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Module 3: Defects, Diffusion and Conduction in Ceramics
Introduction
In this module, we will discuss about migration of the defects which happens via an atomistic process
called as diffusion. Diffusivity of species in the materials is also related to their physical properties
such as electrical conductivity and mobility via Nernst-Einestein relation which we shall derive. As we
shall also see, the conductivity in ceramics is a sum of ionic and electronic conductivity and the ratio
of two determines the applicability of ceramic materials for applications. Subsequently framework will
be established for understanding the temperature dependence of conductivity via a simple atomistic
model leading to the same conclusions as predicted by the diffusivity model. Subsequently we will
look into the conduction in glasses and look at some examples of fast ion conductors, material of
importance for a variety of applications. Presence of charged defects in ceramics also means the
existence of electrical potential gradients, in addition to the chemical gradient, which results in a
unified equation for electrochemical potential. Finally, we will look at a few important applications for
conducting ceramic materials.
The Module contains:
Diffusion
Diffusion Kinetics
Examples of Diffusion in Ceramics
Mobility and Diffusivity
Analogue to the Electrical Properties
Conduction in Ceramics vis-à-vis metallic conductors: General Information
Ionic Conduction: Basic Facts
Ionic and Electronic Conductivity
Characteristics of Ionic Conduction
Theory of Ionic Conduction
Conduction in Glasses
Fast Ion Conductors
Examples of Ionic Conduction
Electrochemical Potential
Nernst Equation and Application of Ionic Conductors
Examples of Ionic Conductors in Engineering Applications
Summary
Suggested Reading:
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Physical Ceramics: Principles for Ceramic Science and Engineering, Y.-M.
Chiang, D. P. Birnie, and W. D. Kingery, Wiley-VCH
Principles of Electronic Ceramics, by L. L. Hench and J. K. West, Wiley
Electroceramics: Materials, Properties, Applications, by A. J. Moulson and J.
M. Herbert, Wiley
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Module 3: Defects, Diffusion and Conduction in Ceramics
Diffusion
3.1 Diffusion
Diffusion causes changes in the microstructures to take place in processes such as sintering,
creep deformation, grain growth etc.
Diffusion is also related to transport of defects or electronic charge carriers giving rise to
electrical conduction in ceramics. Ionic conductors are used in variety of applications such as
chemical and gas sensors, solid electrolytes and fuel cell. For example, an oxygen sensor
made of ceramic ZrO 2 is used in automobiles to optimize the fuel/air ratio in the engines.
Atomic diffusion rates and electrical conductivity are largely governed by defect types and
their concentration where concentration is a function of temperature, partial pressure of
oxygen or pO 2, and composition.
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Module 3: Defects, Diffusion and Diffusion in Ceramics
Diffusion Kinetics
3.2 Diffusion Kinetics
Typically diffusion is explained on the basis of compositional gradients in an alloy which act as
driving force for diffusion. Thermodynamically speaking, this amounts to gradient in the chemical
potential which drives the migration of species from regions of higher chemical potential to lower
chemical potential so that system reaches a chemical equilibrium. The atomic flux as a result of
driving force is expressed in terms of chemical composition gradient, also called as Fick’s law(s).
These laws are briefly explained below. For detailed discussion on diffusion, readers are referred to
standard text books on diffusion .1,2
3.2.1 Fick's First Law of Diffusion
It states that atomic flux, under steady-state conditions, is proportional to the concentration gradient.
It can be stated as
(3.1)
where
J is the diffusion flux with units moles/cm 2-s, and basically means the amount of material
passing through a unit area per unit time;
D is the proportionality constant, called as diffusion coefficient or diffusivity in cm 2 /s;
x is the position in cm; and
c is the concentration in cm 3 .
The negative sign on the R.H.S. indicates that diffusion takes place from regions of higher
concentration to lower concentration i.e. down the concentration gradient. Diffusivity is a temperature
dependent parameter and is expressed as D = D 0 exp (-Q/kT) where Q is the activation energy, k is
Boltzmann's constant and D 0 is the pre-exponential factor in cm 2 /s.
3.2.2 Fick's Second Law of Diffusion :
Strictly speaking it is not a law, but rather a derivation of the first law itself. It predicts how the
concentration changes as a function of time under non-steady state conditions . It can be derived
from Fick's first law easily. For further discussions, one can refer to “Diffusion in Solids”, a classic
book written by Paul Shewmon and published by Wiley or other NPTEL courses related to Phase
Transformations and Diffusion.
(3.2)
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where t is the time in seconds. Other terms are defined above .
1 Diffusion in Solids, Paul Shewmon, Wiley
2 Diffusion in Solids, Martin Glicksman, Wiley-
Interscience
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Module 3: Defects and Diffusion in Ceramics
Diffusion Kinetics
3.2.3 Diffusivity: A Simple Model
Figure 3. 1 Schematic of the planes of
atoms with arrows showing the crossmovement of species
As shown in Figure (3.1), a schematic diagram shows atomic planes, illustrating 1-D diffusion of
species across the planes .
Flux from position (1) to (2) is written as
(3.3)
where n 1 is no. of atoms at position (1) and G is the jump frequency i.e number of atoms jumping
per second (atoms/s)
Similarly, Flux from plane (2) to (1) is expressed as
(3.4)
where n 2 is the number of atoms at (2) and G is the jump frequency in s -1 .
in both the above expressions, factor ½ is there because of equal probability of jump in +x and -x
directions.
Now, the net flux, J, can be calculated as
(3.5)
Concentration is defined as
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(3.6)
if area is considered as unit area (=1) and λ is the distance between two atomic planes.
Figure 3. 2 Schematic diagram showing
concentration gradient between two planes of
atoms
Concentration gradient can be written as (note the minus sign)
(3.7)
Hence, flux can now be expressed as
where D = ½ λ 2 t with unit cm 2 /s in 1-D and can easily show to become D = 1/6 λ 2 t in a 3-D
cubic co-ordination scenario.
In general, diffusivity can be expressed as
(3.9)
where γ is governed by the possible number of jumps at an instant and λ is the jump distance and
is governed by the atomic configuration and crystal structure.
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Module 3: Defects, Diffusion and Conduction in Ceramics
Diffusion Kinetics
3.2.4 Temperature Dependence of Diffusivity
Now, equation (3.9) can further be modified by replacing the jump frequency, G , which, by Boltzman
statistics, is defined as
(3.10)
where ν is the vibration frequency in s -1 , ΔG * is the activation energy of migration in J and k is
Boltzmann Constant (J/K).
Further, ΔG * can be written as
(3.11)
where ΔH * is the enthalpy of migration and ΔS* is the associated entropy change. Now, substitution
of equation (3.10) in equation (3.11) leads to
OR
or
where pre-exponential factor
.
Equation (3.12) explains the thermally activated nature of diffusivity showing an exponential
temperature dependence resulting in significant increase in the diffusivity upon increasing the
temperature.
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Module 3: Defects, Diffusion and Conduction in Ceramics
Examples of Diffusion in Ceramics
3.3 Examples of Diffusion in Ceramics
3.3.1 Diffusion in lightly doped NaCl
NaCl is often used a conducting electrolyte and it a good case for proving an example. Consider the
ease of NaCl containing small amounts of CdCl2 lIn such scenario, for each Cd ion, Cd occupying Na
site with an extra positive charge and a sodium vacancy with one negative charge is created according
to the following defect reaction:
CdCl2 → CdNa • + 2Cl Cl x + VNa '
(3.13)
In addition, NaCl will also have certain intrinsic sodium and chlorine vacancy concentration (VNa ' and
VCl •) due to Schottky dissociation, depending on the temperature.
In such a scenario, the diffusivity of sodium ions is governed by vacancy diffusion and can be worked
out as
(3.14)
where ΔG Na * is the migration free energy for sodium vacancies and V Na ' is the sodium vacancy
concentration. This diffusivity of vacancy is dependent on vacancy’s concentration which is related to the
dopant concentration. However, the diffusivity dependence on temperature shows two regimes as shown
in figure 3.3; low temperature extrinsic regime where vacancy concentration is independent of
temperature and is determined by solute concentration and another high temperature intrinsic regime
where vacancy concentration is dominated by intrinsic thermally degenerated vacancies.
Figure 3. 3 Schematic diagram showing
temperature dependence of diffusivity in
ceramics
3.3.1.1 Low Temperature Regime:
Extrinsic region is dominant where vacancy concentration is constant as it is determined by the solution
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concentration i.e. [V Na '] = [Cd Na •] . The diffusivity is given by
..........................................(3.15)
In this regime, diffusivity exhibits a temperature dependence of
.
3.3.1.2 High Temperature Regime:
In this regime, the vacancy concentration is governed by intrinsic defect creation mechanism with
Schottky defect formation and hence, diffusivity exhibits a steeper slope with higher activation energy
which include not only the energy for defect migration but also for defect creation i.e
(3.16)
At the point of crossover of two regimes, vacancy concentration due to dopants equals the thermally
created vacancies.
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Module 3: Defects, Diffusion and Conduction in Ceramics
Mobility and Diffusivity
3.4 Mobility and Diffusivity
In addition to diffusivity, another useful term to describe conduction in ionic compounds and ceramics
is mobility which is defined as velocity (v) of an entity per unit driving force (F) and is expressed as
The Force F can be defined as either of chemical potential gradient, electrical potential gradient,
interfacial energy gradient, elastic energy gradient or any other similar parameter.
Absolute mobility
(3.18)
Where N A is Avogadro's Number, µ i is chemical potential in J/mol and x is the position in cm.
Chemical mobility
(3.19)
Electrical mobility
(3.20)
For atomic transport, Einstein first pointed out that the most general driving force is the virtual force
that acts on a diffusing atom or species and is due to negative gradient of the chemical potential or
partial molar free energy. It is expressed as
Where µ i is the chemical potential of i and N A is the Avogadro's Number.
Absolute mobility, Bi is given by
(3.22)
To obtain the relation between mobility and diffusivity of species, i, we need to write the flux in a
general form as a product of concentration, c i , and velocity, v i , i.e.
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(3.23)
Now, substituting for Fi from equation (3.21), we have
(3.24)
Now, for an ideal solution with unit activity of species i , chemical potential can be expressed as
(3.25)
where R is the gas constant. So, the change in the chemical potential can be written as
(3.26)
OR
(3.27)
Substituting equation (3.27) into equation (3.24) leads to
(3.28)
If we compare the above equation with Fick's first law, which is equation (3.1), diffusivity of species i
can be written as
(3.29)
The above equation is called Nernst Einstein Equation.
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Module 3: Defects, Diffusion and Conduction in Ceramics
Analogue to the Electrical Properties
3.5 Analogue to the Electrical Properties
Electrical force is given as
(3.30)
where f is potential, E is the electric field, Zi is the atomic number, e is the electronic charge and Zi e
is the total charge on the particle.
Using the above relations, one can write the flux, J i , as
Now, since J i = c i .v i , the velocity can be written as
(3.32)
So, defining electrical mobility, µ i as velocity per unit electric field
(3.33)
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Module 3: Defects, Diffusion and Conduction in Ceramics
Conduction in Ceramics vis-à-vis metallic conductors: General Information
3.6 Conduction in Ceramics vis-à-vis metallic conductors: General
Information
While it is possible to apply Nernst-Einstein's relation to express the motions of both ions and
electrons to state the similarity between electrons and ions, in reality the motion of electrons
is quite different.
In metals, semiconductors and high mobility ceramics, the electrons and holes are considered
as quasi-free particles with a drift velocity, v d , acquired under an applied electric field. The
extent of this drift velocity is governed by various scattering phenomenon such as scattering
from the lattice.
Under an applied field E, the force on an electron is given as
(3.34)
In such a condition, drift velocity increases in such a manner as expressed by Newton's law of
motion:
(3.35)
where m is the mass of carrier, v is velocity and t is the relaxation time.
Under steady-state conditions i.e. when carrier motion reaches a steady state,
(3.36)
and hence mobility, µ, can be defined as
(3.37)
where m* is the effective mass of the carrier. Typically, in metals and semiconductors, the relaxation
time, t, shows a temperature dependence and varies as T-3/2 due to thermal scattering and T3/5 due
to impurity scattering. For details, please consult any book on the basic solid state physics as
mentioned in the Bibliography.
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Module 3: Defects, Diffusion and Conduction in Ceramics
Ionic Conduction: Basic Facts
3.7 Ionic Conduction: Basic Facts
Conduction in ionic solids is not similar to the band transport in metals and semicoductors.
Conduction in these solids is often governed by concentration of impurities, dopants and point
defects.
Conduction in these materials tends to be of polaronic or hopping type which is migration of
charges between either two dissimilarly charged ions or counter migration of ions and defects
such as vacancies.
In ionic solids, proximity of the carriers to the ions results in the formation of a polarized
region i.e., a polaron which may be longer or smaller than size of the unit-cell.
Effective mass, m* , depends on level of interactions between electrons or holes and periodic
lattice potential.
Effective mass can be taken as equal to that of a free electron i.e. me * = mfree electron * . Large
polarons typically depict weak interaction between carries and ions and hence electronic
mobility expressions can be valid for the polaron.
In a number of of ionic systems such as LiNbO3 , COFe2 O 4 , FeO etc., carrier-ion interaction
is strong and these are called as strongly-correlated-systems, In such cases, polarons are of
smaller sizes and the effective mass tends to be larger than that of a free electron.
In such systems, motion of carriers is thermally activated i.e. via hopping mechanism.
For example, Li 2 O doping in NiO under oxidizing conditions gives rise to oxidation of Ni 2+
ions to Ni 3+ ions (write the defect reaction yourself). Mixed presence of Ni ions in +2 and +3
states leads to hopping type conduction of electrons between two states. This conductivity of
NiO increases as Li doping is increased as shown schematically in Figure 3.4.
Ni 3+ acts as a perturbing charge which polarizes the lattice surrounding it leading the
formation of Polaron.
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Figure 3.4 Example of change in resistivity
(ρ) of NiO upon doping with Li2 O
The movement of carrier between these differently charged states can happen by hopping
which is a thermally activated phenomenon.
Conduction in such cases is determined by dopant concentration and hence higher the dopant
concentration (within appropriate limits), higher the conductivity. In contrast, in the band
model, carrier concentration is determined by temperature.
In ionic solids, while carrier concentration is independent of temperature (within extrinsic
region), mobility is strongly affected by temperature as is also obvious from equation (3.33)
and is expressed as
(3.38)
where Ea is the activation or migration energy.
Typically carrier mobilities in strongly interacting ionic systems are of the order of ~0.1 cm 2 /Vs which are about 3-4 orders of magnitude smaller as compared to the weakly interacting
systems where carrier mobilities can be between 100-1000 cm 2 /V-s.
Table 3.1 lists some of the mobility values in oxides and for comparison, other materials are
listed too.
Table 3.1: Approximate Carrier Mobilities at Room Temperature
Mobility ( cm2 /V sec)
Mobility ( cm2 /V sec)
Crystal
Electrons
Holes
Crystal
Electrons
Holes
Diamond
1800
1200
PbS
600
200
Si
1600
400
PbSe
900
700
Ge
3800
1800
PbTe
1700
930
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InSb
10 5
1700
AgCl
50
InAs
23,000
200
KBr ( 100 K
)
100
InP
3400
650
CdTe
600
GaP
150
120
GaAs
8000
AlN
--
10
SnO 2
160
FeO
SrTiO3
6
MnO
Fe 2 O 3
0.1
CoO
TiO2
0.2
NiO
Fe 3 O 4
GaSb
--
2500-4000
~0.1
650
CoFe2 O 4
3000
0.1
…
10 -4
10 -8
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Module 3: Defects, Diffusion and Conduction in Ceramics
Ionic and Electronic Conductivity
3.8 Ionic and Electronic Conductivity
Electrical conductivity (si ) is defined as charge flux per unit electric field with units (O -1 cm -1 ) or S/m.
It can be expressed as
OR
OR
si
(3.39)
where J i is the flux of species i.
For ionic species, we can apply Nernst-Einstein equation, i.e. equation (3.29).
(3.40)
Note:
As we saw earlier that temperature dependence of diffusivity shows two distinct regions: a low
temperature extrinsic region and a high temperature intrinsic region. For the same reasons,
temperature dependence of ionic conductivity also exhibit intrinsic and extrinsic regions. While in
intrinsic regions, conductivity is governed only by defect migration as defect concentration is
independent of temperature, in the extrinsic region, defect concentration is temperature dependent
as well leading higher slope of log s vs 1/T plot consisting of energy for defect creation as well as
defect migration.
3.8.1 Total conductivity and Transference Number
Since all charged species contribute to the electrical conductivity, we can write total conductivity as
(3.41)
Fraction of total conductivity carried by each charged species is called as transference number, ti and
is expressed as
(3.42)
and it is straightforward to see that
(3.43)
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Naturally when for a predominantly electronic conductor,
and for a predominantly ionic conductor
and for mixed conduction
As shown below, Table 3.2 lists the transference numbers for some conducting
oxide ceramics.
Compound
T(°C)
ZrO 2 +7%CaO
>700
0
1.0
10 -4
Na2 O.11Al2 O 3
<800
1 (Na+ )
--
<10 -6
FeO
800
SiO2
--
1.0
ZrO 2 +18%CeO 2
1500
--
0.52
0.48
ZrO 2 +50%CeO 2
--
0.15
0.85
Na2 O.CaO.SiO2 Glass
1 (Na+ )
-
-
0.1 (Ca2+ )
-
0.9
15%(FeO.Fe 2 O 3 )CaO.SiO 2 .Al 2 O 3 Glass
1500
Typically ionic conductors exhibit a temperature dependence of mobility, i.e. mobility is
thermally activated. Typically conductivities of ionic conductors are in the range of 10 -5 to 100
S/m, depending on the temperature and are about 3 - 5 orders of magnitude lower than the
metallic conductors but about 10 - 15 orders of magnitude higher than the ceramic insulators.
Easily reducible oxides such as TiO2 , SnO 2 , ZnO, BaTiO3 and SrTiO3 show an n-type
semiconducting behaviour due to creation of oxygen vacancies and compensating electrons,
whilst easily oxidizable oxides are CoO, FeO types which have cation deficiency and exhibit
p–type behaviour.
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Module 3: Defects, Diffusion and Conduction in Ceramics
Characteristics of Ionic Conduction
3.9 Characteristics of Ionic Conduction
Long range migration of ionic charge carriers, the most mobile species, through the lattice
under application of an electric field for example migration of Na+ ions in soda-silicate
glasses
Dependent on the presence of vacant sites in neighbourhood of mobile defects/ions.
Can occur through grain boundaries such as in polycrystalline ceramics or through the lattice
as in fast ion conductors.
When external field is absent, the thermal energy, kT, is required for counter migration of ions
and vacancies overcoming the migration energy Ea, as shown in Figure 3.5, which is nothing
but process of self diffusion.
Figure 3.5 Schematic diagram
showing the lattice ions and
vacancies in a potential well with
and without applied electric field
In the presence of electric field, the potential energy is tilted to one side leading to higher
driving force for migration towards one side than to another side (see figure 3.4). The field is
small enough not to pull the ions out, rather it is enough to tilt the energy well such that ions
move in one direction preferentially.
Ionic conductivity is promoted by
Small ionic size
Small charge i.e. less Coulomb interaction between ions
Favourable lattice geometry
Cations are usually smaller than anions and hence, they diffuse faster. For example, in
case of NaCl, smaller size of Na + ion (102 pm) as compared to Cl - ions (181 pm)
makes them diffuse faster.
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Module 3: Defects, Diffusion and Conduction in Ceramics
Theory of Ionic Conduction
3.10 Theory of Ionic Conduction
Now, having learnt the characteristics of ionic conduction and to know that ionic conductivity
depends on the diffiusivity, we can now develop a theory for ionic conduction using Boltzmann
statistics and see whether we arrive at the same formalism.
Here, we consider one dimensional movement of ions in a lattice as shown schematically in figure
below. The distance between two ionic sites is d and in the absence of any field, the ions require
to overcome an energy barrier of strength Ea or activation energy.
Figure 3.6 Schematic diagram of the ionic
movement in a lattice under application of a field
In the absence of the any field, the probability of ion moving to its left or right, according to the
Boltzmann statistics, is equal and is given as
(3.44)
Where a is the accommodation coefficient related to the irreversibility of the jump and ν is the
vibrational frequency of the ions. Ea can be considered as activation energy or free energy change
per atom.
Now when the field E is applied and if the ionic charge is z, then the force experience by the ion
would be zeE. As a result for application of this force, the energy levels on the left and right of the
ion would move up and down as show in the figure by a magnitude ½.F.d or ½ zedE, with total
energy change being zeE.d.
Now, the probability that the ion will move to its right is given as
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(3.45)
Similarly, the probability of ion moving to its left would be
(3.46)
The above expressions shows that at any given temperature, the probability of the ion moving to
right is higher than to the left i.e.
. This results in net motion of ions towards the right i.e.
along the direction of the field and as a consequence, they acquire an average drift velocity, v av.
This velocity is given as
(3.47)
Invoking a relation equivalent to
, we get
(3.48)
Now, under normal circumstances and normal field for ceramic conduction, the field are quite small,
a few V per cm and thus the field induced energy change is small as compared to the thermal
energy i.e. 1/2 zeE.d << kT. Using this we can modify above equation into
(3.49)
However, if 1/2 zeE.d >> kT
(3.50)
Now, at low field strengths, assuming the validity of Ohm’s law, we write the current density as
(3.51)
where n is the ionic density per cc and total charge is ze.
Hence,
(3.52)
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We know that P = a v exp [- Ea /kT].
Thus
(3.53)
Since, J = s.E, we can write an expression for conductivity as
(3.54)
This expression has similar form that we worked out using diffusivity and mobility earlier and again
show the similar dependence on temperature as well as activation energy for migration.
The following table shows activation energies for diffusion (kcal/mole) for a few oxides. 4
Table 3: Activation energy for diffusion for selected oxide
ceramics
Composition of Glass (mol%)
Activation Energy
(kCal/mol)
Na 2 O
CaO
Al 2 O
SiO 2
33.3
-
-
66.7
13-14
14-16
15.7
-
12.1
72.2
16.4
15.6
11.0
-
16.1
72.9
15.6
15.1
15.9
11.9
-
72.2
22.0
20.8
14.5
12.3
5.8
67.4
20.2
19.5
Good correspondence between activation energies for diffusion and dc conductivity shows that the
model we just derived is also correct.
Conductivity of many ß' and ß'' -alumina are very high and vary between 10 - 10 -1 (O-cm)-1 over a
temperature range of 300 K to 675 K. These also have typical activation energies between ~3.5-4.5
kCal/mole.
Spinel oxides such as Fe 3 O 4 have conductivities of the order of 0.5 (O-cm)-1 and have very low
activation energies, typically about 0.35 kCal/mole, representing almost temperature independent
behaviour.
Compounds like Y2 O 3 , HfO 2 , SiO2 , Al 2 O 3 are far more insulating and have conductivities in the
range of 10 -5 to 10 -14 (O-cm)-1 in the temperature range of 400 K to 1000 K.
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4 Principles of Electronic Ceramics, L.L. Hench and J.K. West, Wiley,
Table 4.2
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Module 3: Defects, Diffusion and Conduction in Ceramics
Conduction in Glasses
3.11 Conduction in Glasses
Glasses have a random 3-D network and contain glass forming agents such as SiO2 , B2 O 3 ,
and Al 2 O 3
Interstitial positions are occupied by modifier ions such as Na+ , K+ , and Li + or blocking ions
such as Ca2+ and Mg 2+ .
Conductivity increases as the temperature increased as barrier for migration for mobile
modifier ions is easily overcome by thermal activation.
Usually as the size of the modifier ion decreases, the conductivity increases.
As the concentration of blocking ions increases, the conductivity decreases.
3.11.1 Molten Silicates
Molten silicates are of commercial importance as glass melting is typically done in electrical
furnaces using resistive heating of glass.
Electrical conductivity is dependent upon the concentration of alkali ions (R: Na+ , Li + , K+ ).
For R 2 O greater than 22%, highest conductivity is achieved for Li 2 O followed by Na2 O
and then K2 O while for R 2 O < 22%, Na2 O yields the highest conductivity followed by
K2 O and Li 2 O.
Typical electrical resistivity for soda-silicate glasses varies between10 -2 to 50 O-cm and
depends strongly upon the alkali content.
Silicates also contain alkali earth ions such as Ca2+ , Sr2+ etc but the conductivity of alkali
earth containing silicate glasses is about an order of magnitude lower than alkali containing
silicates.
In alkali earth compounds, highest conductivity is obtained by incorporating CaO
followed by MgO, SrO and BaO, however the differences are very small.
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Module 3: Defects, Diffusion and Conduction in Ceramics
Fast Ion Conductors
3.12 Fast Ion Conductors
Materials showing very high conductivities, several orders of magnitude higher than normal
ionic solids
Conductivities greater than 10 -2 (W-cm)-1 at temperatures much lower than melting points
Number of charge carriers is a very large fraction of potentially mobile ions which is
significantly larger than in a normal ionic compound hence ion diffusion is solely governed by
the diffusion energy and hence defect formation energy need not be provided resulting in
overall low activation energy for migration.
Intrinsic FICs
Halides and Chalcogenides of silver and copper, e.g.
- AgI is a silver ion (Ag + )
conductor
Alkali metal conductors such as non-stoichiometric aluminates e.g. ß-Al 2 O 3 or
Na2 O.11Al2 O 3 .
Extrinsic FICs
Oxides with fluorite structure such as ZrO 2 are doped with aliovalent oxides creating a
large number of oxygen vacancies and are called as fast ion conductors such as
Y2 O 3 .ZrO2 . In these materials the enthalpy of migration, ΔH m , can be as low as ~
0.01-0.2.
Table 3.4: Examples of Fast Ion
Conducting Ceramics and Conducting
Ions
Material
Conducting Ions
Alkali ions
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Module 3: Defects, Diffusion and Conduction in Ceramics
Examples of Ionic Conduction
3.13 Examples of Ionic Conduction
3.13.1 Cobalt oxide and Nickel oxide: p-type Conductors
These are easily oxidizable oxides and are prone to metal vacancies (M 1-x O) which govern the
conduction.
The carrier concentration can be expressed as (hint: write the defect reaction, refer to the Defect
Chemistry module)
In these materials, hole mobility is much larger than ionic mobility such that conductivity is essentially
electronic. While Hall measurements can be used to measure the mobility, further conductivity
measurements can yield the carrier concentration in these materials.
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Module 3: Defects, Diffusion and Conduction in Ceramics
Examples of Ionic Conduction
3.13.2 Mixed electronic ionic conduction in MgO
Here we demonstrate mixed ionic conduction in MgO. For working this out, we need to start with certain
parameters. Assuming the temperature is 1600o C, for MgO, we can estimate
concentration of Schottky defect ˜ 1.4*10 11 cm 3
concentration of electronic defects ˜ 3.6*109 cm -3
Vacancies present are VMg '' and V0 '' and VMg '' having higher mobility than V0 '' .
Diffusion coefficient of magnesium vacancies is estimated as
At 1600o C, one can calculate the mobility from the Nernst-Einstein Equation i.e
However, at 1600o C
So, even though the vacancy concentration is larger than electron or hole concentration, conduction is
dominated by electrons or holes simply because their mobility is orders of more than 6 order of magnitude
higher.
However, in reality, MgO is prone to having impurities such as Al 2 O 3 with Al concentrations of the order of
400 mole ppm or 2*1019 cm -3 . Using the defect reaction, one can write
Again we assume that the conduction is via Magnesium vacancies. Hence, Ionic conductivity is
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However, electronic conductivity can be electron or hole dominated depending upon pO 2 You can prove
yourself that electrons dominate under reducing conditions while holes dominate under oxidizing conditions.
For example, in normal atmosphere i.e. air (pO2 = 0.21atm), holes dominate. Hole concentration can be
determined using the following defect reaction:
Here, from the mass conservation, we can see that n h = 2 VMg '' and hence the equilibrium constant is
and hence
Using the above equation, n h can be determined at 1600o C air
So, the total conductivity is
Hence, we can now work out the ionic transference number as
and
The system shows mixed electronic and ionic conduction even when vacancy concentration is
10 5 times higher than electronic ( 2*1019 cm -3 vs 5*1013 cm -3 ).
p-type conduction can change to n-type as the pO 2 decreases.
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Module 3: Defects, Diffusion and Conduction in Ceramics
Examples of Ionic Conduction
3.13.3 Ionic conduction in cubic ZrO2
Zirconia, ZrO 2 , is made stable in cubic form by doping with small amounts, ~8-10%, of aliovalent CaO or type of impurities.
Cubic ZrO 2 is considered as a very good and fast ion conductor. Oxygen vacancies, the prevailing defects in Cubic ZrO 2
are created according to
Small energy of migration (Z i )e of oxygen vacancies results in an unusually high diffusion of oxygen vacancies in ZrO 2 .
Example: Take the case of Ca0.14 ,Zr0.86, O 1.86 . Bandgap of ZrO 2 is ~5.2 eV.
At high temperatures, the diffusion coefficient is given as
From the defect reaction
You can show that 14% molar concentration of CaO in ZrO 2 gives rise to an oxygen vacancy concentration of ~ 3.9*1021
cm -3 .
Now, ionic conductivity can be written as
Mobility can be derived by using Nernst-Einstein equation as the following
Hence, ionic conductivity is
Electronic conductivity is calculated as
At 1823 K, electron concentration, n e , using band model, is estimated to be 1.32*1013 cm -3 and electron mobility, µ e , is
24 cm 2 /V.s
Hence, the electronic conductivity is
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From the above analysis it turns out that s ionic >> s elec .
s ionic increases only until about 12-13% of CaO doping and then decreases due to increased defect interactions.
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Module 3: Defects, Diffusion and Conduction in Ceramics
Examples of Ionic Conduction
3.13.4 Conductivity in SrTiO3
Perovskite oxides SrTiO3 and BaTiO3 have two cation sub-lattices onto which aliovalent ions can
substitute. These oxides can also be easily reduced in reducing atmospheres leading creation of
mobile defects. Typically, Ba/Ti or Sr/Ti is approximately very close to 1. Maximum deviation occurs
at temperatures lower than 1200°C and is of the order of ~100ppm.
To promote n-type conductivity in these materials, solutes such La 3+ , Nb5+ , and Ta 5+ are added to
BaTiO3 , giving rise to shallow donor levels in the bandgap. The defect reaction upon doping with La
or Nb is expressed as
and
Occupancies are determined by the relative atomic sizes.
On the other hand, undoped materials often contain impurities such as Fe 3+ or Al 3+ substituting Ti4+
acting as acceptors. Consider lightly acceptor doped SrTiO3 with Al 2 O 3 as impurity with the
following defect reaction:
In such a situation, either SrO is accepted or TiO2 is rejected to maintain the stoichiometric cation
ratio.
The above equation is ionically compensated reaction. It follows that [Al Ti ' ] = 2[V 0 •] and thus
Electronic compensation is also possible according to
It follows that [Al Ti ' ] = n h and thus
Experiments show that Equation (3) , i.e. ionic compensation is predominant mechanism.
If you are interested, you can refer to the work of Choi and Tuller, Journal of American
Ceramic Society, 71, 201, 1988 for discussion on defects and conduction in Ba-doped SrTiO3
under various conditions.
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Module 3: Defects, Diffusion and Conduction in Ceramics
Electrochemical Potential
3.14 Electrochemical Potential
Usually in ionic systems, both the effects of gradients in chemical and electrical potential on
the mobilities are present.
Non-uniform distribution of charged defects can give rise to internal electric field even in the
absence of external electric field.
Total driving force for mass transport is defined as Electrochemical potential.
The electrochemical potential of spices i , denoted as η i , is the sum of the chemical potential
µ i , and electric potential f acting upon it and can be written as
(3.55)
where Zi is the effective charge and F is Faraday's constant (=96500 Coulomb/mol).
Now, virtual force on the particle is negative gradient of η i i.e.
(3.56)
Flux, J, can be expressed as
OR
(3.57)
This shows that flux, J i , is a function of electric field, (df/dx).
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Module 3: Defects, Diffusion and Conduction in Ceramics
Nernst Equation and Application of Ionic Conductors
3.15 Nernst Equation and Application of Ionic Conductors
We have seen that the electrochemical potential is a combination of chemical potential and electrical
potential.
Applied chemical potential gradient gives rise to a voltage which is useful in the operation of devices
like batteries, fuel cells, sensors.
Ionic transport is essential for the imposed oxygen pressure gradient to establish a gradient in the
oxygen concentration. Simultaneously, electronic conduction must be avoided to prevent short
circuiting which requires ti ˜ 1.
Consider two sides separated by an electrolyte. At equilibrium, the chemical potential of both the
sides is equal, i.e.
(3.58)
Electrical potential across the electrolyte is related to the chemical potential difference by
(3.59)
Chemical potential can be expressed as
(3.60)
Thus,
OR
(3.61)
The above equation is referred to as Nernst Equation.
For partially ionic conduction, equation (3.61) can be modified to
(3.62)
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Module 3: Defects, Diffusion and Conduction in Ceramics
Nernst Equation and Application of Ionic Conductors
3.15.1 Example:
In automotive applications, usually on one side is air i.e., pO 2 is 0.21 atm and the other side is at
pO 2 of 10 -9 - 10 -21 atm. . From the Nernst equation, one can calculate that at 350°C, a voltage
difference, Δf , between 0.25-0.6V will be developed which can be easily monitored and can be used
as the feedback in control system to regulate the fuel supply to the engine.
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Module 3: Defects, Diffusion and Conduction in Ceramics
Examples of Ionic Conductors in Engineering Applications
3.16 Examples of Ionic Conductors in Engineering Applications
Conducting ceramics are used in a variety of applications such as
SiC and MoSi2 as heating elements and electrodes
ZnO and SiC as varistors for circuit protection
YSZ, ß -Alumina as electrolytes in fuel cells and batteries
Materials like YSZ in gas sensing applications
While heating elements and varistors utilize the electronic conduction properties of the
ceramics, fuel cells and sensors typically utilize ionic conductivity characteristics.
Since details of each and every application is beyond the scope of this course, you are
referred to the contents of the book titled Electroceramics, written by A.J. Moulson and
J.M. Herbert and published by Wiley.
In the following sections, we will briefly discuss the major applications of conducting ceramics
viz . three applications: varistors, solid oxide fuel cells and exhaust sensors in automobiles.
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Module 3: Defects, Diffusion and Conduction in Ceramics
Examples of Ionic Conductors in Engineering Applications
3.16.1 Varistors or Voltage Dependent Resistors
Devices offering high resistance at low voltages and low resistance at high voltages.
Used to protect a circuit against sudden upsurges in the voltage or transients (Figure 3.7)
Typically used materials are ZnO and SiC powders bonded in a matrix with an intergranular
layer that is highly resistive. The current-voltage characteristics are governed by the electrical
nature of grain and intergranular layers.
The basic principle of operation is that the material's resistance is high at low voltages and
decreases when voltage is increased, allowing excess currents to pass through thereby
protecting the circuit (Figure 3.8).
Figure 3. 7 Schematic diagram showing use
of varistor in circuit protection
Figure 3. 8 Current (I) - Voltage (V) characteristics of a
varistor
For example, zinc oxide particles of size about 10-50 µm are mixed with other oxides such
as bismuth or manganese oxides which form a highly resistive intergranular layer of thickness
between 1 nm to 1 µm between zinc oxide grains.These IGLs contain defect states whose
role is to trap free electrons from the neighbouring n-type semiconducting zinc oxide grains.
This results in the formation of a space charge depletion layer in the ZnO grains in the region
adjacent to the grain boundaries as shown in the figure. These boundaries between each
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grain and its neighbor forms a junction showing diode like characteristics and allowing flow of
current in one direction only (Figure 3.9).
In the figure, the left-hand grain is forward biased by a voltage VL , and the right side is
reverse biased to VR. The respective barrier heights are f L and f R while the zero biased
barrier height is f 0 . As the voltage bias is increased, f L is decreased and f R is increased,
leading to a lowering of the barrier and an increase in conduction.
The whole mass of material resembles an electrical circuit network with back to back pair of
diodes, with each pair of diode in parallel with many others. Very small current flows when a
small voltage is applied across the electrodes which is mainly caused by reverse leakage
through the diode junctions. When voltages are high enough to cause break down of the
diode junction, large currents flow due to thermionic emission as well as tunneling, as seen in
the above graph.
Figure 3. 9
Schematic of the electronic structure of ZnO
grain and intergranular layer (note the bias on the grains)
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Module 3: Defects, Diffusion and Conduction in Ceramics
Examples of Ionic Conductors in Engineering Applications
3.16.2 Solid Oxide Fuel Cells (SOFC)
SOFCs are useful as electrochemical conversion devices in which oxidation of fuel leads to
production of electricity.
Schematic of such a device is shown in Figure 3.10 where two electrodes, anode and cathode,
are separated by an electrolyte e.g. CSZ (cubic stabilized zirconia with 8-10 mol% Y2 O 3 ).
This device can usually work with an efficiency of ~60% at 1000 ( o C) with an output power of
100 kWh.
The device can be used in a wide range of oxygen partial pressures (~1 atm to 10 -20 atm) and
can work at temperatures between 700-1000( o C) with areal resistance of ~ 15µOm-2 .
Anode material is typically porous Ni and cathode is a porous p-type semiconducting oxide
such as La 1-x Srx MnO 3+y .
One needs to connect many of these devices together to obtain substantial power and for that
purpose, interconnects are used which are typically made of Ca, Mg or Sr doped LaCrO 3 or
stainless steel depending upon the operation temperature.
Figure 3.10 Schematic of (a) a SOFC device and (b) operation of a
SOFC device
In case of H 2 and CO gases as fuels, the following reactions take place at two electrodes, as
shown in Figure 3.10 .
Anode/Electrolyte Interface Reaction :
and
Reaction at the Cathode/Electrolyte Interface
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Overall reactions that occur are:
It must be noted that high ionic conduction through the electrolyte is required to minimize the
resistive losses.
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Module 3: Defects, Diffusion and Conduction in Ceramics
Examples of Ionic Conductors in Engineering Applications
3.16.3 Oxygen Sensors in Automobile Exhaust
Modern automobiles demands good fuel efficiencies and pollution control.
Exhaust gases are NO x gases, CO, Hydrocarbons.
One alternative is to use the catalytic converters, which however do not regulate the fuel
consumption.
The above air/ fuel ratio of ~ 14.5, hydrocarbons and CO are decomposed efficiently but not
NO x , while below 14.5, the reverse is true.
When there are deviations in the air /fuel ratio from the stoichiometric value of 14.5, pO 2 in
the exhaust changes dramatically from ~ 10 -9 to 10 -26 atm. This differential provides a way to
use the activity of oxygen, i.e. to measure the combustion efficiency.
The process is to first measure the pO 2 of the exhaust by using an oxygen sensor. The
difference between the pO 2 of the exhaust and air can be translated into an electrical signal
which is fed as an electrical input to the fuel injection system to optimize the air/fuel ratio.
Oxygen sensors are typically conducting ceramics operating on two principles:
First type is based on Nernst effect where an electrical potential is created when there
is a pO 2 gradient across the sensor.
Second type is where one can measure the variation in the conductivity of the ceramic
vs pO 2 .
Defect concentration varies by orders of magnitude upon changes in background pO 2 and
temperature. This also affects the electrical conductivity of the ceramic, which is generally
proportional to the number of current carriers, dependent upon T and pO 2 .
Figure 3.11 shows the schematic of an exhaust gas sensor using Y2 O 3 stabilized ZrO 2
which gives rise to sufficient voltages under a gradient of oxygen partial pressures
encountered in automobiles which can be used to regulate the fuel/air supply.
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Figure 3.11 (a) Schematic of a automobile exhaust sensors and (b)
potential generated by different oxygen partial pressure on two sides
of the electrolyte.
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Module 3: Defects, Diffusion and Conduction in Ceramics
Summary
Summary
Migration of defects plays a very crucial role in determining conduction in ceramic materials.
Migration of ionic defects such as vacancies or interstitials is basically a diffusive process and is
governed by the law of diffusion. The diffusivity of species is a strongly temperature dependent
parameter with an exponential dependence on the temperature. The diffusivity related to electrical
conductivity is shown by Nernst-Einstein relationship. As a result, conductivity in the ionic
systems also follows a temperature dependent Arhenious type behavior. In ionic systems, to utilize
the ionic conduction of materials, it is essential that ionic conductivity is a dominant contribution
towards the total conductivity which typically happens at either higher temperatures or at reasonably
high doping levels unless the electron concentration is extremely low. The examples of ceramic
materials with high ionic conductivity are materials like fast ion conductors and a few ceramic
glasses. The presence of composition gradients of charged defects also leads to electrical potential
gradients and this when combined with the chemical potential gradients, gives rise to an expression
for electro chemical-potential, called Nernst equation. This equation is often employed in
designing the sensors for use in automotives where differential pO 2 gives rise to a voltage which can
be used as a feedback to regulate the fuel/air supply in the engines. Ceramic conductors are used in
a variety of applications, for example, ZnO as varistors, doped ZrO 2 as solid electrolytes in fuel cells
and as oxygen sensors in the automobile exhaust.
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