ARTICLE pubs.acs.org/JPCC
Anisotropic Phase Boundary Morphology in Nanoscale Olivine Electrode Particles Ming Tang,†,* James F. Belak,† and Milo R. Dorr‡ †
Condensed Matter and Materials Division and ‡Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore, California 94550, United States ABSTRACT: Li-insertion-induced phase transformation in nanoscale olivine particles is studied by phase-field simulations in this paper. We show that the anisotropic growth morphology observed in experiments is thermodynamically controlled by the elastic energy arising from the misfit strain between the Li-rich and Li-poor olivine phases and kinetically influenced by the Li surface-reaction kinetics. The one-dimensional Li diffusivity inherent to the olivine structure is found to kinetically stabilize the phase boundary morphology after Li insertion termintates and facilitate ex-situ observation. Our calculations suggest that examination of the phase boundary morphology provides an effective approach to determine the limiting process of the Li intercalation kinetics in olivine nanoparticles.
’ INTRODUCTION In recent years LiMPO4 (M = Fe, Mn, Co, Ni) olivine compounds1,2 have become important cathode materials for Liion rechargeable batteries with applications ranging from cordless power tools to plug-in hybrid vehicles.3,4 During battery charge/ discharge bulk olivines usually undergo a first-order phase transformation between Li-rich and Li-poor olivine phases (denoted as LFP and FP hereafter) that are separated by a miscibility gap.1,5 Both theoretical calculations6,7 and experiments8 show that Li diffusion in the olivine structure is confined to one-dimensional channels along the [010] axis and the diffusivity in other directions is negligible. Olivine also exhibits strong anisotropy in its composition dependence of lattice parameters. As illustrated in Figure 1a, FePO4 expands along [100] and [010] but contracts in the [001] direction upon Li insertion.9 The linear misfit strain between LFP and FP is largest along [100] (5%) and smallest along [001] (-1.9%). Atomistic calculations by Wang and Ceder10 and Fisher and Islam11 also reveal significant anisotropy in the surface energy of olivines, which often results in faceted, platelike single-crystalline particle morphology, e.g., as prepared by hydrothermal12 and polyol methods.13 The anisotropic transport and misfit-strain properties of the olivine phases are expected to significantly influence the phase morphology evolution during the Li intercalation process. Nevertheless, existing experimental results on the phase boundary morphology in olivines are limited. On the basis of an electron microscopy study of micrometer-sized particle samples, Chen et al.14 proposed that the LFP/FP phase boundary lies in the (100) plane and moves along the [100] axis upon Li insertion/extraction, which is further elaborated by Delmas et al. in a “domino-cascade” model.15 Laffont et al.16 examined LFP/FP two-phase coexistence in nanoscale platelet particles. Using electron energy loss spectroscopy (EELS) they deduce that the phase boundary is not perfectly aligned with (010) r 2011 American Chemical Society
but has an inclination toward the (100) axis. Different from the larger particles studied by Chen et al., few dislocations and cracks were observed in the samples of Laffont et al., indicating that a coherent interface between LFP and FP can be maintained in nanosized particles. Similar findings are also reported by Meethong et al.,9,17 who found that significant coherency stress is accommodated within nanoparticles due to the large volume mismatch between the two olivine phases. Although the effect of misfit strain energy on the phase stability and phase transition kinetics has been considered by various researchers,9,14-17 a detailed understanding of how it influences the phase boundary morphology in olivine particles remains to be achieved. On the modeling side, Singh et al.18 developed an intercalation dynamics model which predicts the LFP/FP phase boundary to propagate perpendicular to the Li-diffusion direction like a “travelling wave”. Their results apply to the surface-reaction-limited Li-insertion process in the absence of elastic energy. Simulation of phase evolution under more general conditions requires a more involved model. In this letter we report a phase-field modeling study of the phase transformation in nanoscale prismatic olivine particles using LiFePO4 as the model system. The anisotropic diffusion and misfit strain properties of the olivine phase are explicitly incorporated in the model. Our simulations confirm the phase boundary morphology observed in ref 12 and reveal the important roles of elastic energy, Li diffusion anisotropy, and surface reaction kinetics in controlling the phase morphology evolution upon Li insertion.
’ MODEL Previously, we developed a phase-field model to study the competing phase transition pathways in nanoscale olivines.19-21 Received: October 7, 2010 Revised: December 11, 2010 Published: March 02, 2011 4922
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The Journal of Physical Chemistry C The model predicts that a metastable crystalline-to-amorphous phase transition may replace the equilibrium crystalline phase transition between the two olivine phases upon delithiating nanosized particles under intermediate electrical overpotentials. A simplified version of this model is employed in this work, in which we focus on phase morphology evolution during the crystalline-to-crystalline transformation and assume that a particle remains perfectly crystalline upon lithiation. Here, a particle state is described by the lithium concentration (or site occupancy fraction) field c(rB) (0 < c < 1), which distinguishes between the LFP (c = 1) and FP (c = 0) olivine phases, and the elastic strain field εij (rB) (i, j = 1, 2, 3) that characterizes the coherency stress in the particle. The free energy of a particle is given by ZZZ k 2 Ftot ¼ fchem ðcÞþfel ðεij ,cÞþ ðrcÞ dV ð1Þ 2 V
In eq 1, the Li-concentration gradient term κ/2(rc)2 contributes to the phase boundary energy. fchem(c) represents the homo-
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geneous free energy density of the stress-free state and is modeled by a regular solution formulation fchem ðcÞ ¼ ff FP ð1-cÞþf LFP cþRT½c ln cþð1-cÞþWc cð1-cÞg=Vm
ð2Þ where Vm is the molar volume of LicFePO4 and T is the temperature. fFP and fLFP are the molar free energies of stoichiometric FP and LFP, respectively, and Wc is the regular solution coefficient that characterizes the lithium solubility behavior in olivine. The existence of a Li miscibility gap between LFP and FP indicates Wc > 0 at room temperature. fel(εij,c) is the elastic energy density arising from the lattice mismatch between LFP and FP. Different from the previous isotropic treatment adopted in refs 19-21, here we explicitly account for olivine’s elasticity and misfit-strain anisotropy in the elastic energy formulation. In the small strain approximation 1 ð3Þ fel ¼ Cijkl εij -ε0ij ðcÞ εkl -ε0kl ðcÞ 2 where Cijkl (i, j, k, l = 1, 2, 3) is the stiffness tensor. As firstprinciple calculations22 show that the difference between the elastic modulus of LFP and FP is not very significant, Cijkl is assumed to be composition-independent here. ε0ij(c) represents the composition-dependent stress-free strain of LicFePO4 due to lattice expansion upon Li insertion. Under Vegard’s law (i.e., the unit cell volume has a linear Li concentration dependence), ε0ij(c) = Δe0ijc, where Δe0ij is the linear misfit strain between stoichiometric LFP and FP. The thermodynamic and elasticity parameters appearing in Ftot have been measured or estimated in the literature and are listed in Table 1. The governing equations for the evolution and equilibrium of the Li concentration and elastic strain fields, which involve variational derivatives of Ftot with respect to c and εij, are derived by applying variational principles to the free energy functional Ftot. The lithium diffusion process in olivine is described by the Cahn-Hilliard equation23 Dc ¼ r 3 ½MLi cð1-cÞrμLi Dt
Figure 1. (a) Schematics of one-dimensional Li diffusion channels along [010] and anisotropic misfit strain between FePO4 and LiFePO4. (b-f) Li concentration field in a 200 50 50 nm3 particle at rescaled time t = 0, 2 105, 8.5 105, 1.6 106, and 1.9 106 upon Li insertion under Δφ = -25 mV and βh = 0.01.
ð4Þ
where MLi is the Li mobility tensor, which is in general a function of Li concentration. For olivine, the most significant component . As of MLi is its diagonal element in the [010] direction, M[010] Li atomistic calculations6,7 show only a moderate difference in Li
Table 1. List of Model Parameters parameters
physical meaning
values
refs/notes
Vm
molar volume of LicFePO4
43.8 cm3/mol
ref 19
T Wc
temperature regular solution coefficient of crystalline LicFePO4
298 K 12 kJ/mol
ref 19
κ
concentration gradient coefficient
5 10-12 J/cm
ref 19
Cijkl
stiffness tensor of LicFePO4
C1111 = 157 GPa, C1122 = 51 GPa,
average of the GGAþU DFT calculation
C1133 = 53 GPa, C2222 = 171 GPa,
results for LFP and FP from ref 22
C2233 = 33 GPa, C1212 = 52 GPa, C2323 = 38 GPa, C1313 = 49 GPa, Cijkl = 0 GPa, otherwise Δe0ij
linear misfit strain between LiFePO4 and FePO4 olivines.
Δe011 = 5% Δe022 = 3.6%
ref 9
Δe033 = -1.9% Δe0ij = 0 (i 6¼ j) 4923
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diffusivity in FP and LFP, M[010] is assumed to be a constant in Li our simulations. μLi is the lithium chemical potential given by δFtot Dfchem Dfel ð5Þ μLi ¼ ¼ þ -kr2 c δc Dc Dc To facilitate direct comparison with experiments, a single-crystalline particle with prismatic morphology (viz. Figure 1a) is considered here. A zero-flux boundary condition, jLi = 0, is imposed on the (010) and (001) particle surfaces. On the (010) surfaces which is active for Li insertion, we apply a flux boundary condition ð010Þ
jLi
electrolyte
¼ βðμLi
-μsurf Li Þ
ð6Þ
where β is the surface reaction rate constant and μsurf Li and are the Li chemical potentials on the (010) surface μelectrolyte Li and in the surrounding electrolyte, respectively. During electrois controlled by the electrical overchemical cycling, μelectrolyte Li = μcoex potential Δφ applied on cathode particles: μelectrolyte Li Li coex FΔφ, where μLi is the equilibrium chemical potential at LFP/ FP two-phase coexistence and F the Faraday constant. A nonzero overpotential Δφ provides the driving force for Li insertion/ extraction and the phase transition between FP and LFP. Equation 6 can be viewed as a linear approximation to the Butler-Volmer equation24 electrolyte
jLi ¼ j0 ðeð1-RÞðμLi
-μsurf Li Þ=Vm RT
electrolyte
-e-RðμLi
-μsurf Li Þ=Vm RT
Þ ð7Þ
at relatively small overpotentials (Δφ = O(10 mV)),25 with the rate constant β j0/VmRT proportional to the exchange current density j0 across the olivine electrode-electrolyte interface. As the relaxation time for diffusion is typically much larger than for stress, mechanical equilibrium is assumed to be maintained within the particle at any time, i.e. Cijkl
Dεkl Dc ¼ Cijkl Δe0kl ði, j, k, l ¼ 1 , 2 , 3Þ Dxj Dxj
ð8Þ
We apply the phase-field microelasticity theory by Wang et al.26 to solve for the stress/strain fields in a particle with traction-free boundary conditions on the free surface. The diffusion and stress equilibrium equations are solved numerically in their dimensionless forms using the length, energy, and time units, l0 = 1 nm, e0 = e0). In rescaled time, our simulation 10-18 J, and t0 = l50/(M[010] Li only through the results depend on the Li diffusivity M[010] Li , which is kept as an dimensionless rate constant βh = βl0/M[010] Li adjustable parameter to study the effect of surface reaction kinetics on phase evolution.
’ RESULTS AND DISCUSSION We simulated the Li-insertion process in a platelet particle of dimensions L[100] L[010] L[001] = 200 nm 50 nm 50 nm, which is comparable to the average particle size reported in ref 16. The particle domain is discretized on a 200 50 50 mesh. A moderate overpotential Δφ = -25 mV is applied to provide driving force for Li insertion into an initially delithiated particle. Our simulations show that the FP f LFP phase transition occurs by nucleation and growth at this overpotential and particle corners are the most favored nucleation sites. A supercritical LFP nucleus is thus placed at a particle corner at the beginning of simulations to initiate LFP growth. As shown in one simulation using βh = 0.01 (Figure 1b-f), the LFP nucleus grows rapidly along [001] but exhibits much slower growth in other directions.
Figure 2. Illustration of a “control simulation” with the misfit strains between LFP and FP set to zero for comparison to the simulations with misfit strains (e.g., Figure 1). Snapshots of Li concentration field in the particle are shown at rescaled time t = (a) 2 105 and (b) 5 105. The same color map as in Figure 1 is used.
It is clear that such a growth anisotropy results directly from the anisotropic misfit strain between LFP and FP, which is much smaller along [001] (-1.9%) compared to [100] (5%) and [010] (3.6%). Elastic energy thereby favors a preferential growth along [001] to reduce the coherency stress at the LFP/FP interface. After an initial fast expansion along [001], LFP growth becomes quasi-two-dimensional in the (001) plane and the LFP/FP phase boundary moves primarily along [100] at later times, as shown in Figure 1d-f. This agrees well with the growth direction suggested by Chen et al.14 and Delmas et al.15 and is analogous to the phase-transformation wave propagation predicted by Singh et al.18 Nevertheless, the phase boundary shows a curved morphology rather than a flat interface as previously perceived. The main segment of the boundary forms an angle θ with the (100) plane and is inclined toward [100], which is consistent with the finding of Laffont et al.16 Near the particle surface, the phase boundary bends and develops a 90 contact angle with the (010) facets. One notable aspect of the phase evolution shown in Figure 1 is that LFP does not form rapidly on the (010) surface even though abundant Li ions are available in the abutting electrolyte. Previously, this phenomenon has been hypothesized to result from kinetic limiting factors such as an ion/electron mobility that is only significant at the LFP/FP interface14,15 or sluggish Li surface insertion kinetics.18 Our study, however, shows that the phase boundary motion in the [100] direction is also thermodynamically limited by the misfit strain energy. Because LFP and FP have the largest linear misfit strain along [100], fast LFP growth along the (010) surface is disfavored by the elastic energy as it will create a phase boundary plane containing the [100] axis. This is well illustrated through a “control” simulation in which we neglect the lattice mismatch between LFP and FP, i.e., no coherency stress is present during the phase transition. As shown in Figure 2, the LFP growth front moves on the (010) surface much more rapidly in the absence of misfit strain and the phase boundary normal becomes much closer to [010]. The effect of misfit strain energy is also visible in Figure 1e-f, which shows that the phase boundary develops a larger inclination angle θ as it approaches the particle edge due to the stress relaxation near the surface. Comparison between Figures 1 and 2 clearly demonstrates the close control the elastic energy exerts over the phase boundary orientation. Consistent with this finding, we found that the coherency stress within the particle reaches a very high level (∼1 GPa) during lithiation, which is a result of the large misfit strain between FP and LFP and the high 4924
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Figure 3. (a and b) “Rest” simulation shows that the phase boundary morphology remains unchanged after relaxing the particle state shown in Figure 1e without Li-insertion flux for t = 1 105 and 4 105. (c and d) Phase boundary morphology at t = 1 105 and 4 105 from a “control” simulation which uses isotropic diffusion coefficients to verify that diffusion anisotropy restricts interface rotation.
elastic moduli (∼100 GPa) of olivines.22 It is remarkable that such large stress can be accommodated by olivine nanoparticles without causing significant cracking in experiments,16,17 while extensive dislocations and fractures were seen in larger, micrometer-sized particles.14 A similar magnitude of stress is also estimated to be present in silicon27 and SnO228 nanowires upon electrochemical cycling. These observations agree well with recent theoretical works by Hu et al.29 and Bhandakkar and Gao,27 which predict crack nucleation and propagation in Liinsertion electrodes to be effectively suppressed by particle size reduction. As most previous experimental characterizations14,16 of the phase morphology in LiFePO4 reported in the literature were carried out after lithiation/delithiation was ended, it is important to understand whether the observations might be clouded by possible structural relaxation following the intercalation process. We examined this issue by performing a simulation in which Li surface flux is terminated when one-half of the Li sites in the particle are occupied (viz. Figure 1e) and phase evolution in the subsequent “rest” period is monitored. As shown in Figure 3a and 3b, the Li concentration field is essentially unchanged after a long time. This shows that the phase boundary becomes “frozen” when Li insertion stops, suggesting that the ex-situ observations represent the in-situ morphology. Nevertheless, the tilted phase boundary in Figure 1e is not a configuration as energetically favorable as the [100] boundary orientation which minimizes the misfit strain energy. In other words, a thermodynamic driving force exists to adjust the boundary orientation upon relaxation. However, this relaxation is so slow that it is nonobservable in our simulation because of the kinetic constraint imposed by the 1D Li diffusivity, which inhibits Li transport along [100] and prevents rotation of boundary plane toward (100). To verify this kinetic limiting effect, the above relaxation simulation was repeated = M[100] = using a hypothetic isotropic Li diffusivity, i.e. M[010] Li Li [001] MLi , to allow Li diffusion in all crystallographic directions. Figure 3c and 3d shows that the phase boundary readily evolves to the (100) plane in the absence of diffusional anisotropy. Therefore, the strong Li diffusion anisotropy inherent to the olivine structure is fundamental for preserving the evolving phase morphology during the phase transition and makes it accessible to ex-situ characterization. In addition to the elastic energy and Li-diffusion anisotropy, the phase boundary orientation is also regulated by the surface reaction kinetics. At a constant overpotential, the dimensionless Li-surface-insertion rate constant , also referred to as the electrochemical Biot number by Cheng et al.,20 characterizes the relative rates of Li surface insertion vs bulk diffusion kinetics. Li intercalation becomes surface reaction controlled as βh f 0
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Figure 4. Li concentration field in a half-lithiated particle upon Li insertion with βh = (a) 10-4 and (b) 1. (c) Dependence of the phase boundary inclination angle θ on the dimensionless Li-surface-insertion rate constant .
and bulk diffusion controlled as βh f ¥. Figures 1e and 4a and 4b compare the phase boundary morphologies from simulations using three different βh values 0.01, 10-4, and 1. In Figure 4c, the boundary inclination angle θ measured at the half-lithiated particle state is plotted against βh used in simulations. A monotonic increase of θ with βh can be seen across a range of seven decades. Such a relation between θ and βh can be understood as follows. The LFP/FP phase boundary inclination is determined by the ratio of the LFP growth rate in the [100] direction relative to that along [010]. While LFP depends on Li bulk diffusion to grow along [010], its expansion along [100] is achieved by incorporation of Li ions into the “empty channels” at the LFP growth front on the (010) surface. With decreasing βh, the Li surface reaction becomes more sluggish compared to bulk diffusion, which limits the [100] growth rate and causes the boundary plane to lie closer to (100). Simulations of the insertion process with βh below 10-5 are CPU limited and not reported here. Nevertheless, the trend shown in Figure 4c indicates a continuous decrease of the boundary inclination angle with further decreasing βh. In the surface-reaction-controlled limit βh f 0, we expect the phase boundary to lie completely in the (100) plane (i.e., θ = 0), as predicted by Singh et al.14 For values of βh > 1, we find that the insertion process becomes bulk diffusion controlled and phase boundary motion has a negligible dependence on βh. In this regime, θ converge to a maximal value θmax < 90, which is determined by thermodynamics rather than interface kinetics. It is an open question whether Li transport in olivine electrodes is mainly limited by surface reaction or bulk diffusion as vastly different Li diffusion coefficients have been reported from theoretical6,7 and experimental31,32 studies in the literature. Nevertheless, understanding the limiting process in the Liinsertion kinetics is important for developing proper strategy to improve electrode performance. Our prediction of the dependence of the phase boundary inclination on βh provides a useful metric to examine this issue. The tilted boundary morphology observed by Laffont et al. (θ ≈ 10-25 estimated from their EELS data) indicates that Li intercalation in their samples is controlled by a combination of surface reaction and bulk diffusion. For olivine particles, surface treatments such as carbon coating33 can significantly enhance the surface transport of charge carriers and modify the surface reaction kinetics. Recently, Kang and Ceder34 reported that formation of a Li4P2O7-like amorphous layer with a thermodynamic equilibrium thickness35 on LiFePO4 particle surface dramatically increases the rate capability of the Li-ion battery, for which the fast ion conductivity of the surface glassy phase is suggested to be responsible. As the nature of this observed rate enhancement is still under debate,36,37 we suggest that a careful comparison of the phase 4925
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The Journal of Physical Chemistry C morphology in particles with and without the amorphous coating may shed more light on the involved mechanism. Such experiments may be complimented by electrochemical tests, as a small β results in increased polarization across the electrodeelectrolyte interface during galvanostatic cycling, which in principle can be isolated from the voltage-capacity curve of olivine electrodes38,39 after carefully considering the effects of diffusion and misfit strain anisotropy.
’ CONCLUSIONS In summary, phase-field simulations of the Li-insertion process in LiFePO4 platelet particles reveal that the elastic energy arising from the misfit strain between the Li-rich and Li-poor olivine phases has a considerable influence on the phase growth direction and morphology during the olivine phase transition. We found that interpretation of ex-situ studies is a reliable determinant of in-situ phase morphology because of the onedimensional Li diffusivity of the olivine phases, which kinetically stabilizes the LiFePO4/FePO4 phase boundary after lithiation stops. We propose that characterization of the phase boundary orientation is an effective way to quantify the relative importance of the Li bulk diffusion and surface reaction processes to the intercalation kinetics.
’ AUTHOR INFORMATION Corresponding Author
*Phone: (925) 424-4157. Fax: (925) 422-6594. E-mail:
[email protected].
’ ACKNOWLEDGMENT This work was performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. M.T. acknowledges the financial support from the Lawrence Postdoctoral Fellowship. He is indebted to W. Craig Carter and Yet-Ming Chiang for a critical reading of the manuscript and many helpful discussions that inspired this work. Computations carried out in this work used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. ’ REFERENCES
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