Module 3: Defects, Diffusion and Conduction in Ceramics Introduction

Objectives_template

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:

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture13/13_1.htm[5/25/2012 3:03:41 PM] Objectives_template

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

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture13/13_1.htm[5/25/2012 3:03:41 PM] Objectives_template

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 ZrO2 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 and composition. 2,

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture13/13_2.htm[5/25/2012 3:03:41 PM] Objectives_template

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/cm2-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 cm2/s;

x is the position in cm; and

c is the concentration in cm3 .

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 = D0 exp (-Q/kT) where Q is the activation energy, k is 2 Boltzmann's constant and D0 is the pre-exponential factor in cm /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)

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture13/13_3.htm[5/25/2012 3:03:41 PM] Objectives_template

where t is the time in seconds. Other terms are defined above.

1Diffusion in Solids, Paul Shewmon, Wiley 2Diffusion in Solids, Martin Glicksman, Wiley- Interscience

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture13/13_3.htm[5/25/2012 3:03:41 PM] Objectives_template

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 cross- movement 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 n1 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)

-1 where n2 is the number of atoms at (2) and G is the jump frequency in s .

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

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture13/13_4.htm[5/25/2012 3:03:41 PM] Objectives_template

(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 λ2 t with unit cm2/s in 1-D and can easily show to become D = 1/6 λ2 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.

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture13/13_4.htm[5/25/2012 3:03:41 PM] Objectives_template

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

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.

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture13/13_5.html[5/25/2012 3:03:41 PM] Objectives_template

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:

• x CdCl2 → CdNa + 2ClCl + 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 ΔGNa* is the migration free energy for sodium vacancies and VNa' 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 file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture14/14_2.html[5/25/2012 3:03:41 PM] Objectives_template

• concentration i.e. [VNa'] = [CdNa ] . 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.

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture14/14_2.html[5/25/2012 3:03:41 PM] Objectives_template

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 NA is Avogadro's Number, µi 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 µi is the chemical potential of i and NA 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, ci, and velocity, vi, i.e.

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture14/14_3.htm[5/25/2012 3:03:42 PM] Objectives_template

(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)

(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.

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture14/14_3.htm[5/25/2012 3:03:42 PM] Objectives_template

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 Zie is the total charge on the particle.

Using the above relations, one can write the flux, Ji , as

Now, since Ji = ci .vi , the velocity can be written as

(3.32)

So, defining electrical mobility, µi as velocity per unit electric field

(3.33)

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture15/15_2.html[5/25/2012 3:03:42 PM] Objectives_template

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, vd , 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.

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture15/15_3.html[5/25/2012 3:03:42 PM] Objectives_template

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, COFe2O4, 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.

2+ For example, Li2O doping in NiO under oxidizing conditions gives rise to oxidation of Ni ions to Ni3+ 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.

Ni3+ acts as a perturbing charge which polarizes the lattice surrounding it leading the formation of Polaron.

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture15/15_4.htm[5/25/2012 3:03:42 PM] Objectives_template

Figure 3.4 Example of change in resistivity (ρ) of NiO upon doping with Li2O

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 cm2/V- s which are about 3-4 orders of magnitude smaller as compared to the weakly interacting systems where carrier mobilities can be between 100-1000 cm2/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

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture15/15_4.htm[5/25/2012 3:03:42 PM] Objectives_template

InSb 105 1700 AgCl 50

KBr ( 100 K InAs 23,000 200 100 )

InP 3400 650 CdTe 600

GaP 150 120 GaAs 8000 3000

AlN -- 10 SnO2 160

FeO SrTiO3 6

0.1 MnO Fe2O3 -- ~0.1 0.1 0.2 CoO TiO2 … NiO Fe3O4

GaSb 2500-4000 650 CoFe2O4 10-4 10-8

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture15/15_4.htm[5/25/2012 3:03:42 PM] Objectives_template

Module 3: Defects, Diffusion and Conduction in Ceramics Ionic and Electronic Conductivity

3.8 Ionic and Electronic Conductivity

-1 -1 Electrical conductivity (si) is defined as charge flux per unit electric field with units (O cm ) or S/m. It can be expressed as

si (3.39)

where Ji 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)

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture15/15_5.html[5/25/2012 3:03:42 PM] Objectives_template

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)

ZrO2+7%CaO >700 0 1.0 10-4

Na2O.11Al2O3 <800 1 (Na+) -- <10-6

FeO 800 SiO2 -- 1.0

ZrO2+18%CeO2 1500 -- 0.52 0.48

ZrO2+50%CeO2 -- 0.15 0.85

Na2O.CaO.SiO2 Glass 1 (Na+) - -

15%(FeO.Fe2O3)CaO.SiO2.Al2O3 Glass 1500 0.1 (Ca2+) - 0.9

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 , SnO2 , 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.

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture15/15_5.html[5/25/2012 3:03:42 PM] Objectives_template

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.

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture16/16_2.htm[5/25/2012 3:03:43 PM] Objectives_template

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

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture16/16_3.html[5/25/2012 3:03:43 PM] Objectives_template

(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, vav. 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)

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture16/16_3.html[5/25/2012 3:03:43 PM] Objectives_template

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)

Na2O CaO Al2O SiO2

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.

-1 Spinel oxides such as Fe3O4 have conductivities of the order of 0.5 (O-cm) and have very low activation energies, typically about 0.35 kCal/mole, representing almost temperature independent behaviour.

Compounds like Y2O3, HfO2, SiO2, Al2O3 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.

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture16/16_3.html[5/25/2012 3:03:43 PM] Objectives_template

4Principles of Electronic Ceramics, L.L. Hench and J.K. West, Wiley, Table 4.2

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture16/16_3.html[5/25/2012 3:03:43 PM] Objectives_template

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 , B2O3 , and Al2O3

Interstitial positions are occupied by modifier ions such as Na+ , K+, and Li+ or blocking ions

such as Ca2+ and Mg2+.

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 R2O greater than 22%, highest conductivity is achieved for Li2O followed by Na2O and then K O while for R O < 22%, Na O yields the highest conductivity followed by 2 2 2 K2O and Li2O.

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.

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture17/17_2.htm[5/25/2012 3:03:43 PM] Objectives_template

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-Al2O3 or Na2O.11Al2O3.

Extrinsic FICs

Oxides with fluorite structure such as ZrO2 are doped with aliovalent oxides creating a large number of oxygen vacancies and are called as fast ion conductors such as Y2O3.ZrO2. In these materials the enthalpy of migration, ΔHm , 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

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture17/17_3.html[5/25/2012 3:03:43 PM] Objectives_template

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 (M1-xO) 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.

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture17/17_4.html[5/25/2012 3:03:44 PM] Objectives_template

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 1600oC, for MgO, we can estimate

concentration of Schottky defect ˜ 1.4*1011cm3

concentration of electronic defects ˜ 3.6*109cm-3

'' '' '' '' Vacancies present are VMg and V0 and VMg having higher mobility than V0 .

Diffusion coefficient of magnesium vacancies is estimated as

At 1600oC, one can calculate the mobility from the Nernst-Einstein Equation i.e

However, at 1600oC

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 Al2O3 with Al concentrations of the order of 400 mole ppm or 2*1019cm-3. Using the defect reaction, one can write

Again we assume that the conduction is via Magnesium vacancies. Hence, Ionic conductivity is

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture17/17_5.html[5/25/2012 3:03:44 PM] Objectives_template

However, electronic conductivity can be electron or hole dominated depending upon pO2 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 nh = 2 VMg and hence the equilibrium constant is

and hence

o Using the above equation, nh can be determined at 1600 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 105 times higher than electronic ( 2*1019cm-3 vs 5*1013cm-3).

p-type conduction can change to n-type as the pO2 decreases.

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture17/17_5.html[5/25/2012 3:03:44 PM] Objectives_template

Module 3: Defects, Diffusion and Conduction in Ceramics 00 Examples of Ionic Conduction

3.13.3 Ionic conduction in cubic ZrO2

Zirconia, ZrO2, is made stable in cubic form by doping with small amounts, ~8-10%, of aliovalent CaO or type of impurities. Cubic ZrO2 is considered as a very good and fast ion conductor. Oxygen vacancies, the prevailing defects in Cubic ZrO2 are created according to

Small energy of migration (Z )e of oxygen vacancies results in an unusually high diffusion of oxygen vacancies in ZrO . i 2

Example: Take the case of Ca ,Zr O . Bandgap of ZrO is ~5.2 eV. 0.14 0.86, 1.86 2 At high temperatures, the diffusion coefficient is given as

From the defect reaction

21 You can show that 14% molar concentration of CaO in ZrO2 gives rise to an oxygen vacancy concentration of ~ 3.9*10 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

13 -3 At 1823 K, electron concentration, ne , using band model, is estimated to be 1.32*10 cm and electron mobility, µe , is 24 cm2/V.s

Hence, the electronic conductivity is

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture17/17_6.html[5/25/2012 3:03:44 PM] Objectives_template

From the above analysis it turns out that sionic >> selec .

sionic increases only until about 12-13% of CaO doping and then decreases due to increased defect interactions.

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture17/17_6.html[5/25/2012 3:03:44 PM] Objectives_template

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 La3+, Nb5+, and Ta5+ are added to BaTiO , giving rise to shallow donor levels in the bandgap. The defect reaction upon doping with La 3 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 Fe3+ or Al 3+ substituting Ti4+ acting as acceptors. Consider lightly acceptor doped SrTiO3 with Al2O3 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 [AlTi ] = 2[V0 ] and thus

Electronic compensation is also possible according to

' It follows that [AlTi ] = nh 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.

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture17/17_7.html[5/25/2012 3:03:44 PM] Objectives_template

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 ηi, is the sum of the chemical potential µi, 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 ηi i.e.

(3.56)

Flux, J, can be expressed as

(3.57)

This shows that flux, Ji , is a function of electric field, (df/dx).

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture18/18_2.htm[5/25/2012 3:03:44 PM] Objectives_template

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,

(3.61)

The above equation is referred to as Nernst Equation.

For partially ionic conduction, equation (3.61) can be modified to

(3.62)

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture18/18_3.html[5/25/2012 3:03:45 PM] Objectives_template

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., pO2 is 0.21 atm and the other side is at -9 -21 pO2 of 10 - 10 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.

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture18/18_4.html[5/25/2012 3:03:45 PM] Objectives_template

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.

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture19/19_2.html[5/25/2012 3:03:45 PM] Objectives_template

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 file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture19/19_3.htm[5/25/2012 3:03:45 PM] Objectives_template

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 fL and fR while the zero biased barrier height is f0. As the voltage bias is increased, fL is decreased and fR 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)

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture19/19_3.htm[5/25/2012 3:03:45 PM] Objectives_template

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% Y O ). 2 3

This device can usually work with an efficiency of ~60% at 1000 (oC) 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(oC) with areal resistance of ~ 15µOm-2 .

Anode material is typically porous Ni and cathode is a porous p-type semiconducting oxide such as La1-x Srx MnO3+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 LaCrO3 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 H2 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

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture19/19_4.html[5/25/2012 3:03:45 PM] Objectives_template

Overall reactions that occur are:

It must be noted that high ionic conduction through the electrolyte is required to minimize the resistive losses.

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture19/19_4.html[5/25/2012 3:03:45 PM] Objectives_template

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 NOx 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

NOx, while below 14.5, the reverse is true.

When there are deviations in the air /fuel ratio from the stoichiometric value of 14.5, pO2 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 pO2 of the exhaust by using an oxygen sensor. The difference between the pO2 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 pO2 gradient across the sensor.

Second type is where one can measure the variation in the conductivity of the ceramic vs pO2 .

Defect concentration varies by orders of magnitude upon changes in background pO2and temperature. This also affects the electrical conductivity of the ceramic, which is generally proportional to the number of current carriers, dependent upon T and pO2 .

Figure 3.11 shows the schematic of an exhaust gas sensor using Y2O3 stabilized ZrO2

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.

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture19/19_5.htm[5/25/2012 3:03:45 PM] Objectives_template

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.

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture19/19_5.htm[5/25/2012 3:03:45 PM] Objectives_template

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 pO2 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 ZrO2 as solid electrolytes in fuel cells and as oxygen sensors in the automobile exhaust.

file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture19/19_6.htm[5/25/2012 3:03:46 PM]