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Electrochimica Acta 56 (2010) 969–976

Contents lists available at ScienceDirect

Electrochimica Acta

journa l homepage: www.e lsev ier .com/ locate /e lec tac ta

odeling the competing phase transition pathways inanoscale olivine electrodes

ing Tanga,∗, W. Craig Carterb, James F. Belaka, Yet-Ming Chiangb

Condensed Matter and Materials Division, Lawrence Livermore National Laboratory, Livermore, CA 94550, United StatesDepartment of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, United States

r t i c l e i n f o

rticle history:eceived 10 July 2010eceived in revised form 1 September 2010ccepted 1 September 2010vailable online 16 September 2010

eywords:ithium-ion battery cathodeslivines

a b s t r a c t

Recent experimental developments reveal that nanoscale lithium iron phosphate (LiFePO4) olivineparticles exhibit very different phase transition behavior from the bulk olivine phase. A crystalline-to-amorphous phase transition has been observed in nanosized particles in competition with the equilibriumphase transition between the lithium-rich and lithium-poor olivine phases. Here we apply a diffuse-interface (phase-field) model to study the kinetics of the different phase transition pathways in nanosizedLiFePO4 particles upon delithiation. We find that the nucleation and growth kinetics of the crystalline-to-crystalline and crystalline-to-amorphous phase transformations are sensitive to the applied electricaloverpotential and particle size, which collectively determine the preferred phase transition pathway.

morphizationverpotentialiffuse-interface model

While the crystalline-to-crystalline phase transition is favored by either faster nucleation or growthkinetics at low or high overpotentials, particle amorphization dominates at intermediate overpoten-tials. Decreasing particle size expands the overpotential region in which amorphization is preferred.The asymmetry in the nucleation energy barriers for amorphization and recrystallization results in aphase transition hysteresis that should promote the accumulation of the amorphous phase in electrodesafter repeated electrochemical cycling. The predicted overpotential- and size-dependent phase transition

ePO4

behavior of nanoscale LiF

. Introduction

Olivine-type LiMPO4 (M = Fe, Mn, Co, Ni) compounds [1,2] havemerged as important positive electrode materials in Li-ion bat-eries for a wide range of applications from cordless power toolso plug-in hybrid vehicles to electric grid storage systems [3–5].t room temperature, a lithium miscibility gap exists in thisroup of compounds [1]. Bulk LiMPO4 undergoes a first-orderransition between a Li-rich and Li-poor olivine phase (denoteds cLFP and cFP hereafter; the qualifiers c and a refer to crys-alline and amorphous) during electrochemical cycling. Recently,he nanoscaling of crystallite sizes, essential for achieving highate capability, has been shown to significantly modify the phaseransition behavior of olivines [6–10]. In particular, the forma-ion of a delithiated amorphous phase (denoted as aFP) has beenbserved in nanoscale LiFePO4 and LiMnPO4 particles [11,12]. This

rystalline-to-amorphous phase transition, cLFP → aFP, competesith the “conventional” crystalline-to-crystalline phase transition,

LFP→cFP, and the competition was found to be significantly influ-nced by both particle size [11] and the magnitude of the electrical

∗ Corresponding author. Tel.: +1 925 424 4157; fax: +1 925 422 6594.E-mail addresses: tang25@llnl.gov, tang01@gmail.com (M. Tang).

013-4686/$ – see front matter. Published by Elsevier Ltd.oi:10.1016/j.electacta.2010.09.027

particles is consistent with experimental observations.Published by Elsevier Ltd.

overpotential applied to olivine particles [12] during electrochem-ical cycling.

In a diffuse-interface (phase-field) thermodynamic model [13],we assessed the driving force and nucleation energy barriersassociated with the two competing phase transitions, cLFP → cFPand cLFP → aFP, in LiFePO4. Our results showed that bulk aFPis metastable relative to cFP under delithiation conditions andcLFP → cFP is the equilibrium phase transition pathway. However,the surface nucleation of aFP upon cLFP particles should be kinet-ically favored since disordered (amorphous or liquid) structuresusually have lower surface energies than their crystalline coun-terparts. This is the underlying reason for melting transitions tostart from surfaces and interfaces (e.g. grain boundaries) and forliquid-like films to develop at surfaces in metals [14], oxides [15]and ice [16] below the bulk melting point (known as “surface pre-melting”). This surface effect becomes increasingly significant asparticle size decreases, exemplified by the suppression of meltingpoint in nano-gold particles [17] and the stabilization of amorphousphase in nanosized ZrO2 [18]. The amorphous phase formation is

also promoted by the glass-forming ability of MPO4, which sug-gests that bulk crystalline and amorphous MPO4 has a relativelysmall free-energy difference.

In addition to particle size, the electrical overpotential appliedto electrode compounds during use is another fundamental param-

9 imica Acta 56 (2010) 969–976

edteoitFfea

ootwumts

ctt[(avsofnnpl

eopv�qTtpevaowc

2

2

ccafgfac

Fig. 1. (a) Schematic illustration of various phase transition pathways that mayoccur upon delithiation of a nanoscale cLFP particle. The crystalline-to-crystallinetransformation cLFP → cFP competes with a crystalline-to-amorphous transforma-tion leading to a metatstable delithiated amorphous phase aFP, which may furthertransform to cFP under high overpotentials. A “surficial amorphous film” (SAF) mayform upon delithiation as an intermediate particle state between the bulk cLFPand aFP phases. (b) Overpotential dependence of the nucleation energy barriers

70 M. Tang et al. / Electroch

ter that controls the phase transition behavior. For LiMPO4, weefine the term “overpotential” as the magnitude by which the elec-rical potential experienced by the active material deviates from thequilibrium potential at cLFP/cFP two-phase coexistence, �coex. Theverpotential �� determines the Li-ion electrochemical potentialn the Li+ reservoir (e.g. the electrolyte surrounding cathode par-icles) as �e

Li = �coexLi − F��, where �coex

Li = −F�coex and F is thearaday constant [19]. A non-zero �� thus supplies the drivingorce for Li insertion or extraction and phase transformations inlectrodes – overpotential is the electrical analogy to undercoolingnd superheating in thermally driven phase transitions.

Although the overpotential is a constantly varied quantity thatften drives electrodes in the far-from-equilibrium real-world usef Li-ion batteries [20], its influence on phase stability and phaseransition behavior of electrode materials appears to have not beenidely appreciated. As the nucleation and growth rates of the prod-ct phase in a first-order transformation are both sensitive to theagnitude of driving force (e.g., see chapters 19 and 20 of Ref. [21]),

he competition between cLFP → cFP and cLFP → aFP pathways areignificantly affected by ��.

In the previous work [13], we report that overpotential hasritical influence on the nucleation kinetics of competing phaseransitions in LiMPO4. Below a critical particle size that is sensitiveo the misfit strain between cLFP and cFP, phase-field calculations13] show that cLFP → aFP has a smaller nucleation energy barrier�Fc→a) than cLFP → cFP (�Fc→c) when the overpotential is raisedbove a critical value ��e, as shown in Fig. 1(b). �Fc→a furtheranishes at a higher characteristic overpotential ��s1, inducingpontaneous amorphous phase nucleation in cLFP particles. More-ver, we predicted that “surficial amorphous films” (SAFs) [22] mayorm on olivine surface under suitable conditions [23]. SAFs areanometer-thick disordered surface layers with equilibrium thick-ess and a composition that differs from the bulk phase. With theresence of SAFs, cLFP may spontaneously amorphize at an even

ower overpotential ��s2 (viz. inset of Fig. 1(b)).In this paper, we apply the phase-field model to examine the

ffect of overpotential and crystallite size on the growth kineticsf competing phase transitions in LiFePO4. Such an extension isarticularly relevant for understanding the competition betweenarious transition pathways at relatively large overpotentials (e.g.,� > ��s1), where critical nuclei form rapidly and their subse-

uent growth into the bulk phase becomes the rate-limiting step.ogether with previous results on the nucleation kinetics [13],he model reveals a non-trivial overpotential dependence of thereferred phase transition pathway in nanoscale olivines duringlectrochemical cycling. Our findings are consistent with and pro-ide explanations for recent experimental observations [11,12],nd underscore the importance of understanding/controlling theverpotential in the real-world use of batteries. In the following,e first describe the phase-field model. Simulation results and dis-

ussion are then presented, followed by conclusions.

. Model

.1. Particle free energy

In the phase-field model, the state of a LiFePO4 particle isharacterized by three field variables: local Li concentration c(�r),rystallinity �(�r) and the radial displacement u(�r) with respect tostress-free particle state. c(�r) (0 < c < 1) is the local occupancy

raction of available Li sites. The crystallinity field �(�r) is a coarse-rained measure [24,25] of the local structural order of the FePO4ramework. We assign � = 1 to a perfectly crystalline structurend � = 0 to a disordered phase. The radial displacement field u(�r)haracterizes the elastic deformation within the particle. As the

for phase transformations cLFP → cFP and cLFP → aFP in a 100-nm-diameter parti-cle upon delithiation. The “Max” curve (dashed line) represents a local maximumin the nucleation energy barrier. See Ref. [13] for the calculation method. Modelparameters used for the calculations are listed in Table 1.

olivine particles in related experiments [11,12] generally haveequiaxed shape, we consider a simple spherical particle shape andapply isotropic approximations to particle properties to make cal-culations as transparent as possible. The field variables thus arefunctions of the radial distance r from the particle center alone.Upon delithiation a phase transformation is postulated to initiateuniformly on particle surface and propagate inwards, similar to theshrinking-core geometry employed in the discharge model of Srini-vasan and Newman [27] (however, this core-shell assumption isquestionable when the driving force for phase transition is small[7]). The total free energy of a LiFePO4 particle is expressed as afunctional of the field variables:

Ftot = 4�R2�(�s) +∫ R

dr4�r2[

�fchem(c, �) + �fel

(u,

du, c, �

)

0 dr

+�2

2

(∂c

∂r

)2

+ 2

2

(∂�

∂r

)2]

(1)

M. Tang et al. / Electrochimica Acta 56 (2010) 969–976 971

Table 1List of model parameters.

Parameters Physical meaning Values

Vm Molar volume of LicFePO4 43.8 cm3/mol�fFP,c→a(�fLFP,c→a) Molar free energy difference between bulk crystalline and amorphous FePO4 (LiFePO4) 6 kJ/mola2

FP(a2LFP) Energy barrier height between bulk crystalline and amorphous FePO4 (LiFePO4) 12 kJ/mol

Wc Regular solution coefficient of crystalline LicFePO4 12 kJ/molWa Regular solution coefficient of amorphous LicFePO4 −12 kJ/molK Isotropic bulk modulus averaged over all crystallographic orientations 93.9 GPaG Isotropic shear modulus averaged over all crystallographic orientations 48.4 GPa

0 in between lithiated and delithiated olivines 0.02

stallin

wti

(

(

�e Orientation-averaged linear misfit stra�2 Concentration gradient coefficient2 Crystallinity gradient coefficient�� Surface tension difference between cry

here R is the particle radius. The formulation of various contribu-ions to the particle free energy in Eq. (1) has been given in detailn [13] and is described briefly below.

1) �fchem(c, �) is the chemical free energy per volume of ahomogeneous system with a uniform Li concentration c andstructural order �. This term is formulated by an extended reg-ular solution model [13,26]:

�fchem(c, �) = 1Vm

{f FP(�)(1 − c) + f LFP(�)c

+RT[c ln c + (1 − c)ln(1 − c)] + [Wcp(1 − �)

+Wap(�)]c(1 − c)} (2)

where Vm is the molar volume of LiFePO4. fFP(�)(fLFP(�)) is themolar free energy of stoichiometric FePO4 (LiFePO4), given bythe following:

f FP(�) = f FP(� = 1) + �f FP,c→ap(�) + a2FP2

�2(1 − �)2

(f LFP(�) = f LFP(� = 1) + �f LFP,c→ap(�) + a2LFP2

�2(1 − �)2)(3)

In Eq. (3), �fFP,c→a is the vitrification free energy of stoichiomet-ric FePO4, a2

FP is the energy barrier height between crystallineand disordered FePO4, and �fLFP,c→a and a2

LFP are their coun-terparts for LiFePO4. p(�) = (1 − �)3(1 + 3� + 6�2) is a smoothinterpolation function between p(� = 0) = 1 and p(� = 1) = 0.

The lithium solubility behavior in the crystalline (amorphous)phase is characterized by the regular solution coefficient Wc

(Wa) in Eq. (2). While the phase-separating olivine phase hasa positive regular solution coefficient Wc > 0 at room tempera-ture, amorphous LicFePO4 was shown to remain a solid solutionwithin the entire Li composition range c = 0–1 (e.g. see Refs.[28,29]), indicating Wa < 0. The regular solution coefficients ofpartially disordered states (0 < � < 1) are interpolated betweenthe crystalline and amorphous phases with p(�).

2) �fel(u, du/dr, c, �) is the elastic strain energy density thatarises from the coherency stress field during the crystalline-to-crystalline phase transformation due to the volume mismatchbetween cLFP and cFP. Li intercalation often results in large,heterogeneous volume changes in electrode compounds. Asdemonstrated in literature [30], the incurred stress field couldhave significant influence on the phase behavior and electro-chemical responses of the electrodes and needs to be taken intoaccount. For small strains, �fel is given by

�fel =(

K

2− G

3

)(du

dr+ 2

u

r− 3e0(c, �)

)2

+ G(

du

dr− e0(c, �)

)2

+ 2G(

u

r− e0(c, �)

)2(4)

5 × 10−12 J/cm10−11 J/cm

e and amorphous LicFePO4 0.2 J/m2

where K and G are the isotropic bulk and shear mod-uli averaged over all crystallographic orientations. e0(c,�) = �e0(c − 1)p(1 − �) is the stress-free strain characterizingthe linear Li concentration dependence of the unit cell volume(i.e. Vegard’s law), where �e0 is the average linear misfit strainbetween cFP and cLFP; p(1 − �) in e0(c,�) ensures a negligiblestress field across the crystalline/amorphous phase boundarywhich has an incoherent interface structure.

(3) (�2/2)(( ∂ c/∂ r))2 and (2/2)(( ∂ �/∂ r))2 are the gradient energiesrelated to the interface energies of crystalline/crystalline andcrystalline/amorphous phase boundaries. These excess energycontributions are only significant in the interface region whereLi concentration and/or crystallinity gradients are present.

(4) �(�s) is the surface energy term. Inspired by Cahn’s critical pointwetting theory [31], the surface energy is postulated to be afunction of the surface crystallinity �s ≡ �(r = R):

�(�s) = ���2s + �(�s = 0) (5)

where �(�s = 0) is the surface energy of amorphous structureand �� is the surface energy difference between the crystallineand amorphous phases. As disordered structures have lowersurface energy than crystalline phases, i.e. �� > 0, �(�s) is amonotonically increasing function of �s.

The material parameters involved in Eqs. (1)–(5) have beenassessed for LiFePO4 in Ref. [13] and are summarized in Table 1.

2.2. Kinetic equations

To simulate the spatio-temporal evolution of phases in an ini-tially crystalline LiFePO4 particle upon delithiation, we model Lidiffusion within the particle with the Cahn–Hilliard equation [32]:

∂c

∂t= ∇ · [MLic(1 − c)∇�Li] (6)

where the lithium mobility MLi is related to the lithium diffusioncoefficient DLi in the dilute limit: MLi = DLiVm/(RT). �Li is the localLi chemical potential given by

�Li = ıFtot

ıc= ∂fchem

∂c+ ∂fel

∂c− �2∇2c (7)

The evolution of local structural order �(r) within the particle ispostulated to follow the Allen–Cahn kinetics [33]:

∂�

∂t= −Mam

ıFtot

ı�= −Mam

(∂fchem

∂�+ ∂fel

∂�− 2∇2�

)(8)

where Mam is the mobility of the crystalline/amorphous interface;

its value is related to thermally activated processes of atomic reori-entation at the interface.

As the relaxation of the elastic stress field is typically much fasterthan the evolution of concentration and crystallinity fields duringphase transitions, mechanical equilibrium, ıFtot/ı�u = 0, is assumed

972 M. Tang et al. / Electrochimica Acta 56 (2010) 969–976

0 10 20 30 40 50

0

0.2

0.4

0.6

0.8

1

25 30 35 40 45 500.2

0. 4

0. 6

0. 8

1

25 30 35 40 45 50

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50

0

0.2

0.4

0.6

0.8

1(a) (b)

0 s

0 s

0 s

0 s

22.6 s

7.35 s

1.70 s

2263 s

2263 s

735 s

735 s

surfaceparticle particle

surface

(nm)r (nm)r

Li c

once

ntra

tion c

Cry

stal

linity

η

F -diamo V (t =f

tp

c

Ice(At�bvkb

(T�i

ig. 2. Snapshots of lithium concentration and crystallinity fields within a 100-nmverpotential: (a) �� = 50 mV (t = 0, 7.35, 22.6, 57, 283, and 735 s) and (b) �� = 80 mor the two simulations are provided in Supplemental Information.

o be maintained within the particle throughout the delithiationrocess.

Eqs. (6)–(8) are completed by boundary conditions at the parti-le surface and center:

JLi

∣∣r=R

= ˇ(�Li(r = R) − �eLi) ≡ ˇ(�Li(r = R) − �coex

Li + F��) (9)

∂c

∂r

∣∣∣∣r=R

= 0 (10)

∂�

∂r

∣∣∣∣r=R

+ 2�� �s

2= 0 (11)

∂c

∂r

∣∣∣∣r=0

= ∂3c

∂r3

∣∣∣∣r=0

= ∂�

∂r

∣∣∣∣r=0

= 0 (12)

n particular, the flux boundary condition (Eq. (9)) implements aonstant overpotential �� on the particle surface, analogous to thexperimental situation in potentiostatic intermittent titration testsPITT) [34] when the ohmic drop within the electrode is negligible.ccording to Eq. (9), the Li surface flux JLi

∣∣r=R

is proportional tohe difference between Li chemical potentials at particle surface,

Li(r = R), and the surrounding electrolyte, �eLi, which is determined

y the overpotential ��. Eq. (9) may be viewed as the linearizedersion of the Butler–Volmer equation [35] or the Arrhenius surfaceinetic equation employed in the Li intercalation dynamics modely Singh et al. [36].

We performed the delithiation simulation by solving Eqs.6)–(12) numerically with a semi-implicit finite difference scheme.ypical grid spacing and time step size used in simulations arex = 0.5 nm and �t = 50 �s, which have been tested for stabil-

ty and convergence of numerical calculations. In addition to the

eter particle as a function of time during the delithiation process under constant0, 1.70, 7.35, 22.6, and 2263 s), using Mam/MLi = 1 nm−2. Additional animation files

thermodynamic parameters listed in Table 1, a Li diffusion coeffi-cient DLi = 10−14 cm2/s [10] is used in the simulations, which givesMLi = 1.8 × 10−16 cm5/(J s). As the lithium mobility in the amor-phous phase is not currently known, this MLi value is assigned toboth crystalline (cLFP and cFP) and amorphous (aFP) bulk phases.A large rate coefficient ˇ is chosen to ensure that Li surfaceextraction does not impose kinetic limitations on lithium inter-calation (this approximation is consistent with the experimentalresults of Meethong et al. [10] which indicate a bulk-diffusion-controlled insertion kinetics). To the best of our knowledge, thecrystalline/amorphous interface mobility Mam has not been mea-sured or evaluated for LiFePO4. Nevertheless, the effect of itsmagnitude on the growth kinetics has been examined in simula-tions and we found that it has limited influence on the overall phasetransition behavior, which will be described in details in the nextsection.

3. Results and discussion

When a cLFP particle is delithiated at �� > ��s1 (Fig. 1b), theamorphous phase nucleates spontaneously at particle surface andits growth into the bulk becomes the rate-limiting step. To studythe phase transformation kinetics in this growth-limited regime,we simulated the time-dependent delithiation of an initially crys-talline particle. Fig. 2 (and also Supplemental Information) shows

such simulations at two representative overpotentials for a 100-nm-diameter particle (��s1 = 31.7 mV), using Mam/MLi = 1 nm−2. At�� = 50 mV, an aFP nucleus, characterized by its low crystallinity(small �) and depleted Li concentration, rapidly forms at the particlesurface when delithiation starts and subsequently grows inwards

M. Tang et al. / Electrochimica Acta 56 (2010) 969–976 973

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

130 mV

40 mV

50 mV

60 mV

70 mV

80 mV

Li Surface Concentration

Surf

ace

Cry

stal

linit

y

time

aFP

cFP LFPc

Fdt

aiaaanAvafscteip

iMacfiaaesdsctiecdi

ioatLtl

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0.01

0.05

0.1

0.5

0.8

20

Li Surface Concentration

Surf

ace

Cry

stal

linit

y

time

FPa

cFP LFPc

ig. 3. Evolution of the surface Li concentration and crystallinity of a 100-nm-iameter particle upon delithiation under different overpotentials. The inset showshe evolution trajectories at the beginning of the delithiation process.

t the expense of cLFP [Fig. 2(a)]. The entire particle is transformednto the bulk amorphous phase with a Li concentration cLi = 0.36t the end of simulation. At a higher overpotential �� = 80 mV,n amorphous nucleus also forms spontaneously on the surfacet the beginning of the delithiation. At this potential, however, theucleus does not grow into a bulk phase with further delithiation.s shown in Fig. 2(b), the surface crystallinity reached a minimalalue at t ≈ 7.4 s, but returned to higher values afterwards. Themorphous nucleus eventually diminishes, although particle sur-ace still retains a certain degree of structural disorder due to theurface energy effect (Eq. (5)). While the surface region recovers itsrystallinity after transient disordering, the Li surface concentra-ion decreases monotonically with time. A fully delithiated nucleusventually emerges at the surface and further grows into the bulkn the absence of structural change, producing a cLFP → cFP bulkhase transition.

The overpotential-dependent growth behavior depicted in Fig. 2s also representative of simulations at other particle sizes and

am/MLi values. A main finding is that the spontaneous aFP nucle-tion at �� > ��s1 does not always lead to bulk amorphization; therystalline-to-crystalline bulk phase transition reemerges at suf-ciently large overpotential, despite the formation of a transientmorphous surface nucleus. The sensitivity of the final delithi-ted particle state to �� is illustrated in Fig. 3, in which the timevolution of the surface state (i.e., Li surface concentration cs andurface crystallinity �s) of a 100-nm-diameter particle is plotted forelithiation processes under various overpotentials. At the end ofimulations, the surface state reflects the bulk phase to which theLFP particle is transformed. Detailed numerical calculations showhat the particle delithiation kinetics exhibit a bifurcation behav-or at a critical overpotential ��c = 67.1 mV: the particle surfacevolves to a fully disordered structure as does the bulk of the parti-le at �� < ��c. When �� > ��c, however, the evolution follows aifferent path to a fully delithiated, relatively ordered surface state;

n this case a cLFP → cFP transformation occurs.The observed change of the phase-transition pathway upon

ncreasing �� can be understood by inspecting the effects that theverpotential has on the two kinetic processes of lithium diffusionnd structural disordering. For Li diffusion, the applied overpo-

ential influences the Li chemical potential difference betweeniFePO4 particles and the surrounding electrolyte. As shown byhe boundary condition Eq. (9), a higher overpotential brings aarger Li surface-flux out of the particle upon initial delithiation.

Fig. 4. Evolution of the surface Li concentration and crystallinity of a 100-nm-diameter particle upon delithiation under �� = 65 mV with different values of themobility ratio Mam/MLi being used in simulations.

The increased Li surface-flux in turn creates a larger Li chemical-potential gradient across the surface layer and accelerates lithiumdiffusion within the particle (Eq. (6)). On the other hand, the rate ofstructural disordering leading to amorphization, governed by Eqs.(8) and (11), is not directly affected by changes in ��. Lithium dif-fusion is thus expected to become increasingly more facile relativeto the structural disordering process as �� increases. This is illus-trated in the inset of Fig. 3, in which the enhanced rate differencebetween the evolution of Li surface concentration and of surfacecrystallinity at higher overpotentials can be clearly seen. As aFP dif-fers from cLFP in both crystallinity and Li concentration, its growthinto a cLFP particle requires “synchronized” lithium diffusion andstructural disordering across the crystalline/amorphous interface.However, the coupling between the two processes cannot be main-tained at sufficiently large overpotential values, whereas structuraldisordering cannot keep up with the much faster lithium diffu-sion. The kinetic disparity between the two processes induces aswitch to the crystalline transition pathway cLFP → cFP that can beaccommodated by Li diffusion alone.

Fig. 4 illustrates the influence of the crystalline/amorphousinterface-mobility Mam on the phase-transformation kinetics. Asthe phase-transition behavior of nanoscale olivines is controlledby the rate difference between structural disordering and Li diffu-sion kinetics, the preferred transition pathway is not affected by theabsolute mobility values Mam and MLi, but by their relative magni-tudes, Mam/MLi. Fig. 4 shows the evolution of surface crystallinityand composition in a 100-nm-diameter particle under the sameoverpotential �� = 65 mV but at different values of Mam/MLi. Forrelative small Mam/MLi (0.01–0.5 nm−2), the structural disorderingprocess is impeded kinetically and the cLFP → cFP transforma-tion occurs (i.e. the bifurcation overpotenital, ��c, is less than65 mV for Mam/MLi ≤ 0.5 nm−2). The transition pathway switchesto cLFP→aFP at higher Mam/MLi due to increased structural disor-dering kinetics, i.e., ��c is above 65 mV for Mam/MLi ≥ 0.8 nm−2.Similar to the overpotential effect shown in Fig. 3, there is abifurcation in the phase-transition pathway at a critical mobilityratio between 0.5 and 0.8 nm−2. A larger crystalline/amorphousinterface-mobility delays the onset of the crystalline-to-crystallinetransformation to a higher overpotential. We calculated the

mobility-ratio dependence of the bifurcation overpotential ��c.Fig. 5 shows that ��c increases monotonically with Mam/MLi ata given particle size. However, the magnitude of ��c does notgrow indefinitely as Mam/MLi approaches infinity, but saturates at a

974 M. Tang et al. / Electrochimica

0 10 20 30 40 50 60 70 80

40

50

60

70

80

= 25 nm

= 50 nm

= 15 nm

R

R

R

Fig. 5. Dependence of the calculated critical overpotential ��c on the mobility ratioMam/MLi at particle size R = 50 nm (�), 25 nm (�) and 15 nm (�). Calculations showthat ��c saturates at a size-dependent overpotential value ��max

c as Mam/MLi → ∞.

10 20 30 40 50 60 70 800

20

40

60

80

100

Particle Radius (nm)

Ove

rpot

enti

al

(

mV

)

by smaller nucleation

by smaller nucleation

energy barrier

energy barrier

by fast Li diffusionfavored

favored LFP w/o SAF spontaneously amorphizes

favored

R

c c

c c

c ac

LFP w/ SAF spontaneously amorphizesc

F max

�tP

fi�tmcopp

croiw

TL

ig. 6. Particle size dependence of the characteristic overpotentials, ��c , ��e ,�s1 and ��s2. The plot predicts the dominant phase transition pathway as a func-

ion of overpotential and particle size upon delithiating a crystalline LiFePO4 particle.art of the figure is adapted from Refs. [12,13].

nite, size-dependent maximal value, denoted as ��maxc . For �� >

�maxc , a cLFP particle undergoes the crystalline-to-crystalline

ransformation regardless of the magnitude of Mam. Although theobility of the crystalline/amorphous interface in LiFePO4 is not

urrently known and is needed for determination of the bifurcationverpotential ��c, ��max

c serves as an upper bound to the over-otential below which cLFP → aFP is the predominant transitionathway upon delithiation.

In Fig. 6, we plot ��maxc against particle radius along with the

haracteristic overpotentials obtained from nucleation energy bar-

ier calculations [13]. Table 2 lists the physical meaning of eachf the characteristic overpotentials derived from the model. Fig. 6s a phase transition map showing the preferred transition path-

ay upon delithiation. It predicts that the crystalline-to-crystalline

able 2ist of characteristic overpotentials.

Symbol Physical meaning

��e �� > ��e , cLFP → aFP has a smaller nucleation energybarrier than cLFP → cFP

��s1 �� > ��s1, aFP has a vanishing nucleation energy barrierin a cLFP particle without surficial amorphous film (SAF)

��s2 �� > ��s2, aFP has a vanishing nucleation energy barrierin a cLFP particle with SAF

��c �� > ��c , cLFP → cFP replaces cLFP→aFP as the preferredphase transition pathway because of its faster growthkinetics.

��maxc ≡��c(Mam/MLi = ∞), upper bound estimate of ��c

Acta 56 (2010) 969–976

transformation cLFP → cFP is preferred at low overpotentials(<��e) due to a smaller nucleation barrier than that for amor-phization, and again at high overpotentials (> ��max

c ) due to fastLi diffusion kinetics. Within an intermediate overpotential region(��e < �� < ��max

c ), the crystalline-to-amorphous transforma-tion cLFP → aFP has a smaller and even vanishing nucleation barrier,and is the predominant phase transition pathway. As shown inFig. 6, ��max

c is relatively constant at particle sizes R > 15 nm andrises sharply below 15 nm, while ��e decreases with decreas-ing particle size and vanishes at R < 35 nm. Thus the intermediateoverpotential window expands significantly below R ≈ 50 nm, sug-gesting an increased propensity for amorphization in nanosizedolivine particles upon delithiation.

The predictions of the model are consistent with recent exper-imental investigations of the overpotential-dependent phase-transformation pathway in nanoscale olivines. Kao et al. [12]studied phase evolution in two undoped LiFePO4 powders (of meanspherical diameter 113 and 34 nm, respectively) using in situ syn-chrotron X-ray diffraction performed during cycling of Li half-cells.Upon potentiostatic charging of the 113 nm particle sample, theyobserved a dominant cLFP → cFP transformation behavior at low(<20 mV) and high (>75 mV) overpotentials. However, as predictedby the model, a pronounced formation of noncrystalline phase wasseen at intermediate overpotentials. This perhaps counter-intuitiveobservation can be understood from the overpotential-dependentnucleation and growth kinetics of the two competing phase tran-sition pathways explained here. The experimentally estimatedcritical overpotenitals ��e and ��max

c are also in reasonableagreement with the predictions of the model. A similar overpo-tential dependence was seen in the 34 nm particle sample, but thecrystalline-to-crystalline transformation was not significant evenat low overpotentials, which is consistent with the predicted par-ticle size dependence of ��e in Fig. 6.

The good agreement between experiment and model is perhapssomewhat fortuitous given that the numerical values for certainof the parameters in Table 1 are not known with great accu-racy, and given the simplifying assumptions used in the model.These include isotropic approximations of particle shape, sur-face/interface energy, Li diffusivity in olivine phases and phasetransformation geometry (i.e. the shrinking-core geometry [27]).A refined model that incorporates the anisotropy of variousolivine properties would provide more precise predictions. Nev-ertheless, the current model captures the competition betweenthe crystalline-to-crystalline and crystalline-to-amorphous phasetransformations, and provides reasonable estimates of the range ofoverpotentials and particle sizes over which such competition isexpected to be significant. It provides a basis for interpretation ofexperimental results.

We now discuss the stability of the delithiated amorphousphase. Although aFP is kinetically preferred at intermediateoverpotentials, it is still metastable relative to cFP [13], and recrys-tallization into cFP is thermodynamically favored. However, theformation of crystalline nuclei in amorphous particles is disfa-vored by the larger surface energy of the crystalline phase. Thusthe nucleation kinetics of aFP → cFP may be more sluggish than theconverse crystalline-to-amorphous phase transition. As illustratedin Fig. 7, the nucleation-energy barrier for aFP→cFP decreases withincreasing �� at a much slower rate than that for cLFP→aFP. In[11], Meethong et al. observed a long relaxation time (>200 h) forthe open-circuit voltage (OCV) of lithium half-cells made from the34 nm LiFePO4 particles after charging terminates. This is consis-

tent with slow recrystallization of the amorphous phase formedupon delithiation.

Fig. 7 indicates that the kinetics of the aFP → cFP transformationwill also be overpotential-dependent and may become significantat high overpotentials beyond 100 mV, due to a shrinking nucle-

M. Tang et al. / Electrochimica

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

1.2x 10

−5

Overpotential (mV)

ca

c a

Fig. 7. Overpotential dependence of the nucleation energy barriers for cLFP → aFPaTg

awafcattttw

slheromesiotant

i[stfisw

mobbtd

(1997) 1188.

nd the recrystallization transformation aFP → cFP in a 100-nm-diameter particle.he nucleation energy barrier of aFP → cFP is estimated by using a shrinking-coreeometry.

tion barrier. However, unlike the reverse nucleation of cLFP → aFP,hich is barrierless at �� > ��s1, the aFP → cFP transition retainsnon-zero nucleation energy-barrier even as �� → ∞. There-

ore, we expect a hysteresis in the phase transformation uponharge/discharge cycling: while the amorphization is spontaneousbove ��s1, the reverse crystallization process always requireshermal activation. This asymmetry should lead to the accumula-ion of aFP after repeated cycling. Indeed, in galvanostatic cyclingests [12], Kao et al. observed a continuous reduction of the crys-alline phase fraction (cLFP + cFP) in a 113 nm particle sample,hich is even more pronounced in a 34 nm particle sample.

The overpotential-dependent competition between phase tran-ition pathways in olivines has significant implications forithium-ion battery operation. In emerging applications such asybrid electric (HEV), plug-in hybrid electric (PHEV), and batterylectric vehicles (BEV), Li-ion batteries will be subjected to a wideange of duty cycles that invariably produce widely fluctuatingverpotentials. The overpotentials to which the active electrodeaterials are subjected also will depend strongly on cell and

lectrode design. Significant hysteresis in the phase state of thetorage compounds is likely. Furthermore, since each electrodes a large assembly of particles, variations in overpotential mayccur between particles due to differences in local ohmic resis-ance. Multiple phase transition pathways (cLFP → cFP, cLFP → aFPnd aFP → cFP) may occur simultaneously and produce inhomoge-eous phase states. The present model could be further extendedo treat such conditions.

Recent experimental studies [6,9,37] show that lithium solubil-ty in olivines increases with decreasing particle size. It is suggested6,9,37] that crystalline LicFePO4 may remain as a single-phaseolid solution throughout lithiation/delithiation in small nanocrys-allites. The competition between this alternative pathway and therst-order phase transformations (cLFP → cFP and cLFP → aFP) con-idered here may also be size- and overpotential-dependent andill be assessed in a following study.

Although we have focused on olivines, clearly, other electrodeaterials can experience similar phenomena. The characteristics

f LiMPO4 that lead to overpotential-dependent phase transition

ehavior are: (1) good glass-forming capability; (2) Li immisci-ility between the crystalline phases (presence of a crystallineransition); (3) large Li solubility in the amorphous phase; and (4)iffering surface energies between phases.

Acta 56 (2010) 969–976 975

These features are not unique to LiMPO4 and can be found inelectrodes of much current interest including Li2FeSiO4 orthosil-icates [38] and Si [39,40], in which crystalline-to-amorphousphase transitions during electrochemical cycling have also beenobserved. We suggest that amorphization in these materials willalso be overpotential-dependent. From a thermodynamic perspec-tive, the crystalline-to-amorphous phase transformation reportedhere is also analogous to the solid-state amorphization phenomenaobserved in many metallic systems [41–43]. There, the thermo-dynamically favored crystalline-to-crystalline phase transition isalso suppressed by a large nucleation barrier and gives way to theformation of a metastable amorphous phase.

4. Conclusions

We have shown that the electrical overpotential appliedto electrode-active materials is a fundamental parameter thatinfluences the kinetic competition between available phase tran-sition pathways, including crystalline-to-crystalline, crystalline-toamorphous and amorphous-to-crystalline transformations. Thenucleation and growth kinetics of competing transition pathwaysin LiFePO4 particles have been assessed using a phase-field modelthat takes into account the chemical, strain, gradient and surfaceenergy contributions. Calculations predict that the crystalline-to-crystalline phase transition between the lithiated and delithiatedolivine phases is preferred at low overpotentials due to a smallernucleation energy barrier and also at high overpotentials due toits fast growth kinetics. A crystalline-to-amorphous transition isthe preferred transition pathway upon delithiation at intermedi-ate overpotentials. As particle size decreases, the overpotentialrange over which amorphization dominates is broadened. Differ-ences in the surface energies of amorphous and crystalline phaseswill in general lead to asymmetric behavior wherein the nucle-ation barrier is larger in one phase transition direction than thereverse. In olivines, recrystallization of the amorphous phase hasa larger nucleation energy barrier than does amorphization, dueto the lower surface energy of the amorphous phase. Therefore,repeated charge/discharge cycling may result in gradual loss ofcrystallinity. Experimental observations on the overpotential- andsize-dependence of phase transformation behavior in nanoscaleLiFePO4 can be explained by the model.

Acknowledgements

The research work of MT and JFB was performed under theauspices of the U.S. Department of Energy by Lawrence LivermoreNational Laboratory under Contract No. DE-AC52-07NA27344. MTis grateful for the financial support by the Lawrence PostdoctoralFellowship. WCC and YMC acknowledge the financial support ofU.S. Department of Energy Grant No. DE-SC0002626. We thankNonglak Meethong and Yu-Hua Kao for many helpful discussions.We also acknowledge helpful comments from the anonymousreviewers.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.electacta.2010.09.027.

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