Letter pubs.acs.org/NanoLett

Lithium Potential Variations for Metastable Materials: Case Study of Nanocrystalline and Amorphous LiFePO4 Changbao Zhu,† Xiaoke Mu,‡ Jelena Popovic,† Katja Weichert,† Peter A. van Aken,‡ Yan Yu,†,§ and Joachim Maier*,† †

Max Planck Institute for Solid State Research, Heisenbergstr. 1, Stuttgart, 70569, Germany Max Planck Institute for Intelligent Systems, Heisenbergstr. 3, Stuttgart, 70569, Germany § School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, China ‡

S Supporting Information *

ABSTRACT: Much attention has been paid to metastable materials in the lithium battery field, especially to nanocrystalline and amorphous materials. Nonetheless, fundamental issues such as lithium potential variations have not been pertinently addressed. Using LiFePO4 as a model system, we inspect such lithium potential variations for various lithium storage modes and evaluate them thermodynamically. The conclusions of this work are essential for an adequate understanding of the behavior of electrode materials and even helpful in the search for new energy materials. KEYWORDS: Lithium potential, metastable materials, nanocrystalline, amorphous, LiFePO4, cathode materials

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one considers single phase storage whereby lithium is absorbed in a nanocrystalline or amorphous phase changing its composition. As we see also here the LiξFePO4 system provides a pertinent example. Nanocrystalline LiFePO4 has attracted much attention due to its excellent performance.12−20 Thermodynamically, one can regard the system as quasi-binary, as the FePO4 units remain intact on lithiation, and one can treat it as a constrained equilibrium binary system (LiX). This is because the FePO4 part and also the inherent defect chemistry remains invariant during these processes. The same argument applies for the treatment of frozen size or frozen structure. For nanomaterials, capillary pressure effects give rise to variations of the standard chemical potential of the compound, components, and defects. Ignoring stress effects and concentrating on capillarity, the excess term of the standard potential is given by (2γ/r)v, with γ and r being effective surface tension and effective radius of the particles, and v being the (partial) molar volume of compound, component, or defect, respectively. (We refer to crystallographic averages, i.e., to mean γ and r values; for the Wulff-shape this point is meaningless as then the ratio γ/r is invariant with respect to orientation.) As the volume of a compound is always positive, its chemical potential (μLiX) is always increased by size reduction (if no other effects than capillarity come into question), a well-known consequence being the reduced melting temperature. As far as the partial molar volume of a component (cf. μLi) and its effect on the

lthough great efforts have been devoted to battery research, surprisingly, a number of fundamental issues related to batteries and electrode materials have not been adequately tackled. One such issue is the effect of size on the equilibrium potential of cells, i.e., on the lithium chemical potential, involving nanocrystalline and amorphous materials. A major focus of present research is on nanocrystalline and nanoporous materials. This is essentially but not solely due to the decreased transport length. Such size effects can lead to increased rate performance1 and even to a change of the reaction mechanisms.2−4 The first established size effect for LiFePO4 reported was the size-dependent lithium miscibility gap in nanoscale LiFePO4.5 A usually not intended but necessarily occurring consequence is the modified lithium potential, i.e., equilibrium cell potential.5−7 Such variation of the voltage though can in certain cases even be highly desired in terms of performance.8 One of the few examples that inspect such effects systematically is the treatment of excess potentials of nanocrystalline and amorphous RuO2 generated via a conversion reaction.9,10 That particular work was concerned with the effects of size on a two-phase equilibrium with one phase being macroscopic and crystalline, while the other was nanocrystalline or amorphous. As far as thermodynamic complexity is concerned, it is the simplest case. (The simplified treatment given in ref 10 overestimates the effects in the single phase region as made clear by the more detailed treatment in ref 11. See also the Appendix.) More complex but very realistic cases are two- or multiphase equilibria where two or more phases are simultaneously nanocrystalline or amorphous. The least clear case is met if © XXXX American Chemical Society

Received: June 27, 2014 Revised: August 11, 2014

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Table 1. Excess Lithium Potential for Various Metastable Cases (Colon Indicates Two Phase Coexistence)a nano 1 (single phase) (2γ1/r1)vLi,1 amorphous 1 (single phase) ex ex μex Li = μi + μn

nano 1:nano 2

nano 1:macro 2

(2γ1/r1)v1 − (2γ2/r2)v2 amorphous 1:nano 2

amorphous 1:amorphous 2

ΔmG1 − ΔmG2 ≃ (2γ1/r0)v1 − (2γ2/r0)v2

ΔmG1 − (2γ2/r2)v2 ≃ (2γ1/r0)v1 − (2γ2/r2)v2

(2γ1/r1)v1 amorphous 1:macro 2 ΔmG1 ≃ (2γ1/r0)v1

γ1,2 = surface tension of phase 1, 2; v1,2 = mole volume of phase 1, 2; vLi,1= partial molar volume of Li in phase 1; r1,2 = particle size of phase 1, 2; r0 = ex atomistic size; ΔmG1,2 = melting free enthalpy of phase 1, 2; μex i = excess chemical potential of Li interstitial; μn = excess chemical potential of electrons. a

Figure 1. TEM images (a, c) and HRTEM micrographs (b, d) for amorphous LiFePO4 (a, b) and nanocrystalline LiFePO4 (c, d). The inset in b shows the SAED pattern for amorphous LiFePO4, which displays concentric electron diffraction rings characteristic of amorphous materials.

electromotive force (EMF) or defect (cf. μdefect, and its effect on disorder mass action constants) is concerned, the sign is not definite. If typical values for γ, r, and v are taken, the absolute values of the excess cell voltages are in the range of 1 to 100 mV. A special consideration is required for amorphous materials. If the chemical potential of the compound is referred to, its excess value may be roughly approximated by a similar (2γ/r)v term but with r being atomistic.9,11 For the evaluation of excess terms, it is not only important to what entity we refer; it also makes a great difference whether we refer to a two (or multi) phase regime or to a single phase regime. In the first case, the chemical potential of lithium and, hence, the EMF can be expressed by the standard values of the phases, the molar volumes of which are positive, giving rise to a positive (2γ/r)v term. If (as in the RuO2/LixRuO2 case), only

one phase exhibits an excess term; the fact whether the excess EMF is positive or negative depends on whether it is the Lipoor or the Li-rich phase that is metastable.10,11 If both phases are metastable, then the difference between the two excess terms matters. This reduces to a positive or negative difference in the γv product if the nanocrystallinity is preserved during phase change and to the difference in γ, if two coexisting phase with similar volumes are concerned, what is rather expected for multinary materials, as the volume increment of Li is comparatively small. If the excess EMF in a single phase (α) situation is concerned, we have to refer to the partial molar volume of Li in this phase (νLi(α)); hence it is (2γα/rα)vLi(α) that is to be looked at (not (2γα/rα)v(α) as above), a term that may even become negative (if phase α volume shrinks on Liaddition). Knowing that Li is incorporated as Li+ interstitials B

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Figure 2. (a) Typical 50% DOD and 50% SOC with relaxation curves, leading to thermodynamic stable EMF values for various particle sizes in the two-phase regime. (b) The plot of EMF vs 1/r. (Note that the EMF values are obtained by 50% DOD and 50% SOC with relaxation, as shown in part a.) The black curve refers to r = dx/2, and the red curve refers to r = dT/2. Both curves show that the surface tension varies for the various particle sizes.

(or by filling vacancies) and excess electrons (or by filling electron holes), it is their individual partial molar volumes that need to be considered. While the amorphous case can be considered as the limit of nanocrystallinity whenever we address chemical potential of phases, in the single phase situation where we refer to the chemical potential of Li within the phase, we have to address the excess potentials of ionic and electronic defects in the amorphous matrix. A pertinent treatment of the various cases is given in ref 11. For a brief account, the reader is referred to the Appendix. The relevant expressions of the excess chemical potential and hence of the excess cell potential are summarized in Table 1. In order to experimentally tackle the problem, we synthesize nanocrystalline LiFePO4 of various sizes as well as the amorphous phase. Amorphous LiFePO4 and FePO4 are prepared by a spontaneous precipitation method21 as described in greater detail in the Experimental Section. The amorphicity of LiξFePO4(0 ≤ ξ ≤ 1) was confirmed by X-ray diffraction (XRD) (Figure S1 in the Supporting Information). The structural evolution with increasing heating temperature (heated in Ar/H2) was investigated by the same technique (Figure S1b). When the temperature was less than 400 °C, the material stayed amorphous, while for a heating temperature of 550 °C, well-crystalline and phase-pure LiFePO4 with a ∼150 nm particle size was obtained. LiFePO4 of controlled nanosizes was synthesized by the surfactant-assisted polyol method.22 The phase purities of the differently sized nanocrystals were confirmed by XRD patterns (Figure S2). Figure 1 shows TEM images and HRTEM micrographs for both amorphous LiFePO4 and nanocrystalline LiFePO4. The inset in Figure 1b refers to a SAED pattern for LiFePO4, indicating amorphicity. The particle sizes for amorphous LiFePO4 are around 50 nm. The typical morphology of nanocrystalline LiFePO4 can be recognized in Figure 1c and d. The thickness of the oblongshaped nanoparticles was varied from 10 to 20 nm, while the length was varied from 20 to 50 nm. As far as the elucidation of an effective size is concerned, for the smaller sizes (smaller than 100 nm and obtained by polyol method) we used two procedures. (i) We calculated sizes according to the powder pattern refinement by Topas 4 software (dx). (ii) We calculated sizes from TEM images by defining-spherical-shaped mean particles that exhibit the same area or the same volume as the oblong-shaped real particles. This typical size distribution is

given in the Supporting Information as well (Figure S8). For thermodynamic reasons (cf. capillarity) we prefer the sphericalshaped mean particles that exhibit the same area (dT) rather than the same volume even though the differences are negligible. Note that TEM and XRD observed sizes are almost identical; for example, the particles with the sizes of 44, 37, and 29 nm (dx) are corresponding to the sizes of 44, 35, 28 nm (dT), respectively. Nonetheless, we use for the interpretation both dT and dx data to be on the safe side. For the particles larger than 100 nm, the values are obtained from TEM or SEM images. Let us focus on the lithium potential variations in nanocrystalline LiFePO4 first and stress again that, for the definition of μLi in a matrix, metastability is not an exclusion factor. (Indeed, metastability of one sublattice in a multicomponent crystal is the usual case as we typically face, cf. frozen-in defects in the anionic or the nonlithium cationic sublattice). Decisive criteria for μLi to be defined are the reversibility of Li in/excorporation and immobility of the rest (size, shape, anion sublattice, etc.). Discharge curves for nanocrystalline LiFePO4 with different particle sizes can be seen in the Supporting Information (Figure S10). Note that all of the samples are without carbon coating, which results in lower capacity and perceptible polarization especially for the micrometer-sized sample. The capacity is less satisfactory for the smaller sized LiFePO4 samples (dT = 28 and 35 nm) owing to the fact that the particles were covered with remaining organic molecules from the preparation method, hence worsening electronic and ionic transport.23 The presence of a surface layer was confirmed by both HRTEM (Figures 1d) and FTIR (Figure S3). Figure S11 shows the galvanostatic intermittent titration (GITT) curves for samples with sizes of 150, 44, and 29 nm, respectively. Even though a very low current density (0.05 C) was used for stepwise discharging (1 h) the samples followed by a 10 h waiting time, the relaxation time is not sufficient to attain equilibrium state. For that reason, as far as crystalline materials are concerned, we concentrate in this work on the two-phase regime (confirmed by XRD measurements of the half-discharge samples, Supporting Information, Figure S4), the open circuit voltage (OCV) values of which we could reliably measure. Here we considered samples characterized by 50% state of charge (SOC) and 50% depth of discharge (DOD) followed by a very long time relaxation (normally 3−5 days), which should allow C

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particle sizes, a significant OCV decrease is observed. In this range, the plot EMF vs 1/r (r = dx/2) is roughly linear, yet not compatible with a proportionality to 1/r according to the capillary term, as it should extrapolate to the macroscopic value for 1/r tending to 0. This semiquantitative conclusion is equally valid if we refer to r = dT/2 (see Figure 2b). It is also valid if the substantial scatter and anisotropy of the particles and particle distribution is taken into account. If the excess EMF is considered and the respective values for r and v are used, significantly different Δγ-values are obtained. In all cases, the difference of the surface tension of LiFePO4 and FePO4, i.e., γLFP − γFP is obviously positive (Eex < 0). The interpretation of the magnitude depends on the surface tension (i.e., surface chemistry) as well as on the morphology. The obtained values for (r = dx/2 and r = dT/2) are summarized in Table 2. The compilation clearly indicates that besides morphology the variation of surface chemistry is a dominating factor.6 While the difference of surface tensions for bulk and 150 nm LiFePO4 can be naturally neglected, the surface tension difference and presumably the surface tension itself increase drastically with particle size reduction. Note that unlike for 44 nm (dT), the samples of smaller sizes exhibit an organic coating of varying size and compactness. The low Δγ values are most probably due to the fact that the differences of two similar surface tensions need to be considered. This is, e.g., the case for a digital storage mechanism.20 In a core−shell mechanism, overall values are expected that refer to the surface tension of the shell phase, while the internal interfacial tension may be comparatively small.24 (For a more detailed consideration of course

for far-reaching relaxation of the polarization effects. This is confirmed by Figure 2a. The measurement details for the cell voltage can be found in the Experimental Section. Table 2 gives Table 2. OCV and Surface Tension for LiFePO4 with Different Particle Sizes dT (dx) (nm)

OCV−50% SOC (V)

OCV−50% DOD (V)

OCV (V)

28 (29) 35 (37) 44 (44) 150 1000

3.408 3.414 3.421 3.427 3.427

3.373 3.410 3.421 3.427 3.425

3.391 3.412 3.421 3.427 3.426

Δγ (J m−2) 0.55 (0.57) 0.29 (0.31) 0.15 (0.15)

the differences between relaxation after charge and relaxation after discharge. The proximity of the values implies the possibility of a thermodynamic interpretation. Averaging of charge and discharge values gives a reliable approximation for the reversible equilibrium cell voltage. All of the so-obtained OCV values for different sizes are summarized in Table 2 (Note that the error bar of the OCV values is only around 3 mV). In the case of nanocrystalline LiFePO4:FePO4, we refer to an excess emf (OCV) described by Δ(2γvphase/r) (Table 1). Normally, the lithium potential variation is not expected to be pronounced if both phases are affected by the capillary effect in a similar manner. As can be seen in Table 2, the OCV values are almost constant in the size range larger than ∼100 nm. They refer to the macroscopically expected value. However, for smaller

Figure 3. (a) Charge and discharge profiles for amorphous LiFePO4 heated at 300 °C for the first three cycles at a 0.01C rate. (b) Typical PITT measurement on discharge process for amorphous LiFePO4. The single phase regime for lithium insertion is confirmed by diffusion-like current relation profiles. (c) OCV curves obtained from GITT for heterosite FePO4 and amorphous FePO4 in the single phase regime region (Note that, for nanocrystalline LiFePO4, only the first point is in the single phase regime). The error range takes account of the difference between charge and discharge as well as possible insufficient relaxation in GITT process. (Figure S9). (d) OCV curves obtained from GITT for amorphous FePO4 and trigonal FePO4 as a function of different lithium content in the single phase regime. D

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regime. The respective GITT curves are almost parallel. Note that now OCV values for the amorphous FePO4 are higher than those for trigonal FePO4 for the whole lithium content. In order to interpret the OCV data, even in terms of the sign of Eex, it is crucial to decide on about what matrix structure (short-range order) we have to refer to if we speak of amorphous materials. FePO4 has various polymorphs, among them heterosite FePO4 (the counterpart of LiFePO4, with the orthorhombic space group Pnma), and trigonal FePO4 (with the trigonal space group P3121). Heterosite FePO4 is stable in air and transforms at 620 °C to trigonal FePO4.28 Notably, amorphous FePO4 transforms to trigonal FePO4 when sintered at 550 °C in air (see Figure S5). At a first glance, the parallelity of the curves in Figure 3 suggest the short-range order in the amorphous case to be trigonal and not orthorhombic, yet detailed studies show contrary. The pair distribution function (PDF) presents the density of existing atomic pairs versus the pair separation in the investigated material, statistically providing insight into the local structural information. By this method, we are able to investigate short-range order of amorphous FePO4, and compare to other polymorphs, e.g. heterosite FePO4 (note as Pnma) and trigonal FePO4 (note as P3121). The stronger similarity between the PDF determined from experimental electron diffraction data (PDF-exp) and the calculated PDF from the orthorhombic structure (PDF-Pnma) in Figure 4,

also the detailed geometry need to be considered.) The fact that Δγ is largest for the smaller size may be due to the coating. Note that an increased value on coating is not in contradiction as for wetting the surface tension of the coating itself needs to be considered: for the simplest case of a planar coating, it is only required that the interfacial tension γphase/coating be smaller than the sum of γphase and γcoating. Given all the complexity, the analysis shows that size effects on OCV for nanocrystalline materials can be manifold, and it is even not at all surprising that positive excess voltages have been found.5,7 Now we can investigate the lithium potential variations for amorphous LiFePO4. The charge and discharge profiles for amorphous LiFePO4 were shown for a 0.01C rate (1C = 170 mA/g) within an electrochemical window of 2−4 V for the first three cycles (Figure 3a, information on the rate dependence is given in the Supporting Information, Figure S12). Even at such low current, only ∼0.6 lithium can be reversibly intercalated. The profiles of charge and discharge curves for amorphous LiFePO4 are very similar to the amorphous FePO4 previously reported in the literature.25 Sloped charge and discharge curves instead of a voltage plateau are observed, which suggests single phase lithium intercalation. As, in thermodynamic terms, amorphous material may be viewed as the extreme case of nanocrystallinity, and this fits into the general observation that size reduction shrinks the miscibility gap. The single phase situation is further confirmed by a potentiostatic intermittent titration test (PITT), which displays diffusion-like current relaxation (red curves in Figure 3b) over the whole composition range for charge−discharge process,26 indicating an absence of any miscibility gap for the amorphous LiFePO4. In such amorphous LiξFePO4, we have to consider the excess chemical potentials of Li-defects and electronic defects explicitly. Owing to the crystallographic fluctuations in the amorphous matrix and the possibility of finding well-suited environments, we generally can assume the μ0-values of the ionic defects (vacancies and interstitials) to be decreased; however it might be energetically more difficult to introduce electronic defects owing to a loss of delocalization (i.e., μ0values of the electronic defects increase), making the sign of μLi0 (= μ0Li+ + μ0e−) undecided. Unlike a two-phase regime in nanocrystalline LiFePO4, we apparently deal with a single-phase system in the case of amorphous LiFePO4. The comparison of lithium potential between the amorphous and crystalline LiFePO4 will only be quantitatively meaningful if both of them are in the single phase regime. For nanocrystalline LiFePO4, the single phases Li1−yFePO4 and LixFePO4 only exist in the very narrow range (x, y < 0.1) near the two end members. For example, for 40 nm LiFePO4, a solid solution confined by the single phase Li0.074FePO4 and Li0.891FePO4 end-members was experimentally confirmed.27 Figure 3c displays the OCV curves obtained from GITT for crystalline LiFePO4 (dT = 44 nm) and amorphous LiFePO4. Note that only the first point of crystalline LiξFePO4 belongs to the single phase regime. It can yet be clearly seen that the values for the amorphous LiξFePO4 are lower by around 200 mV, which means that the introduction of metastability would increase the lithium potential. On the other hand, Figure 3d shows OCV curves obtained from GITT for amorphous FePO4 and trigonal FePO4 (obtained from heating amorphous FePO4 at 550 °C) as a function of different lithium content. Here the deviation is opposite, unlike in Figure 3c, both curves in Figure 3d (amorphous FePO4 and trigonal FePO4) are in the single phase

Figure 4. Comparison between PDFs obtained from experiment data (black), Pnma-type structure (green), and P3121-type structure (red).

reveals that the amorphous FePO4 is characterized by local atom arrangements of Pnma-type structure with an average dimension of 6 Å (detailed information can be seen in the Supporting Information, Figure S6 and Figure S7). The most remarkable features are as follows: first, the peak at 3.73 Å in the PDF-exp (the correlated peak in PDF-Pnma is at 3.79 Å) corresponds to the Fe−Fe separation, which is associated with corner sharing of the FeO6 octahedra in the Pnma-type structure, whereas the P3121-type structure does not contain such an arrangement. Instead, it exhibits a 4.53 Å Fe−Fe separation reflected by an additional shoulder at around 4.53 Å. Second, the larger separation between the first two pronounced peaks in the PDF-exp reveals that the P and Fe atoms in the amorphous material have rather different oxygen environments, corresponding to PO4 tetrahedra and FeO6 octahedra. This is identical with the Pnma-type structure of orthorhombic FePO4 but different from the P3121-type structure (i.e., PO4 tetrahedra E

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route. An aqueous solution containing Li, Fe, and P in stoichiometric amounts was first prepared at room temperature by dissolving Fe(NO3)3·9H2O and LiH2PO4. The light yellow solution was stirred under constant magnetic stirring and heated to slowly evaporate the water. The solid precipitate (light yellow-brown color) was milled and then heated at 350 °C in reducing atmosphere (95% argon and 5% hydrogen) for 2 h to decompose the salts. The resulting red-brown powder was milled again and isostatically pressed to small pellets (thickness ∼1.5 mm and diameter ∼7 mm). The pellets were sintered in argon/hydrogen atmosphere at 600 °C for 12 h, where they turned to a gray color. Crystalline FePO4 with a trigonal structure was obtained by heating the amorphous FePO4 at 550 °C for 12 h in the air. XRD was performed using Cu−Kα radiation in Phillips PW 3020 diffractometer and diffraction data collected at 0.02° step width over 2θ range from 10° to 90°. Energy-filtered electron diffraction was performed with a Zeiss EM912 Omega microscope (Zeiss, Oberkochen, Germany) operated at 120 kV, where a 10 eV energy-selecting slit centered at 0 eV energy loss was used to avoid contribution of electrons suffering inelastic scattering. The effective recording angle reached 3.2 Å−1, above which noise predominated the electron diffraction signals. Pair distribution functions (PDF) were obtained by applying a Fourier transformation to the experimental diffraction pattern. PDFs of the Pnma and P3121 structure was obtained by Fourier transforming electron diffraction profiles calculated from crystalline Pnma and P3121 models, respectively. For the electrode preparation, LiFePO4 (70 wt %), carbon black (20 wt %, Super-P, Timcal), and poly(vinylidene fluoride) binder (10 wt %, Aldrich) in N-methylpyrrolidone were mixed into a homogeneous slurry. The obtained homogeneous slurry was pasted on an Al foil, followed by drying in a vacuum oven for 12 h at 80 °C. Electrochemical test cells (Swagelok-type) were assembled in an argon-filled glovebox (O2 ≤ 0.1 ppm, H2O ≤ 3 ppm) with the coated Al disk as working electrode, lithium metal foil as the counter/reference electrode, and 1 M solution of LiPF6 in a 1:1 vol/vol mixture of ethylene carbonate and diethyl carbonate as the electrolyte (Novolyte technologies). Glass fiber (Whatman) was used as separator. Potentiostatic intermittent titration testing (PITT) was performed as follows: a “staircase” voltage profile with a 5 mV voltage increment/decrement (each titration was stopped when the current reached around ∼C/20) was applied, and the response of current vs time was recorded at each constant potential. Galvanostatic intermittent titration techniques (GITT) is performed as follows: the 1/20 C current was used for charging and discharging the samples for 1 h followed by a 10−30 h waiting time leading to the relaxation to equilibrium state as a function of different lithium contents. The stable EMF values for nanocrystalline LiFePO4 with various particle sizes in two-phase regime is done as follows: the battery at first is carried on one full charge−discharge cycle, and then it is discharged to 50% DOD followed by sufficient relaxation (normally 3−5 days) until the stable value. After the discharge relaxation process, the battery is further discharged to 2.5 V, and then one fully charge−discharge cycle is performed. Afterward, it is charged to 50% SOC with sufficient relaxation until reaching a stable value. Finally, the average value from OCV-charge and OCV-discharge has been taken as a reversible EMF value.

along with FeO4 tetrahedra) of trigonal FePO4. Hence, according to these results, we have to direct our OCV discussion to the difference between the OCV - value of amorphous and heterosite FePO4. This can only be meaningfully done for the single left-hand side data point (Figure 3c), where in both cases we refer to single phase storage. The two other data points necessarily overestimate the difference, and extrapolation into the hypothetical single phase situation may lead to rather parallel curves (as for the trigonal FePO4, Figure 3d). At any rate, it is safe to conclude that Eex < 0 and μLiex > 0, i.e. that Li-accommodation in the amorphous matrix is more difficult than in the crystalline phase. As mentioned above, we should, owing to the size fluctuations in the amorphous material, assume an easier accommodation of Li+. Hence, we can conclude that it is the increased difficulty of accommodating electronic carriers owing to the loss of periodicity that generates the excess value. Finally, let us sum up our main findings in this communication. For nanocrystalline LiFePO4 in the twophase regime, the OCV values decrease with decreasing particle sizes; the effects continuously increase with increased capillary pressure. Surface chemistry is very crucial for lithium potential variations in this situation. In the single phase regime, decreased OCV values are observed for amorphous LiFePO4 as well. The probable explanation is the increased lithium potential because of increased chemical potential of the electronic defects. All of the conclusions here are of significant importance in order to arrive at a deeper understanding for metastable LiFePO4, one of the most promising cathodes in the battery field at the moment. In spite of the necessity of more systematic research in order to deconvolute the various control parameters, the revealed features of lithium potential variations for metastable materials in different storage modes are crucial for understanding the behavior of other electrode materials and maybe even helpful in the search for new energy materials. Experimental Section. Amorphous FePO4 was prepared as follows:29 an equimolar water solution of Fe(NH4)2(SO4)2· 6H2O and NH4H2PO4 were mixed in a 1:1 volume ratio. The proper amount of H2O2 was added to the solution under vigorous stirring. The white precipitate was collected by repeated washing and centrifugation with distilled water 3 times. After drying in the oven at 400 °C for 2 h, amorphous FePO4 was obtained. Amorphous LiFePO4 was prepared by chemical lithiation of amorphous FePO4 by LiI in the acetonitrile solution, stirred for 24 h, and collected by centrifugation. Heating amorphous LiFePO4 in Ar/H2 (95:5) atmosphere at 550 °C for 2 h yielded the crystalline LiFePO4. Nanocrystalline LiFePO4 with different sizes were prepared by the surfactant-assisted polyol method:22,30 A stoichiometric equimolar ratio of Li-CH 3 COO, Fe-(CH3 COO) 2 , and NH4H2PO4 was dissolved/dispersed in a tetraethylene glycol (TEG) and oleylamine (OL) mixture with different volume ratios. The solution was heated at 320 °C (controlled by the thermocouple dipped in the solution) in a round-bottom flask with magnetic stirring attached to a refluxing condenser. The resultant LiFePO4 nanorods were collected by repeated washing and centrifugation with ethanol and acetone for 3 times, respectively, followed by drying in a vacuum oven at 80 °C for 20 h. The size can be controlled by the different experimental parameters, such as amount of surfactant, concentration of precursor, etc.22 Non-carbon-coated pristine LiFePO4 with a particle size of 1 μm was obtained from a straightforward aqueous solution F

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APPENDIX If phase thermodynamics is considered two different definitions of Gibbs phase energy are used, which are differently connected with component potential variations (e.g., Li for LiX): Gibbs energy of a mixture (here denoted as g) per total number of moles (nx + nLi) and Gibbs energy of a mixture (denoted as Γ) per mole number of the invariant component (nx for LiξX). (For more details see Appendix in ref 11.) In both representations, the double tangent method gives the compositions of coexistence in the two phase regime. For the g−x notation, μ is obtained from the intercept of the double tangent to g(x) with x = 1 axis, while in the Γ−ξ representation, μ is the slope of the double tangent. If one is to represent the free energy of the metastable phase, the question arises how gex and Γex depend on composition. (We assume here a constant μ0ex = (2γ/r)v; see below.) The simplest assumption is met if these functions are shifted by a constant value. As far as gex is concerned, this is a good approximation, if vLi = vLiξX/(1 + ξ), i.e., if vLi is equal to the mean molar volume (denoted as VLiξX in ref 10). (i.e., vLi = vFePO4, but vLi = vLiFePO4/2). Then gex corresponds to μex Li . If vLi is small compared to vLiξX, then not gex but Γex is fairly constant and μex Li ≃ 0. In fact for LiFePO4, vLi ≃ vLiX/10. This can be quantitatively seen if the detailed functions are inspected. Reference 11 gives for LiξX ≡ LiM+δX 2γ [(1 − x)vLiξX + (1 − (M + 1)(1 − x))vLi] g ex = r With x = nLi/(nLi + nX) and (1 − x) = (1/(1 + M + δ)); 2γ Γ ex = [vLiξX + (ξ − M )vLi] r with ξ = M + δ. The above approximations obviously lead to the above conclusions. Furthermore, we can draw the following conclusions: (a) In a nonvariant system, as it is the case for LiFePO4:FePO4, μLi and hence the EMF can be expressed by ex ex the phase potentials according to μex Li = μLiFePO4 − μFePO4. The rhs values are directly given by gex or Γex of the respective phases. For small x, ξ values [not too large vLi and not too small difference between (2γ1/r1)v1 and (2γ2/r2)v2, if 1 and 2 refer to the stoichiometric phases], the vLi terms can be ex,0 ex,0 neglected, corresponding to μex Li ≃ μLiFePO4 −μFePO4, i.e., referring to the stoichiometric phases. This corresponds to the usual approach of calculating the EMF from the tabulated ΔGf0 values. (b) If we refer however to the univariant case (single phase storage), we have to refer to μLi(ξ) directly. Then the vLi term can not be neglected as is clearly seen if (dΓex/dξ) is considered. So we have to inspect 2γ vLi μLiex = r Specially, this equation has to be exploited if the excess EMF is to be considered within the nanocrystalline LiFePO4 or FePO4 phases. If Li is incorporated as interstitials and conduction electrons, as it is the case for FePO4, μex = (2γ/r)(vi + vn). As far as the amorphous situation is concerned, setting virtually r = atomistic is not a viable approach here, as vi and vn are no longer well-defined quantities. We can state μLiex = μiex + μnex

and try to energetically and entropically interpret the situation for the excess ions and electrons in the amorphous matrix. Because of the inconsistencies in the literature, let us take up the above discussion about g and Γ. As shown in ref 11 for LiM+δX = LiξX g = μLi0 X (1 − x) + μLi (x(1 + M ) − M ) ξ

=

μLi0 X (1 ξ

− x) + μLi δ(1 − x)

0,ex As shown above gex = μLi is a good approximation if vLi = ξX (vLiξX/(1 + ξ)). As this is not fulfilled for FePO4 or RuO2,10 a constant shift of the g-representation overestimated the thermodynamic effects. Yet we also see here that gex = μ0exLiξX is a realistic approximation at the stoichiometric point (δ = 0) if M = 0 (i.e., FePO4 or RuO210). For Γ it is valid that11

Γ = μLi0 X + δμLi = μLi0 X + ξ

ξ

⎛ 1 ⎞ ⎜ − M − 1⎟μLi ⎝1 − x ⎠

As stated above Γex = μ0,ex LiξX is a good approximation for vLi ≪ vLiξX. We see here that it is generally a good approximation near the stoichiometric point, for all M (unlike gex), i.e., also for LiFePO4 or LiRuO2.10 It is revealing to exploit the relations for the coexistence case LiFePO4:FePO4 for which μLi ≃ μL0 − μF0 (L, F are short for LiFePO4 and FePO4). Then gL = (1 − 2x)μF 0 + xμL 0

As it should be, we get the same result for gF. The extrapolation of g to x = 1 yields μLi, i.e. μ0L − μ0F. If we assume the artificial subcase μ0,ex = μ0,ex = μ0,ex, the excess L F potential disappears. It is important to see that gex = μ0,ex(1 − x) is increasingly less increased if the compound becomes Liricher. (Yet for the two-phase region the treatment in ref 10, where x ≃ 0 is precise.) Otherwise the difference in the intercept at x = 1 would not disappear. This is different for Γ; here an equal upward shift occurs. Note that Γ = μLi0 X + δμLi = μLi0 X + ξ

ξ

⎛ 1 ⎞ ⎜ − M − 1⎟μLi ⎝1 − x ⎠

and in the two-phase region for both L and F we find Γ = 2μF0 − μL0 + (δ + 1)(μL0 − μF0 )

with

dΓ = μL0 − μF0 = μLi dδ

for our subcase Γex = μ0,ex. Note again that we refer to the twophase system. Another point refers to the x-dependence of μ0,ex, in particular of the molar volume. As pointed out in ref 10 (Appendix), the criterion for its independence is identical with the criterion derived above for the fulfillment of gex = (2γ/r)vLiξX, namely, vLi = (vLiξX/(1 + ξ)) = VLiξX, where VLiξX is the mean molar volume (analogous to g above).

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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