Table Of ContentZ.Phys. Chem. 219 (2005) 89–103
by Oldenbourg Wissenschaftsverlag, München
AC and DC Conductivity in Nano- and
Microcrystalline Li O:B O Composites:
2 2 3
Experimental Results and Theoretical Models
∗ ∗
By Sylvio Indris1, Paul Heitjans1, , Markus Ulrich2, and Armin Bunde2,
1 InstitutfürPhysikalischeChemieundElektrochemie,UniversitätHannover,
Callinstraße3–3a,30167Hannover,Germany
2 InstitutfürTheoretischePhysikIII,Justus-Liebig-UniversitätGießen,
Heinrich-Buff-Ring16,35392Gießen,Germany
Dedicated to Prof. Dr.Klaus Funke on the occasion of his 60th birthday
(Received August 10, 2004; accepted in revised form November 10, 2004)
NanocrystallineComposites/LiIonConductor/ImpedanceSpectroscopy/
PercolationModels
We report on impedance measurements of nano- and microcrystalline composites of the
Li ion conductor Li O and the ionic insulator B O as well as their interpretation in
2 2 3
the frame of percolation models. In the experimental part, besides the dc conductivity
and its dependence on composition and temperature (i.e. its activation energy) also the
ac conductivity and its dependence on composition, temperature and frequency (i.e. the
conductivityexponent)arepresented.Strikingdifferencesbetweenthe nanocrystallineand
the correspondingmicrocrystallinecompositeswere found. Deviations of the ac from the
dc results can be explained by the fact that the experiments probe ion dynamics on
different time and thus length scales. In the theoretical part, a continuum percolation
model, a brick-layer type bond percolation approach and a Voronoi construction are
alternatively used to model the dc behaviour. Based merely on the largely different
volume fractions of the interfaces between ionic conductor and insulator grains in the
nano- and microcrystalline composites, good overall agreement with the experimental dc
results is obtained.The high critical insulatorcontent above which the experimentalcon-
ductivityvanishesinthenanocrystallinecompositessuggeststhepresenceofanadditional
Li diffusion passageway of nanometer length in the interface between nanocrystalline
insulator grains.
* Correspondingauthors.
E-mail:[email protected];[email protected]
90 S.Indrisetal.
1. Introduction
Nanocrystalline materials are compacted, polycrystalline materials with crys-
tallite sizes in the range of typically 5 to 50nm[1–3]. They often show en-
hanced diffusivity when compared to their coarse grained counterparts [4–6].
Inionicmaterials,theenhanceddiffusivityleadstoanincreasedionicconduc-
tivity which is attractive with respect to possible applications in battery sys-
tems,fuelcellsorsensors.Hereweareinterestedinnanocrystallinecomposite
materials, where in addition to the interfaces between like grains interfaces
between unlike grains exist, which can lead to the formation of ionic space
chargelayersandthustoanevenmoreenhancedconductivity[7–10].Besides
thevariationofthegrainsizethechangingofthecompositionofsuchmateri-
alsoffersanadditional degreeoffreedomtotailorthemacroscopicproperties
of these composites. This was shown for the dc conductivity of nanocrys-
talline (1−x)Li O:xB O composites where (except for x close to one) the
2 2 3
increased conductivity oftheinterfaces betweenunlike Li OandB O grains
2 2 3
leads to an enhancement of the overall dc conductivity by a factor of about
10 compared to pure Li O[11]. In this paper we compare the dc and ac con-
2
ductivity of such composites to get more insight into the short-range motion
of the Li ions and to elucidate the influence of the morphology of the net-
work of fast diffusion pathways formed by the different interfaces in these
samples. Furthermore, we present various theoretical approaches, by which
the dependence of the dc conductivity on the composition of these materi-
als can be described, and discuss the advantages and disadvantages of these
models.
2. Samplepreparationandexperimentalsetup
Thenanocrystalline oxidesLi OandB O werepreparedbyhigh-energy ball
2 2 3
millingfromthecoarsegrainedsourcematerials[12,13].Allpreparationsteps
were done in dry argon atmosphere to prevent contact of the samples with
water or CO . The air tight sample holder for the conductivity measurements
2
was loaded with the sample and sealed in a glove box filled with dry argon.
X-ray diffraction and transmission electron microscopic measurements show
that milling for8his sufficient toreduce thegrain size downtoabout 20nm.
Differentialthermalanalysisshowedthatthenanocrystallinematerialsaresta-
bleattemperaturesuptoabout500K.Athighertemperaturesgraingrowthsets
in. For comparison we also prepared the microcrystalline counterparts from
the unmilled source materials. The composites were prepared by mixing the
appropriate amounts of the two components and compacted under uniaxial
pressureof1GPa.Contactsforconductivity measurementswerepressedfrom
platinumpowdertothepelletswiththesamepressure.Thecylindricalsamples
had a diameter of 8mm and a thickness of typically 1mm. The ac conduc-
tivity ofthecomposites was measured withaHP4192impedance analyzer in
ACandDCConductivityinNano-andMicrocrystallineLi O:B O Composites... 91
2 2 3
Fig.1.Realpartoftheconductivityvs.frequencyfornanocrystalline(1−x)Li O:xB O
2 2 3
for x=0.6. The corresponding temperatures are 489K, 478K, 468K, 457K, 446K,
436K,425K,414Kand404K(toptobottom).
thefrequency rangefrom5Hzto13MHzattemperatures between 300Kand
500K.
3. Experimentalresults
Fig.1shows the real part ofthe conductivity of nanocrystalline (1−x)Li O:
2
xB O for x =0.6 in the frequency range from 5Hz to 1MHz for tempera-
2 3
tures between 404K and 489K. The characteristic features of these spectra
which are common to all samples are (i) a plateau at low frequencies which
reflects the dc conductivity of the samples and (ii) a dispersive region at high
frequencieswheretheconductivitycanberoughlydescribedbyapowerlaw
σ(ω)∝ωn (1)
with0<n<1.Bothregionsareseparatedbyacross-overregime.Thisoverall
behaviourhasbeenfoundformanyionicconductors andisdiscussedindetail
in, e.g., [14–19]. In the following we study how the dc conductivity σ (ex-
dc
tracted fromthelow-frequency plateau) andtheacconductivity σ (extracted
ac
atafrequencyν=ω/2π=100kHz)dependonthecomposition.Isothermsfor
433KaredisplayedinFig.2,whereσ andσ areplottedasafunctionofboth
dc ac
insulatormolecontentxandinsulatorvolumefraction p; pandxarerelatedby
p=αx/(1−x+αx)whereαistheratioofthemolevolumesoftheinsulator
andtheionicconductor(about1.91here).Foreachcomposition,themeasure-
mentsweredoneontwoindependentlypreparedsamples.Forcomparison,we
92 S.Indrisetal.
Fig.2.Dc(circles) andac(squares) conductivities ofmicrocrystalline (full symbols)and
nanocrystalline(opensymbols)(1−x)Li O:xB O compositesvs.insulatorvolumefrac-
2 2 3
tionat433K.Dashedlinesaredrawntoguidetheeye.Thearrowsindicatethecomposi-
tionswherethedcconductivity ofthemicro-andnanocrystalline materialsshowasharp
dropandfallbelowthedetectionlimit.
alsoperformedmeasurements onthecorresponding microcrystalline samples.
Thedata points in the graphs showthe weighted average with weighted error
bars.Thedashedlinesareguidestotheeye.
Let us first concentrate on the dc conductivity. For pure Li O, i.e. x =
2
p=0, the dc conductivity of the microcrystalline and the nanocrystalline
samples coincide. When Li O is successively substituted by B O , the two
2 2 3
systemsbehaveverydifferent.Inthemicrocrystalline samples,thedcconduc-
tivity decreases monotonically with x, while in the nanocrystalline samples,
the dc conductivity first increases and reaches a maximum at x ≈0.6 where
theconductivityisaboutoneorderofmagnitudelargerthanthatofpureLi O.
2
Further increase of the insulator content leads to a decrease of the conduc-
tivity. At x=0.95, finally, the conductivity has dropped below the detection
limit.Wehaveshownbefore[11,20],thattheoverallbehaviour(includingthe
differences between nano- andmicrocrystalline samples) can beexplained by
the assumption of an enhanced conductivity at the interfaces between unlike
grains.Evenmoreremarkablethantheincreaseofthedcconductivitywithin-
creasinginsulatorcontentisthefactthat,startingfromthepureinsulatorB O ,
2 3
onlyatinyvolumefractionofLi Oisneededtoobtainadcconductivitywhich
2
isconsiderablyhigherthanthedcconductivityofpureLi O.
2
ACandDCConductivityinNano-andMicrocrystallineLi O:B O Composites... 93
2 2 3
Next we consider the ac conductivity σ , which is represented by the
ac
squaresinFig.2.Forthenanocrystalline samples,σ showsroughlythesame
ac
behavior asσ ,but the increase of σ at small insulator content is morepro-
dc dc
nounced than the increase of σ . Of course, σ is always above σ . For the
ac ac dc
microcrystalline samples, in contrast, both quantities behave very different.
While σ decreases monotonically, falling below the detection limit above
dc
x=0.5resp. p=0.65,σ diminishesonlyslightlybetweenx=0andx=0.1.
ac
Betweenx=0.1andx=0.7,σ isroughlyconstant,anddecreasesslightlyfor
ac
x approaching1.
The different behavior of σ and σ in the microcrystalline system can
ac dc
be understood by percolation theory[21]. The dc conductivity is a result of
thelong-rangemotionoftheions,whichcantakeplaceonlywhenapercolat-
ing conductive cluster exists. Hence, σ is zero when, above the percolation
dc
threshold,thisclusternolongerexists.Ontheotherhand,σ probestheionic
ac
motion on shorter length scales, and thus is not connected with the existence
of a percolating conductive cluster. As a result, σ depends only slightly on
ac
thecomposition andshowsaremarkable decrease onlyatverylarge insulator
contents.
Nextwediscussthebehavioroftheactivationenergies E ,whichdescribe
A
thetemperature dependence ofσ andσ .Ingeneral,theactivation energyis
ac dc
relatedtotheconductivitybytheArrheniusrelation
(cid:1) (cid:2)
E
σT = Aexp − A , (2)
k T
B
where A is the pre-exponential factor, k the Boltzmann constant and T the
B
temperature. Fig.3 shows E , for both the dc and ac conductivities of both
A
micro- and nanocrystalline samples, as a function of the insulator content.
Again, the data points represent the average of two samples with equal com-
positions. For both the dc and ac conductivity, the activation energies of the
nanocrystalline samples show only a weak dependence on the insulator con-
tent,with Enano≈1eVforthedcconductivityand Enano=0.6...0.7eVforthe
A,dc A,ac
ac conductivity. This is in contrast with the behaviour of the microcrystalline
samples where the activation energy of σ increases from Emicro≈0.9eV at
dc A,dc
x =0 to Emicro≈1.2eV at x=0.5 and the activation energy of σ decreases
A,dc ac
from Emicro≈0.75eV at x=0 to Emicro≈0.4eV at x≈0.95. The fact that in
A,ac A,dc
allcasestheactivationenergyoftheacconductivityislowerthanthatofthedc
conductivityagainreflectsthattheacconductivityprobesdynamicsonshorter
timeandthuslengthscales.
Fig.4 finally shows the effective conductivity exponent n (see Eq.(1)) of
the micro- and nanocrystalline samples, extracted from the dispersive region,
as afunction oftheinsulator content x.Inthe microcrystalline samples, n in-
creases with x, i.e. the dispersion becomes stronger with x. This reflects the
behaviorseeninFig.2wherethedifferencebetweentheacandthedcconduc-
94 S.Indrisetal.
Fig.3.Activation energies ofthedc(circles) andac(squares) conductivity ofmicrocrys-
talline(fullsymbols)andnanocrystalline(opensymbols)(1−x)Li O:xB O composites
2 2 3
vs.insulatorvolumefraction.
Fig.4. Exponent n (cf. Eq.(1)) for the ac conductivity of microcrystalline ((cid:1)) and
nanocrystalline ((cid:2))(1−x)Li O:xB O compositesvs.insulatorvolumefraction.
2 2 3
ACandDCConductivityinNano-andMicrocrystallineLi O:B O Composites... 95
2 2 3
Fig.5. Continuum percolation model with insulating spheres (radius R) dispersed in an
ionic conductor (conductivity σ ) and a highly conducting interface (conductivity σ ,
B A
thickness λ),after[23].Thefiguresshowthetwocriticalinsulator contents where(a)in-
finitehighlyconductingpathwaysareformedand(b)thesepathwaysaredisrupted.
tivityincreaseswithx.Inthenanocrystalline samples,incontrast,n decreases
slightlywithxagaininaccordancewithFig.2wherethedifferencebetweenac
anddcconductivitydecreaseswithx.
4. Models
4.1Continuumpercolationmodel
The increase of the ionic conductivity at intermediate insulator contents for
the nanocrystalline composites clearly shows that the interfaces between the
twocomponentsareresponsiblefortheconductivityenhancement.Todescribe
thedependence ofthedcconductivity ofthecomposites ontheinsulator con-
tent p, we first employed a continuum percolation model[22,23] which is
sketchedinFig.5.Theinsulating particles arerepresentedbysphereswithra-
dius R. Around these insulating particles a highly conducting interface with
width λ and ionic conductivity σ is created. The remaining volume repre-
A
sents the ionically conducting phase which has a conductivity σ . We con-
B
sider an enhancement factor τ =σ /σ of 100 and an interface thickness
A B
λ=1nm. The grain radii R have been determined by transmission elec-
tron microscopy and X-ray diffraction[12,13] and are roughly 5µm for the
microcrystalline composites and 10nm for the nanocrystalline composites.
The overall conductivity of the system was then calculated by an effective
medium approximation[11]. The results are shown in Fig.6, where they are
also compared with the experimental results for the dc conductivity. A good
overall agreement is found. However, the effective coordination numbers z
used in this approach as fit parameters to reproduce the experimental re-
sults (z=7 for the microsystem and z=59 for the nanosystem) can hardly
berationalized.
96 S.Indrisetal.
Fig.6.Resultsofthecontinuum percolation approach (dashed lines)comparedtotheex-
perimental results for the dc conductivity at 433K, after[11]. The arrows indicate the
compositionswherethedcconductivities fallbelowthedetectionlimit.
4.2Brick-layertypepercolationmodel
In an attempt to understand the experimental results on a more microscopic
basis, we have considered a brick-layer type model where both, micro- and
nanocrystalline compositesaretreatedonthesamefooting[20].Inthemodel,
westartwithacubicboxofsizeL3thatisdividedintoalargenumberofsmall
cubeswithequalvolumesa3.Eachofthesmallcubesisregardedasagrainof
the composite. With a given probability p the cubes are supposed to be insu-
lating B O grains. Thus the volume fraction of the Li O grains is 1−p. By
2 3 2
definition,conductinggrainsareconnectedwhentheyhaveonecornerincom-
mon.Fig.7(a) showsthe largest cluster ofLi Ograins that connects opposite
2
facesofthelargeboxclosetothepercolation threshold,at p=0.9.Againwe
assume a highly conducting interface of width λ between insulating and ion-
ically conducting particles. Next we replace the small cubes in Fig7(a), that
represent the Li O grains, by a bond lattice sketched in Fig.7(b). The bonds
2
represent the ionic conduction (i) inside the grain (Σ ), (ii) along the inter-
0
facearea(Σ )and(iii)alongtheinterfaceedges(Σ ).Thelengthofeachbond
1 2
is a/2. The cross section each bond represents is (a−λ)2 for the Σ bonds,
0
(a−λ)λ for the Σ bonds and λ2 for the Σ bonds. Having this in mind,
1 2
the conductance of each bond can be calculated easily. We assume that (i) in
the bulk of the insulating B O grains, the specific conductivity is zero, that
2 3
(ii) in the bulk of the conducting Li O grains as well as in the interfaces be-
2
tweenthem,thespecificconductivityisσ ,andthat(iii)Li Ograinsincontact
B 2
with a B O grain share a highly conducting interface with specific conduc-
2 3
ACandDCConductivityinNano-andMicrocrystallineLi O:B O Composites... 97
2 2 3
Fig.7. (a) The largest cluster of insulating particles for the brick-layer type model in
three dimensions at the critical concentration. (b) A single grain and the bonds assigned
to conduction in the interior of the grains, along the sides and along the edges of the
grain[20].
tivity σ =τσ . The enhancement factor τ is assumed to be of the order of
A B
102−103.
Finally, for calculating the total conductivity of the composite we map
the problem onto the corresponding diffusion problem by defining appropri-
ate jump rates (proportional to the bond conductances) along the bonds. For
given values of τ, a, λ and p, we then determine the mean square displace-
ment(cid:4)r2(t)(cid:5)ofmanyrandomwalksasafunctionoftimetonthelargestcluster
of each model system, average over all of them, and obtain the diffusion co-
efficient D=lim (cid:4)r2(t)(cid:5)/6t, which is related to the dc conductivity by the
t→∞
Nernst–Einsteinequation(seee.g.[24])
q2ND
σ = . (3)
dc k T
B
Here, q is the ion charge, N is the number density of mobile ions, k is the
B
BoltzmannconstantandT thetemperature.
Thenumerical results forthedcconductivity vs.insulator volumecontent
p are shown in Fig.8 for various grain sizes a and enhancement factors τ.
In all model calculations, we assumed a fixed interface thickness λ=1nm.
The figure shows that this microscopic model, which treats nano- and mi-
crocrystalline samples in exactly the same way (the only difference is the
size of the grains) is able to reproduce all qualitative features of the experi-
mental results (cf. Fig.2). One feature cannot be reproduced by the model,
however, namely the very high insulator concentration where the conductiv-
ity drops to zero. We will come back to this point at the end of the next
section.
98 S.Indrisetal.
Fig.8. Numerical results of the normalized dc conductivity σ(p)/σ(0) vs. insulator vol-
ume fraction p in the brick-layer type percolation model for different grain sizes a and
enhancement factors τ=σ /σ . In all cases the interface thickness λ=1nm is fixed.
A B
Nanocrystalline grains: (cid:8) a=10nm, τ =200; (cid:2) a=10nm, τ =100; a=20nm,
τ=200; (cid:3) a=20nm, τ=100; Microcrystalline grains: •a=10µm, τ=200; (cid:1) a=
10µm,τ=100;(cid:4)a=20µm,τ=200;(cid:5)a=20µm,τ=100[20].
Fig.9.Polycrystalline composite materialcreated byVoronoiconstruction intwodimen-
sions.Darkgreyareasrepresent theionicconductor grainsandlightgreyareasrepresent
theinsulatorgrains.
4.3Voronoiconstruction
To get a more realistic structure of the composites in our model descrip-
tion (compared to the over-simplified cubic arrangement) we used a Voronoi
approach[25],seeFig.9foratwo-dimensionalsketch.2000seedswhichrep-
resentthecenters ofthegrains havebeenplacedrandomlyinsideavolumeof
1503 lattice sites. The borders of the grains are defined by the planes perpen-
