Collective Phase Transition Dynamics in Microarray Composite

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Collective Phase Transition Dynamics in Microarray Composite LixFePO4 Electrodes Tracked by in Situ Electrochemical Quartz Crystal Admittance Mikhael D. Levi,*,† Sergey Sigalov,† Gregory Salitra,† Prasant Nayak,† Doron Aurbach,*,† Leonid Daikhin,‡ Emilie Perre,§ and Volker Presser§,∥ †

Department of Chemistry, Bar-Ilan University, Ramat-Gan 52900, Israel School of Chemistry, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Ramat Aviv, 69978, Israel § INM − Leibniz-Institute for New Materials, D-66123 Saarbrücken, Germany ∥ Department of Materials Science and Engineering, Saarland University, D-66123 Saarbrücken, Germany ‡

S Supporting Information *

ABSTRACT: A novel approach to tracking intercalation-induced phase transitions in Li-ion battery materials demonstrated herein consists of simultaneous analysis of intercalation charge and the accompanying mechanical (geometric) changes in a microarray electrode composed of LixFePO4 intercalation particles probed by the electrochemical quartz-crystal admittance (EQCA) method. A recently elaborated approach to population dynamics of active (phase-transforming) nanoparticles1 has been used here for modeling current transients applying small potential steps to LixFePO4 electrodes. The number fraction of (phase) transformed particles thus calculated was directly compared with the changes in the effective thickness and permeability length of the electrode coating derived by EQCA. Geometric changes of thin active mass originating from different molar volumes of the parent and transformed phase result in nonuniform deformations of intercalation particles. This study confirms the collective behavior of LixFePO4 intercalation particles during electrochemically induced phase transition. The use of EQCA as a highly precise and sensitive probe of mass and geometric changes in the electrode layer of intercalation particles paves the way for dynamic in situ studies of nonuniform intercalation particles deformations which can hardly be assessed by other available techniques.

1. INTRODUCTION

density of screening electrons, interaction of elastic dipoles through a strain-mediated field in the host matrix upon intercalation,3 and/or volume change induced elastic interactions of intercalation particles linked through a concentration-dependent stress and strain field.6 These interactions have been therefore regarded as a thermodynamic origin of the hysteresis on charging/discharging curves of intercalation electrodes.3,5−7) The transition between low- and high-density phases (separated by a maximum on the free-energy curve) was considered in the framework of two alternative relaxation mechanisms such as the electronic and/or ionic interfacial charge transfer, or the passage over the energetic barrier by means of a fluctuation process (e.g., small droplets (nuclei) formation).5,8 Note that this approach was elaborated for macroscopic electrodes in the form of thin films or porous electrode layers composed of individual intercalation particles. The equilibrium electrochemical characteristics of such porous

Electrochemically induced phase transition is a physicochemical phenomenon, often appearing during charging/discharging of various energy storage materials such as electronically conducting and conventional redox-polymers2 or a vast class of Li-ion battery (LIB) electrodes.3,4 A general understanding of the nature of phase transition in these electrodes was initially reached on the basis of (mean-field) lattice gas models with highly attractive short-range electron−ion interactions in the host bulk when guest ions are extracted/inserted during the electrode’s charge/discharge.3 The concentration-independent interaction constant enters the expressions of nonequilibrium electron-ion free energy of the current collector/electrode layer/solution system, and hence of the related chemical potential. A maximum on the compositional dependence of the free-energy curve resulting in N- and S-shape dependences of the chemical potential vs composition and charging degree vs potential, respectively, appears when exceeding a certain critical value of the interaction constant.5 Highly attractive short-range interactions have been suggested to be a consequence of a combination of several effects such as Friedel oscillations in the © 2013 American Chemical Society

Received: April 13, 2013 Revised: June 29, 2013 Published: July 3, 2013 15505

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characteristics of the intercalation particles. The feasibility and usefulness of this technique for this purpose have been recently demonstrated.25 Extending the EQCA approach to small potential perturbations of LixFePO4 electrodes, we apply the statistical theory of activated intercalation particles1 not only to describe the related current transients, but also for their direct comparison with the simultaneously measured changes of the geometric characteristics of the electrodes’ active mass (i.e., a thin layer of particles). Specifically, the emphasis here is on how the number of transformed particles (Nt) affects the geometric characteristics of the electrodes’ active mass due to collective changes in the particles’ volume and the resulting particle deformation. Such important in situ information cannot be easily assessed by any technique used so far for the characterization of LIB electrodes.

electrodes (e.g., capacity) were considered as the arithmetic average of that for the individual intercalation particles.9 At a larger deviation from equilibrium, using, for example, relatively high currents during galvanostatic charging/discharging, finite electronic resistance of the composite electrode, together with finite ionic resistance of the pores filled with electrolyte solution significantly affected transport characteristics in accordance with a simple porous electrode model.10−12 Recently, evidence was provided that such a simplified approach to the characteristics of porous composite electrodes breaks down when the chemical potential of individual intercalation particles becomes a non-monotonous function of the composition (the situation typical of first-order phase transition).3−8 Under such conditions the system consisting, for example, of a microarray of parallel arranged intercalation particles demonstrates “collective” dynamic behavior: the multiparticle electrode is not charged coherently, that is, simultaneously in all particles, but rather sequentially, particleby-particle, once the turning point on the chemical potential vs composition curve is passed.13,14 The main consequence of this collective behavior of interconnected intercalation particles is that the probability of finding the particles with two coexisting phases is smaller than to observe a system of homogeneously charged single-phase particles with higher and lower Li-ion contents.13 This conclusion was in good agreement with high resolution transmission electron microscopy studies (HR TEM) of LixFePO4 electrodes delithiated to obtain Li0.5FePO4 (a transport of one-half electron and Li ion per LiFePO4), where most of the intercalation particles were found in either completely lithiated or delithiated form (Li1FePO4 and FePO4, respectively).15,16 At the same time as the work of refs 13 and 14 a “mosaic instability” of interconnected intercalation particles was discovered17 that can be understood as “the equilibrium state for a collection of particles that can exchange lithium” charged sequentially when the chemical potential entered the miscibility gap (for details see excellent recent review;18 comparison between the mosaic multiparticle and the single intercalation particle behaviors with focus on the coherency strain as a factor determining either phase separation or its suppression is presented in full detail in ref 17). Recent statistical kinetics model of porous electrodes consisting of phase-transforming nanoparticles in porous electrodes1 precludes evolution of the electrodes’ charge following a classical Kolmogorov-Johnson-Mehl-Avrami (KJMA) nucleation mechanism,19−21 according to which the nuclei of the new phase are randomly distributed in a large volume of the initial (pristine) phase activated by small fluctuation of concentration of the inserted ions. The statistical model of interconnected intercalation nanoparticles deals with time evolution of the fraction of the intercalation particles with active phase boundaries, Na, and the remaining fraction of transformed particles, Nt, described as exponential functions of time with two parameters, namely, the activation rate, n, and the rate of filling active particles with Li-ions, m.1 Depending on the relation between n and m, current transients of different shapes can be statistically interpreted including the nonmonotonous current transients with maxima, which were previously ascribed to the KJMA mechanism.8,22−24 In this article, we supplement the study of nucleation dynamics in LixFePO4 electrodes by application of the electrochemical quartz crystal admittance (EQCA) method which, in addition to tracking the intercalation charge, also allows continuous monitoring of the mechanical (geometrical)

2. ESSENTIAL FEATURES OF THE EQCA TECHNIQUE We applied EQCA as a suitable charge/mechanical probe of microarray LiFePO4 electrodes composed of one/a few layers of single intercalation particles in direct contact with (or very close to) the current collector. Figure 1 (bottom) presents a

Figure 1. Typical SEM image of the thin microarray electrodes used herein (bottom). Sketch of the electrode containing also PVdF binder (depicted by small yellow circles) is shown at the top of the figure.

scanning electron microscopy (SEM) image of such an electrode. The sketch in the top of this figure shows that one/a few layer(s) of intercalation particles are rigidly attached with a polyvinyl difluoride (PVdF) binder (shown as yellow circles) to the surface of a thin gold current collector on a quartz crystal sensor. It is important that PVdF is insoluble in the electrolyte solution used as it does not swell in contact with aqueous solutions, ensuring the rigidity of the entire electrode’s coating. The use of microarray coating consisting of one/a few layer(s) of intercalation particles has two advantages over 15506

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in the admittance model by periodical changes of h (E) and ξ (E) as was already demonstrated in our recent paper.25 A variation of the electrode’s potential results in either lithiation or delithiation of the bulk electrode materials during their discharge and charge, respectively. When molar volumes of initial and final phases differ substantially, a phase transformation of intercalation particles is accompanied by nonuniform deformation of these intercalation particles since they are clamped rigidly to the current collector and to each other by the PVdF binder. The difference in the mechanical properties of a composite microarray coating attached to the quartz crystal surface by PVdF binder from that of ideally free (nonclamped) individual particles stems from the fact that the binder may fully or only partially allow the particles to deform, making the resultant deformation highly nonuniform. This has already been reported in earlier studies as the problem of free and clamped surfaces in the ion insertion induced development of elastic deformations in composite LIB electrodes.3 Most important is that these nonuniform deformations are strictly reproducible during consecutive cycling, tracked by EQCA in the form of potential dependent variation of the effective thickness and permeability of the microarray electrodes. Naturally, when the same intercalation reaction occurs in a LiFePO4 monocrystal, the significant variation of the molar phase volumes results directly in a drastic increase of porosity of the emerging phase with a lower molar volume (i.e., FePO4). This process can be tracked by in situ optical microscopy where cracks are also clearly seen in microscopic images.30 The above consideration together with reproducible experimental results obtained in our previous paper25 suggests that EQCA can portray intercalation phenomena in microarray electrodes via the quantities h(E) and ξ(E), or h(t) and ξ (t), depending on the mode of applied electrochemical perturbation. Thus, EQCA relates to those aspects of electrochemistry of LIB materials (such as nonuniform deformations of the electrodes’ active mass, in situ viscoelastic behavior of binders in the composite electrodes) which are not covered by the techniques known so far. The probe-beam deflection31 or the advanced multibeam optical sensor wafer-curvature system32 are used for direct determination of stress development in continuous film coatings on glass substrates or Si-wafers, respectively, rather than for the study of porous composite electrodes.

typical thick porous electrodes’ coatings studied by conventional electrochemical techniques: (i) The loading mass density as is exemplified below in the Experimental section is typically 250-times smaller than that of porous electrodes ensuring a much better resolution of electrochemical response. (ii) EQCA deals with three potential-dependent parameters of Li-ion insertion/extraction reaction in LiFePO4 electrodes measured in parallel, that is, recording intercalation/deintercalation charge, in addition to two mechanical parameters of the microarray electrode layer, namely, its effective thickness, h, and intrinsic permeability, ξ2, probed by EQCA.25 The EQCA method has an extremely high sensitivity both in gravimetric sensing and in tracking oscillation energy damping pathways via shifts in the resonance frequency, Δf (real component of admittance), and in the resonance peak width, Γ (imaginary component of admittance), respectively; see excellent latest review of this methodology in ref 26. The experimentally measured quantities, Δf and Γ, are transformed into a pair of geometric parameters, namely, effective thickness, h, and permeability length, ξ, ascribed to the hydrodynamic solid−liquid layer consisted of a microarray of intercalation particles with solution between them.27,28 These global geometric parameters are determined with the use of an admittance model for a microarray of intercalation particles based on exact analytical solution to the related solid−liquid interactions (SI, eqs S1 and S2, respectively, and refs 27 and 28). The resisting force from non-homogeneous solid acting on the liquid in the layer depends on the permeability length, ξ. Hence, it presents an important characteristic lateral length of the layer. A link between the permeability and the layer’s porosity, ϕ, is given by the empirical Kozeny-Carman equation:29 ξ2 ∝ r2ϕ3/(1 − ϕ)2, where r is the characteristic size of inhomogeneity, and the electrode’s active mass porosity, ϕ, is understood as a volume fraction of the liquid phase in the interfacial intercalation particles−solution layer. It can be easily seen from this equation that high porosity results in an enormous increase in permeability accompanied by viscous dissipation of oscillation energy in the electrode layer, whereas a low porosity drastically decreases the permeability so that the trapped solution contributes to a measured resonance frequency change only. The initial rough/porous structure of the electrode’s microarray coating is of a biographical origin: it is created during gas-assisted spraying of a dilute slurry of intercalation particles mixed with PVdF binder as detailed in the Experimental section. This structure is believed to be preserved when the coated crystal is transported from air into contact with the electrolyte solution since the binder does not swell in the aqueous solutions used herein. The rough/porous structure of the uncharged microarray electrode can be described in terms of hydrodynamic spectroscopy.28 In brief, the frequency shift and the resonance peak width are measured in a series of liquids with different velocity decay length, δ. Application of the hydrodynamic admittance model which portraits the rough/ porous electrodes’ microarray layers, allows determination of global geometric parameters, h0 and ξ0, where index 0 denotes initial values of both the effective layer’s thickness and the permeability length already existing at open circuit potential before application of any electrochemical perturbation to the electrode. Application of a periodical electrochemical signal to the microarray electrodes results in periodical variations of the structure of the electrode’s active mass (a thin layer) reflected

3. ESSENTIAL EQUATIONS OF THE STATISTICAL MODEL OF ACTIVATED AND TRANSFORMED PARTICLES USED FOR TREATING THE CURRENT TRANSIENTS We used a statistical model of interconnected intercalation particles as initially developed in a recent paper.1 The fractions of the activated intercalation particles, Na, and of the transformed particles, Nt, are given by eqs 1 and 2, respectively: Na = C1 exp( −mt ) + C2 exp( −nt ) Nt = 1 − C1 exp( −mt ) −

m C2 exp( −nt ) n

(1) (2)

n denotes the rate of activation of the intercalation particles (equivalent to the probability of formation of active particles per untransformed particles per unit time), whereas m is the averaged filling rate over all active nanoparticles, regarded as the probability to find a fully transformed particle per active nanoparticles per unit time.1 15507

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Figure 2. Li-ion extraction/insertion from/to LixFePO4 electrodes tracked by galvanostatic cycling (I = 25 μA) (A) and by application of small potential steps during charge (panel B) and discharge (panel C); horizontal green and blue lines denote the values of the applied potential.

The pre-exponential factors are defined through m and n and the initial values (i.e., at time t = 0) of Na(0) = No and Nt (0) = N1:1 C1 =

N0m + (N1 − 1)n m−n

(3)

C2 =

(1 − N1 − N0)n m−n

(4)

AgCl/KCl(sat.) was observed. For this reference, equilibrium was reached with respect to SO42− anions in the electrolyte solution. The following simple calculation clearly demonstrates the microarray character of our electrode coating (in addition to SEM image shown in Figure 1). The loading mass of the LiFePO4 composite coating in our experiments was 0.05 mg per electrochemically active crystal surface, or 0.04 mg/cm2. A typical composite LiFePO4 electrode characterized in ref 1 had a mass loading of 5 mg per current collector with 8 mm diameter, or 10 mg/cm2. The ratio of loading masses of the porous and our microarray single-particle electrode coatings is thus as high as 250:1. Such a small mass of our microarray of one/a few layer(s) particles composite electrodes ensures the best resolution of the electrodes’ titration using small potential steps.

Since the active particles that are consisted essentially of the pristine phase are transformed (at a rate equal to m) to the final phase, the transient current, I, measured during application of a potential step, is proportional to Na with the proportionality constant i ̃ assigned to an averaged reaction current over all the active intercalation particles of the electrode.1

4. EXPERIMENTAL SECTION For preparing composite microarray electrodes we used commercially available carbon-coated LiFePO4 powder (Sud Chemie, now Clariant Ltd.). SEM images of a typical pristine electrode and XRD patterns of the related powder are shown in Figure 1 and Figure S1 of the Supporting Information, respectively. The BET specific surface area (determined via nitrogen gas sorption at 77 K) was 19 m2/g which is indicative of a particle size significantly smaller than 1 μm. Gas-assisted spraying of the composite slurries onto the heated surface of 1in. 5 MHz Maxtek crystals was similar to that described elsewhere and results in composite coatings consisted of microarray of one/a few layer(s) of individual particles or their aggregates.25 Aqueous solutions of Li2SO4 were used in which PVdF does not swell. As-prepared LixFePO4 electrodes revealed high electrochemical reversibility in the aqueous Li salt solutions.25,33 All EQCA measurements were carried out with the use of an E5100A High-Speed Network Analyzer in combination with a Schlumberger 1287 electrochemical interface driven by LabView 9.0 software. The latter was employed for data acquisition and processing. During the measurements, the cells were kept at a constant temperature of 25.0 ± 0.2 °C. Other details on the experimental setup and the data processing were similar to that already reported.25,34 Potentials were measured and reported versus a Ag/AgCl/KCl(sat.) reference electrode. In controlled experiments with mixtures of Li2SO4 and Na2SO4, a Hg/ Hg2SO4 sat./0.5 M Na2SO4 reference electrode was used: a constant potential difference −0.460 ± 0.001 V against Ag/

5. RESULTS AND DISCUSSION 5.1. Electrochemical Characterization of LiFePO4 Microarray Electrode Coating. Galvanostatic charge− discharge cycling (Figure 2A) of the LixFePO4 electrodes measured at 4C rate results in a specific electrode’s capacity around 140 mAh/g with a small voltage hysteresis of 40 mV between the charge and discharge plateaus. The shape of the galvanostatic plateau indicates that the Li-ion extraction/ insertion reaction proceeds via a first-order phase transition with a characteristic bump (“overshoot” of the galvanostatic plateau)35 during Li-ion extraction. A significant asymmetry in the shape of the galvanostatic curves during charge and discharge should be noted. The asymmetry becomes even more pronounced when the electrodes are charged by small potential steps (Figure 2B−C): it is seen that Li-ion extraction proceeds, basically as a continuous process during a single (10 mV height) potential step, from 0.21 to 0.22 V. However, clearly a single step of 10 mV is not large enough to continuously stimulate the Li-ion insertion process. Two further 10 mV steps are required to almost fully lithiate the electrode. Thus, the time for lithiation was about twice as long as that required for delithiation. The following information is important for understanding the nature of the cathodic current transients in Figure 2C. Performing the electrode’s titration with an amplitude of 10 mV, we noticed that in the repeated measurements, a first long cathodic current transient always corresponded to a potential of 0.19 V with an accuracy of several mV. When we applied a 15508

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Figure 3. Raw EQCA data (the components of quartz crystal admittance, Δfexp and ΔΓ) for the galvanostatic charging/discharging (A) and the incremental charging/discharging with a small potential amplitude (B and C, respectively); the related electrochemical responses are shown in Figure 2A−C. Δf mass was calculated from the intercalation charge using Sauerbrey’s equation. Only major potentials steps are indicated in panels B and C.

discharging of the electrode (the related curves are presented in Figure 2A−C). Figure 3A shows the time dependence of a fraction of the experimental frequency shift, Δf mass, calculated from the intercalation charge, Q, using Sauerbrey equation, Δf mass = −[(Q·Mi)/(F × 106)] × 56.6, where Mi is the mass of the inserted/extracted Li-ions ions, and the sensitivity factor of the AT-cut quartz crystal at 25 °C was taken as 56.6 Hz cm2/ μg. Similar calculations were carried out to determine Δf mass of the electrodes during their potentiostatic titration for the evaluation of the difference, (Δfexp − Δf mass), which together with the measured Γ were translated into two global geometric parameters, namely, the effective electrode thickness and the permeability length, h and ξ, respectively. Both these two parameters are functions of the potential due to the intercalation reaction. The initial values of h0 and ξ0 of the uncharged electrodes were determined in different solvents with variable velocity decay length (hydrodynamic spectroscopy) as reported,25 whereas the potential dependence of h and ξ caused by the volume changes of the intercalating particles during Li-ion insertion/extraction were computed via admit-

smaller amplitude step from 0.190 to 0.185 V, the driving force was not enough to initiate the appearance of LiFePO4 nuclei at a reasonable rate. An extremely low rate of the phase transition, requiring long-term measurements, resulted in this case in an enhanced contribution of the parasitic reactions to the measured current, imposing practical limit for the potentialstep experiments with small amplitudes aggravated by a huge solution to microarray electrode mass ratio, in the cell suitable for EQCA measurements. To the best of our knowledge, the cathodic capacity limitation during titration of LixFePO4 similar to that shown in Figure 2c has never been previously reported, and no reports on the potential step experiments with microarray or single particles LiFePO4 electrodes have appeared in the literature so far. The physical origin of this effect is detailed later, after discussion of the anodic current transients in terms of the statistical model of interconnected intercalation particles. 5.2. EQCA Characterizations. Figure 3A−C shows the quartz crystal admittance components measured in parallel to the galvanostatic and potentiostatic incremental charging/ 15509

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Figure 4. Effective thickness, h, and permeability length, ξ, of a thin LixFePO4 electrode (A and B, respectively) obtained with the use of eqs S1 and S2 from raw admittance data of the galvanostatic experiment shown in Figure 3A.

Figure 5. Fraction of the active particles, Na, obtained with the use of eqs 1−4, fitted to dimensionless transient current I/Ip during an anodic potential step from 0.21 to 0.22 V (A), and the fraction of the transformed particles, Nt as compared to changes in two geometric parameters of the electrode’s active mass (a thin layer), h and ξ, measured by EQCA during the same potential step (B). Panel C provides a further treatment of the results shown in panel B by comparing the first derivatives of Nt′, h′, and ξ′ obtained by numerical differentiation of Nt, h, and ξ, respectively. The best fitting for the statistical model parameters used, No, N1, n, and m, is listed in Table 1. 15510

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for in situ dynamic studies of particle deformations and the resulting geometric changes of composite ion insertion electrodes. The case of the cathodic potential steps (during which FePO4 is lithiated) is even more surprising than the single-step continuous anodic process because the intercalation reaction proceeds clearly in three separate potential steps. The microarray LixFePO4 electrodes, with their active mass about 250-times smaller than that used typically in porous composite electrodes, ensures the closest approach to equilibrium conditions of charging and discharging. This situation essentially prevents the formation of large Ohmic potential drops in the solution and within the active mass of the electrode. This circumstance, however, cannot be the origin of asymmetric charge−discharge behavior of LixFePO4 electrodes. Our composite microarray electrodes have a rather broad particle size distribution (see the SEM image at the bottom of Figure 1) so that insertion of Li-ions after application of a small potential step occurs first into the smallest particles rather than into the larger ones.19 At longer times, activation of larger particles may occur, but a higher potential step is required to support a reasonably high cathodic current. So, why does this factor not play a role during the continuous (anodic) process of Li-ion extraction from the electrode? We believe that the rate of appearance of the active particles during charge and discharge is different because of the following reason: the phase having a smaller unit-cell volume (i.e., FePO4) tends to be located in the core of the particles rather than at the periphery.36 However, the tendency of the larger unit-cell phase (i.e., LiFePO4) to be located at the periphery of the particles is in conflict with filling the core of the particles with LiFePO4 nuclei, which should energetically promote insertion of Li-ions closer to these nuclei, making the kinetics of lithiation more limited compared to that of delithiation.36 This effect is not accounted for by the statistical model of particle activation; however, the model can capture this possibility through the fitted parameter n (see Table 1 and the related discussion below).

tance modeling with the use of eqs S1 and S2. As an example, Figures 4A and 4B present h and ξ as a function of time for the galvanostatic curves shown in Figure 3A. Similar h and ξ were also obtained for potential step experiments in the direction of both charge and discharge. In order to conveniently compare the fraction of transferred intercalation particles calculated by the statistical model with the related potential-induced changes in h and ξ, the latter are further presented as a function of time in the dimensionless form: Δh(t)/Δh∞, and Δξ (t)/Δξ∞, where subscript ∞ in Δh∞ and Δξ∞ denotes the change of the effective height of the electrode layer and its permeability length at the end of the potential step when the current drops to a very low value. 5.3. Modeling Current Transients and Their Relation to Changes of the Geometric Parameters of the Thin Electrodes’ Active Masses. Application of the statistical model (i.e., the set of eqs 1−4) to the anodic current transient during a potential step from 0.21 to 0.22 V results in quite a good fitting of the current transient (see solid circles versus solid line in Figure 5A) in the vicinity of the peak and continuing to the charging time equal to t/t∞ = 0.25. Indeed as predicted in ref 1 when n < m (here 0.00023 s−1 and 0.00048 s−1, respectively), the population of active particles reaches a peak value decaying thereafter exponentially at longer times (the best fitted value of No was equal to 0.14: a nonzero value of N0 can be reasonably explained by incomplete equilibrium in the electrode’s thin active mass).1 A somewhat less rapid decay of the experimental current at t/t∞ > 0.25 may originate from a broad particle size distribution (more details on this issue is provided later when discussing the cathodic potential steps). We expect a correlation between the fraction of the transformed particles and the changes in the effective layer thickness, h, and permeability length, ξ, since the transformed particles have different molar volume compared to the initial (untransformed) particles. Figure 5B shows that such a correlation indeed is observed. It is seen that, up to the value of t/t∞ = 0.07, the rate of changing the particle population fraction, Nt, and the geometric characteristic of the layer, h and ξ, are roughly equal, implying an approximately linear dependence between them. At larger values of phase transformation, h and especially ξ, vary more slowly than Nt, implying first of all that the changes of the geometric parameters of the active mass are no longer strictly proportional to the fraction of the transformed particles, the latter being clamped by the rigid PVdF binder, revealing nonuniform deformations. We cannot exclude that this primary effect is further complicated by a broad particle size distribution (see later). Fine differences between the time-dependent quantities Nt, h, and ξ during the entire anodic potential step are clearly seen in the time dependences of their first derivatives, representing the rate of changing of these quantities; see Figure 5C. It is seen that the rates of change of both geometric characteristics of the electrodes reach maxima and start to decay earlier compared to that for the appearance of transformed particles, again emphasizing the role of the nonuniform deformations which control the extent of volume and permeability changes in the active mass of the electrode. An interesting feature is observed within an intermediate time range (0.16 < t/t∞ < 0.24) when a slight deflection point on the Nt′ curve becomes much stronger for h′, and even changes the sign for ξ′. This clearly shows that EQCA has the capability of amplifying even small changes in the rate of appearance of the transformed particles and hence may serve as a powerful tool

Table 1. Parameters of the Statistical Model N1, N0, m, and n Obtained by Fitting Eqs 1−4 to the Experimental Current Transients during Charge and Discharge As Indicated parameters potential step

N1

0.21→0.22 V

0

0.19→0.18 V 0.18→0.17 V 0.17→0.16 V

0 0.2 0.68

N0

m (s‑1)

Li-ion extraction 0.14 0.00023 Li-ion insertion 0.29 0.00040 0.10 0.00040 0.10 0.00018

n (s‑1) 0.00048 0.00027 0.00037 0.00040

The fact that our LixFePO4 microarray electrodes underwent the discharge process during three separate cathodic potential steps (see Figure 2C) is a challenge for the statistical model, since at the beginning of the second and third consecutive potential steps, the fraction of the transformed particles, N1, is no longer zero but attains certain values which can be assessed independently by chronoamperometry. The ratio of the cathodic charges under the three cathodic current transients during an intermittent titration of the electrode is close to 0.26:0.48:0.26. This gives an estimation of N1 as 0 for the first step, 0.26 for the second step, and 0.74 for the third step. The best fitted values of N1 were found to be 15511

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Figure 6. Fraction of active particles, Na, obtained with the use of eqs 1−4, fitted to a dimensionless transient current I/Ip during three cathodic potential steps as indicated in panel A, and the fraction of transformed particles, Nt, as compared to the changes in two geometric parameters of the electrode’s active mass, h and ξ, measured by EQCA during potential steps from 0.19 to 0.18 V (B) and from 0.17 to 0.16 V (C). The best fitted statistical model parameters used, N0, N1, n, and m, are listed in Table 1. In panel A the solid curves relate to the experimental currents whereas the solid circles represent the best fit of the statistical model.

probability of particle activation, which nevertheless can be effectively captured through four fitted model parameters.

0, 0.20, and 0.68 for the three consecutive steps (see Table 1 and Figure 6A). The statistical model fitted to the experimental current transients resulted in good results for the first two steps when t/t∞ did not exceed 0.17 (Figure 6A). However, surprisingly, the third (last) step was accurately fitted to the statistical model along its entire time domain with the initial value of the transformed phase N1 = 0.68 (close to the related experimental value). The difference in quality of fitting the model to the three cathodic steps can be rationalized assuming a broad particle size distribution. As mentioned above, the first cathodic step is assumed to relate to transformation of smaller particles (m is larger for the first cathodic steps compared to the last one: see Table 1), whereas the tail of the transient (with a worse fitting quality) may relate to the particles of the larger size. In contrast, the last cathodic step relates to transformation of all the available large particles (the smaller ones have been already transformed earlier), so that the quality of the fitting is much better and is observed within the entire time domain of the third potential step. Interestingly, the better fit of the last cathodic transient to the statistical model also shows a much better agreement between Nt, h, and ξ (see Figure 6C). Finally, we realize that application of the statistical model to a collection of intercalation particles with different particle sizes inevitably transforms the fitted model parameters into some effective quantities, which also depend on the features of the particle activation during lithiation and delithiation. It is seen from Table 1 that the fitted value of n for the first cathodic step appears to be smaller than for the subsequent cathodic steps and for the entire anodic step. This can be reasonably linked to a more facile activation of particles on their periphery during Liion extraction and, in contrast, slower particle activation in their core during first Li-ion insertion as already discussed.36 In summary, the method of LixFePO4 synthesis, particle size distribution, extent of structure defectiveness, and even the mode of electrode’s layer preparation, all may affect the

6. CONCLUSIONS Despite the enormous amount of work devoted so far to LiFePO4 electrodes, there are still notable differences/ ambiguities in understanding phase transition processes of this unique and important LIB cathode material. The ambiguity extends also to whether or not the KJMA nucleation and growth model is valid for describing charge/discharge processes of LixFePO4 electrodes; compare refs 23−25 with refs 1,15,16,18, and 19. Recent studies of mosaic instability in LiFePO4,18,19 as well as analysis of the compositional dependence of the chemical potential of a collection of intercalation particles within the unstable branch of the freeenergy curve,15,16 present evidence for the collective behavior of the intercalation particles in the composite electrodes during their charging and discharging. Implementation of this idea for modeling current transients during potential-step experiments with intercalation particles experiencing phase transformation has been carried out within the framework of a simple statistical model of electrodes comprising collection of nanoparticles.1 A similar approach was used in the present work. Our quartz crystal admittance characterizations of microarray composite LixFePO4 electrodes are in good agreement with this model. We observed a significant asymmetry between the charge and discharge of LixFePO4 electrodes, originated presumably from a broad particle size distribution in the microarray electrodes. The last cathodic transient related apparently to Li-ion insertion into the largest particles in the electrode microarray demonstrates agreement with the statistical model along the entire time domain of the potential step, taking into account the nonzero initial value of the transformed particle fraction. A comparison between the fraction of the transformed particles and changes in the geometric characteristics of the electrode layer during charge and discharge opens the door for highly 15512

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(10) Meyers, J. P.; Doyle, M.; Darling, R. M.; Newman, J. J. Electrochem. Soc. 2000, 147, 2930. (11) Levi, M. D.; Aurbach, D. Impedance of a Single Intercalation Particle and of Non-Homogeneous, Multilayered Porous Composite Electrodes for Li-ion Batteries. J. Phys. Chem. B 2004, 108, 11693− 11703. (12) Levi, M. D.; Aurbach, D. Distinction between Energetic Inhomogeneity and Geometric Non-Uniformity of Ion. J. Phys. Chem. B 2005, 109, 2763−2773. (13) Dreyer, W.; Jamnik, J.; Guhlke, C.; Huth, R.; Moskon, J.; Gaberscek, M. The thermodynamic origin of hysteresis in insertion batteries. Nat. Mater. 2010, 9, 448−453. (14) Dreyer, D.; Guhlke, C.; Huth, R. The behavior of a manyparticle electrode in a lithium-ion battery. Phys. D 2011, 240, 1008− 1019. (15) Chueh, W. C.; Gabaly, F. E.; Sugar, J. D.; Bartelt, N. C.; McDaniel, A. H.; Fenton, K. R.; Zavadil, K. R.; Tyliszczak, T.; Lai, Wei; McCarty, K. F. Intercalation Pathway in Many-Particle LiFePO4 Electrode Revealed by Nanoscale State-of-Charge Mapping. Nano Lett. 2013, 13, 866−872. (16) Brunetti, G.; Robert, D.; Bayle-Guillemaud, P.; Rouvi, J. L.; Rauch, E. F.; Martin, J. F.; Colin, J. F.; Bertin, F.; Cayron, C. Confirmation of the Domino-Cascade Model by LiFePO4/FePO4 Precession Electron Diffraction. Chem. Mater. 2011, 23, 4515−4524. (17) Cogswell, D. A.; Bazant, M. Z. Coherency Strain and the Kinetics of Phase Separation in LiFePO4 Nanoparticles. ACS Nano 2012, 6, 2215−2225. (18) Bazant, M. Z. Theory of Chemical Kinetics and Charge Transfer based on Nonequilibrium Thermodynamics. Acc. Chem. Res. 2013, 46, 1144−1160. (19) Kolmogorov, A.N. On the statistical theory of metal crystallization (1937), in Selected works of A.N. Kolmogorov; Kluwer Academic: Dordrecht, 1991; Vol. II, p 188. (20) Avrami, M. Kinetics of phase change. I. General theory. J. Chem. Phys. 1939, 7, 1103−1112. (21) Avrami, M. Kinetics of phase change. II. Transformation-time relations for random distribution of nuclei. J. Chem. Phys. 1940, 8, 212−224. (22) Allen, J. L.; Jow, T. R. Wolfenstine, Kinetic Study of the Electrochemical FePO4 to LiFePO4 Phase Transition. J. Chem. Mater. 2007, 19, 2108−2111. (23) Allen, J. L.; Jow, T. R.; Wolfenstine, J. Analysis of the FePO4 to LiFePO4 phase transition. J. Solid State Electrochem. 2008, 12, 1031− 1033. (24) Oyama, G.; Yamada, Y.; Natsui, R.-i.; Nishimura, S.-i.; Yamada, A. Kinetics of Nucleation and Growth in Two-Phase Electrochemical Reaction of LixFePO4. J. Phys. Chem. C 2012, 116, 7306−7311. (25) Levi, M. D.; Sigalov, S.; Salitra, G.; Elazari, R.; Aurbach, D.; Daikhin, L.; Presser, V. In Situ Tracking of Ion Insertion in Iron Phosphate Olivine Electrodes via Electrochemical Quartz Crystal Admittance. J. Phys. Chem. C 2013, 117, 1247−1256. (26) Hillman, A. R. The EQCM: Electrogravimetry with a Light Touch. J. Solid State Electrochem. 2011, 15, 1647−1660. (27) Daikhin, L.; Urbakh, M. Effect of surface film structure on the quartz crystal microbalance response in liquids. Langmuir 1996, 12, 6354−6360. (28) Daikhin, L.; Levi, M. D.; Sigalov, S.; Salitra, G.; Aurbach, D. Quartz Crystal Impedance Response of Nonhomogenous Composite Electrodes in Contact with Liquids. Anal. Chem. 2011, 83, 9614−9621. (29) Sahimi, M. Flow and Transport in Porous Media and Fractured Rock; VCH: Weinheim, 1995. (30) Weichert, K.; Sigle, W.; van Aken, P. A.; Jamnik, J.; Zhu, C.; Amin, R. Acartürk, T.; Starke, U.; Maier, J. Phase boundary propagation in large LiFePO4 single crystals on delithiation. J. Am. Chem. Soc. 2012, 134, 2988−2992. (31) Chung, K. Y.; Kim, K.-B. Investigation of Structural Fatigue in Spinel Electrodes Using In Situ Laser Probe Beam Deflection Technique. J. Electrochem. Soc. 2002, 149, A79−A85.

precise and sensitive dynamic studies of nonuniform deformations of intercalation particles hardly assessed by other available techniques so far.

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ASSOCIATED CONTENT

S Supporting Information *

XRD characterization of LiFePO4. Hydrodynamic modeling of EQCA responses of composite intercalation electrodes. This material is available free of charge via the Internet at http:// pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (M. D. Levi), [email protected]. ac.il (D. Aurbach). Author Contributions

S.S. and P.N. prepared the electrodes and performed the EQCA measurements. G.S. and V.P. performed ex situ characterizations of the electrode materials. L.D. performed the hydrodynamic admittance modeling and the related calculations. M.L. and D.A. coordinated the experimental and the theoretical work. M.L., G.S., L.D., E.P., V.P., and D.A. jointly wrote the paper. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful to Prof. Dr. A. Bund and Dr. I. Efimov (Technical University of Ilmenau, Germany) for their valuable advices during performing of this work. V.P. and E.P. acknowledge funding from the German Federal Ministry for Research and Education (BMBF) in support of the nanoEES3D project (award number 03EK3013) as part of the strategic funding initiative energy storage framework. V.P. also thanks Prof. Eduard Arzt for his continuing support.

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REFERENCES

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