable to yield useful closed-circuit data that might otherwise be used as a rich source of information regarding the dynamics of mass transport in the solution. A steady-state current (potentiostatic polarization) method which accounts for solution nonideality can also be used to estimate the transference number of ions in concentrated solutions.44−46 However, similar to the semi-infinite diffusion method, it requires prior knowledge of the solute diffusion coefficient and thermodynamic factor. Recently, nuclear magnetic resonance (NMR) spectroscopy has been used to carry out in situ space- and time-resolved imaging of species concentrations which can then be combined with a mathematical model to estimate the solute diffusion coefficient and transference number.32,33 However, MRI-based techniques require elaborate and expensive instrumentation, which is not routinely available in electrochemistry laboratories. A novel method was recently introduced47 to estimate the transport properties of concentrated binary electrolytes that, unlike conventional methods, is able to make use of the data obtained from both the closed-circuit and open-circuit operation modes. The current paper provides a deeper insight into this new method. A general mathematical model is presented which is applicable to a variety of electrolyte systems such as Li-, Na-, Zn-, and Mg-based electrolytes. The method is applied to characterize transport properties of two Li-based nonaqueous electrolytes and compared with conventional electrochemical routines for transport property estimation. Finally, a sensitivity analysis of the kinetic and design parameters is conducted.

3. METHODOLOGY The proposed electrolyte characterization method involves the application of a rectangular current waveform to the WE and CE and monitoring the resulting electric potential of the electrolyte across the two reference electrodes (i.e., potential sensors). The sensors are isolated from the WE and CE and are connected to an external circuit dedicated to potential measurement. No net current flows through the sensors except for the negligible current from the potentiometer. This fourelectrode cell provides direct access to both the closed-circuit and open-circuit voltage in a manner not distorted by the main and side reaction kinetics or double-layer effects at the working and counter electrodes. Figure 2 shows a typical input current

2. ELECTROCHEMICAL CELL DESIGN The experimental setup involves a cylindrical electrochemical cell consisting of counter and working electrodes located at the two ends and two annular potential sensors (PSs) placed midway between the counter (CE) and working (WE) electrodes. The cell is made cylindrical to reduce the dimensionality of the computational domain from 3D to 2D axisymmetric. Figure 1 shows an exploded view of the cell. The cell consists of a 7-layer stack of ring−stainless steel connector, electrically wired PTFE spacer, PS,1, PTFE spacer, PS,2, electrically wired PTFE spacer, and stainless steel connector. The potential sensors are wired to the stainless steel rings surrounding the WE and CE current collectors through the electrically wired PS-CE and PS-WE spacers. The hollow space within the ring stack is filled with a separator made from grade D Whatman borosilicate glass microfiber filter (GF). The separator helps to confine the solution within the working volume of the cell and to minimize convection that may otherwise happen due to external forces (e.g., vibrations).30 The inner diameter of each PS annulus is identical to the inner diameter of the PTFE spacers and the outer diameter of the WE and CE disks and of the separator. The spacing between PS,1 and WE and between PS,2 and CE is chosen to be identical and is controlled by means of the electrically wired PTFE spacers. Disks of 12.98 mm diameter are punched out of a 200 μm thick Li metal foil as the CE/WE. The PS annuli are prepared from the same Li foil. The thickness of the Li rings is measured to be 191 μm after the experiment is conducted. All of the nonmetallic cell parts (i.e., spacers as well as the top,

Figure 2. Typical potential difference between the two sensors (upper plot) in response to a rectangular current waveform applied to the WE and CE (lower plot).

waveform used in this technique and the corresponding voltage response measured across the potential sensors. Transport properties are estimated all at once by fitting a mathematical model to the potential difference between the two sensors during (closed-circuit) and after (open-circuit) galvanostatic polarization. As shown in section 6, the recorded data exhibit unique features each of which can be unambiguously attributed to one of the three independent transport properties: (i) abrupt change of voltage at the outset and end of the closed-circuit period is determined solely by ionic migration (i.e., ionic conductivity of the electrolyte), (ii) exponential decay during the open-circuit stage is controlled by diffusion (i.e., chemical diffusion coefficient), and (iii) transient response during the closed-circuit stage is attributed to both migration and diffusion (i.e., transference number). In addition to the estimation of electrolyte transport properties, the four-electrode cell also should allow the simultaneous investigation of reaction kinetics at the WE and CE by also monitoring their potentials with respect to the potential sensors. This, however, is beyond the scope of this manuscript and warrants a separate evaluation. 4114


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4. EXPERIMENTAL SECTION The cell was assembled entirely in an argon-filled glovebox ( ti to t = ti (i.e., linear with respect to the so-called “reduced time” τ = t /( t − t i ) where ti is the interruption time) to obtain a voltage value at the interruption time suitable for transference number estimation.42 This linear extrapolation is valid provided that the system perfectly satisfies the semiinfinite condition and that the concentration polarization Γ is constant (i.e., valid for sufficiently small concentration differences).43 The same assumptions underlie the linearity of potential versus It1/2 as It1/2 approaches zero. The slope of the i i

Figure 6. Flowchart of the methodology proposed in this study for the estimation of transport and thermodynamic properties of concentrated binary electrolytes.

The fitting process is carried out according to the above guidelines only for one galvanostatic pulse experiment (0.151 mA cm−2 and 0.226 mA cm−2 for LiPF6/EC/DEC and LiPF6/ EC/DMC, respectively), and the resulting parameter estimates are used to simulate the response of the cell at other applied currents to test the robustness of the method. The fitting procedure described above is demonstrated by performing a sensitivity analysis on the transport properties of the LiPF6/ EC/DEC system, as shown in Figure 7. The ionic conductivity shifts the CCV upward or downward (Figure 7a); chemical diffusion coefficient (i.e., + with constant t0+) affects the slope of the semilogarithmic OCV (Figure 7b); and the transference 4122


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Figure 7. Sensitivity of the model to (a) bulk ionic conductivity of the solution, (b) diffusion coefficient with respect to thermodynamic driving force, and (c) and (d) transference number. (c) and (d) share the same legends that are shown in (d). Model parameters are given in Tables 1 and 2 except those varied (see legend).

potential−It1/2 curve is at the heart of the semi-infinite diffusion i method, and any error in obtaining its value (i.e., whether it is generated at the voltage measurement step, extrapolation step, or the linear regression step) will propagate into the final estimates of the transference number, diffusion coefficient (thermodynamic and chemical), and thermodynamic factor. The semi-infinite diffusion method is therefore limited to to comply with narrow experimental conditions of small It1/2 i the underlying semi-infinite condition and constant Γ. However, these conditions are difficult to meet in practice since large potential signals are required for the measurements not to be distorted by noise. The new four-electrode method being proposed bypasses the need for semi-infinite diffusion experiments and the complications associated with parameter estimation described above by directly analyzing the closed-circuit data. In fact, a plot of the is open-circuit potential after current interruption versus It1/2 i not required to estimate the transference number. Instead, it is obtained by fitting the model to the closed-circuit voltage−time data. Measurement of CCV data at the potential sensors essentially obviates the need for any extrapolation required in the conventional semi-infinite diffusion method. Furthermore, continuous measurement of CCV data in the proposed method provides as many reliable data points as the potentiostat/ galvanostat can record. The abundance of the reliable CCV data points compared to the fewer number of data points used in the conventional semi-infinite diffusion method significantly reduces the error associated with curve fitting. Moreover, relaxation of the core assumptions of the semi-infinite diffusion experiment and numerical solution of the transport model make possible the analysis of data at larger t. in the conventional semi-infinite The condition of small It1/2 i diffusion method is not only required to satisfy the semi-infinite

diffusion condition but also to comply with the assumption that the variations of transport properties and thermodynamic factor with concentration are insignificant (i.e., constant solution properties at their bulk values). The same assumption is considered in the restricted diffusion method which is valid at sufficiently large times after the current interruption when the concentration profile flattens out. The chemical diffusion coefficient so estimated corresponds accurately to the bulk concentration and is termed the “differential” chemical diffusion coefficient.36 In the case of the four-electrode method, the assumption of constant properties is not required. Numerical solution of the complete transport model permits concentration-dependent transport properties to be estimated if electrolyte samples at different concentrations are prepared and separately examined in the four-electrode cell. Such a comprehensive analysis is beyond the scope of this paper and will be presented in a future publication. As such, the assumption of constant properties made here is merely for the sake of simplicity. 6.2. Convection. Faradaic convection (i.e., convective transport of species induced by charge-transfer reactions at the working and counter electrodes) is accounted for in the model by solving for the mass-average velocity. However, the diffusion of species is referred to the solvent velocity when inverting the force-explicit Stefan−Maxwell equations eq 7 into flux expressions eqs 8 and 9. This formulation resembles that used in most electrochemical battery models to describe the potential losses associated with species transport in the electrolyte.17 Nonetheless, in almost all of these battery models, the solvent velocity is set to zero, and faradaic convection is neglected to simplify the model. Despite their accuracy, the transport and thermodynamic properties determined with 4123


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allowed to be nonzero (dashed-dotted lines) using the same set of transport properties estimated for the LiPF6/EC/DEC system for the case of v0 ≠ 0 (Table 2). Exclusion of the convective terms from the flux expressions causes a small overestimation of the rate of relaxation of the concentration profile, i.e., leads to a steeper decay of voltage with time in Figure 8b. This leads to a decrease in the parameter estimation of + by ∼11.1% and increase in t0+ by ∼19.7% when v0 = 0 compared to that obtained with v0 ≠ 0. Thus, neglect of faradaic convection leads to an underestimate of + and an overestimate of t0+. Such a significant discrepancy suggests that faradaic convection may not be negligible in electrolytic solutions regularly used in Li-ion batteries. Therefore, it is important that the electrochemical model in which these electrolyte transport properties are to be used be consistent with the method from which they were obtained in the first place. The foundation of the conventional restricted and semiinfinite diffusion methods is critically discussed and revisited to include convection effects in refs 43 and 53. The system is assumed to exist in a state of mechanical equilibrium where only external forces can cause a pressure gradient in the solution. Neglecting the effect of gravity, the only body force is related to the resistive interaction between the liquid solution and the solid matrix in the porous separator. This resistive force is manifested in the permeability that appears in Darcy’s law implemented in the model.55 The momentum equation is used here solely to close the system of equations in the 2D model. It involves the calculation of pressure gradient which is commonly assumed not to affect mass transport even in concentrated solutions.16,52 However, it is of value to estimate the extent of pressure variations within the cell during the galvanostatic polarization experiments conducted in this research. To this end, the permeability of

convection included are not compatible with an electrochemical battery model where convective mass transport is ignored. Figure 8 shows simulated voltage responses to a galvanostatic pulse assuming the solvent velocity is zero (dashed lines) or

Figure 8. Simulation results for LiPF6/EC/DEC at Iapp = 0.2 mA excluding (dashed lines) or including (dashed-dotted lines) faradaic convection. (a) Potential difference between the two PSs versus t1/2 and (b) semilogarithmic plot of potential difference versus t. The set of transport properties is estimated based on the model with v0 ≠ 0 (Table 2).

Figure 9. (a) Electric potential profile along the centerline (dashed lines) and at the wall (solid lines) of the cell as a function of dimensionless axial distance z/L from the WE and (b) dimensionless concentration c/cini profile at the center (dashed lines) and at the wall of the cell at Iapp = 0.2 mA for the LiPF6/EC/DEC system. Insets: enlarged views of the regions of the curves within the red boxes; dotted lines are simulation results when k0PS = 7 × 10−5 mol0.5 m−0.5 s−1, i.e., ten times larger than its value used for the other two sets of simulations. 4124


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negligible impact on the concentration and potential profiles for the two systems examined in this paper. However, depending on the kinetic parameters (i.e., k0PS and βPS) of the potential sensors and geometric design of the cell (i.e., sensor width δ and cell radius Rc) the bipolar effect may influence the potential difference between the two PSs. Sensitivity of the model to these parameters is investigated by varying one parameter at a time while keeping the other parameters constant, as shown in Figures 10 and 11. As shown in Figure 10a, although the rate

the separator is determined using the modified Gebart relation, an empirical equation that has been shown to be accurate for highly porous binder-free fibrous media58 ⎛ 1 − 0.0743 ⎞2.31 Κ 0.491 1 = − ⎜ ⎟ ⎝ ⎠ 1−ε Y2

(41)

where Y is the fiber radius. The validity of using a scalar value for the permeability instead of a tensor for isotropic fibrous porous media was demonstrated in ref 58. The fiber radius is estimated to be ∼1 μm from a SEM micrograph of Whatman filter grade GF/C59 (GF/C is less permeable than GF/D). With this value of Y and the known porosity of the separator (Table 1), the permeability of the separator is calculated from the modified Gebart relation to be 3.96 × 10−12 m2. The maximum pressure difference along the cell filled with the LiPF6/EC/DEC solution at the applied current of Iapp = 0.2 mA is ∼4.4 × 10−3 Pa (calculated from the simulation results) which is very small as expected and is unlikely to have a significant impact on the thermodynamics of the solution. 6.3. Bipolar Effect. A bipolar effect can take place when local charge-transfer reactions occur on an isolated conductor (i.e., not connected to a power supply) with a finite length in contact with an electrolytic solution that is exposed to an electric field. The potential sensors in the four-electrode cell act as bipolar electrodes on which reduction and oxidation reactions occur simultaneously at their cathodic and anodic poles, respectively, in such a way that the net faradaic current over the surface of the electrode is zero. The production and consumption of the reacting species perturb the concentration profile in the vicinity of the sensors which may introduce a systematic error in the parameter estimation if not included in the model. Due to the small applied current, the bipolar effect is expected to be very small in the four-electrode cell; however, its impact on the recorded potentials must be carefully investigated to gain insight into the cell design prior to considering possible simplifications. Figure 9 shows the simulated potential and Li concentration profiles in the axial direction along the centerline (dashed lines) and at the cell wall (solid lines) at different times. During the early stages of galvanostatic polarization (t = 100 s), semiinfinite diffusion conditions prevail, and the Li concentration is uniform along the length of the cell except for the edges adjacent to the WE and CE. At the other end of the process, near-linear concentration potential profiles are attained as the system reaches the steady state (i.e., at t = 27 900 s). The profiles along the cell wall perfectly match those at the centerline except for two narrow regions coinciding with the potential sensors. It can be seen in the insets that the concentration and potential profiles adjacent to the sensors deviate from those at the center of the cell (i.e., away from electrodes) negatively on the left side (cathodic pole) and positively on the right side (anodic pole) of the electrode. Thus, the bipolar effect reduces the gradient of the solution electric potential across the electrode, i.e., smaller local field amplitude, and hence is self-damping. The potential measured by the sensor l and the equilibrium locus zeq,l coincides with the inflection point of the concentration and potential profiles (i.e., ∂2c/∂z2 = 0 and ∂2Φ/∂z2 = 0, respectively) alongside the sensor where the local reaction rate is zero, i.e., i·r ̂ = 0 at r = Rc and z = zeq,l. However, the overriding result shown in Figure 9 is that the bipolar electrochemical reaction at the potential sensors has

Figure 10. Model sensitivity to (a) reaction rate constant and (b) charge-transfer coefficient for Iapp = 0.2 mA applied to the LiPF6/EC/ DEC system. Model parameters are given in Tables 1 and 2 except for those varied (see legends).

constant is varied by an order of magnitude, no visible change is observed in the simulated potential difference between the potential sensors. Although the bipolar effect is more pronounced for an electrode with a faster kinetics (dotted

Figure 11. Effects of (a) cell radius and (b) potential sensor width on the potential difference between the two sensors for Iapp = 0.2 mA applied to the LiPF6/EC/DEC system. Model parameters are given in Tables 1 and 2 except for those varied (see legends). 4125


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performed at different concentrations and temperatures where manual fitting becomes cumbersome. The positions of the PSs determine the magnitude of the potential difference between them and the time when the effect of the concentration gradient begins to appear at the onset of the closed-circuit (Figure 13a) and when the open-circuit

lines in Figure 9), it is insufficient to alter the measured voltage. Moreover, the equilibrium locus zeq,l where the sensor selects the solution potential, is independent of the reaction rate constant according to eq S.4 if local perturbations are ignored. Simulations also remain unchanged when the charge-transfer coefficient is varied. It is shown in Supporting Information that the variation of βPS does not lead to a significant displacement of the equilibrium locus even under a constant electric field four times as large as that at the centerline of the cell at t = 8 h. It implies that no alteration of the simulated voltages occurs by changing the charge-transfer coefficient. The same argument is valid if local perturbations of the electrolyte concentration profile are taken into account as they reduce the electric field locally. Further discussion of the effect of the charge-transfer coefficient is presented in the Supporting Information. In brief, the four-electrode measurements of the two electrolyte systems studied are insensitive to charge-transfer kinetics at the sensors. The design parameters Rc and δ are examined by increasing or reducing them by a factor of 4 relative to their values in the actual experimental setup. The results shown in Figure 11a and b again indicate that these changes have no effect on the voltage response. Altogether, under the operating condition of interest to electrolyte characterization studies (i.e., small applied current), local perturbations are not strong enough to cause the reaction kinetics to effectively influence the measurements in a four-electrode cell with the design parameter values examined here. Figure 12 shows a comparison between the

Figure 13. Simulated potential difference between the two PSs in a cell with different spacing between WE/CE and the PSs (Iapp = 0.2 mA applied to the LiPF6/EC/DEC system). Model parameters are given in Tables 1 and 2 except for those varied (see legend).

potential decay becomes linear on a semilogarithmic plot (Figure 13b); the farther the PSs are from each other, the larger the potential difference and the more sensitive it is to the transport properties assuming the cell length, and the operating conditions remain unchanged. A cell with a larger spacing between the PSs provides as much useful data at a given applied current as one with a smaller spacing but operating at a larger current. However, the applied current cannot be too large in order to maintain the underlying assumptions including negligible heat of reaction at the electrodes and heat of mixing in the bulk (especially important at low-temperature measurements), negligible free convection due to temperature nonuniformity, and negligible displacement of the WE/solution and CE/solution interfaces due to electroplating/stripping. Moreover, a larger applied current during the closed-circuit portion of the pulse leads to a longer relaxation time when the current is switched off, making the system more susceptible to external vibrations. On the other hand, if the sensors are moved too close to the CE and WE, the constant voltage region at the onset of the closedcircuit phase shrinks and makes estimation of the ionic conductivity more difficult although not impossible. The ionic conductivity can be obtained by analyzing its contribution over the entire CCV response. Nonetheless, the nonzero electrode width and technical barriers in the cell assembly itself limit the minimum spacing between the WE/CE surface and the potential measurement spot in the PS, i.e., zeq,l. It should be noted that no theoretical constraint on the location of the PSs exists since the model is solved numerically.36 Altogether, the cell arrangement used in this research (i.e., cell length, sensor spacing, and sensor width) and the operating condition (i.e.,

Figure 12. Comparison between the potential differences between the PSs obtained using the 2D (dashed line) and the 1D models accounting for (triangles) and ignoring (circles) convective transport in the 1D model (Iapp = 0.2 mA applied to the LiPF6/EC/DEC system).

simulations of the 2D model and a 1D version of the same model (i.e., obviously with no bipolar effect) with convection included (triangles) and excluded but compensated for by using the corresponding values of + and t0+ in Table 2 (circles). It turns out that the results of the two models overlap almost perfectly under the operating conditions and for the cell design of this study. Therefore, we conclude that the 2D model presented here can be reduced to a 1D model without a significant loss of accuracy. A 1D model is computationally less expensive and easier to implement than the 2D model and can be more easily coupled with an appropriate optimization routine for statistical fitting if a comprehensive analysis is to be 4126


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ties if experimental data at different concentrations are available. The proposed technique benefits considerably from the measurement of closed-circuit data for estimation of the transference number. The use of more data points than normally is the case with the conventional semi-infinite method should ensure a high level of accuracy and reliability of the estimated transference number. The measurement of the diffusion coefficient closely follows the restricted diffusion experiment. Nonetheless, the three transport properties are obtained simultaneously for the same sample from a single experiment in the same electrochemical cell under less strict operating conditions. These factors eliminate uncertainties associated with combining data obtained from separate experiments, as done in the conventional methods. The method was demonstrated by applying it to two Li-based liquid binary electrolytes, but it is equally applicable to any other binary electrolytes dissolved in aqueous, nonaqueous, or polymeric solvents.

the applied current waveform) proved to be a good trade-off between the quality of data and practical considerations of the method.

7. CONCLUSIONS In this paper, a four-electrode-cell method for the characterization of concentrated binary electrolytes has been demonstrated and compared with the conventional electrochemical methods. The cornerstone principle of the method is the use of two reference electrodes (i.e., potential sensors) in an electrochemical cell in addition to the working and counter electrodes. Variations in the electric potential of the electrolyte are recorded across the two sensors, while a rectangular current pulse is applied to the working and counter electrodes. The four-electrode cell enables data obtained during both the closed-circuit and open-circuit portions of the pulse to be used in a manner not distorted by the main and side reactions or double-layer effects at the working and counter electrodes. The recorded data exhibit unique features each of which can be unambiguously attributed to one of the three independent transport properties: (i) constant voltage at the onset is contributed solely by ionic migration (i.e., ionic conductivity of the electrolyte), (ii) exponential decay of open-circuit data with time is controlled by diffusion (i.e., chemical diffusion coefficient), and (iii) the transient portion of closed-circuit data is attributed to both migration and diffusion (i.e., transference number). A general 2D axisymmetric model based on concentrated solution theory that takes into account the bipolar effect at the sensors located at the cell wall was developed. A single galvanostatic polarization pulse in combination with concentration-cell experiments was shown to be sufficient to uniquely estimate the transport properties and thermodynamic factor. The model was compared with experimental data via a manual parameter search routine in accordance with the orthogonal features of the data. To investigate the effect of faradaic convection, the model was used to estimate transport properties with and without the convective term included in the flux expressions. The thermodynamic diffusion coefficient + and the transference number t0+ were estimated to differ by ∼11% and ∼20%, respectively, depending on whether or not convection is included. Therefore, the inclusion of this effect is warranted when using these parameter estimates in an end-use electrochemical model to ensure that its assumptions are compatible with those of the parameter estimation method. Although neglected in the mass transport model, the pressure drop along the cell length was calculated according to Darcy’s law to be of the order of 10−3 Pa, which is small enough for the isobaric assumption to hold. Sensitivity analysis of the reaction kinetics at the sensor electrodes and the geometric parameters of the cell suggest that the bipolar effect is not strong enough to interfere with the potential measurements under the operating conditions relevant to the electrolyte characterization experiments. This is further confirmed by the simulations of a 1D model (where no bipolar effect can be included) that yield identical results to those obtained from the original 2D model. This also demonstrates that the 1D model can be used as the basis for parameter estimation rather than the 2D model. This is advantageous since solution of the 1D model is computationally inexpensive and can be coupled with an optimization routine to estimate concentration-dependent transport proper-

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.6b11501. Dependence of potential sensor equilibrium locus on electrode kinetic properties (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +1 519 888 4567 x 33415. ORCID

M. Fowler: 0000-0002-9761-0336 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Support for this work was provided by the Natural Sciences and Engineering Research Council of Canada (NSERC, Project RGPIN-170912) and General Motors Co., Automotive Partnerships Canada (APC, Project APCPJ 395996-09). The authors thank Bert Habicher for his kind assistance in fabricating the four-electrode cell.

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LIST OF SYMBOLS A, symbol for single salt c, concentration of the electrolyte in the solution, mol m−3 ci, concentration of species i in the solution, mol m−3 cT, total concentration of solution, mol m−3 cq, concentration of reacting species in q = PS, WE,CE, mol m−3 + , diffusion coefficient of electrolyte based on a thermodynamic driving force, m−2 s−1 +ij , Stefan−Maxwell diffusion coefficients, m−2 s−1 D, chemical diffusion coefficient on a molar basis, m−2 s−1 E, electric field applied to electrolyte, V m−1 e−, symbol for electron F, Faraday’s constant, C mol−1 f i, molar activity coefficient of species i in solution f±, mean molar activity coefficient of electrolyte i, ionic current density, A m−2

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Iapp, applied current, A Iq, current density normal to the surface of electrode q = PS, WE, CE, A m−2 k0q, charge-transfer reaction rate constant q = PS, WE, CE, molβq m1−3βq s−1 L, cell length, m Mzi i, symbol for species i in solution Mi, molar weight of species i, g mol−1 Me, molar weight of electrolyte, g mol−1 M0, molar weight of solvent, g mol−1 N, total number of species in solution Ni, molar flux density of species i n, number of electrons involved in electrode reaction p, pressure, Pa R, universal gas constant, J mol−1 K−1 Rc, cell radius, m Y , borosilicate glass fiber radius, m r, radial distance from the centerline of the cell, m r̂, radial component unit vector si, stoichiometric coefficient of species i in electrode reaction T, temperature, K t, time, s t0i , transference number of species i with respect to the solvent velocity Uq, equilibrium potential of electrode q = PS, WE, CE vs reference electrode used to define solution electric potential, V V̅ i, partial molar volume of species i in solution, m3 mol−1 V̅ e, partial molar volume of electrolyte in solution, m3 mol−1 vi, velocity of species i in solution, m s−1 v, mass-average velocity of solution, m s−1 yi, mole fraction of species i in solution z, axial distance along centerline, m ẑ, axial component unit vector zi, charge number of species i in solution zeq,l, equilibrium locus between cathodic and anodic poles of the potential sensor l, m

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Greek

βq, charge-transfer coefficient q = PS, WE, CE Γ, concentration polarization, V δ, potential sensor width, m ηPS,l, overpotential at interface between solution and potential sensor l, V κ, ionic conductivity, S m−1 K, permeability of separator, m2 μ, dynamic viscosity of solution, Pa s μi, chemical potential of species i in solution, J mol−1 μe, chemical potential of electrolyte, J mol−1 μeθ, chemical potential of electrolyte at the secondary reference state θ, J mol−1 ν, total number of moles of ions into which one mole of salt dissociates νi, number of moles of ion i produced by salt dissolution ρ, solution density, kg m−3 σ, spacing between working/counter electrode and potential sensor, m Φ, electric potential of solution, V ΦWE, electric potential of WE, V ΦCE, electric potential of CE, V ΦPS,l, electric potential of potential sensor l, V 4128


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Method of the Four-Electrode Electrochemical Cell for the

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