Lecture 5 - Course Notes

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Lecture 5

Tannaz

Javadi

October 7, 2013

Ceramic Science 4RO3

Magnetic properties

Susceptability

(
χ
)
:

similar

to

dielectric

materials,

is

a

parameter

which

expresses

magnetic

response

of

electron

in

a

material

to

the

applied

magnetic

field

and

is

a

dimensionless

quantity
.


M: induced magnetization

H: magnetic field

Diamagnetic
:




Materials

with

small

negative

susceptibility

(
χ
m

<

0
)



χ
m

in

Superconductors

are

-
1

(Perfect

Diamagnet
)
.



Atoms

have

no

net

magnetic

moments




Exposed

to

a

field,

a

negative

magnetization

is

produced

(
inherent

effect
)



The

susceptibility

is

temperature

independent

o

quartz

(SiO
2
)
:

-
0
.
62
x
10
-
8

m
3
/kg
,

Calcite

(CaCO
3
)
:

-
0
.
48
x
10
-
8

m
3
/kg
,

water
:

-
0
.
90
x
10
-
8

m
3
/kg



Materials

with

positive

susceptibility

are

either

paramagnetic,

ferromagnetic

or

ferrismagnetic

(
χ
m

>

1
)
.


Paramagnetism




Atoms

have

a

permanent

non
-
zero

net

magnetic

moment

due

to

the

sum

of

orbital

and

spin

magnetic

moments
.




The

magnetic

moments

randomly

orientated

due

to

thermal

fluctuations

when

there

is

no

magnetic

field
.




The

moments

align

parallel

to

the

field

when

magnetic

field

is

applied
.



Susceptibility

is

positive

but

very

small

for

paramagnetic

materials

o

Montmorillonite

(clay)

13
x
10
-
8

m
3
/kg

Ferromagnetism



The

magnetic

moments

in

a

ferromagnet

aligned

parallel

to

each

other

under

the

influence

of

a

magnetic

field
.



These

moments

will

then

remain

parallel

when

a

magnetic

field

is

not

applied

(unlike

the

moments

in

a

paramagnet)




Above

T
c
,

the

Curie

temperature,

all

ferromagnetic

materials

become

paramagnetic
.



Because

thermal

energy

is

large

enough

to

overcome

the

cooperative

ordering

of

the

magnetic

moments
.


The
Curie temperature
is an intrinsic
property and is a diagnostic
parameter that can be used for
mineral identification

saturation magnetization goes to zero



The

maximum

induced

magnetic

moment

that

can

be

obtained

in

a

magnetic

field

(
H
sat
)
;

beyond

this

field

no

further

increase

in

magnetization

occurs
.




An

intrinsic

property,

independent

of

particle

size

but

dependent

on

temperature
.

Saturation Magnetization

Ferromagnetic hysteresis loop



Domain

that

has

a

direction

closest

to

that

of

the

applied

field

grows

at

the

expense

of

the

other

domains
.




Such

growth

occurs

by

motion

of

the

domain

walls
.




Initially

domain

wall

motion

is

reversible,

and

if

the

applied

field

is

removed

the

magnetisation

will

return

to

the

initial

demagnetised

state
.




In

this

region

the

magnetisation

curve

is

reversible

and

therefore

does

not

show

hysteresis

(OB)

Ferromagnetic hysteresis loop


(Point

B)
:

domains

aligned

in

the

direction

of

applied

field

and




saturation

magnetization,

M
S
(Point

B),

domains

completely

aligned

(
-

M
S

in

the

opposite

direction)
.



Ferromagnetic

Vs
.

Ferroelectric
:



In

ferromagnetic

materials,

favourably

oriented

domain

growth

at

the

expense

of

unfavourably

oriented

domains
.



In

ferroelectric

materials,

favourably

oriented

domains

nucleate

and

grow
.



(Point

C)
:

the

field

is

reduced

to

zero,



the

domains

do

not

come

back

to

their

configuration

in

the

virgin

state




a

net

magnetization

in

the

absence

of

field

called

remnant

magnetization,

M
r

(

-
M
r

in

the

opposite

direction)
.



(Point

D)
:

magnetization

bring

back

to

zero,




an

extra

field

in

the

opposite

direction

is

applied

called

coercive

field,

-

H
c
.

(+
H
c

in

the

opposite

direction)
.





hard

magnet

has

large

coercivity




soft

magnet

when

coercivity

is

small
.


Antiferromagnetism



Without

an

applied

field,

adjacent

magnetic

moments

(electron

spins

associated

with

magnetic

atoms)

align

anti
-
parallel

to

each

other
.



Adjacent

magnetic

moments

are

equal

in

magnitude

and

opposite

therefore

there

is

no

overall

magnetisation
.



This

occurs

below

a

particular

temperature,

called

Néel

temperature

(T
N
)

above

which

the

material

behaves

as

a

paramagnet
.



Ilmenite

Ferrimagnetism



Antiparallel

alignment

of

moments

at

particular

atomic

sites



Most

of

these

materials

consist

of

cations

of

two

or

more

types

(i
.
e
.

magnetic

moment

of

one

crystal

sub
-
lattice

is

anti
-
parallel

to

the

other)




The

aligned

magnetic

moments

are

not

of

the

same

size
.



An

overall

magnetisation

is

produced

(net

magnetization

is

not

equal

to

zero)

but

not

all

the

magnetic

moments

may

give

a

positive

contribution

to

the

overall

magnetisation
.

Example for
Ferrimagnetic

material



Magnetite

(Fe
3
O
4
)



Spinel

structure




The

large

oxygen

ions

are

close

packed

in

a

cubic

arrangement

and

the

smaller

Fe

ions

fill

in

the

gaps
.




The

gaps

come

in

two

flavors
,

(two

magnetic

sublattices
)


tetrahedral

site

(A)
:

Fe

ion

is

surrounded

by

four

oxygens



octahedral

site

(B)
:

Fe

ion

is

surrounded

by

six

oxygens




The

spins

on

the

A

sublattice

are

antiparallel

to

those

on

the

B

sublattice

A periodic table showing the elements and the
types of magnetism at room temperature

Diffusion



Migration

of

the

defects

which

happens

via

an

atomistic

process

called

as

diffusion
.




Diffusion

causes

changes

in

the

microstructures

(sintering,

creep

deformation,

grain

growth)




Diffusion

is

also

related

to

transport

of

defects

or

electronic

charge

carriers

(
electrical

conductivity

and

mobility
)



The

electrical

conductivity

in

ceramics

is

a

sum

of

ionic

and

electronic

conductivity



o

Ionic conductors applications:



chemical and gas sensors,



solid electrolytes



fuel cell (optimization the fuel/air ratio in the engines
by using an oxygen sensor made of ceramic ZrO
2

in
automobiles)

Atomic

diffusion

rates

&

electrical

conductivity

Defect types


Defect Concentration (T, PO
2
, Comp.)

Electrical conductivity



Fick's

First Law of Diffusion

Diffusion Kinetics

J
:

diffusion

flux

(moles/m
2
-
s),

and

basically

means

the

amount

of

material

passing

through

a

unit

area

per

unit

time
;



D
:

diffusion

coefficient

or

diffusivity

in

m
2
/s
;


x
: the position in m.


C
:

the

concentration

in

m
3

.

D = D
0
exp (
-
Q/
kT
)

Q
:

activation

energy,



K
:

Boltzmann's

constant



D
0
:

pre
-
exponential

factor

in

m
2
/s
.



Diffusivity is a
temperature dependent
parameter



Fick's

Second Law of Diffusion

Diffusion Kinetics

o

It predicts how the concentration changes as a function of time
under non
-
steady state conditions

t: is the time in seconds

Driving force:
the chemical potential which drives the
migration of species from regions of higher chemical
potential to lower chemical potential

Schematic

of

the

planes

of

atoms

with

arrows

showing

the

cross
-
movement

of

species

In general, diffusivity can be expressed as

γ: is governed by the
possible number of jumps at
an instant

λ: is the jump distance and is
governed by the atomic
configuration and crystal
structure.

Temperature Dependence of Diffusivity

Γ
: the jump frequency (# of atoms jump/ s)

ν: the vibration frequency (s
-
1
)

ΔG
*
: the activation energy of migration (J)

k: Boltzmann Constant (J/K).



exponential temperature dependence
resulting in significant increase in the
diffusivity upon increasing the
temperature

Examples of Diffusion in Ceramics

Diffusion in lightly doped
NaCl

(
NaCl

containing small amounts of CdCl
2
)

CdCl
2


Cd
Na


+ 2Cl
Cl
x

+
V
Na




In

addition,

NaCl

will

also

have

certain

intrinsic

sodium

and

chlorine

vacancy

concentration

(
V
Na


and

V
Cl

)

due

to

Schottky

dissociation
,

depending

on

the

temperature
.


the

diffusivity

of

sodium

ions

is

governed

by

vacancy

diffusion
;

Vacancy

concentrations

depends

on

dopant

concentrations

ΔG
Na
*

is

the

migration

free

energy

for

sodium

vacancies



[
V
Na
‘]


is

the

sodium

vacancy

concentration

The diffusivity dependence on temperature shows two regimes



low

temperature

extrinsic

regime

where

vacancy

concentration

is

independent

of

temperature

and

is

determined

by

solute

concentration



high

temperature

intrinsic

diffusivity

exhibits

a

steeper

slope

with

higher

activation

energy

which

include

not

only

the

energy

for

defect

migration

but

also

for

defect

creation

Extrinsic region
is
dominant where
vacancy
concentration is
constant as it is
determined by
the solution
concentration i.e.
[
V
Na
'] = [
Cd
Na

] .

the vacancy
concentration is
governed by
thermally intrinsic
defect creation
mechanism (
Schottky

defect formation
)

Defect migration

Defect migration & Creation

Mobility

Conduction in
ionic compounds

Diffusivity


Mobility: velocity (
ν
) of an entity per unit driving
force (F);

F can be defined as either of chemical potential gradient or electrical potential gradient.



The

most

general

driving

force

for

atomic

transport
:

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
.


Where
μ
i

is the chemical potential of
i

and N
A

is the Avogadro's Number

Absolute mobility, B
i

is given by

Relation between mobility and diffusion



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.

substituting for
F
i

for an ideal solution with unit activity of species
i


R is the gas constant. So, the change in the
chemical potential can be written as

Relation between mobility and diffusion

compare the above equation with
Fick's

first law, diffusivity of species
i

can be written as

The above equation is called
Nernst Einstein Equation

Analogue to the Electrical Properties

Electrical force is given as

φ is potential, E is the electric field,
Z
i

is the atomic number,
e
is the electronic
charge and
Z
i
e

is the total charge on the particle.

Now, since
J
i

=
c
i

.v
i

, the velocity can be written as

So, defining electrical mobility,
μ
i

as velocity per unit electric field

Using the above relations, one can write the flux,
J
i

, as



electrons are free particles with a drift velocity,
v
d
, under an applied
electric field. F=
ZeE

F: the force on an electron




various scattering phenomenon control drift velocity.



under an applied E, drift velocity increases as expressed by Newton's
law of motion

Electronic conductivity


m is the mass of carrier,


v
is velocity and
τ

is the relaxation time.

Under steady state conditions

And mobility, μ, is

m
*

is the effective mass of the carrier
(the mass that it seems to have when
responding to forces)

The

relaxation

time,

τ,

in

metals

and

semiconductors

shows

a

temperature

dependence


o

τ ̴T
-
3
/
2

due to thermal
scattering

o

τ ̴ T
3
/
5

due to impurity
scattering.

Ionic Conduction: Basic Facts


Conduction

in

ionic

solids

is

often

governed

by

concentration

of

impurities,

dopants

and

point

defects
.



Conduction

happens

through

hopping

type

which

is

migration

of

charges

between

either

two

dissimilarly

charged

ions

or

counter

migration

of

ions

and

vacancies
.



NiO

doped

with

Li
2
O

under

oxidizing

conditions

gives

rise

to

oxidation

of

Ni
2
+

ions

to

Ni
3
+

ions
.

Mixed

presence

of

Ni

ions

in

+
2

and

+
3

states

leads

to

hopping

type

conduction

of

electrons

between

two

states
.




Thermally

activated

phenomenon

(
Mobility

of

charge

carriers

and

hopping
)



The

higher

the

dopant

concentration

(within

appropriate

limits),

the

higher

the

conductivity



Carrier

concentration

is

independent

of

temperature

(within

extrinsic

region)

and

mobility

is

strongly

affected

by

temperature

Ionic and Electronic Conductivity

Electrical

conductivity

(
σ
i
)

is

defined

as

charge

flux

per

unit

electric

field

with

units


-
1
cm
-
1
)

or

S/m
.

It

can

be

expressed

as


J
i

is the flux of species
i



For ionic species, we can apply
Nernst
-
Einstein equation


Similar

to

diffusivity,

temperature

dependence

of

ionic

conductivity

also

exhibit

extrinsic

and

intrinsic

regions

at

low

and

high

temperature,

respectively
.



J
i

=
c
i

.v
i


Total conductivity and Transference Number

Since all charged species contribute to the electrical conductivity, we
can write total conductivity as

Fraction of total conductivity carried by each charged species is called
as transference number,
t
i

and is expressed as

and it is straightforward to see that


electronic conductor



ionic conductor



mixed conduction


Long range migration of ionic charge carriers, the most
mobile species, through the lattice under application of an
electric field (
e.g. 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.


Characteristics of Ionic Conduction


When external field is absent, the thermal energy,
kT
, is required for
counter migration of ions and vacancies overcoming the migration
energy Ea, which is nothing but process of self diffusion.











I
n 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.


Characteristics of Ionic Conduction

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


Characteristics of Ionic Conduction

Theory of Ionic Conduction



β
'

and β
''
-
alumina
:
very high conductive ceramics (
10
-

10
-
1

(
Ω.cm
)
-
1
,
@300 K to 675 K
) Typical activation energies: ~3.5
-
4.5
kCal
/mole.




Spinel

oxides (Fe
3
O
4
):

0.5 (
Ω.cm
)
-
1

and have very low activation energies (0.35
kCal
/mole),
representing almost temperature independent behaviour.




Y
2
O
3
, HfO
2
, SiO
2
, Al
2
O
3
: insulating ceramics (10
-
5

to 10
-
14

(
Ω.cm
)
-
1
,
@ 400 K to 1000 K

n: ionic density,

α: irreversibility of the jump



Metals



Ce

~ constant



µe decreases as temp increases



σ decreases as temp increases





Semiconductors & insulators



Ce

increases as temp increases (dominates)



σ increases as temp increases




Ionic conduction



Ci

is either constant (extrinsic) or increases as temp
increases (intrinsic)



µ
i

increases as temp increases (diffusion)



σ increases as temp increases

Conducting ceramics are used in a variety of applications
such as:



SiC

and MoSi
2

as heating elements and electrodes



ZnO

and
SiC

as
varistors

(A semiconductor diode with
resistance dependent on the applied voltage) for circuit
protection



YSZ, β
-
Alumina as electrolytes in fuel cells and
batteries



Materials like YSZ in gas sensing applications

Examples of Ionic Conductors in Engineering
Applications



Exhibit a sudden drop in
electrical resistance to exactly
zero when cooled below room
temperature.




Happens at a specific
temperature called critical
temperature, T
C
.




The phenomenon was
discovered by Heike
Kamerlingh

Onnes

in
1911
when he was
studying properties of mercury
at liquid helium temperatures.

Superconductors

Scal i ng behavi or of mi xed
-
state hal l effect i n MgB
2

thi n fi l ms,
Physi ca

C: Superconducti vi ty, Soon
-
Gi l l Jung et. al., v 450, 2006

Inset shows a magnified view near a
superconducting transition region.



elemental metals & metallic alloys: (
Sn
, Al, niobium nitride,
niobium
-
titanium, and niobium
-
germanium alloys), most of
these are superconducting at temperatures below 30 K.




In 1986 Bednorz &
Müller

demonstrated
superconductivity in a
perovskite

structured lanthanum based
cuprate

oxide (La
2
Cu
2
O
4

) which showed a T
C

of 35 K.




chemical substitution in
perovskite

cuprates

increases the
transition temperatures to 77 K and beyond.

Superconductors

o

YBa
2
Cu
3
O
7
-
x
(YBCO):T
C

~92 K


highest T
C

when they are slightly oxygen deficient (
x

= 0.15)
Superconductivity disappears at
x

≈ 0.6, (structure of YBCO changes from
orthorhombic to tetragonal)

Meissner

Effect


The

magnetic

field

is

completely

expelled

from

the

interior

of

the

superconductor,

when

it

is

placed

within

a

magnetic

field
.




Application

magnetically levitated trains

Maglevs