Similarities and Differences between Potential-Step and Impedance Methods for Determining Diffusion Coefficients of Lithium in Active Electrode Materials

If V(t) is the cell voltage as a function of time in a PITT experiment, we will denote its Fourier transform as

In general, we will use a tilde to denote the Fourier transform of any function of time. For conventional PITT experiments, the voltage is given as

The Fourier transform of the current response is given by

where Z(ω) is the impedance and A(ω) is the inverse of Z(ω), that is, the admittance. To calculate the current response, we employ the inverse Fourier transform

The second of equations 4 is obtained by noting that is the complex conjugate of , a fact which holds for any real-valued function of time. It should be noted that numerical evaluation of the integrals in equation 4 can be very challenging. The current discontinuity when the voltage is stepped results in a very slow convergence of the integral at large frequencies. One of the standard tools to evaluate such integrals is the Fast Fourier Transform (see below). It is also helpful to look at special cases, for which the exact answer is known, so that it can be compared to the numerical results to assess accuracy. An example of this is given below as well.

Several other issues also impact the accuracy that is possible in converting PITT data to impedance data via Fourier transform and vice versa. First of all, impedance can only be measured over a finite frequency range, whereas, at least in principle, arbitrarily high and low frequencies are needed for the inverse Fourier transform. For our experiments, the impedance was measured within the range of ω_low/2π = 5 mHz to ω_high/2π = 100 kHz, where ω is understood to be in radians per second. We would like to know if this frequency range is large enough to recover the current by means of equation 4. Two different issues determine the answer to this question. The first is the logging rate of current data in the PITT experiment, and the second is the characteristic frequencies for the different transient effects that occur in the cell after the voltage step.

The impedance spectrum and PITT response of a graphite half-cell with lithium counter and reference electrodes, which we shall consider in this work, have been analyzed in some detail.¹ A characteristic set of material values, taken from the literature, were used for simulations, and numerical calculations of impedance were then done and shown in Figure 1 of Reference 1. The Nyquist plot of impedance, taken from Reference 1 and based on the values shown in Table I, is shown in Figure 1 of this paper as the green curve. Below it is a plot of minus the imaginary part of impedance as a function of frequency, in order to illustrate the frequency ranges in which various effects take place. One sees that the relaxation of the double layer occurs in the highest frequency range. Below this range, the double-layer is in quasi-steady state, and it is only the charge-transfer resistance, combined with ohmic resistance in the solid and electrolyte phases, that is observed.^1–5 Salt diffusion impacts impedance in the intermediate range. Below this range, the salt diffusion is also in quasi-steady state, and its contribution to impedance appears as a multiplicative factor on the ohmic resistance in the electrolyte.¹ In the lowest range, intercalate diffusion of lithium in graphite is occurring. Note that there is some overlap in the ranges in which salt and lithium intercalate diffusion are occurring.

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Since the primary interest in PITT experiments is not the interfacial capacitance, a low data logging rate for current can be used. When this is done, relaxation of the double layer cannot be observed in the resulting current data, because the current is already in quasi-steady state by the time the first point after the voltage step is logged. Calculations by Ong and Newman⁴ indicate that double-layer relaxation occurs on the order of 20 ms. Reference 1 shows that the salt diffusion has usually reached quasi-steady state after about 30 seconds. The frequency range in which salt diffusion impacts impedance is easily within access of conventional impedance measurements. The intercalate diffusion starts to dominate at frequencies below 1 to 5 mHz, which is a problem, because impedance measurements below this frequency, though desirable, are difficult and time consuming to obtain using conventional instrumentation. The range between 5 mHz and 100 kHz can capture the first two transient effects but it is questionable how accurately it can reproduce the transient behavior in a PITT experiment resulting from intercalate lithium diffusion. It is thus helpful to look at a special test case, where exact solutions to the integrals in equation 4 are known, in order to assess the contributions of below 5 mHz to the inverse Fourier transform. Such a case, described below, is given by the dashed red line in the the Nyquist plot shown in Figure 1.

Simplified models, which ignore voltage losses occurring in the electrolyte, as well as ohmic losses in the solid phase, and focus on lithium diffusion losses in the solid phase, have been used to extract solid-phase diffusion coefficients for lithium in host materials.^6,7 These models usually assume, in addition, that double-layer relaxation is in quasi-steady state and can thus be ignored. Such models can be applied, for example, to very thin electrodes constructed with very large intercalate particles. The appendix of Reference 8 derives a set of inequalities that must be satisfied by the electrode parameters for such a model to be valid. However, we will argue here that this model also provides a useful first approximation in situations where electrolyte and solid-phase ohmic losses are not negligible. As will be seen below, it is only necessary to lump these losses together with the charge-transfer resistance in order to fit the model to an impedance spectrum.

The electrode performance can be characterized by a set of resistances, each of which corresponds to a different transport mechanism in the electrode. The kinetic, or charge-transfer, resistance R_k and the intercalate diffusion resistance R_d were defined in Reference 1 as

Necessary definitions are given in The List of Symbols placed near the end of this paper. In particular, the Biot number

is defined as the ratio of intercalate diffusion resistance to charge-transfer resistance. The impedance of the simplified model¹ is given as

Shown in Figure 1 are two curves. The solid green curve (labeled "unsimplified") is the numerical solution to the full system of equations as described in Reference 1. The dashed red curve (labeled "simplified") corresponds to equation 7. For both cases, the parameter values are given in Table I. For the simplified model, the charge-transfer resistance R_k was increased to the value of about 15.84 Ohm-cm² so as to match the full spectrum as closely as possible in the low frequency regime, where intercalate diffusion is significant. (This also results in a changed value of B = 0.199.) The charge-transfer resistance must be increased for the simplified model to better represent the full model, as it now represents a lumped value that includes electrolyte and ohmic solid phase resistance as well as charge-transfer resistance. For an example of this approach, see equation 3 of Levi et al.⁹

Two different expressions for the current i(t) after a voltage step, corresponding to the impedance in equation 7, are available.^1,6 At short times, the dimensionless current is given as

where the dimensionless time is given as

where t has the units of seconds, and the time constant has been taken from the values in Table I. At long times, it is given as

λ₁ = 0.7575 is the smallest positive solution to the eigenvalue equation

A plot of F_PITT(τ) is shown in Figure 2. There is an overlap region of times, in which both solutions agree so well with each other that they cannot be distinguished in the resolution of the plot. For τ ⩽ 0.15, we use the short time solution, and for τ > 0.15, we use the long time solution. The point τ = 0.15 was chosen to minimize the difference between the two solutions. Note this means that, for the majority of the time we are interested in for most applications (τ > 0.15), we are using an exponential decay rate to describe the current.

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One can show¹⁰ that

In the above, δ(ω) is the Dirac delta function. Since the admittance satisfies A(ω = 0) = 0, it follows from equation 3 that

In dimensionless form, the lower limit imposed on integration due to impedance measurements is , corresponding to the frequency 5 mHz. To assess the errors inherent in restricting the integral in this way, we make a plot of F_PITT, which is equal to the inverse Fourier transform of the impedance equation 7 via equation 4, and compare it to a plot of

Equation 14 is the inverse Fourier transform when the admittance values at frequencies less than are set to zero, since they were not measured. We do the calculation in this way, because it avoids evaluating the Fourier integral, equation 4, at high frequencies, where convergence is difficult. Figure 2 shows a comparison of F_PITT with equation 14. It seems that most of the frequency content in the PITT data is at or below 5 mHz. To further illustrate this, we compare F_PITT with the function

This is the inverse Fourier transform, where the upper frequency limit is now restricted to 5 mHz. The comparison is shown in Figure 3.

Figure 3 also shows plots of the inverse Fourier transform of , as given in equation 13, when lower frequency limit is set to 0.1 mHz or to 0.01 mHz ( or ) instead of . The comparison shows that one needs impedance information down to at least 0.01 mHz in order to be able to approximate F_PITT(τ) with an inverse Fourier transform.

This then is an important difference between PITT and impedance measurements. It is very difficult to make accurate impedance measurements in the low frequency range that is impacting PITT measurements. In light of this, the question arises as to whether or not PITT data can be used to extract low frequency impedance data when direct measurement seems to be too difficult or cumbersome. Again, the same issues seem to impact the answer to this question. In order to do a Fourier transform of the PITT data, one needs data at infinite times, both positive and negative. If one restricts oneself to a finite time range, what impact will this have on the measurements? To answer this question, we look at the dimensionless time period 0 ⩽ τ ⩽ 2 for the above electrode, as shown in Figure 2. We will assume that outside this range of dimensionless time the current is zero. The Fourier transform of this then becomes

Figure 4 shows a comparison of dimensionless impedance, based on a calculated Fourier transform of current

with the exact answer

It appears that the fact that the PITT data has not gotten close to zero at τ = 2(cf. Figure 2) may be influencing the accuracy of the result. Figure 5 shows the same plot, when the current integrals are evaluated out to τ = 5. Here one sees that the agreement is much better. It appears then that one must extend the PITT data out to the point where the current almost becomes zero before one can expect to accurately obtain Fourier transforms.

In summary, most of the spectral content in a PITT experiment is at frequencies below 5 mHz. Because impedance data at such low frequencies is so difficult and time consuming to measure, one should not expect to be able to use impedance measurements and the inverse Fourier transform to simulate PITT currents. Conversely, in order to use PITT measurements to calculate impedance data, one needs to extend the measurements out long enough so that the remaining current is a very small fraction of the initial current. For the parameter values we have used in these simulations, the required time for that is more than 5000 seconds. In addition, one would have to demonstrate the repeatability of these PITT measurements and show that the voltage step was taken only after sufficient time had passed so that the entire electrode, including the interiors of all of its intercalation particles, was equilibrated.

The situation becomes different, however, if the voltage step occurs not once, but rather in a periodic fashion with period T. In such situations, the voltage can be expanded as a Fourier series, and the smallest non-zero frequency that will arise in this expansion is 1/T Hz. The current response will also become periodic, but it approaches periodicity only in the limit of many cycles. We were thus led to consider voltage stepping that occurs with a period of T = 200 seconds, corresponding to a lowest frequency of 5 mHz for impedance. The periodic voltage function used in the experiments to be described had the form

The switching time of T/4 was chosen somewhat arbitrarily; the intention is to give the electrode enough time to substantially re-equilibrate before starting the next cycle. We can write the series expansion

Note that, because current and voltage are both real-valued, it follows that

where the overbar on the right of the above equation denotes complex conjugation. In addition, since A(ω = 0) = 0, it follows from equation 3 that , so that the dimensionless periodic current is then given as

where

Conversely, if the impedance is to be determined, the Fourier coefficients for current satisfy

for n > 0 and Ts = t, so

At this point a note of caution must be given because, from the choice of V(t) given in equation 19, it follows that when n is divisible by 4. Under these circumstances, equation 25 cannot be used to determine impedance, because both the numerator and the denominator on the right hand side vanish.

Figure 6 shows two estimates of the dimensionless current F_PITT(τ), calculated from the series approximation based on the first 100 and first 1000 terms of equation 22, assuming B = 0.199. The impedance in equation 22 is given in equation 7. Clearly most of the spectral content in F_PITT(τ) is contained in the first 100 harmonics, but higher order frequencies are needed to capture the behavior at the voltage discontinuities. For a period T = 200 seconds, with the lowest frequency of 5 mHz, the 100^th harmonic has a frequency of 0.5 Hz and the 1000^th harmonic has a frequency of 5 Hz. Measurement of impedance data at these upper frequency limits is easily done. The decay rate shown in Figure 6 is quite small, but the voltage switch occurs after only 50 seconds, whereas, in contrast, the dimensioned time scale shown in Figure 2 is 2000 seconds.

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The Discrete Fourier Transform (DFT) and the Fast Fourier Transform (FFT)

Fourier coefficients for periodic data of the type given in the previous section are usually calculated by means of the discrete Fourier transform. The data is assumed to be sampled at a rate Δt = T/N, for some large number N, and the periodic function is discretized in the form

The actual measured data can be linearly interpolated, if its values at these particular times are not available. The transform and inverse transform of this data are given as

The Fast Fourier Transform is an algorithm that evaluates both the DFT and its inverse in a fast and efficient manner. Note that

If Nis an odd integer, it follows that one can rewrite the second of equations 27 as

When using standard algorithms for the DFT and the FFT, it helps to keep in mind that

The formula 29 has the most direct relation to the Fourier series for continuous data, as described below, and it has the most direct connection to impedance data.

One can show that, for |n| ≪ (N − 1)/2, the equation

is a very good approximation, where is the Fourier series coefficient for either positive or negative n, as defined in the previous section. If N₁ is large enough so that the first N₁terms of the Fourier series, for positive and negative indexes, accurately approximate the entire series, the DFT will also give an accurate representation of the first N₁ Fourier coefficients, as long as (N − 1)/2 ≫ N₁. This, in turn, means that the DFT and equation 31 can be used to relate impedance values to time-series data using equation 24, again under the constraint that |n| ≪ (N − 1)/2. In practice, the user usually chooses N large enough so that the resulting values F_PITT(k), as calculated from impedance data, do not change with further increases in N. The estimates given in the previous section serve as in aid in choosing the size of N₁ and thus also of N. Equation 31 should only be used when |n| ≪ (N − 1)/2. Outside of this range, the approximation can become very inaccurate.

Electrochemical measurements were performed in Ar filled glove box by using a Solartron FRA 1260 frequency response analyzer combined with a Solartron model 1287 electrochemical interface at room temperature. A three-electrode cell¹¹ was assembled by employing lithium foil (Sigma-Aldrich, 99.99%) as counter electrode and reference electrode. A graphitic electrode (cut from the anode of a commercial cell used for traction applications), with an area of approximately 1.43 cm² was used as working electrode. The electrolyte solution was 1 M LiPF₆ in ethylene carbonate (EC) and diethyl carbonate (DEC) with a volume ratio of 1:2. Before electrochemical measurement, the assembled cell was rested for 3 hours and then galvanostatically cycled at C/10 rate between 1.0 V and 0.05 V in order to form a stable SEI (solid electrolyte interface) layer on the electrode surface.

The EIS measurement and periodic potential step measurement were conducted at three different open-circuit voltages: 0.114 V, 0.198 V, and 0.968 V, corresponding approximately to 60%, 20% and 0% state of charge. At each state-of-charge, EIS was first measured using a potential-excitation amplitude of 5 mV. The frequency selected was between 100 kHz and 5 mHz, with 10 measurements per decade. After a half hour relaxation, the rest cell voltage was stepped by −5 mV for 50 s and was immediately followed by a potential step of 5 mV with a 150 s duration. The potential steps at 50 s and 150 s were repeated alternately for 5 times without any relaxation in between. The current was measured as a function of time, at a sampling rate of approximately 0.1 s.

Construction of a PITT response from EIS data

Figure 7 shows Nyquist plots of the measured impedance of the half-cell at the three different open-circuit voltages (OCVs) and States of Charge (SOC). In the DFT analysis, N = 40001, and the impedance data were linearly interpolated between the measured frequency points. The discrete time values for transformed current are then given at intervals of T/N = Δt = 200/40001 ≈ 5 ms, but only every 20-th data point was plotted for the current calculated based on impedance. This corresponds to time intervals of about 0.1 s, which was the approximate sampling rate of the data. The periodic voltage steps were input into the calculation in discrete form and a DFT was used to calculate the discrete transform of the voltage. The impedance was then used to calculate the discrete transform of the current, based on knowledge of the impedance, using equation 24. As already noted, it is important to restrict n to the range − (N − 1)/2 ⩽ n ⩽ (N − 1)/2 when calculating the frequencies at which impedance must be used to calculate . For example, equations 21, 24 and 29 imply that F_PITT(k) will be real valued, whereas this will not generally be the case if the second of equations 27 replaces equation 29. Once was calculated, it was inverse transformed back into a discrete time series i_n. The results were then compared with measured currents for the first five voltage cycles, as described above.

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Figure 8 shows a plot of the first five cycles at an OCV of 0.114 V. One sees that the current response to the voltage has not yet reached a periodic steady state in the first five cycles, since the curves for each cycle do not lie on top of each other. As previously noted, the Fourier coefficient

for any periodic current response, because the admittance A(ω = 0) = 0. This means that the average current per cycle should be zero. The fact that the current has not reached a periodic steady state can be seen from a plot of the average current per cycle as a function of cycle number, shown in Figure 9. This plot suggests that one could correct the measured currents by subtracting off their average values to make them closer to the periodic currents they would eventually become after many cycles. It is likely that the decay rate to a periodic solution is slower at lower frequencies and, if this is true, subtracting off the average current would eliminate the slowest decaying component in these measurements. The curve labeled "measured 5-avg" in Figure 8 does exactly this, and approximates very closely the "simulated" curve, which was calculated using impedance data.

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Figure 10 and Figure 11 show corresponding plots at OCV values of 0.198 V and 0.968 V. The corrected curves for the fifth cycle give very good agreement with the curves calculated based on impedance measurements at the OCV's of 0.114 V and 0.968 V. Although the agreement at 0.198 V is not quite as good, it still indicates a surprising level of agreement. One last aspect of this data should be mentioned. The average currents at OCV = 0.968 V are positive, instead of negative, and are increasing with each cycle. This seems to contradict the assertion that the average currents decay to zero as the cycle number increases. Because the amplitude of the voltage perturbation is so small, the system response should be linear. It follows that the average current over each cycle should be unchanged if one removes all higher frequency components in the voltage signal and leaves only the component at zero frequency. This would correspond to a PITT experiment in which the voltage is stepped by ΔV/4 and remains there for an infinite time. Since the current response to such a voltage step (ΔV = −5 mV) should be negative, the average currents should also be negative. We are thus forced to conclude that there is a small offset error in where zero current lies in these measurements. Judging by the average measured current at OCV = 0.968 V for the fifth cycle, this offset error could be as small as 3 μA.

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Construction of an EIS response from PITT data

We turn now to the task of using periodic PITT data to calculate impedance data. The procedure consists of using the formula

where is obtained by applying the DFT to the measured current data, which is again linearly interpolated to produce the discrete time series i_n. Measured voltage data corresponding to the measured current data were not recorded, and the corresponding discrete time series of voltage V_n must be chosen carefully before transforming it, if one is to avoid introducing artifacts in this calculation. The current was measured at time intervals of approximately 0.1 second, and then linearly interpolated to produce a time series of N = 40001 points at evenly spaced time intervals of about 0.005 s. Best results for impedance were obtained when the voltage was first set at the same measured times taken for the current data and then linearly interpolated in the same fashion as was the current. This means that both current and voltage have ramps of about 0.1 seconds when switching occurs, and it is important to make sure that the ramps for both current and voltage coincide in time.

As previously noted, in theory the voltage transform when n is divisible by 4, so that theoretically as well. However, because neither the voltage nor the current switched instantaneously, these harmonics did not vanish exactly, but instead only became too small to use in calculating impedance. Impedance values were thus calculated only for those harmonics where n was not divisible by 4.

Figure 12 shows a comparison of the measured impedance to the impedance calculated by transforming the fifth current cycle in the periodic PITT experiment at an OCV of 0.114 V. Also shown in the same figure are plots of the real and imaginary parts of impedance as functions of frequency for all five cycles as well as for the measured impedance. The calculations of impedance are clearly evolving with each successive cycle, since the current has not yet reached steady state. Furthermore, the calculations show more oscillation at lower frequencies in the earlier cycles. In the frequency range above 1 Hz, the calculations are saturated by noise, which is not surprising, since the sampling rate for current was approximately 0.1 s. A small difference at low frequencies between the measured impedance and the one calculated from the fifth current cycle is apparent, and it is interesting to calculate periodic PITT current from both of these impedances in order to see how big the corresponding differences are in the time domain. The measured impedance was linearly interpolated at the frequencies and, at those frequencies where n is divisible by 4, the calculated impedance at was linearly interpolated between its values at and . A special case was made when n = 0, where was set to the mean of the measured current for the fifth cycle in order to better match the measured current, as previously discussed. Figure 13 shows a comparison between the measured current and the currents calculated from both the measured and the calculated impedance. The comparison indicates that the differences that can be seen between the measured and calculated impedance become almost indistinguishable, when impedance is converted to a periodic PITT response. Plots of measured and calculated impedance data at the OCV of 0.198 V are shown in Figure 14, and at the OCV of 0.968 V are shown in Figure 15.

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Before discussing the comparisons between calculated and measured impedance and periodic PITT data, some general discussion of the measured impedance spectra seems warranted. Figure 16 shows the imaginary part of measured impedance plotted as a function of frequency for all three OCVs, so that one can determine what frequency ranges correspond to the major features. One sees that the major features at each OCV occur in approximately the same frequency ranges. The lower frequency bound of the first loop occurs at approximately 100 Hz, and this is what one would expect from double layer relaxation at the particle interfaces. The lower frequency bound of the next loop occurs somewhere between 0.1 and 1 Hz, which corresponds well to the frequency at which salt diffusion should start to reach quasi-steady state.¹ However, there is a significant difference in the size of this second loop at 0.968 V as compared to the other two voltages. We know of no reason why the size of the salt diffusion loop should exhibit such a large dependence on SOC, and this raises questions about the proper interpretation of this feature. Other authors^16–18 have speculated that one of the loops is a relaxation of capacitance across an SEI, but not enough seems to be known about transport across an SEI to be able to prove or disprove such a theory. Further investigation is needed to better understand the cause of this feature.

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The low frequency ends of all three spectra should be dominated by solid-phase diffusion, but here again a problem emerges, because the phase angle exhibited by the Warburg behavior in the experimental data is often different from 45 degrees, whereas the phase angle predicted by models is 45 degrees, at least before the Nyquist plot turns up to become vertical. This behavior is seen, not only in predictions from the simple model given in equation 7, but also in predictions by more sophisticated models which include other transport effects that were neglected in the simplified model.¹ There are many possible explanations for such behavior, as discussed by Hirschorn et al.¹⁹ The porous electrode structure that is assumed in standard porous electrode theory assumes particles sizes that are much smaller than the electrode thickness, whereas the ratio of particle diameter to electrode thickness for these electrodes is on the order of 1/5. Ong and Yang²⁰ attribute such behavior to the use of a lithium counter electrode, and they present data using a graphite cell for both working and counter electrodes, which exhibits 45 degree behavior. In light of the difficulties that this phase angle presents when trying to match models to data, further research into this issue is needed.

Of the three OCVs considered above, the deviation of the phase angle from 45 degrees is most significant at 0.968 V. For illustrative purposes only, we will present a fit of the impedance model given by equation 7 (the simplified impedance expression) to the impedance data at 0.968 V. The impedance model can then be transformed into simulations of the periodic PITT data so that one can compare the accuracy of the model in both the time and the frequency domains.

Figure 17 shows a fit of this model to the measured impedance data. The large phase angle of the measured data forces one to set the resistance R_k close to zero, if the model and measured impedances are to come close to each other at the lowest measured frequency. Both the model and the measured impedance data were then converted via Fourier transform into simulations of periodic PITT data. In doing this, 40001 points were used in the DFT, in order to maximize the accuracy of the transformation, but the resultant currents were only plotted at intervals of 0.1 s, in order to better compare to the measured data sampling rate. Figure 18 shows the current density calculated from the measured impedance, from the model impedance, and the measured current density. In spite of the large discrepancies between the model fit and the impedance data (Figure 17), both do a surprisingly good job at predicting the measured periodic PITT current density. However, a close look at the region from 48 to 55 seconds, where a voltage step occurs, shows quite a different picture. Here one sees that initially the model predictions are very inaccurate, but they converge close to the measured current density (and the prediction based on measured impedance) within about 1 second. This is not surprising, because the largest discrepancy between the impedance model and the data is occurring at frequencies above 1 Hz, and the fit was chosen to match the measured impedance in the range of 5 mHz.

Although the model and measured impedance match approximately at low frequencies, Figure 17 shows that the two curves in the Nyquist plot are already starting to diverge at the lowest frequency shown (5 mHz). This is due to the different slopes of each curve. If one had impedance data at significantly lower frequencies than 5 mHz, the Nyquist plot comparing model and measured impedance could be extended and would presumably show an even greater divergence of the two. The lower frequency impedance data would enable a comparison to periodic PITT data at periods greater than 200 seconds, and the divergence of the model and measured impedance data at low frequencies would presumably also be detectable at longer times in the PITT data. However, at these large times, the current densities may be so small that the differences would be difficult to determine. Li et al.⁶ compared the long-time model (equation 10) to data at times scales of 2000 s and the short-time model (equation 8) to data at much shorter time scales. This procedure was also done using a different model,²¹ where Levi et al.⁷ found that the long-time and short-time fits did not match at SOCs greater than about 0.7, although agreement was obtained at lower SOCs.

Finally, we note that both impedance and PITT measurements should be done starting with an equilibrated electrode, and true equilibrium is difficult to attain in lithium batteries. This can be seen from the discernible hysteresis in OCVs measured at low C-rates during lithiation and delithiation. Since the measurements reported here have a frequency content greater than or equal to 5 mHz, an approximate equilibrium is sufficient to perform this analysis, provided that the residual changes in the rest state are occurring at frequencies much lower than 5 mHz. The close agreement in the comparisons presented in Figures 8–11 provides a form of empirical validation that the rest states were sufficiently close to equilibrium.

In theory, PITT and impedance measurements contain the same information and can be transformed into each other using the Fourier transform. In practice, these measurements exhibit differences. For the simple example we have used for illustration, the main frequency content in a PITT experiment lies in the range from 0.01 to 5 mHz, and it is very difficult and time consuming to measure impedance at these low frequencies. The two different types of experiments become more comparable if one restricts attention to periodic PITT measurements, where the voltage is cycled with a period T that is short enough so that the lowest frequencies appearing in the measured PITT data are in the range which can also be measured directly by EIS. For this purpose, we have chosen a period of 200 seconds, with a fundamental frequency of 5 mHz. The tool for analyzing periodic PITT data is no longer the Fourier transform, but rather the Fourier series and the discrete Fourier transform. It is possible to then convert impedance measurements into periodic PITT data, which can be compared with actual periodic PITT measurements, and conversely to convert periodic PITT measurements into impedance data, which can be compared with actual impedance measurements. We have done this for a graphite half-cell at three different states of charge: zero SOC (OCV = 0.968 V), 20% SOC (OCV = 0.198 V) and 60% SOC (OCV = 0.114 V). For each state of charge, the periodic PITT data was cycled 5 times, and it was observed that, even after five cycles, the data had still not reached a completely periodic steady state. In particular, the average current over each cycle is nonzero, although for a completely periodic cycle it should be equal to zero. By correcting for this discrepancy, however, the impedance data was able to reproduce the periodic PITT data quite well, and for two of the three cases, it reproduced it almost to within the resolution of the plots.

When converting periodic PITT measurements into impedance data, the sampling rate of the PITT measurements plays a critical role. For the sampling rate of approximately 0.1 s used in these measurements, the impedance data can be reproduced for frequencies between about 5 mHz (the fundamental frequency) and 1 Hz. At higher frequencies, the interpolation of the sampled data introduces a large amount of noise into the calculations. Here again, one sees that the impedance calculations evolve with each cycle, due to non-periodicity of the PITT measurements, and earlier cycles exhibit more noise than later cycles. Even at frequencies for which the signal to noise ratio is low, some differences can be observed between the calculated and measured impedance. In this context, care was also taken to make sure that the impedance measurements at low frequencies were taken after at least five cycles, but no difference was observed between the impedance measurements at five and at fewer cycles. Although some differences were observed at 0.114 V OCV between the measured impedance and the impedance calculated from the periodic PITT current, when each of these impedances was converted back into a periodic PITT response, the differences became almost indistinguishable. We note that this apparently greater sensitivity to small differences in the impedance domain can place a higher standard for data reproducibility on impedance data than on PITT data.

The use of a simple model^1,6 (see also equation 7) to fit solid phase diffusion coefficients to measured data illustrates some of the pitfalls of this process. For the example considered (0.968 V OCV), the model was able to reproduce the periodic PITT data quite well, even though the match between model and measured impedance showed a poor level of agreement. The disagreement between periodic PITT data, measured and model, only becomes apparent within approximately the first second after a voltage step occurs, because the largest discrepancies in the frequency domain are occurring at frequencies higher than 1 Hz. When matching data in the time domain to a model, there is a tendency to focus primarily on errors that occur on the largest time scale of the data, in this case 200 s. In fact, this is precisely what a standard least squares algorithm will do, if the data points are uniformly distributed throughout the entire measuring time. Impedance data, however, gives a more uniform weighting of errors on very different time scales, and comparisons of this type can sometimes expose fundamental errors in the data fitting process. For the example considered here, the problem stems from the fact that the constant phase region of the measured data in the frequency range where solid phase diffusion is dominant varies significantly from the 45 degrees predicted by the model. As a result of this, it becomes very difficult to match the model to data over a large frequency range. One way to try to detect this problem, when working with non-periodic PITT data, is to try to match the model over two very different time scales, one short term and one very long term, to insure consistency between results from the two different time scales. A better understanding of the causes of this phase-angle deviation from 45 degrees could potentially enable more accurate fitting of diffusion coefficients to measured data.