William&H.&Woodford,&Yet1Ming&Chiang,&and&W.&Craig&Carter&

Department&of&Materials&Science&and&Engineering,&MIT&

“Whiskers,&Wires,&and&Walls”&Workshop&–&NIST&

30&September&2010&

*J.R.&Wilson,&et&al.,&J.#Power#Sources#(2010),&doi:10.1016/j.jpowsour.2010.04.066&

!

5#

5

LiCoO2

*#
V

=

−∆μLi e

highμLi low

μLi
V = −∆μLi

e

high μLi

Fresh#Cell#

H.&Wang,&Y.1I.&Jang,&B.&Huang,&D.&Sadoway,&and&Y.1M.&Chiang,&J.#Electrochem.#Soc.,&146&(2)&4731480&(1999).&

low μLi
V = −∆μLi

e

high μLi

Fresh#Cell# 50#Cycles#

H.&Wang,&Y.1I.&Jang,&B.&Huang,&D.&Sadoway,&and&Y.1M.&Chiang,&J.#Electrochem.#Soc.,&146&(2)&4731480&(1999).&

low μLi
LiCoO2 LiMn2O4 Si
LiCoO2 → Li0.5CoO2

∆V = +1.9%

LiMn2O4 → Mn2O4

∆V = −7.3%

LiCoO2

Si → Li4.4Si

∆V = +311%

LiMn2O4 Si
R.E.&García,&et#al.,&J.#Electrochem.#Soc.#152[1]&A2551A263&(2005)&

Concentration& Voltage& Stress&
Concentration& Voltage& Stress&

R.E.&García,&et#al.,&J.#Electrochem.#Soc.#152[1]&A2551A263&(2005)&
R.E.&García,&et#al.,&J.#Electrochem.#Soc.#152[1]&A2551A263&(2005)&

Concentration& Voltage& Stress&
Critical#

combinations#of#

particle#size#and#

rate#

Need&fracture&toughness&

measurements!&

W.H.&Woodford,#et#al.,#J.#Electrochem.#Soc.,&157&[10]&A10521A1059&(2010)&
Critical#

combinations#of#

particle#size#and#

rate#

Need&fracture&toughness&

measurements!&

W.H.&Woodford,#et#al.,#J.#Electrochem.#Soc.,&157&[10]&A10521A1059&(2010)&

What#About#Polycrystals?#

##########################Phase#Transformations?#
EBSD&orientation&maps&

for&LiCoO2

polycrystalline&particles&

in&composite&(Lishen)&

electrodes&

J.R.&Wilson,&et&al.,&J.#Power#Sources#(2010),&doi:

10.1016/j.jpowsour.2010.04.066&
Goal:&&Develop&Models&

for&Electrochemical&

Shock&that&incorporate&

thisµstructural

complexity&and&address&

(anisotropic)&phase&

transformations&
2D&Voronoi&Construction&

 2N&Random&numbers&per&grain&(xi,yi)&

Random&Grain&Orientations&

 Use&OOF2’s&EulerABG&Convention&

 3&Random&numbers&per&grain&for&

(αi,βi,γi)&

 All&done&in&Mathematica
Each&grain&is&numbered&

Each&boundary&is&numbered&and&we&

know&which&grains&it&separates&

 &&&&&&&&&&&&&&&&&&&&&&&&for&opposite&direction&

along&same&edge&

 Exclude&Boundary&Edges&from&

Analysis&

We&know&the&location&of&each&triple&

point

n → −n

!50#

50#
X in LiXMn2O4

a

!

 ̊A

"

3.0% shrinkage c

!

 ̊A

"

12.3% expansion

T.&Ohzuku,&M.&&Kitagawa,&and&T.&Hirai,&J.#Electrochem.#Soc.&137[3]&769–775&(1990)&

LiMn2O4 + Li+ + e− → Li2Mn2O4

cubic tetragonal

Start#with#end#point#analysis...#(Analogies#to#thermal#shock)#
Assumptions:##Plane&Stress&Condition;&Zero1Displacement&normal&to&external&boundaries&

Note:&&Stresses&here&are&referred&to&the&the&OOF/Lab&frame&coordinates&
Use&tensor&transformation&to&extract&grain&boundary&normal&stress&

!50#

50#

σnn
!50#

50#

σnn

Red&line:&Linear&interpolation&of&raw&data&from&OOF2&

Blue&line:&Calculated&with&cubic&polynomial&fit&(fast!)&

σ(x) = a + bx + cx2 + dx3
x

y

K: Stress-Intensity Factor

r

σlocal

a

∼ K

√r

Still#have#r!1/2

singularity!#

K = 2

√πa

! a

0

σnn(x!)

" x!

a − x! dx!

σnn(x)

G.C.&Sih,&Handbook#of#Stress#Intensity#Factors,&Lehigh&University&Press&(1973)&

K > KIC
x

y

K: Stress-Intensity Factor

r

σlocal

a

∼ K

√r

Still#have#r!1/2

singularity!#

K = 2

√πa

! a

0

σnn(x!)

" x!

a − x! dx!

σnn(x)

G.C.&Sih,&Handbook#of#Stress#Intensity#Factors,&Lehigh&University&Press&(1973)&

K > KIC
x

y

K: Stress-Intensity Factor

r

σlocal

a

∼ K

√r

Still#have#r!1/2

singularity!#

K = 2

√πa

! a

0

σnn(x!)

" x!

a − x! dx!

σnn(x)

G.C.&Sih,&Handbook#of#Stress#Intensity#Factors,&Lehigh&University&Press&(1973)&

K > KIC
Method:#

1)&&Choose&an&edge&&

Remember,&they&have&been&numbered...&

2)&&Choose&a&random&flaw&size&from&

a&flaw&size&distribution&

Need&flaw&distribution&information!&

3)&&Choose&a&random&position&along&

the&edge&

4)&&Calculate&K&using&polynomial&

model&for&stress&distribution&
 Flaws&are¬$Weibull&distributed

J.&D.&Poloniecki&and&T.&R.&Wilshaw,&Nature,&229&2261227&(1971)&&&A.&De&S.&Jayatilaka&and&K.&Trustrum,#J.#Mater.#Sci.,&12#142611430&(1977)&

f(a) = cn−1

(n − 2)!a−ne−c/a f:&probability&density&

a:&flaw&size&

c:&scale¶meter&

n:&width¶meter&

c/n ∼ a−n

Is#this#distribution#characteristic#of#

the#underlying#microstructure?#
xˆ yˆ

zˆ

!c1!c2 Grain#Orientation

Triple#Points#

Grain#Size#

Porosity#
Calculate&Stress&Intensity&

Flaw1Size&Relations&for&All&

Grain&Boundaries&

Extract&Failure&

Probabilities&and&Extreme&

Values&Statistics&

Finite&Element&

Stress&Calculations&

in&OOF2&

Generate&Random&

Voronoi&Microstructure&

in&Mathematica

Extract&Local&Stresses&

Along&Each&Grain&

Boundary&

Populate&the&

Microstructure&with&

Random&Flaws&
Calculate&Stress&Intensity&

Flaw1Size&Relations&for&All&

Grain&Boundaries&

Extract&Failure&

Probabilities&and&Extreme&

Values&Statistics&

Finite&Element&

Stress&Calculations&

in&OOF2&

Generate&Random&

Voronoi&Microstructure&

in&Mathematica

Extract&Local&Stresses&

Along&Each&Grain&

Boundary&

Populate&the&

Microstructure&with&

Random&Flaws&

Microstructure&matters&for&electrochemical&shock&

–&opportunity#for#design#

 Mathematica&+&OOF2&is&a&convenient,&powerful&tool&

set&

This&is&a&snapshot&of¤t&development&

Potential&Future&Directions:&

 Introducing&Second#Phases&(or&porosity)&

 Electrochemical&Kinetics#

Experiments&–&X1Ray&Tomography?&
 National&Science&Foundation&Graduate&

Research&Fellowship&

 Dept.&of&Energy,&Basic&Energy&Science,&Award&

No.&DE1SC0002633&