Characterizing Slurry Electrodes Using Electrochemical Impedance Spectroscopy

Techniques for interpreting electrochemical impedance spectroscopy of different ﬂowing slurry electrodes conﬁgurations are presented based upon models developed for macrohomogeneous porous electrodes. These models are discussed with regards to three different slurry systems; particles in deionized water, in supporting electrolyte without redox active species (akin to electrochemical ﬂow capacitors), and in electrolytes supporting aqueous redox couples (akin to redox ﬂow batteries). Through investigating each of these systems, the individual properties of a slurry can be determined. It was found that traditional overpotential descriptions, (ohmic, activation, and mass transfer) were insufﬁcient to fully describe the impedance and polarization of the slurry electrodes. An overpotential due to the distributed current distribution in the slurry electrode was considered in the frequency range of activation overpotentialsthatdependsontheexchangecurrentdensityandtheratiooftheelectronicandionicconductivities.Inslurryelectrodes madewithmulti-wallcarbonnanotubeparticlessupportingtheferric/ferrousredoxcouple,thedistributedoverpotentialwasfoundtobeaboutthesameorderofmagnitudeastheactivationoverpotentialandthetotalvoltaicefﬁciencywasover80%at200mA/cm 2 . © The Author(s) 2015. Published by ECS. This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 License (CC BY, http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse of the work in any medium, provided the original work is properly cited. [DOI: 10.1149

Slurry electrodes are inherently dynamic. Particles are constantly moving and are constantly making and breaking contact with each other. Percolation theory describes a critical loading, f c , above which there are enough particles in the slurry that, at any point in time, a continuous network of particles is formed that spans the distance from one boundary to another. [26][27] It is due to these networks that slurries are able to conduct electrons out into the electrode and away from the current collector to utilize the high surface area of the particles for electrochemical processes. At a single instant in time, slurry electrodes resemble porous electrodes such as stationary carbon felts [28][29][30][31] or packed beds; [32][33][34][35][36] bicontinuous electrodes with both electronic and ionic conductivity.
There are many advantages to using slurry electrodes over conventional stationary electrodes. The use of a slurry allows the electrode (1) surface area to be scaled independently of the separator area, (2) to be easily manufactured and replaced without disassembling the cell, (3) and to be easily recycled with filtration.
Electrochemical impedance spectroscopy (EIS) is a versatile technique employed to investigate electrochemical systems. With an understanding of the system, equivalent circuit models can be developed that describe the EIS response and can be used to determine individual properties. These equivalent circuit models have been developed for a variety of macrohomogeneous porous electrodes. [28][29][30][31] This paper will study how these EIS equivalent circuit models can be applied to * Electrochemical Society Active Member. * * Electrochemical Society Student Member. * * * Electrochemical Society Fellow. z E-mail: tyler.petek@case.edu; tyler.petek@gmail.com slurry electrodes in order to better characterize their electrochemical properties leading to better engineered electrodes.

Experimental
Materials.-Three different carbon particles were investigated and their physical properties are described in Table II. Asbury Carbons (Asbury, NJ, USA) graciously provided 230 U, a natural flake carbon with an average diameter of 20 μm, thickness of 100 nm, a reported surface area of 6.5 m 2 /g, and a carbon purity of 99.4%. Asbury Carbons also provided Nano27; carbon platelets with an average diameter of 100 nm, thickness of 2.5-10 nm, a reported surface area of 250 m 2 /g, and a carbon purity of > 99%. MWCNTs were purchased from Nanostructured and Amorphous Materials, Inc. (Houston, TX, USA). These MWCNTs have an OD of 50-80 nm, length of 10 μm, reported surface area of 40 m 2 /g, and a carbon purity of > 95%. For these studies, all carbon particles were used as received. Before tests were performed, particles were mixed by hand in electrolyte and then pumped through the system for at least 30 min.
All salts used in the subsequent experiments were reagent grade and were obtained from Fisher Scientific (USA). All solutions were made with DI water (> 18 M -cm). When using ferrous salts, solutions were made under a continuous nitrogen blanket and experiments were performed with the electrolyte under a continuous nitrogen purge to mitigate the air oxidation of ferrous to ferric iron. 37,38 All tests were conducted at room temperature (20-25 • C).
It was observed that when the volume fraction of the solid particles was above some critical concentration, there was no longer enough liquid in the slurry to fully wet all of the particles. When the volume fraction of solids rose above this critical volume fraction, the slurry transitioned quite quickly to a semi-solid gel. At this point, the slurry is no longer able to flow readily. This viscosity critical volume fraction, f cv , is given in Table II and described in detail by Youssry et al. 12 Electrochemical studies.-Electrochemical impedance spectroscopy (EIS) was performed using a Solartron 1280B potentiostat (Ametek, UK). Each test was performed at the open circuit potential (OCV) with a sinusoidal perturbation of 10 mV amplitude over the frequency range 20 kHz-0.2 Hz. The EIS experiments were controlled using ZPlot (Scribner, USA). Cell polarization and an oscilloscope (Tektronix, USA) were used in congress to ensure that the response of the system under perturbation remained linear. The ionic conductivity of the electrolytes used has been characterized using a glass conductivity cell with platinized platinum electrodes. This apparatus has a cell constant of 200 cm −1 over a wide ionic conductivity range encompassing all results presented in this paper. The resistance of the electrolyte was measured in this cell with EIS. The real limit of the impedance at high frequency, R HF , of the ionically conducting liquid can be used to find the ionic conductivity, shown in Equation 1, where κ is the ionic conductivity and δ/A is the cell constant (200 cm −1 ). Table III shows the ionic conductivity of relevant electrolytes found using this technique.
The electrical conductivities of the slurry particles were measured dry in a pellet press. This device is constructed of two 1.27 cm diameter aluminum rods set inside a clear polycarbonate sheath, shown in Figure 1. An 850 mg sample of particles was placed between the two aluminum rods (inside the polycarbonate sheath) and was then compressed with a Carver manual press (Wabash, IN, USA) between 1.5-3.5 MPa. EIS was performed between the two aluminum rods while the particles were under compression at intervals of 350 kPa. This technique is similar to those used by Marinho et al. 39 The measured resistance is used in Equation 2 to determine the electronic conductivity of the particles, σ. The thickness of the pellet under compression, δ, was calculated as the difference of the distance between the press plates and the length of the aluminum rods without any particles present. A is the cross-sectional area of the aluminum rods. Every time a new particle was investigated using this technique, the aluminum rods were cleaned and degreased with methanol and then hexanes before the high frequency resistance was measured without any particles present. These steps were repeated until the resistance between the two rods was < 5 × 10 −4 and was subsequently considered negligible.
The electrical conductivity of the slurry electrodes were characterized while flowing in a tubular cell, designated the "flowing conductivity cell", for suspensions in DI water with no added salts. This cell was constructed from a polycarbonate tube with 1.3 cm inner diameter and a length of 13 cm. Four pieces of platinum wire, each with a diameter of 1 mm, were situated in the tube so that EIS could be performed on the flowing slurry. The two platinum wires that served as the reference electrodes were 1 cm apart and the two platinum wires that served as the working and the counter electrodes were 8 cm apart. All four wires protruded into the flow < 1 mm and were situated on the same side of the tubular cell as shown in Figure 2. The electrical conductivity of the slurry was determined using Equation 2 and the measured real high frequency resistance of each slurry electrode. To determine the cell constant of the flowing conductivity cell, δ/A in Equation 2, EIS was performed on solutions with known ionic conductivities (1-100 mS/cm) made with varying concentrations of HCl.  The cell constant was found to be 0.76 cm −1 over this conductivity range.
All other electrochemical experiments were conducted in an inhouse channel cell through which the slurry electrodes flow. This "channel cell" was defined by two CPVC flow fields containing straight rectangular channels with a width of 1 cm and length of 12 cm (in the direction of flow) and is described by Figure 3. The current collectors were 1 cm wide and 6.725 cm long (A = 6.725 cm 2 ) graphite plates that were centered in the middle of the CPVC channel. The gap between the current collect of each channel and the separator was either 0.1 cm or 0.2 cm. A peristaltic pump (Cole-Parmer, USA) was used for pumping the slurries. In all experiments, the slurry was pumped vertically up through the electrochemical channel cell. The inlet and exit ports of the channel cell were at roughly 30 • angles (with respect to the current collector face) to minimize any clogging. Current was collected from this cell through two brass plates that were pressed into the back of the graphite current collectors. The cell was held together with stainless steel endplates. Teflon sheets were used to insulate the endplates and Grafoil (Graftech, USA) was used to minimize contact resistances between the brass plates and the graphite plates. The cell assembly was compressed with 8 bolts that were tightened to 90 in-lbs of torque. The reservoirs used to hold the slurry that was pumped through the channel cell were Nalgene bottles modified to be bottom drawn. Unless otherwise noted, all materials were purchased from McMaster-Carr (Cleveland, OH, USA).
The cell constant of the channel cell was determined by performing EIS on solutions with known conductivities (1-100 mS/cm) made with varying concentrations of HCl without a separator between the two cell halves. The cell constant was found to be 0.029 cm −1 and 0.058 cm −1 for cells with 0.2 cm and 0.4 cm total channel depths, respectively. This is in good agreement with the theoretical values of 0.030 cm −1 and 0.059 cm −1 determined from the cell geometry.
For the experiments in which a membrane was used to separate the two channels of the channel cell, Nafion 1035 (Ion Power, DE, USA) was used. The Nafion was pretreated by boiling in 5% H 2 O 2 and then in DI water. The membranes were equilibrated in electrolyte prior to use. Each step was for at least one hour. The swelled thickness of Nafion 1035 was measured to be about 100 μm.

Analytical EIS Models
EIS of a bicontinuous electrode.-An equivalent circuit model of a slurry electrode, ignoring any dynamic nature, is shown in Figure 4 for an electrode between a current collector (left) and an ionically conductive separator (right). Because the dynamic nature is ignored, this equivalent circuit is akin to those of typical porous electrodes. [28][29][30][31] In this figure; R e is the resistance of a finite length of the electronic phase (including any particle-to-particle contact resistance); R ion is the resistance of a finite length of the ionically conductive liquid phase; ζ is the impedance of the interface between the slurry electronic and ionic phases; R c-e is the contact resistance (if any) between the current collector and particles in solution; and ζ c-i is the impedance of the interface between the ionic phase and the current collector. ζ c-i and ζ will always be of the same form (e.g. a capacitor or a capacitor and resistor in parallel) for the systems studied here but the former applies to the real active area of the current collector while the latter corresponds to the active area of the slurry solid phase. From the carbon particle loading in the slurry and the reported active area, the predicted area of the slurry electrode should be on the order of 10 4 -10 7 cm 2 ; at least 2-5 orders of magnitude greater than that of the current collector. For the studies presented in this paper, the impedance associated with the current collector interface will be ignored.
Equation 3a and Equation 3b describe the impedance, Z [ ], of a bicontinuous electrode, ignoring the current collector interface, as developed for macrohomogeneous porous electrodes. [28][29][30][31] Assuming the system is macrohomogeneous allows the electronic and ionic phases to be described by length average resistances, r e and r ion , respectively, described by Equation 4. Any particle-to-particle contact resistance is included in r e . Z = r e r ion r e + r ion δ + r e r ion r e + r ion  Equations 3a-3b consist of four different parameters: the length averaged resistance of the electronic and ionic phases, r e and r ion [ /cm]; the impedance of the interaction of the electronic and ionic phases,Ẑ ζ [ · cm]; and the electrode length in the direction of current flow, δ [cm]. The ionic phase impedance and electrode length can easily be measured independently. EIS can be used, as discussed below, to separate the remaining two parameters depending on the system being studied. (In this paper, the area impedance, EIS of slurry electrodes with ionically conducting electrolyte in the absence of a redox couple.-Salt containing electrolytes with suspended particles have both ionic and electronic conductivity. Slurries such as these, in the absence of Faradiac current, are akin to the electrochemical flow capacitor. [20][21][22][23][24][25] Without a redox couple present, current may only pass from the electronic to the ionic phase through the charging of the double layer capacitance of the solid particles. The impedance of this interaction is described by Equation 5 and is a function of the frequency of the perturbation, ω [s −1 ], and the length averaged capacitance of the solid phase double layer, c dl [F/cm]. This length averaged capacitance is related to the specific capacitance, c dl [F/cm 2 ], by the specific area of the particles active to the electrochemical process per particle volume,ȧ [cm 2 /cm 3 ], the current collector area, A [cm 2 ], and the particle volume fraction, f ≥ f c . In the absence of flow, the equivalent circuit model for an electrochemical slurry capacitor electrode with an ionically conducting separator (the right side) terminated by a current collector (left side) is shown in Figure 5. Figure 6 shows the typical Nyquist plot of the For these models, r e − = 1.75 /cm, r ion = 1.145 /cm, c dl = 15F/cm, δ = 0.1 cm each channel, and R mem = 0.5 · cm 2 . For the model without a separator, 0.5 · cm 2 was added to the real impedance so that the R HF was congruent between the two models. full impedance model of the static electrochemical slurry capacitor, Equations 3-5, in the electrochemical channel cell with and without separators (single channel with thickness of 2δ, or a channel with thickness δ on either side of a separator). A real resistance equal to the separator impedance, 0.5 · cm 2 , was added to the system without a separator to make the high frequency impedance between the two models congruent. In both cases, the real impedance increases as the frequency of the potential perturbation decreases.
At the high frequency limit, Equation 5 approaches zero and, therefore, Equation 3a reduces to Equation 6; the parallel combination of the ionic and electronic paths. As the frequency decreases, the Nyquist plot of the system with an ionically conductive separator tends to negative infinity in the imaginary axis while the Nyquist plot of the system without a separator has a finite low frequency limit. This low frequency behavior exists because without a separator the electronic phase conduction can complete the circuit between the working and counter electrode. The depression of this semi-circle, and the ≈45 • region in the system with a separator, are due to the distributed nature of the electrode, discussed below.
The EIS model presented in Figure 5 and Figure 6 describes a stationary two-phase electrode in the absence of faradaic currents. Under non-stationary conditions, there exists additional current that results from the flow of the particles. 23 The current arises because particles with charged double-layers leave the cell at the outlet thereby permitting uncharged particles (that are hence receptive to accepting current) to enter at the inlet. This current could potentially be modeled in the context of Figure 5 with the inclusion of a volume flow rate dependent source/sink. 23 However, this advective capacitive current (also called "hydraulic current" 23 ) has multi-dimensional characteristics (e.g. boundary layers of overpotential that grow as the slurry flow proceeds downstream along the electrode 24,25 ) that make analysis with 1-D equivalent circuit models problematic. Thus, impedance analysis of these situations is difficult.
However, it is possible to estimate the impedance response at the high and low frequency limits. At high frequencies (relative to the residence time of the slurry in the cell), the slurry behaves as a quasistatic solid matrix, and therefore has the same EIS behavior as in the static case (although perhaps with an altered electronic conductivity due to shear-rate effects 12 ). At low frequency, the current response tends toward the steady-state, DC charging value that can be predicted using available models. 24,25 Thus, even in the situation with a separator present, the Nyquist plot still returns to the real axis at low frequency (instead of having its imaginary component diverge to negative infinity as in the stationary case). The evolution of the Nyquist plot at intermediate frequencies is difficult to ascertain though as the advection-diffusion of surface charge combined with sinusoidal perturbations requires time-dependent, two-dimensional numerical simulations of the governing slurry electrode equations. 24.25 This approach is beyond the scope of this work.
EIS of slurry electrodes supporting Faradaic current.-Slurry electrodes containing electroactive redox species have been of significant interest for electrochemical device applications including energy storage and water treatment. An equivalent circuit representing a slurry electrode, ionically terminated, that supports a redox reaction is shown in Figure 7. The impedance of the liquid-solid interface in the presence of a redox couple is described by Equation 7. In this equation,r ct [ · cm] is the length specific charge transfer resistance.
Z ζ =r ct 1 + jωr ct c dl [7] The advected currents from the previous section also simultaneously exist along with the Faradic reaction currents for slurry electrodes in the presence of redox active species. However, if the kinetics are fast enough and the flow rates are slow enough, the advected capacitive current is generally small 23 and can be ignored. Therefore, the subsequent studies do not include any advective effects.
The impedance of slurry electrodes supporting redox couples are the same whether ionically or electronically terminated due to the presence of the charge transfer resistance. At high frequencies, Equation 7 approaches zero and the total impedance, Equation 3a, reduces to that of the parallel combination of the ionic and electronic resistance, Equation 6. At low frequencies, Equation 7 approaches the impedance of the charge transfer resistance. Because the charge transfer resistance is real and finite, it can bridge the solid phase and the ionic phase at low frequencies.
The first term of Equation 3a represents the parallel combination of the conductive paths and is real, finite, and independent of the voltage perturbation. The second and third terms of Equation 3a, however, are a combination of the electronic, ionic, and charge transfer resistances that encompass the distributed nature of the electrode. At all finite frequencies, the second and third term of Equation 3a, combined with Equation 7, have both real and imaginary components. Figure 8 shows an example Nyquist representation of the impedance of a slurry electrode. In this figure, the specific contributions of the second and third term of Equation 3a are shown. The constant real impedance of the first term is added to the second term while the separator impedance is added to the third term. Doing this only shifts the second and third term in the x-axis of Figure 8. As Figure 8 shows, both the second and third term have unique contributions to the "distributed shape" of the overall impedance response. While the general shape of the total impedance of the redox active slurry electrode (the solid line in Figure  8) is similar to that of the electronically terminated electrochemical flow capacitor (the dashed line in Figure 6), the loop trending back to the real axis at low frequencies is in fact due to the redox couple active at the electrolyte-solid interface, highlighting the importance of applying the correct model to the system being investigated. Figure 9 investigates the effect of the charge transfer resistance on Equation 3a for a slurry electrode supporting a redox reaction. At high frequencies, a near 45 • feature is present in the Nyquist plot for all values of the charge transfer resistance. As the frequency decreases, the real impedance increases and the Nyquist plot begins to roll toward the real axis forming a feature similar to the charge transfer loop of traditional Randles circuits. 31 However, as the charge Figure 8. The Nyquist representation of a slurry electrode supporting a redox reaction. The charge transfer resistance isr ct = 0.01 · cm. All other parameters are the same as in Figure 6.  Figure 6. The square, triangle, and circle represent 100, 1, and 0.01 Hz, respectively. The diamonds represent 0.3 Hz for curves f (r ct = 1.0 · cm) and g (r ct = 10 · cm). transfer resistance increases (the reaction kinetics become slower), the low frequency feature begins to separate from the high frequency feature.
As Equation 3a and Figure 9 indicate, the total impedance of a slurry electrode approaching low frequencies is more than the sum of the high frequency resistance and the charge transfer resistance typical of Randles circuits. The additional impedance will be described as a distributed resistance, R dist . The total impedance of a slurry electrode at the low frequency, or DC, limit can be described by Equation 8. Using the model parameter for the length specific charge transfer resistance, the total charge transfer resistance of a single electrode can be described by Equation 9. [8] R ct =r ct /δ [9] The distributed resistance is a result of the specific current distribution that arises when current can flow through both electronic and ionic phases of the slurry electrode. When charge transfer resistance exists, the current distribution is not that which minimizes ohmic losses, but rather is the distribution that minimizes the total overpotential losses. The additional losses associated with this current distribution are termed the distributed resistance.
Combining the low frequency approximations of Equation 3 with Equations 7-9 allows for the distributed resistance of the slurry electrode to be determined. Figure 10 shows how the distributed resistance depends upon the charge transfer resistance. As the charge transfer resistance decreases, the distributed resistance also decreases. This occurs because, asr ct approaches zero, it shorts out the solid-liquid interface and the impedance of the slurry electrode approaches Equation 6 at all frequencies. Conversely, as the charge transfer resistance increases, the slurry electrode behaves more and more like an electrochemical flow capacitor. At this limit, the distributed resistance approaches a constant value. This maximum distributed impedance (R dist for the capacitor) is a function of the ratio of electronic resistance to ionic resistance and is described in Figure 11. The distributed resistance, normalized by the high frequency resistance, is at a minimum value of 1/3 when the electronic resistance equals the ionic resistance. As the electronic and ionic resistances move further apart, the normalized distributed resistance continues to grow. It should be noted that the high frequency "foot" in Figure 9 does not encompass the distributed impedance. As shown in Figure 8, both the second and third terms of Equation 3a contribute to the distributed resistance.

Results and Discussion
Maximum slurry electronic conductivity.-All slurry electrodes investigated in this study are made of electrically conductive particles suspended in an ionically conductive electrolyte. Before attempting to separate the electronic conductivity from the ionic conductivity in a dynamic flowing slurry, the electronic conductivity was first investigated in a system without electrolyte.
The maximum electronic conductivity occurs when the volume fraction of the liquid phase is zero. However, a "slurry" without electrolyte is not flowable. Therefore, the electrical conductivity of the slurry particles, in the absence of any electrolyte, was measured in the pellet press. The results are shown in Table IV and were found to be weak functions of applied compression over the range tested. Each particle type exhibited conductivity 10 3 times lower than that of pure graphite. 39 This decrease in conductivity can be attributed to particle-to-particle contact resistances and packing efficiencies. 39 In the flowing slurry electrodes, the particles are under no significant force to pack. Even at the lowest pressures, the dry pressed pellet conductivity will over-estimate the electrical conductivity of the flowing Figure 11. The ratio of the distributed resistance to the high frequency resistance as a function of the ratio of the electronic resistance to the ionic resistance. In this figure,r ct = 10 10 , e.g. R dist is at the maximum value for each R e /R ion . slurry electrode. However, it is interesting to note that the conductivity of the packed particles is similar for the three particles considered, despite their different size and shapes.
Slurries in DI water.-Evaluating slurries where the liquid contains no supporting ions (DI water) is a useful diagnostic system to isolate the electronic conductivity. This slurry has negligible ionic conductivity and the EIS response, without an ionic separator, is a single point on the real axis of the Nyquist plot and can be modeled as a pure resistor. The electronic conductivity, σ, can be found using the measured resistance, R, and the cell geometry, δ/A, as in Equation 2. Figure 12 shows the electronic conductivity of slurry electrodes with each particle at varying volume fractions in DI water as determined in the flowing conductivity cell. The conductivity quickly rises with increasing loading of particles above some critical percolation concentration, f c , where the volume fraction of particles is high enough to create fully percolating conductive networks. The critical percolation concentration and the rate at which the conductivity rises as a function of increased loading is a factor of particle shape and particle-to-particle interactions. 26,27,40 As a flowing slurry, the particles exhibit electrical conductivity 10 2 -10 3 times smaller than the dry packed pellets. This again can be attributed to particle-to-particle contact resistances and a further decrease in packing efficiencies. For the following electrochemical experiments, a single reservoir containing the slurry/electrolyte was used to feed both sides of the electrochemical channel cell (divided by a separator; Nafion 1035). The draw from the reservoir was split and fed into two separate heads on the peristaltic pump before entering the cell so that the volumetric flow rate on either side of the cell was directly controlled. After leaving Figure 12. Conductivity of slurries in DI water as a function of the solid phase volume fraction. Conductivity determined from the high frequency real intercept of the impedance for each particle in the flowing conductivity cell.

EIS of slurry electrodes supporting Fe
) unless CC License in place (see abstract). ecsdl.org/site/terms_use address. Redistribution subject to ECS terms of use (see 207.241.231.81 Downloaded on 2018-07-19 to IP the electrochemical cell, the slurry from both sides of the cell were combined into one stream before reentering the reservoir. The volume of the reservoir used was typically 700 mL. This experimental design, deemed a symmetric cell configuration, greatly simplifies the analysis of slurry or porous electrodes in the presence of a redox couple. 45 Using a symmetric cell configuration ensures that the composition of the electrolyte will remain constant throughout all electrochemical tests. Ferric ions are reduced to ferrous ions at the cathode at the same rate that ferrous ions are oxidized at the anode. Since both streams are remixed, the reservoir composition does not change. The other advantage to the symmetric cell configuration is the simplified EIS analysis. The positive and negative slurry entering the cell should be identical. Also, because the ferrous/ferric couple is facile and symmetric (similar polarization whether reducing or oxidizing), the total overpotential of each electrode should be identical. This allows both slurry electrodes in the cell to be modeled by one circuit, Figure 7, instead of two on either side of a membrane and therefore reduces the number of fit parameters significantly. To model the entire symmetric electrochemical cell, Figure 7 is modeled in series with the separator resistance and hardware inductance. It is necessary to include the hardware inductance because, at ∼10 −7 H, the curves have a finite positive slope when the imaginary impedance is zero.
The ferric/ferrous couple was investigated on slurry electrodes made with MWCNTs in the symmetric channel cell. Focus was put on these particles due to their higher electronic conductivity while suspended in a flowing slurry as shown in Figure 12 and due to their smaller primary particle size. For application in continuously circulating slurry electrodes, as opposed to pulse flow, 11 the stability of the slurry is primarily related to the settling of particles. With the MWCNTs in stagnant slurries, no appreciable settling was observed for at least two hours. However, with the larger particles, settling was visibly observed in shorter time frames (<30 minutes). Also, with the MWCNTs, the pumping action has been observed to be enough to keep the slurry well mixed for well over a week without any noticeable settling in the reservoir. Figure 13 shows  frequency regions to the impedance response dominated by different processes. At high frequencies, above 10 3 Hz, the impedance is dominated by the hardware inductance and the parallel ohmic resistance, Equation 6. At intermediate frequencies, 10-10 3 Hz, the characteristics of the charge transfer reaction are present (akin to Figure 8). The third region in the impedance plots of Figure 13 are effects due to mass transfer and the advective current at frequencies below 10 2 Hz. Of these three regions, only the effects due to mass transfer and advective current should depend on the flow rate of the slurry electrode; impedance due to both decreases with increasing flow rate. Comparing the EIS results as a function of flow rate (curves b-d in Figure 13) to the analytical model shows that there is non-negligible low frequency effects at all flow rates investigated. Since the impedance models developed earlier do not account for mass transfer or hydraulic current, the model, Figure 7 in series with hardware inductance and membrane resistance, was only applied to the data range above 10 2 Hz. Over this range, the impedance of curves b-d are near identical; confirming that mass transfer and the advective current are only significant at lower frequencies. When the impedance model was fit to the data, the ionic phase resistance, estimated using Equation 1 and Table III, and the separator resistance, 0.5 · cm 2 corresponding to Nafion 1035 in the iron electrolyte, were fixed. The change in ionic conductivity due to the volume of electrolyte displaced by slurry particles should be negligible (< 5%) 47 at 5 vol% MWCNTs. Table V shows the fitting results of the impedance model to the data presented in Figure 13. The electronic conductivities of the slurry electrodes estimated in Table V match very well with the results shown in Figure 12. While the difference in electrolyte between Figure 12 and Figure 13 can change the particle-to-particle interactions, 26,27,40 the fact that the electronic conductivity was similar in both electrolytes is thought to be due to the high aspect ratio of the MWCNTs which force long range percolation. In this case, the aspect ratio of the MWCNTs is more important to the conductivity than the interactions between particles. Also, the good agreement between the electronic conductivities in Table V and Figure 12 implies that the contact resistance between the particles and the current collector is negligible. Table V shows the charge transfer resistance and the double layer capacitance of each fit in Figure 13. The symmetric configuration allows the entire cell to be modeled by one equivalent circuit (Figure 7) in series with a membrane; each electrode should have the same overlapping EIS response. The total capacitance predicted by this analysis is only half that of each electrode, C dl = 2C dl,total , because they are two identical capacitances in series. Conversely, the total charge transfer resistance is twice the charge transfer resistance of each electrode, 2R ct = R ct,total ; two identical resistances in series. It should be noted that the double layer capacitance values in Table V are estimates that are included simply for complete disclosure of the model fits applied in Figure 13. Determining these values is inherently difficult when a fast redox couple is present due to the fact that the capacitive effects never truly dominate the impedance response (the magnitude of the phase angle is never above four degrees) and the limited frequency range to which the model is applied.
Direct current operation of a Fe 2+/3+ slurry electrode.-The EIS analysis developed above can be used to predict the performance of Table V. Results from the impedance model as shown in Figure 7 in series with hardware inductance and membrane resistance, applied to the data shown in Figure 13. For these fits, the membrane (Nafion 1035) resistance is fixed at 0.5 · cm 2 and the ionic phase conductivity is fixed at 130 mS/cm corresponding to the electrolyte as indicated in Table III  a slurry electrode with a redox reaction at a constant current. Using the results of Table V, Figure 10, and Figure 11, the different contributions to the overall polarization can be investigated. Figure 14 shows the polarization of an individual slurry electrode operated at 550 mL/min with 5.8 vol% MWCNTs in 1.0 M FeCl 2 , 1.0 M FeCl 3 , and 1.0 M NH 4 Cl in the symmetric cell. Assuming the oxidation and reduction reactions occur at a similar rate, the overpotential of each electrode should be equal. The overpotential of the entire cell is then the sum of both electrodes and the membrane. All overpotentials are modeled as linear and are defined by Ohm's law (using the resistances in Table V, Figure 10, and Figure 11). The linear model is justified by the polarization behavior seen in Figure 14. In addition, the charge transfer overpotentials (see below) have been observed to be < 20 mV (curve f in Figure 14 and Figure 13); this implies that the kinetics can safely be assumed to be linear. The low frequency overpotential is the combined effects due to mass transfer and the advective current. These were calculated as the difference between the low frequency limit of the experimental data and the model fits in Figure 13. Table VI shows the overpotential breakdown of the symmetric ferric/ferrous cell at 200 mA/cm 2 . Based upon the analysis presented above, some interpretation of the effects of flow rate and channel dimensions can be afforded. As Equation 2 predicts, doubling the channel gap doubles the ohmic overpotential while the flow rate has no effect on the high frequency ohmic behavior. These effects are both seen in the results in Table VI for η Slurry-IR . The active area, also unaffected by flow rate, can increase as the volume of the slurry in the cell increases, as shown in Equations 10 and 11, and can there- fore cause the charge transfer overpotential to decrease as the gap is increased. This is seen in the results for η CT . The low frequency overpotential is a combination of the mass transfer and the advective current effects. Both of these overpotentials decrease with increased linear velocities. Increasing the channel gap with the same volumetric flow rate decreases the linear velocity and will therefore increase the low frequency overpotentials. Again, this can be seen in Table VI for η LF . As indicated by Figure 10 and Figure 11, the distributed overpotential is a function of both the charge transfer resistance and the ohmic resistances. Because increasing the channel gap can increase the active area (increasing the double layer capacitance and decreasing the charge transfer overpotential), the distributed overpotential may decrease. However, increasing the channel gap also increases the ohmic overpotentials which increases the distributed resistance. In Table VI, the distributed overpotential is seen to increase with increasing channel gap, suggesting the ohmic factors are more significant. Using this information, the voltaic efficiency of a MWCNT slurry electrode supporting the Fe 2+/3+ redox reaction can be estimated using Equation 10. The standard potential of the ferric/ferrous couple, E o , is +0.77 V vs RHE. 42 A slurry with 5.8 vol% MWCNT flowing through a 1 mm gap at 200 mL/min operating at 200 mA/cm 2 has 82% voltaic efficiency.

Conclusions
Techniques for interpreting electrochemical impedance spectroscopy results to characterize different flowing slurry electrodes configurations were presented based upon models developed for porous electrodes. These models were discussed with regards to three different slurry systems; particles in deionized water, in supporting electrolyte without redox active species (electrochemical flow capacitor), and in electrolytes with redox couples (redox flow batteries). Through investigating each of these systems, the individual properties of slurry electrodes were determined. It was found that traditional overpotential descriptions, (ohmic, activation, and mass transfer) were insufficient to fully describe the impedance and polarization of the slurry electrodes.
In addition to the traditional overpotentials associated with electrochemical electrodes, there exists an overpotential due to the distributed nature of the current between the electronic and ionic phases of the slurry. This overpotential is a significant contribution to the cell polarization on the order of both the charge transfer and mass transfer overpotentials. It was found that this distributed overpotential is a function of the slurry electrode charge transfer resistance as well as the ratio of the electronic and ionic phase conductivities. Reducing the charge transfer resistance of the slurry electrodes to < 10 −4 · cm 2 will lower the distributed overpotential significantly (over a factor of three for the MWCNT slurries).
With the ferric/ferrous redox couple, MWCNT slurry electrodes were found to operate at over 80% voltaic efficiency at 200 mA/cm 2 . The electrode polarization was dominated by the ohmic losses in the slurry electrode. By increasing the slurry electronic conductivity, the performance of these electrodes can be significantly improved.