Microstructures on Material Properties
W. Craig Carter
Abstract|Methods for and computed results of
includ-ingthephysicsandspatialattributesofmicrostructuresare
presentedforanumberofmaterialsapplicationsindevices.
The researchin ourgroup includes applicationsof
compu-tationofmacroscopicresponseofmaterialmicrostructures,
thedevelopmentofmethodsforcalculatingmicrostructural
evolution,and themorphologicalstabilityofstructures. In
thisreview,researchhighlightsarepresentedforparticular
methodsforcomputingtheresponsein: 1)ferroelectric
ma-terialsfor actuatordevices; 2) coarse-grainingofatomistic
dataforsimulationsofmicrostructuralevolutionduring
pro-cessing; 3)periodic and non-periodic photonic composites;
and4)re-chargeablebatterymicrostructures.
I. Introduction
M
ICROSTRUCTURES in uencemanymaterial
prop-erties and thereforehavesigni cantpractical
appli-cationsintheapplicationofmaterialsindevicesand
struc-tures. In the generalcase, a materialmicrostructure is a
complexensemble ofcrystalline materialsin alarge
num-beroforientations, secondphases,interfaces,and defects.
The complexity of microstructures and their description
is the originof the diÆculty of including their e ect into
the design of materials. Furthermore, practical problems
ofmicrostructuraldesignforenhancedmaterialproperties
is protracted by thelack of suÆcient quantitative
under-standing ofthe role ofmicrostructure in material
proper-ties. Ourgroupconsistsofoneprofessor,twopostdocs,six
MITand oneSMA graduatestudent. 1
Wedoresearchon
prediction, bothcomputationaland theoretical, of the
ef-fectofmaterialmicrostructuresonmacroscopicproperties
andontheevolutionofmicrostructure. Ourresearch
pro-gram at MIT focuses on thequanti cation of
microstruc-ture,predictionofmaterialpropertiesfrommicrostructural
data, and thee ect of materials processing on the
evolu-tionofmicrostructuresinmaterials. Wedevelop
computa-tional methods of computing that include microstructure
andthesemethodsaredemonstratedtoSMAstudentsand
usedin theirresearch.
Thisreportfocussesonthoseaspectsofourresearch
pro-gramthatisdirectlyrelatedtoourSMAactivities.
Signif-icantly moredetailed aspectsof particularresearchtopics
The work was supported by the Singapore-MIT Alliance, the
National Institute of Standards and Technology grant number
70NANB1H0031, the National Science Foundation grant number
010252.
W. Craig Carter is the Lord Foundation Associate Professor of
Materials Science at MIT, Room 13-5034, 77 Massachusetts Ave.,
Cambridge, MA 02139-4307 USA, 617-253-6048 ccarter@mit.edu
http://prue.mit.edu/~ccarter/.
1
W.CraigCarter;RajeshRaghavanandAndrewReid;Catherine
Bishop,EdwinGarcia,MartinMaldovan,EllenSiem,Andrew
Taka-can be found in the reports of R.E. Garc
ia and M.
Mal-dovanofourresearchgroup.
II. Microstructural Effectsin Piezoelectric
Materials
In a piezoelectric material, polarization develops as a
response to a stress eld. This polarization eld is in a
directionthatisrelatedtothecrystallographicaxesofthe
underlyingpiezoelectricmaterialandtheorientationofthe
appliedstress. Similarly,anappliedelectric eldwillcause
apiezoelectricmaterialto developastress-freestrainthat
dependsoncrystallographicorientationaswell.
Many devices, such asactuators and sensors, are
com-posed ofpolycrystallinepiezoelectricmaterialsand
there-fore the induced electric elds and stresses of
individ-ualgrains areinterrelated throughthepolycrystalline
mi-crostructure.
It is possible|through processing|to create materials
in which many of thegrains in apolycrystallinematerial
tendto shareacommoncrystallographicaxis. The
distri-bution of orientationsiscalled the polycrystallinetexture
of a microstructure; a highly textured material will have
a high probability of grain alignment abouta texture
di-rectionandanuntexturedmaterialwillhavenocorrelated
alignment.
Onepracticalquestionabouttextureinpiezoelectric
ma-terials might be, \Is there a particular microstructure or
texturethatresultsin anoptimalpiezoelectricresponse?"
Theanswertothispracticalquestioniscomplicatedbythe
factthattheelasticityislocallycoupledtothepolarization
eldinamannerthatdependsonthelocalcrystallographic
orientation;thereforetheanswermustaccountfore ectof
microstructure. Figure1isanillustrationofthis.
We haveextended apublic domain nite element code
(OOF: http://www.ctcms.nist.gov/oof/) that was
devel-oped to treat the thermoelastic behaviors of
microstruc-turestotreatthecaseofthecoupledpolarizationand
ther-moelastic behavior. The extension simultaneously solves
the electrostatic Maxwell'sequation and the elastic
equi-libriumequationsusingorientation-dependentpiezoelectric
andcompliancetensorsfromarbitrarymicrostructural
im-ageswith speci able boundaryconditions. The code uses
a niteelementformulationandhasbeenadaptedtosolve
nonlinearpiezoelectricresponse.
Tocharacterizethee ectofthemicrostructuraltexture,
we have applied the code to a series of computer
gener-ated microstructures based on a Voronoi tessellation on
P(
ξ)
ξ
Cook and Jaffe "Piezoelectric Ceramics"
. Academic Press Inc. London, 1971
∆V
−10.51 MPa
10.67 MPa
σ1
+
σ2
+
σ3
250
µm
100
µm
Figure 1 Illustration of the e ect of microstructure on a
polycrystalline piezoelectric material. A polycrystalline material,
suchasthemicrographoftetragonalsymmetryBaTiO
3
intheupper
left,willhaveaprobabilityofhavingitsuniquec-axisoriented by
an angle with respect to a texture-direction. Such probabilities
aretypicallycharacterizedbytheMarch-Dollasedistribution(upper
right). A representative microstructure (center) with an applied
voltage will develop microstructural stresses and a macroscopic
strain. The bottom gure is the result of a simulation using the
March-Dollasefunctionasthe probabilitygeneratoraMonteCarlo
instanceoftherepresentativemicrostructure.Boththemacroscopic
straincanbecalculatedasastatisticalmeasureofthecharacteristic
piezoelectricresponseofthemicrostructureandtheresidualstresses
canbeusedtoevaluatethemechanicalreliabilityofthematerial.
is randomly assigned a crystallographic orientation from
a parameterized texture orientation distribution as
illus-trated in Figure 1 and then assigned elastic and
piezo-electric properties associated with that orientation. Each
simulatedmicrostructureisplacedinanelectric eld with
stress-freeboundaryconditionsandthemacroscopic
piezo-electric coeÆcients can be determined directly from the
averagestrain. An exampleoftheresultsofalargesetof
simulationsforaPZN-PTpiezoelectricmaterialappearsin
Figure2.
Twopapersthat describethetheoreticalandnumerical
aspects and numerical simulations of microstructures are
−0.1
0.1
0.3
0.5
0.7
0.9
1.1
−1.0
0.0
1.0
2.0
3.0
4.0
b=1
r=0.7
1.5×10
-4
-1.5×10
-4
εxy
1.5×10
-4
-1.5×10
-4
b=0.5
r=1
εxy
250
µm
250
µm
250
µm
250
µm
MRD = 1
MRD = 2.9
Figure 2 Results of a large number of simulations of the
polycrystalline piezoelectric response against a texture parameter.
Theordinateisthed15macroscopicpiezoelectriccoeÆcient
normal-ized by the valuefor single crystalPZN-PT.The abscissa,r, isa
measure of the crystallographictexture|if r = 0 the c-axes have
randomdirectionsandifr=1c-axesareperfectlyalignedwiththe
directionofthe applied eld. Becausepiezoelectriccrystalsdo not
have an inversion center, another texture parameter b is included
thatrepresentsthe probabilitythatthe positivec-axisisaligned in
thedirectionoftheapplied eld.
Thedatashowthatthe d15piezoelectriccomponentisenhanced in
untexturedmaterials.Thisisaresultofthecontributionofthelarger
d
33 andd
31
modesfrommisalignedgrains. Anumerical
representa-tionoftheinducedmicroscopicshearstressintheas-poledcondition
isincludedinthe gurefortwodi erentinstancesoftexture.
Such calculations can beused to predictaverage valuesof
macro-scopicresponseaswellastheirvariation.
III. Coarse-Grainingof Atomic Datafor
Continuum Simulations
Evenwithcurrentfastcomputationalprocessors,itisnot
possibletoperformatomisticcalculationsonanamountof
material that is typically occupied by materials in
com-mercial use|or over time-scales that approach the order
oftypicalmaterialsprocessingconditions. Formany
appli-cations orsolutionsto commercial materials design
prob-lems,itisnecessarytoresorttocontinuumapproximation
formaterialscomputation.
Nevertheless, discrete atomistic calculationsdo provide
useful resultson verysmall scales. Therefore itis
poten-tially useful to consider methods that extract data from
discrete computations and coarse-grain that data sothat
it can be incorporated into a continuum calculation at a
largerscale.
Oneexampleofapotentiallyusefulcontinuummodelis
crystallo-are calculated from a variationalprinciple from a
contin-uum representation of energies. Atomistic computations
resultin discretedatasets representingequilibrium
struc-turesandenergies.
We have developed a coarse-grainingmethod for
map-pingdiscretedatatoacontinuousstructuralorder
param-eter. Thismethodisintendedtoprovideausefuland
con-sistent method of utilizing structural data from
molecu-lar simulations in continuum models, such as the phase
eld model. Themethod is basedon alocal averagingof
the variationof aVoronoitessellationof the atomic
posi-tionsfromtheVoronoitessellationofaperfectcrystal(the
Wigner-Seitz cell). The coarse graining method is
invari-anttocoordinateframerotation. Themethodisillustrated
withasimpletwo-dimensionalexamplein Figure3.
-6
-4
-2
0
2
4
6
x(
σ)
0.92
0.94
0.96
0.98
1.00
1.02
η
η Profile at (x,3) for 0.1σ Mesh
Boundary
Signature
Figure 3 Illustration of coarse-graining of structural
infor-mation in two-dimensions. Data from a molecular dynamics
simulation in the form of (x;y)-coordinates is used as the basis
for a Voronoi tessellation. Each tessellation contains information
about the localized departures from perfect crystallinity. A local
averaging on geometric aspects of the tessellations results in a
characteristic signature of the grain boundary in the molecular
dynamicssimulation.
The method is directly applicable to an averaging of
three-dimensional data (Figure 4). Calculatedresults
in-dicate that acontinuous structuralparameteris obtained
that has grain boundary characteristics similar to phase
eld models of grain boundaries. Comparisons to other
coarse-grainingmeasuresofstructurearediscussedaswell
asapplicationstoexperimentaldatasets. Apaperonthis
workhasbeensubmitted[4].
IV. Structure Dependence of Optical Band
Gaps inPhotonic Crystals
Theuseofstructuredmicro-compositesofmaterialswith
dielectric contrast is emergingas a viablemethod of
pro-ducingopticalmaterialswithtunableopticalmaterials. In
particular,photoniccrystallaminateswithlargedielectric
contrasthavebeenshowntohaveopticalstatesthatadmit
nopropagation|i.e.,photonicbandgaps[5]. Such
materi-alsareomnidirectionalre ectorsforradiationwithenergy
within thephotonicbandgaps.
Thedesign ofnew photoniccrystals withspeci c
prop-ertieswilldependontoolsthat cancorrelatethephotonic
band structure with the microcomposite architecture, an
understandingofhowstructuraldefectswilla ectthe
pho-tonic bandstructure,andonprocessingtechniquesto
cre-ate architectures that have been determined to have
ad-0
10
20
30
z (Å)
0.92
0.94
0.96
0.98
1.00
η
cube r=1
sphere r=1
cube r=2
sphere r=2
η Profile at (2,2,z) for 0.25Å Mesh
Figure 4 Example of structural coarse-graining method
ap-pliedtoamoleculardynamicssimulationofa5grainboundaryin
silicon. Thecoarse-grainedmeasureiscomparabletothestructural
predictionsfromthephase eldmodel.
We havebeeninvestigating numericalmethods of
solv-ing Maxwell's equations that can be eÆciently applied to
photoniccompositesforpredictivecorrelationbetween
mi-crostructureandphotonicproperties.
FourierorBlochwavemethodsareveryeÆcientfor
cal-culatingbandstructureinin nitelyandperfectlyperiodic
structures. In simple nite one-dimensionalstructures, it
ispossibletoformulatetheopticalpropagationwith
trans-fer matrix methods where each matrix is correlated with
the opticalproperties of each material. The matricesare
coupledthoughtheirproximatespatialdomains. Wehave
extended this transfer matrix method to materials with
anisotropicdielectricpropertiesand haveinvestigated the
numerical convergence of the transfer matrix method to
the Fouriersolution with the number of layers in a
pho-toniclaminate.
Resultsthat illustrate the e ect ofanisotropyare
illus-tratedinFigures5and6.
For the cases that have been investigated, the
trans-fer matrix method convergesto within 1% of the Fourier
method after about ten layers. This indicates that nite
stacksapproximatein nitematerials.
Apublicationisbeingpreparedontheresultspresented
abovebytheFourierandtransfermatrixmethod[6].
Photonicstructuresthatarenotperfectlyperiodicor
-nite cannot be modelled easily with the Fourier method
architec-8
10
12
14
16
18
0.2
0.4
0.6
0.8
8 10 12 14 16 18 0.2 0.4 0.6 0.8 1λ (
µm)
6
8
10
12
14
16
18
0.2
0.4
0.6
0.8
8 10 12 14 16 18 0.2 0.4 0.6 0.8 1R
M
R
E
λ (µm)
6
8
10
12
14
16
18
0.2
0.4
0.6
.8
8 10 12 14 16 18 0.2 0.4 0.6 0.8 1R
M
R
E
λ
(µ
m)
80°
45°
0°
εa
εb
wa
wb
Figure 5 Illustration of the transfer matrix method applied
to a photonic stack for the isotropic case. Each plot shows for
eachpolarization(electric eldtransversetopropagationdirection,
TE; magnetic eld transverse, TM) the calculated re ectivity as
a function of the incidentplanewaveangle. Eachof the cubesto
the left of illustrated photonic stack is oriented to be parallel to
the propagation direction; the cross-hatched region indicates the
wavelengths at which full re ectivity exists for that propagation
direction. For omnidirectionalre ectivity, aband gap (re ectivity
equal to unity) must exist for all propagation directions. The
calculation is obtained for thirty layers with dielectric constants
1
= 21:16o(o isthe permittivityofvacuum),
2
=2:56o with
thicknessesh1=0:1mandh2=1:6m.
Inthese cases, Maxwell's equationscan besolvedona
-nite element mesh with properties that are isomorphicto
photonicmedia.
Wehavedevelopeda nite elementmethodthat canbe
applied to arbitraryphotonic composites and have
calcu-lated thephotonicbandgaps in triplyperiodic
bicontinu-ouscompositewithinterfacesofuniformcurvaturesuchas
thosefoundin blockcopolymers[7].
Additionally, itwill beimportantto understandthe
ef-fectofdefects onphotonicproperties. Figure 7illustrates
a nite elementresult applied to a nite photoniccrystal
composed ofa square latticeof cylindrical rods that
pos-sess dielectric largerdielectric constant than themedium
in which itisembedded.
We are currently working on methods of using
image-basedmeshingtechniques tofacilitaterapidassessmentof
photonicarchitectures.
V. Microstructural Effectsin Battery Design
Themacroscopicresponseofabatteryandthusitsvalue
depends of the transport of charged species within the
microstructures of the battery components.
Tradition-ally,battery designinvolvesacompromisebetweenpower
andenergycapacity|theneedforrapidtransportofionic
6
8
10
12
14
16
18
0.2
0.4
0.6
.8
8 10 12 14 16 18 0.2 0.4 0.6 0.8 18
10
12
14
16
18
0.2
0.4
0.6
0.8
8 10 12 14 16 18 0.2 0.4 0.6 0.8 16
8
10
12
14
16
18
0.2
0.4
0.6
0.8
8 10 12 14 16 18 0.2 0.4 0.6 0.8 1λ
(µ
m)
R
M
R
E
λ (µm)
R
M
R
E
λ
(µ
m)
80°
45°
0°
wa
wb
ε
11
a
ε
22
a
ε
11
b
ε
22
b
Figure 6 Illustration of the transfer matrix method applied
to a photonic stack for the case of anisotropic materials. The
calculation is obtained for thirty layers with dielectric constants
11 1 = 21:16o, 22 1 = 9:00o, 33 1 = 21:16o, and 11 2 = 2:56o, 22 2 = 2:56 o , 33 2 =2:56 o
, andwiththicknesses h
1
=0:1mand
h
2
=1:6m. Inthiscase,anisotropyreducestherangeofre ective
wavelengths which isconsistent with the reduced contrast inthe
22-directioncomparedtotheresultsinFig.5.
storageofthoseionsforenergycapacity.
Thecurrentparadigmforunderstandingthecoupled
ef-fectsof material transportand thetime-dependent
devel-opmentofavoltagepotentialinabatteryisderivedfroma
mean- eldrepresentationofamicrostructure.Questionsas
to whetheranoptimizedmicrostructurefortheparticular
demands ofdevice exists, andwhether sucha
microstruc-turecanbedesignedwilldependonadetailed
understand-ingofmicrostructurale ectsinbatteries.
Mean eld calculations cannot capture potentially
im-portant e ects of localized electric shielding orstress
de-velopment duringuse and re-chargingcyclesin abattery.
Wehavedevelopedamethod that usesmicroscopic
infor-mation and fundamental material properties to calculate
themacroscopic response ofelectro-activedevices such as
re-chargeablelithium ionbatteries. An illustrationof the
microstructuralmodellingforalithiumionbatteryappears
inFigure 8.
In the case of re-chargeable lithium ion batteries, the
local electronic elds are coupled to transport properties
in themicrostructuresoftheelectrolyte,thecathode,and
the anode. Microstructural e ects on the charging and
discharging behavior of batteries are determined from a
similarmicrostructuralconstruction. Initialresultsonthe
e ects of particle volume fraction, particle size
distribu-tion and galvanostatic discharge rate on the macroscopic
powerandenergydensityofthesystemaresummarizedin
Figure 7 Illustration of the amplitude of an in incident
plane wave on a nite photonic crystal containing an extended
defect. Thismethodwillbeusedtostudythewaveguideproperties
ofengineeredphotonicdefects.
particlesthatareupstream(towardstheanodeside)ofthe
battery have a shielding e ect on particles that are
fur-ther from the lithium source. This shielding e ect limits
the powercapacityof the battery fora xed energy
den-sity. For microstructures that have the same fraction of
particulate cathodic phase, the power capacity should be
enhancedastheshieldinge ectisreduced. Thesimulation
for the regular array of particles illustrates how reduced
shielding may enablethe production ofbatteries that
ex-ceed the properties predictedby empirical correlationsof
standardbatteries.
It maybepossibleto usethese microstructural
simula-tionstodevelopadditionalmicrostructuraldesignelements
suchasgradedparticlesizes orhierarchicalstructures.
A paper reporting the results of these microstructural
electrochemicaltransportsimulationsisinpreparation[8].
VI. Summary
Microstructural computationsare complicated notonly
because many physical phenomena are coupled, but also
becausethecouplingdependsonthecomplexspatial
corre-lations betweencrystallography,materialsproperties,and
defects thatcompriseamicrostructure.
Becausemicrostructuraldesignisapotentiallypowerful
meanstooptimizematerialsforuseindevices,thepotential
impactofmicrostructuralmodellingjusti esthe
computa-tional diÆcultiesthatmaybeencountered.
The simulations described in this report illustrate the
anode
cathode
separator
resistive load
I
o
−e
composite cathode
separator
I
o
+Li
lithium foil
Figure 8 Illustration of microstructural modeling of the
electro-active components of a lithiumion battery. The battery
iscomposed ofa lithium foil that actsas a sourceof lithiumions
at highchemicalpotential. Theseparatorise ectivelyan osmotic
membraneforlithiumionsthathas poorconductivityofelectrons.
Thecathodemicrostructureconsistsofamicrostructureofparticles
that actassinks forthe lithium ions. The gureon the leftisan
example representation of the cathode particulate microstructure
usedinacalculationofthecoupledchemicalandelectricaltransport
equationsintherepresentativemicrostructure.
Energy Density
Power Density
×
Figure 9 Illustration of the e ect of microstructure on
battery performance. With averaged microstructure, battery
performance istypically represented by the so-called Ragone plot
that empirically correlates the mitigating e ect of power density
on energy storage densityfor batteries. Theinset microstructures
aretheresultsofsimulationsofthe electrochemicalresponsewhere
yellowindicatesalargeconcentrationoflithiumionsduringbattery
usage.
References
[1] R.EdwinGarcia, W.CraigCarter,andStephen Langer, \The
e ectoftextureonthemacroscopicpropertiesofpolycrystalline
ferroelectrics. part 1: Formulationand numerical
implementa-tion,"inpreparationforJ.Amer.Ceram.Soc.
[2] R.EdwinGarcia, W.CraigCarter,andStephen Langer, \The
e ectoftextureonthemacroscopicpropertiesofpolycrystalline
ferroelectrics.part2: ApplicationtoBaTiO
3
modelof grainboundaries," Physica D, vol.140,pp. 141{150,
2000.
[4] C.M.BishopandW.C.Carter,\Relatingatomisticgrain
bound-arysimulationresultstothephase- eldmodel," submittedtoJ.
Comp.Mater.Sci.,2001.
[5] FinkY.etal., \Adielectricomnidirectionalre ector," Science,
vol.282,pp.1679{1682,1998.
[6] M.MaldovanandW.C.Carter, \Calculationofphotonicband
gapsindoublyperiodiccompositesofanistropicmedia,"in
prepa-ration.
[7] M. Maldovan, A. Urbas, N. Yufa, W. C. Carter, and E. L.
Thomas, \Abicontinuous networkphotonic crystal ina block
copolymersystem," SubmittingtoPhys.Rev.B.
[8] R. E.Garc
ia, C.M. Bishop, W.C.Carter Y.-M.Chiang, and
S.A.Langer,\Modellingofmicrostructurale ectsincoupled
ir-reversiblethermodynamicsystemswithapplicationsto
