Dislocation based stresses during electrochemical cycling and phasetransformation in lithium-ion batteries
Pankaj Dhiman, Hsiao-Ying Shadow Huang⁎
Mechanical and Aerospace Engineering Department, North Carolina State University, R3158 Engineering Building 3, Campus Box 7910, 911 Oval Drive, Raleigh, NC27695, United States
Lithium iron phosphate (LiFePO4) contains defects that play an important role in structural degradation ofbatteries. Linear defects called dislocations are introduced inside the lithium ion battery material during fab-rication process accompanied by distortion produced stress fields around them. The present study deals withthese mechanical stresses around dislocations in lithium iron phosphate and the change in stress field duringphase transformation of lithium iron phosphate electrode material. A model consisting of multiple dislocationsinside a lithium iron phosphate material incorporating anisotropic material properties is used to calculate stressfields using linear elastic theory and the superposition method. The stress fields around dislocations during phasetransformation of lithium-iron phosphate are numerically calculated by incorporating the anisotropic propertiesof the material. The change in electrochemical behaviour of material due to change in stress field during phasetransformation is also studied, where a modified electrochemical kinetics equation (i.e., Butler Volmer equation)is derived and used to account for dislocation induced stresses during the reversible cyclic voltammatery of thelithium iron phosphate. The results shows the stress inside material does not remain constant during phasetransformation and its variations are dislocation orientation dependent. In addition, the result shows that thepresence of stress fields around dislocations changes the electrochemical behaviour of the material as suggestedby the shift in the cyclic voltammograms. The effect of increasing scan rate on cyclic voltammogram is alsostudied for lithium iron phosphate. The results show that the increase in current at peaks is independent of theorientation of dislocations studied. Moreover, the decrease in current corresponding to a particular overvoltagevalue before anodic peak and increase in current after the anodic peak is found to be somehow proportional tothe scan rate. Increased scan rates show increased deviation of current from a cyclic voltammogram for materialin which there is no phase transformation. The results provide an insight into how presence of defects and phasetransformation changes the electrochemical behaviour of the material. It is concluded that the combined effect ofthe stresses induced around dislocations during phase transformation and high scan rate can be used for mod-ifying battery materials for various applications by changing electrochemistry of electrodes. The present studyincorporates electrochemistry, defects and phase transformation into one battery chemistry and thus is im-portant in our understanding of the Li-ion batteries.
1. Introduction
Lithium ion batteries having high gravimetric and volumetric en-ergy density [1] have made possible the realization of electric vehicles(EVs), hybrid electric vehicles (HEVs) and plug in hybrid electric ve-hicles (PHEVs) reducing harmful environmental effects of internalcombustion engines. Due to their high energy density, lithium ionbatteries are also being used excessively in portable electronics in-cluding mobile phones and laptops. Lithium Cobalt Oxide (LiCoO2) isutilized in most portable electronics as the cathode material due to its
high energy density [2] but possesses safety risks [3] due to cobaltbeing toxic and LiCoO2 having low thermal stability [4]. Overcomingthe limitations of LiCoO2, an alternate cathode material was proposedby Goodenough in 1996 [5]. Despite having lower energy density thanLiCoO2 does, lithium iron phosphate also known as LFP is a highlycommercial used cathode material because of its longer life cycle,thermal stability, flat discharge voltage (∼3.6 V vs Li+/Li0) and lowprice. Despite all these advantages, lithium iron phosphate suffers fromlow electrical conductivity [6]. However, electrical conductivity of li-thium iron phosphate is shown to improve by doping of lithium iron
https://doi.org/10.1016/j.commatsci.2019.109275Received 22 July 2019; Received in revised form 6 September 2019; Accepted 7 September 2019
phosphate with alien atoms, surface coatings or changing crystal mor-phology [7]. Characteristics of lithium iron phosphate such as highenergy density and environmentally safe accompanied with long lifecycle thus makes it a promising cathode material that has already beenexploited in the batteries.
It has been suggested that cathode materials like lithium ironphosphate contains defects that may be formed during fabrication at afinite temperature due to a number of reasons. Previous research hasshown how defects in lithium iron phosphate can be engineered tocontrol material properties. Li et al. [8] evaluated the coupled effects ofsurface and dislocation mechanisms on diffusion induced stress inspherical nanoparticle electrode. They showed that the tensile stressesin spherical electrode are weakened by the coupled effect and evenconverted from tensile to compressive. Thus, the effect was considereduseful suppressing cracks nucleation and propagation caused by theinternal damage during battery charging and discharging cycles. Liet al. [9] analytically modelled the effect of dislocations on diffusioninduced stresses in a cylindrical lithium ion battery electrode. Theyexplained the combined effect of diffusion induced stresses and dis-locations on preventing the crack nucleation and propagation undergalvanostatic or potentiostatic conditions. Ulvestad et al. [10] studiedthe topological defect dynamics in operando battery nanoparticlesusing Bragg coherent diffractive imaging and reported the nucleation oflithium rich phase near the dislocations during the structural phasetransformation. They also found out the material elastic properties to beconsiderably different from the bulk (i.e., negative Poison’s ratio) in thevicinity of dislocations at high voltage. The research supported the ideathat dislocations can be used to tailor material properties.
The aim of this research is to study the dislocation-based stressesinside lithium iron phosphate during electrochemical cycling and phasetransformation. The work is continuation of the work done by Huangand Wang [11] in which the effect of orientation of dislocations onstress field was studied in a lithium iron phosphate material. A modelconsisting of multiple dislocations inside a lithium iron phosphatematerial incorporating anisotropic material properties was used tocalculate stress fields using linear elastic theory and superpositionmethod. The stress fields around dislocations during phase transfor-mation of lithium-iron phosphate were numerically calculated in-corporating the anisotropic properties of the material. The change inscan-rate dependent electrochemical behaviour of material due tochange in stress field during phase transformation transformation oflithium-iron phosphate was also studied, where a modified electro-chemical kinetics equation (i.e., Butler Volmer equation) is derived andused to account for dislocation induced stresses during the reversiblecyclic voltammetry of the lithium iron phosphate. The effect of in-creasing scan rate on cyclic voltammogram was also studied for lithiumiron phosphate.
2. Methods
2.1. Stresses during phase transformation
The cathode material lithium iron phosphate in study has orthor-hombic crystal structure and anisotropic elastic constants as given inMaxisch and Ceder (2006) [12]. LiFePO4 exhibits a two-phase systemwith a single lithium poor phase called heterosite (FePO4) and lithiumrich phase called triphylite (LiFePO4). Both phases are olivine typeorthorhombic structures with the Pnma space group although the latticeconstants for both the phases differ and contribute to ∼7% volumetriccontraction from triphylite when heterosite is formed. The phaseboundary between FePO4 and LiFePO4 is coherent which forms whentwo crystals match perfectly at the interface plane. An analytical solu-tion of dislocation based stress field using anisotropic property wasgiven by Indenbom and Lothe [13]. The solution held good for or-thorhombic materials such as lithium iron phosphate where anisotropicproperties cannot be ignored, and elastic constants varied in different
directions. The solution for stress fields of a dislocation with arbitraryBurgers vector was given by Eq. (2-1).
= −
⎧
⎨
⎪⎪
⎩
⎪⎪
+ + + −
+ −
− + + + −
⎫
⎬
⎪⎪
⎭
⎪⎪
−−
⎧
⎨
⎪⎪
⎩
⎪⎪
+ + + −
+ −
− + + + −
⎫
⎬
⎪⎪
⎭
⎪⎪
σb λ C Cπq t C C ϕ
C C C C x y λ C y C
C C x λ xy
C C C C C C C x y C C
λ y
b λ C Cπq t C C ϕ
C C C C λ xy C x C
C C λ y x y
C C C C C C C λ xy C C
x
( ¯ )4 ¯ sin
[( ¯ ) ]
( ¯ )( )
[( ¯ 2 ¯ ) ¯
]
( ¯ )4 ¯ sin
[( ¯ ) ]
( ¯ )( )
[( ¯ 2 ¯ ) ¯
]
ij
x
ij ij
CC
y
ij ij
CC
12 112 2
11 66
11 11 12 122 2
663
12
11 123 2 2
11 12 122
12 66 11 662
11 66
2 3
12 112 2
11 66
22 11 12 662 2
663
12
11 122 3 2
11 12 122
12 66 11 662 2
11 66
3
ij
ij
22
22
11
11
(2-1)
where ⎜ ⎟= = = ⎛⎝
⎞⎠
= + +
− + −C C C λ C C ϕ
q x xyλ ϕ y λ
¯ ( ) , ( / ) , cos ,
2 cos
ijC C C C
C C11 221/2
11 221/4 1
21 2 ¯
2 ¯
2 2 2 2
122 12 66 11
2
11 66
and = − +t x xyλ ϕ y λ2 cos2 2 2 2. Cij (i, j = 1, …, 6) in Eq. (2-1) referredto the elastic constants of an orthorhombic material, x and y were theposition coordinates at which the stress was to be calculated and bx andby were the Burgers vectors.
The stresses around dislocations depend on the elasticity of thematerial being considered. The lithium-poor and lithium-rich an phasesformed during phase transformation of lithium iron phosphate havedifferent material properties and thus have different elasticity. For ourstudy, the change in elasticity of the material during phase transfor-mation was taken into account by the amount of phase transformationthat occurred in the material at any instant during electrochemicalreaction. Elastic constants of the electrode material change as the li-thium iron phosphate progressively under-went phase transformation.The change in elastic constant at any instant of phase transformationwas given by Eq. (2-2).
= − +C x C x C[ ] (1 )[ ] [ ] ,ij p ij FePO p ij LiFePO4 4 (2-2)
where xp was the amount of phase transformation and varies from 0 to 1during phase transformation. C[ ]ij FePO4 referred to the elastic constantsof lithium poor phase, C[ ]ij LiFePO4referred to the elastic constants of li-thium rich phase, and C[ ]ij referred to the elastic constants of theelectrode material with xp amount of phase transformation from li-thium-poor phase to lithium-rich phase. The xp equaled to 0 indicatedthat no phase transformation had taken place in material whereas xpequaled to 1 indicated that all of the FePO4 has changed to LiFePO4 andcorresponded to 100% phase transformation. The elastic constants forLiFePO4 and FePO4 were taken from Maxisch and Ceder (2006) [12].
The stresses at any instant of time during phase transformation werecalculated by Eq. (2-1) by utilizing in the values of elastic constantobtained from Eq. (2-2) using the values provided from Maxisch andCeder (2006) [12]. Since there existed multiple dislocations simulta-neously in any material, we incorporated two dislocations in our model.Thus, the stresses inside lithium iron phosphate due to the occurrenceof two simultaneous dislocations with arbitrary Burgers vector werecalculated by the superposition principle. The stress at any point inmaterial was equal to the sum of the stresses caused by an individualdislocation at that point considering the existence of that dislocationonly in the material. Since multiple dislocations existed simultaneouslyin a material, we could incorporate many dislocations corresponding toa real-world material. But we chose 2 dislocations only to keep ourmodel simple with less computational time. Also, two dislocationsprovided a better visual understanding of how the stress field changes
P. Dhiman and H.-Y.S. Huang Computational Materials Science 171 (2020) 109275
2
during the phase transformation as seen in the results shown later. Themodel dimensions of lithium iron phosphate with lattice constants oflithium iron phosphate taken from [14] were 100L×60L with 60 unitcells of lithium iron phosphate where L=10 Å. An actual lithium ironphosphate particle was reported to be several hundred nanometres [15]and thus could be represented conveniently by our model. Also themodel size was big enough to avoid dislocation cores of 4L×4L [16]where linear elastic theory fails. The model size of 100L×60L alsodepicted the overall stress distribution without any boundary effects.Mathematica (Wolfram Research, Champaign, IL) [17] was used tonumerically calculate the stresses given by Eq. (2-1) and obtain thestress values and distributions. The stress fields were numerically cal-culated for two dislocations during phase transformation of lithium ironphosphate. The two dislocations were fixed in their locations but dif-fered in their orientation with respect to each other. The first disloca-tion was located at (x, y)= (20L, −12L) whereas the second disloca-tion was located at (x, y)= (40L, 24L) in the model. The orientation ofone of the dislocation labelled dislocation 2 was fixed with Burgervector bx= 1 and by= 0 while the orientation of the other dislocationlabelled Dislocation 1 was varied by varying its Burgers vector.
2.2. Derivation of modified Butler-Volmer equation
The Butler-Volmer Equation provides the most fundamental re-lationship between current and applied potential but lacks the effect ofmechanical stresses inside the electrode. For this propose the effect ofhydrostatic stress caused by dislocations was incorporated in the Butler-Volmer equation. The modified Butler-Volmer equation similar to Luet al. [18] given by Eq. (2-5) was then used to simulate cyclic vol-tammetry of lithium iron phosphate having dislocations. We startedwith chemical equilibrium states without mechanical stresses where theactivation energies of cathodic and anodic reactions are identical, i.e,
= =≠ ≠ ≠G G GΔ Δ Δc a0 0 0 . The chemical equilibrium could be broken by theapplied electric potential and mechanical stress. If the applied electricpotential was changed by = − ′E E EΔ 0 , then the free energy of theoxidized state would be lowered relative to the reduce state by F EΔ .For every mole of lithium intercalation, the increase in elastic energy isequal to the work done by the mechanical stress, i.e., = −W σΔ Ωe h
where σh is the hydrostatic stress inside the material, Ω is the partialmolar volume of lithium iron phosphate. Thus the change in free energyof the reduced state relative to the oxidized state is +F E WΔ Δ e. Thischange in total free energy contributes to the change in activation en-ergies of cathodic and anodic reactions according to the charge transfercoefficients α and (1-α):
= + += − − − −
≠ ≠
≠ ≠
G G αF E α WG G α F E α W
Δ Δ Δ ΔΔ Δ (1 ) Δ (1 )Δ
c e
a e
0
0 (2-3)
Assuming the rate constants have an Arrhenius form [19]:
= −
= −
≠
≠
k A exp G RT
k A exp G RT
( Δ / )
( Δ / )f f c
b b a (2-4)
and proceeding in a regular manner as in deriving Butler-Volmerequation, the modified Butler Volmer equation was obtained:
= ⎛
⎝⎜
⎡⎣⎢− − − ⎤
⎦⎥ −
− − − ⎞
⎠⎟
∗
′
∗
′
i
i C tC
α F E E σRT
C tC
α F E E σRT
(0, ) exp ( ) Ω (0, )
exp[(1 ) ( ) Ω
O
O
h R
R
h
0
0
0
(2-5)
Here R is the universal gas constant, T is the temperature, α is thecharge transfer coefficient, ∗CO and ∗CR are the bulk concentration ofoxidized and reduced species, − ′E E0 is the applied overpotential, σh isthe hydrostatic stress inside the material, Ω is the partial molar volume
of lithium iron phosphate and io is the exchange current density.
2.3. Phase transformation during electrochemical cycling
The over potential and phase transformation in storage electrodeswas related by Tang et al. [20] with applications to nanoscale olivines.The paper served as a base for the research in studying the effect ofmechanical stress in cyclic voltammetry (CV). During CV, the appliedover-potential changed the phase of the lithium iron phosphate whichvaried the hydrostatic stress around edge or screw dislocations in theparticle. This change in hydrostatic stress changed the current as perthe modified Butler-Volmer equation. A phase transition from crystal-line LiFePO4 to crystalline FePO4 is thermodynamically favourable atΔφ (overpotential) > 0. However, such transitions need to overcomeenergy barriers (or activation energies) and the probability of thethermally activated transition is proportional to exp(−ΔF/kT). Tanget al. [20] calculated the variation of activation energy ΔF with in-creasing overpotential for transformation of a particle of r= 1 µm ofcrystalline LiFePO4 to crystalline FePO4. The activation energy de-creased with increased overpotential and evolved toward zero at acritical overpotential Δφc= 30.283mV. Above this overpotential, thephase transformation was barrier less and occurred spontaneously. Theactivation energy, ΔF decreased almost linearly with increased over-potential up to the point where it became zero. Due to thermally acti-vated transition being proportional to exp(−ΔF/kT), the phase trans-formation of crystalline LiFePO4 to crystalline FePO4 was considered tobe exponential. Thus, a sigmoid function was used in our model to showthe variation of phase transformation from 0 to Δφc= 30.283mV.
A relation between mechanical stresses and phase transformationwas developed first followed by a relation between applied overvoltageand phase transformation. The effect of changing stress field arounddislocations during phase transformation was then studied by com-bining the knowledge of the two. The hydrostatic stress around dis-location was then calculated using Eq. (2-6) at any instance of cyclicvoltammetry corresponding to an amount of phase transformationhappened at that instant.
=+ +
σσ σ σ
3.h
xx yy zz
(2-6)
Based on Eq. (2-1), the stress field could be obtained. The inverse ofcosine would results in complex variables by using the orthotropicelastic constant. In this study, the imaginary component of the hydro-static stress was negligible to avoid numerical issues while simulatingcyclic voltammograms. The hydrostatic stress thus calculated at anyinstance of cyclic voltammetry corresponding to a particular amount ofphase transformation was then used in the modified Butler Volmerequation (Eq. (2-5)) to simulate the cyclic voltammogram.
3. Results and discussion
Fig. 1 showed the results of the stress evaluation of stress fieldduring phase transformation with dislocation 1 having Burgers vectorbx=1, by=0 and dislocation 2 with Burgers vector bx=1, by=0(configuration 1). Thus, both the dislocations had the same orientationsby having the same Burgers vectors in this case. The amount of phasetransformations tabulated against stresses in Fig. 1 were chosen tocorrespond with xp=0.2, xp=0.6 and xp=0.9. These values werechosen because they showed the maximum difference in stress fieldsduring phase transformation and represent the phenomenon visually.As shown in the Fig. 1, the stresses increased with increased amount ofphase transformation. Thus, as the phase transformation from FePO4 toLiFePO4 inside the cathode material started proceeding, the materialalso started experiencing more dislocation-based stresses. The increasein the stresses was proportional to the density of dislocations inside thematerial. It was also seen from Fig. 1 that σxx showed the most variationduring the phase transformation whereas σyy showed the least
P. Dhiman and H.-Y.S. Huang Computational Materials Science 171 (2020) 109275
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Fig. 1. Stress field of 1st type dislocation configuration during phase transformation, where dislocation 1 with Burgers vectors bx= 1, by= 0 and dislocation 2 withBurgers vector bx= 1, by= 0. xp was the amount of phase transformation and varies from 0 to 1 during phase transformation.
Fig. 2. Stress field of 2nd type dislocation configuration during phase transformation, where dislocation 1 with Burgers vectors bx= 1, by= 0.6 and dislocation 2with Burgers vector bx= 1, by= 0. xp was the amount of phase transformation and varies from 0 to 1 during phase transformation.
P. Dhiman and H.-Y.S. Huang Computational Materials Science 171 (2020) 109275
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variation. Fig. 2 showed the results of the stress evaluation of differentstresses during phase transformation with dislocation 1 having Burgersvector bx=1, by=0.6 and dislocation 2 with Burger vector bx=1,by=0 (configuration 2). Thus, both the dislocations do not had thesame orientation anymore in this case. The dislocation 2 was fixed inthis orientation while the orientation of dislocation 1 was changed. Asshown in the Fig. 2, the stresses in this case also increased with in-creased amount of phase transformation. It was also seen from Fig. 2that all the stresses increased considerably during phase transformationas more and more FePO4 changed to LiFePO4 compared to Fig. 1 whereσxx showed the most variation during the phase transformation and σyyshowed the least variation and almost did not change. Fig. 3 showed theresults of the stress evaluation of different stresses during phasetransformation with dislocation 1 having Burgers vector bx=0, by= -0.6 and dislocation 2 with Burger vector bx=1, by=0 (configuration3). Compared to Figs. 1 and 2, the stresses during phase transformationin Fig. 3 did not increase as FePO4 was progressively changed toLiFePO4. On the contrary, they decreased because of the particularorientation in which the two dislocations were oriented. Thus, it can beargued the increase or decrease in stresses around dislocations dependson the orientation of dislocations. Also comparing Figs. 1 and 2, it wasobserved that the change in stress in a particular direction was alsodependent on orientation of dislocation. Fig. 1 showed that σxx, σzz andσxy changed significantly during the phase transformation whereas σyyalmost did not change. However, the orientation of two dislocations inFig. 2 corresponded to significant changes in all components of stresses.
The CV curves obtained using the modified Butler-Volmer equation(Eq. (2-5)) with partial molar volume of lithium iron phosphate taken tobe in the order of 10−4 were shown in Fig. 4 for three different or-ientations of two dislocations. The partial molar volume of lithium ironphosphate was taken to be in the order of 10−4 to account for themechanical stresses associated with the introduction of dislocations inthe material. It was seen that partial molar volume of the order of 10−5
or lower had no significant shift in the shape of the CV curve with the
introduction of dislocations in the material. Therefore, any value largerthat 10−4 would further alter the CV curves, i.e., electrochemical be-haviors of the electrodes. The region of phase transformation wasshown in grey color inside the CV curve and it was seen that the stresschanges inside the material during phase transformation due to whichthe CV curves for phase transformation and without phase transfor-mation were different. A typical CV curve for an electrochemically re-versible reaction without any stresses was also shown alongside withblue dotted lines for comparison. It can be concluded that the disloca-tions can be useful for changing the characteristics of CV curve and thuscan be used for designing batteries for various applications by changingthe electrochemistry of electrode battery accordingly. For example, itwas observed that the current decreased for the same overpotential dueto the stresses induced by dislocations for certain overpotential range(between anodic peak and zero overpotential for forward scan). Thereverse effect of increasing current for the same overvoltage can becreated if the nature of stress (compressive or tensile) was changed byvarying orientation of dislocations. Thus, low current or high currentcan be obtained for the same voltage in a CV curve depending on dif-ference usages. Comparing Fig. 4(a)–(c), it was observed that the re-duction in current due to phase transformation in anodic scan (beforeanodic peak) for the same overpotential was more obvious in Fig. 4(b)than one shown in Fig. 4(a). This was attributed to the particular or-ientation of the dislocations corresponding to the cyclic voltammo-grams considered. In addition, comparing Fig. 4(b) and (c), this re-duction in current was almost the same for a particular overpotentialbecause both CV curves corresponded to the dislocations oriented insuch a way that they produced almost equal average hydrostatic stressinside the material. Thus, three important observations were made fromFig. 4:
1. The presence of dislocations changed the electrochemical perfor-mance of electrodes dramatically which were revealed by the shiftof the shape of the CV curve, as shown by comparing the blue dotted
Fig. 3. Stress field of 3rd type dislocation configuration during phase transformation, where dislocation 1 with Burgers vectors bx= 0, by=−0.6 and dislocation 2with Burgers vector bx= 1, by= 0. xp was the amount of phase transformation and varies from 0 to 1 during phase transformation.
P. Dhiman and H.-Y.S. Huang Computational Materials Science 171 (2020) 109275
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curves (without stress) and red dashed curves (with stress) in Fig. 4.This change in shape is attributed to the presence of stresses arounddislocations.
2. Phase transformation inside lithium iron phosphate electrodehaving dislocations caused further change in electrochemical per-formance of the electrode as seen by comparing red dashed curves(without phase transformation) and black solid curves (with phasetransformation) in Fig. 4. As soon as the phase transformationstarted inside the material due to applied overvoltage, the stressfield around dislocations changed along with its electrochemicalbehavior. The more the phase transformation occurred by increasingoverpotential, the current reduced more with respect to its valuewithout having any phase transformation up to the anodic peak andthen increased beyond the anodic peak. The reverse phenomenonhappened for the cathodic peak.
3. The cyclic voltammograms with dislocation oriented as shown inFig. 4(b) and (c) had almost identical CV curves because both CVcurves corresponded to the dislocations oriented in such a way thatthey produced almost equal average hydrostatic stress inside thematerial. Thus, it can be concluded that different orientation ofdislocations can be used to change the electrochemical behavior ofthe electrode in the same way. This is due to the nearly same hy-drostatic stress produced by the two orientation of dislocations ir-respective of having different stress fields for the individual com-ponents of stress.
Thus, it can be concluded from Fig. 4 that presence and orientationof two dislocations during phase transformation were the two primaryfactors that affect the electrochemical behaviour of the material duringbattery charge-discharge cycles.
Fig. 5 showed the effect of increasing the scan rate of the forwardand backward scan on the CV curve. It was seen that increasing the scanrate increased the current for the applied over-voltage for all the threedifferent orientations of the dislocations studied. The dashed lines inFig. 5 showed the CV curves without phase transformation and solidlines showed CV curves with phase transformation corresponding to aparticular scan rate. The key observations from Fig. 5 were as follows:
1. The increase in current for a particular overvoltage was a phe-nomenon that was independent of the orientation of dislocationsconsidered in our model.
2. The decrease in current corresponding to a particular overvoltagevalue before anodic peak and increase in current after the anodicpeak was somehow proportional to the scan rate.
3. Increased scan rates showed increased deviation of current from aCV curve for material in which there was no phase transformation.
The observed increase in the current for increased scan rate can beexplained in terms of diffusion layer size. The diffusion layer grewmuch farther from the electrode with a low voltage scan rate than in ahigh voltage scan rate. Thus, the flux to the electrode surface wasconsiderably smaller at slow scan rates compared to one with the fasterscan rates and hence the current was also smaller at slow scan rates. Itwas also seen that the linear increase or decrease in scan rate did notcause linear variation in the current for the same overpotential. Thiswas due to the inherent nature of the Butler-Volmer equation in whichthe overpotential terms was raised exponentially and any change inoverpotential at any scan rate consequently did not cause linear var-iation in current.
It is known that when two or more dislocations are close to each
Fig. 4. CV curve for 3 different orientations of two dislocations with in lithium iron phosphate material. (a) Dislocation 1 with Burgers vectors bx= 1, by= 0 anddislocation 2 with Burgers vector bx= 1, by= 0. (b) Dislocation 1 with Burgers vectors bx= 1, by= 0.6 and dislocation 2 with Burgers vector bx= 1, by= 0. (c)Dislocation 1 with Burgers vectors bx= 0, by=−0.6 and dislocation 2 with Burgers vector bx= 1, by= 0.
Fig. 5. Effect of scan rate on CV curve for three different orientations of two dislocations in lithium iron phosphate electrode material. (a) Dislocation 1 with Burgersvectors bx= 1, by= 0 and dislocation 2 with Burgers vector bx= 1, by= 0. (b) Dislocation 1 with Burgers vectors bx= 1, by= 0.6 and dislocation 2 with Burgersvector bx= 1, by= 0. (c) Dislocation 1 with Burgers vectors bx= 0, by= -0.6 and dislocation 2 with Burgers vector bx= 1, by= 0.
P. Dhiman and H.-Y.S. Huang Computational Materials Science 171 (2020) 109275
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other, forces of attraction or repulsion occur to reduce the total elasticenergy. We have published the calculations of the force field resultedfrom interactions of dislocations [11], where forces [F]= [σ][b]× [n];the force vector was determined by the stress field [σ] of dislocation 1and Burgers vectors [b] of dislocation 2, and [n] was the normal di-rection of the plane. We have observed that for two parallel edge dis-locations, the closer they were, the stronger the attractive or repulsiveforces were between them, suggesting the dislocations tended to reducethe total elastic energy by repelling each other. As for two parallel edgedislocations with the same Burgers vectors or opposite Burgers vectors,similar but opposite effects occurred, and it was suggested that twodislocations with opposite signs tended to meet each other and can-celled out the forces between them. Moreover, we could speculate thatdislocations could move relatively due to forces of attraction or repul-sion. In this case, the distance between two dislocations would conse-quently change, and the stress fields between them could be easily re-calculated by reassigning the position coordinates (i.e., x and y) in Eq.(2-1).
4. Conclusion
The research dealt with studying the effect of dislocation-basedstresses during phase transformation of a lithium iron phosphatecathode on a cyclic voltammogram. A model consisting of two dis-locations inside a lithium iron phosphate material was used to calculatestress fields using linear elastic theory and the superposition method.The stress fields around dislocations during phase transformation ofLiFePO4 were then numerically simulated incorporating the anisotropicproperties of the material. The results showed that the stress insidematerial increase or decrease during phase transformation and did notremain constant. The variations in stress depended on the orientation ofthe dislocations. While in one case, some dislocations were oriented insuch a way that they increased the overall stress as opposed to the othercase in which they were oriented to decrease the overall stress. Also, itwas observed that the amount of increase of a particular component ofstress during phase transformation was dependent on orientation ofdislocations. While one orientation showed significant increase in allcomponents of stress, the other orientation corresponded to significantincrease only in some of the components with some remaining nearlyconstant throughout the phase transformation. A modified Butler-Volmer equation was used in simulating the reversible cyclic voltam-mogram of the lithium iron phosphate. The effect of stress caused bydislocations during phase transformation was also incorporated. It wasobserved that the presence of dislocations changed the electrochemicalbehaviour of the electrode material by shifting the CV curves. Thischange in electrochemical behaviour was attributed to the presence ofstresses around dislocations. The current reduced significantly withrespect to its value without having any phase transformation up to theanodic peak and then increased beyond the anodic peak. The reversephenomenon also happened for the cathodic peak. It was also con-cluded that different orientations of dislocations can be used to changethe electrochemical behavior of the electrode in same way. This wasdue to the nearly same hydrostatic stress produced by the two or-ientation of dislocations irrespective of having different stress fields forthe individual components of stress. The effect of increasing scan ratewas also studied on cyclic voltammogram for lithium iron phosphate.The increase in current at peaks was found independent of the or-ientation of dislocations considered. Also, the decrease in current cor-responding to a particular overvoltage value before an anodic peak andincrease in current after the anodic peak was found to be somehowproportional to the scan rate. Increased scan rates showed increaseddeviation of current from a CV curve for material in which there was nophase transformation. The results provided an insight into how pre-sence of defects and phase transformation changes the electrochemicalbehaviour of the material. It was concluded that the combined effect ofthe stresses induced around dislocations and their orientation during
phase transformation can be used for creating batteries with electrodeshaving desired electrochemical behavior at particular overvoltage.Thus, defects inside battery electrodes can be engineered to change theelectrochemical behaviour of the material, an example of which was theorientation and dislocation density considered in this work. Furtherstudy extending this work should include developing and using a si-milar model to understand the effect of dislocation-based stressesduring phase transformation on electrochemical behaviour of anotherbattery material chemistry, e.g., lithium nickel manganese cobalt oxide(NMC). Also, more than two dislocations can be considered in futurework corresponding to high dislocation density in the material.
The raw data required to reproduce these findings are available todownload from https://huang.wordpress.ncsu.edu. The processed datarequired to reproduce these findings are available to download fromhttps://huang.wordpress.ncsu.edu.
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