Chapter 3 Crystal Defects

8/10/2019 Chapter 3 Crystal Defects

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Chapter 3 Crystal defects

Point defects (fig. 3.1)

1. Vacancy very important for diffusion2. Self -interstitial atom

3. Frenkel defect a pair of vacancy and self-interstitial site

4. Schottky defect a pair of cation and anion vacancies (because of constraint of chargeneutrality)

5. Substitutional impurity atom

6. Interstitial impurity atom

Interstitial atom usually does not exist as asingle atom on an intersitial site but rather 2atoms share a lattice site which have 2possibility for the arrangement (fig. 3.2)

- dumbbell

- crowdion (pack in the densely packedcrystallographic direction)


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Thermodynamics of point defects

n no. of vacancies in thermodynamics equilibrium

N total no. of atoms (sites)

n/N = ca - concentration of vacancies

The first principle of thermodynamics: Q = dU + pdV (3.1)

Q - the amount of absorbed heatdU - a change of internal energy (the absorbed heat changes to internal energy)

pdV work of expansion of the system at a given temperature

The second principle of thermodynamics:

S the entropy

T

QdS

T

QdS

>

T

QdS

=

Spontaneous/irrevesible

Equilibrium/reversible

From 3.1 and 3.2 0+ TdSpdVdU

(3.2)

(3.3)

From Gibbs free energy, G

0+ TdSpdVdU

From Gibbs free energy, G TSpVuG += (3.4)

Then VdppdVSdTTdSdUdG ++= (3.5)


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At constant pressure p and constant temperature T

dp and dT = 0 0+= pdVTdSdUdG (3.6)

At equilibrium, dG=0

and H= enthalpy0=+= TdSdHTdSpdVdUdG (3.7)

If n vacancies are generated by moving

atoms in the volume to the surface (fig. 3.3),the free energy change will be

)( cv

v

v

f SnSTnHG +=(3.8)

Hfv the formation enthalpy of a vacancy

Svv The change of the vibration entropy per vacancy

Sc the configurational entropy can be calculated by

n the possible configuration of n vacancies on N lattice sites

k Boltzmann constant = 8.62 x 10-5 eV/K

Sc = k lnn (3.9)

!)!(

!

.....3.2.1

)1)....(2)(1(

nnN

N

n

nNNNNn

=

+= (3.10)


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f(x) = x!

If x > 5 , then ln x ~ x ln x - x

Free energy of formation of a vacancy, Gfv

Insert 3.14 in 3.12

At equilibrium, with n vacancies (from

eq. 3.7 and 3.8)

0)( ==dn

dSTTSHdn

Gd cvv

v

f(3.12)

ln x ~ x ln x - x(3.11)

nN

nk

dn

dSc

= ln

(3.13)

n


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From table 3.1

Vacancy concentration of metal (independent of material) is about 10-4 (cva = n/N = 10-4)

Hfv vacancy generation enthalpy

Svv vibration entropy

From eq 3.16a: cva

depend on T , if T 0, cva

0. But in real situation vacancies always exist inmetal because of their vanishing mobility at low temperature.

Therefore, there always a small concentration of vacancies remains in the crystal.

The free energy of formation of an interstitial atom is much larger (by a factor of 3) than for the

vacancy. Therefore, interstitial atoms do no occur in thermodynamic equilibrium.

Au Al Cu W Cd

Hfv

[ev] 0.94 0.66 1.27 3.6 0.41

Svv

[k ] 0.7 0.7 2.4 2.0 0.4Cv

a [10-4] 7.2 9.4 2.0 1.0 5.0


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3.2.3 Experimental evidence of point defects

- Point defects are perturbations of the ideal crystal

- Cause changes in physical properties, electrical resistivity

Specimen

Heat at high temp, Tq

Quench

-Number of lattice point defects increase

-Point defects increase resistivity increase

-Number of lattice point defects remain the same as

at high temperature

-Resistivity remains high

an increase in resistivity (resistivity different between specimen before and after heating)

- the resistivity increases per point defect

ca - the concentration of point defects

)exp()exp(. kT

H

k

S

NNc

v

f

v

fa

== (3.17) proportional to ca

Hfv enthalpy of formation

- determined by measuring the increase in residual resistivity after quenching from different temperature

- increase in resistivity does not yield information on type of point defect, because both interstitials and

vacancies will increase the electrical resistivity


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An experiment: vacancies are the dominant point defects in thermal equilibrium.

- measure thermal expansion and lattice parameter change as a function of temperature in gold.

- if vacancies are formed, atoms will be moved from the crystal interior to the surfaces as shown in fig. 3.3.

V0 - the volume of crystal

VD - the change in volume if there are n vacancies in the crystal

If one vacancy is formed,

- the volume will be increase by (volume of one atom which equal to one vacancy)- the atoms around the vacancy will slightly relax in the free volume so that the volume per defect,

VD is decreased by Vvr(Vvr


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If N - no. of lattice sites N = V0/

The vacancies concentration

If V0 - volume of crystal V0 = m. Vecm - no. of elementary cells

n - no. of vacancies n/m = no. of vacancies per unit cell

The volume change of a unit cell by n vacancies in the total volume Vec = (n/m). Vvr

0

3

0

3

0

00

3)(..

a

a

a

aa

V

Vm

V

Vnecvr

+

=

=

)(3)(.

0000000 a

a

L

L

V

n

V

Vn

V

Vn

V

Vn

V

V vrvrvr

=

=

+=

(3.19b)

(3.20a)

)(3a

a

L

L

N

nc

a

v

== (3.20b)

For vacancies 0)(


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Simmons and Balluffi measured L and a of gold rod found that- L and a function of temperature- Vacancies are formed in thermodynamic equilibrium

- The enthalpy formation of a vacancy can be calculated from eq. 3.20b and 3.16. and agree very wellwith results form measurements of the electrical resistivity.

- Point defects can be generated by irradiation of a material with high energy elementary particles, for

instance by electron, protons, neutrons or heavy ions.

- If a material is irradiated by electrons with an energy beyond a certain threshhold energy (eg. 400 kV

for copper), one Frenkel pair will be produced per electron.

- These defects occur in nuclear reactors and damage the material of the reactor container.


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3.3 Dislocations

A dislocation is characterized by

- The line element, s, : the unit vector tangential to the dislocation line

if dislocation line is curved, s will change along dislocation line

- The Burger vector, b, : the length and direction of the vector by which the two parts of the crystalabove and below the plane of motion of the dislocation are displaced with respect to each other

Edge dislocation Screw dislocation Mixed dislocation


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b remain constant

s change along dislocation lineThe character of a dislocation can change along its line.

If be the edge component of a dislocation

bs the screw component of a dislocation

A dislocation are defined by the orientation of the Burgersvector with regard to the line element. (fig. 3.11)

bs = s.(b.s) = (|b|.cos).s 3.22a

be = s x (b x s) = (|b|.cos).n 3.22b

Dislocation properties Edge dislocation Screw dislocation

Direction parallel to b perpendicular to b

Angle between

dislocation l ine and b perpendicular parallelDirection of dislocation line and b parallel perpendicular

Mechanism of changing slip plane climb cross-slip

n is the unit vector perpendicular to the dislocation line the plane defined bys and b.


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Dislocation loop

- a dislocation can occur in a closed loop in crystal

- closed dislocation loops are mainly generate duringplastic deformation


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Dislocation can move.

- The plane along which the dislocation line is displaced is called the glide plane or slip plane and its normalm is defined

m = s x b 3.23

- For screw dislocation: s // b - do not have a defined slip plane and can change the slip plane

- For edge and mixed dislocation: - have defined slip plane

- the slip plane can be changed only by absorption or emission of pointdefects (climb), eg vacancies.

- the absorbed vacancies come from the crystal.

- dislocations are considered as the sinks for vacancies


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Dislocation density, ,- the total length of the dislocation lines per unit volume

- the dimension - m/m3

= m-2

- if assume that all dislocations are straight and parallel,then dislocation density, , is related to the dislocationspacing (d)

=1/d2

- dislocation density can be determined by edge pits butdifficult after plastic deformation

Thermodynamically, the annealed crystal should free from dislocationsbecause the engergy per atom of a dislocation line is comparable to theenergy of formation of an interstitial atom (around 5 eV for Cu).

Therefore, the concentration of dislocation atoms, cda, along dislocation line

should be negligibly small.

If the spacing of atoms along the dislocation line is b, the concentration ofdislocated atoms for a dislocation density

Because of the high enthalpy of formation, cda, and should be negligibly

small in thermal equilibrium. But in carefully annealed crystal, the

dislocation density in the order of 1010

m-2

.

2

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b

b

bca

d

==


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3.4 Grain boundaries

To study grain boundaries requires an extensive mathematical description. For 2 dimensions, 4 parametersare needed to defined mathematically. Fig. 3.20

a rotation angle: orientation different between the adjacent crystal (orientation relationship)

the spatial orientation of the grain boundary plan (grain boundary orientation)

t the translation vector composes of t1 and t2

(the displacement of the 2 crystals respect to each other)

For 3 dimension (real case), 8 parameters are needed. The properties (energy and mobility) of a grain boundaryare a function of all 8 parameters. 5 of 8 parameters can be influenced externally. To determine dependency of

grain boundary properties, such as mobility, it would necessary to vary only one parameter but the other 4parameters need to be fixed.

The grain boundary energy as function of the orientation of the grain boundary plane is commonly represented inthe Wulff plot.The Wulff plot is constructed by drawing a line from the origin of the coordinate system in thedirection of the grain boundary normal with a length corresponding to the grain boundary energy. (fig. 3.21)


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The points closer to the origin represent low energy grain boundaries, such as twin boundaries. The grainboundary tend to align itself parallel to such orientations at least piecewise to minimize its total energy(facetting)

To study grain boundary: A transformation which has to be applied to the crystals to make both crystallattices coincide.

If a common origin is assumed: a transformation is a pure rotation. (fig. 3.22)

- a twist boundary: the grain boundary plane is perpendicular to the rotation axis

- a tilt boundary: the grain boundary is parallel to the rotation axis

+ symmetric tilt (the adjacent crystals are mirror images of each other (mirror plane=grain

boundary plane)+ asymmetric tilt

A symmetric tilt boundary, the grain boundaryplane has equivalent Miller indicies with

respect to both adjacent crystal. It ratherdifficult to consider asymmetric boundary.Mostly the asymmetric boundary istreated as a deviation from the symmetricposition


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3.4.2 Atomic structure of grain boundaries

3.4.2.1 Low angle grain boundaries

Low angle grain boundary the misorientation between adjacent grains is small, the boundary is entirely

comprised of a periodic crystal dislocation arrangement.

Symmetric til t boundary consists of

- a set of edge dislocations (Burgers vectorb )

- the dislocation spacing D which decreases withincreasing rotation angle,

=

2sin2

D

b


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For asymmetric low angle tile boundaries

at least 2 sets of edge dislocations are required, the

Burgers vectors of which are perpendicular to each

other. The fraction of dislocations of the second set

increases with increasing deviation from the

symmetric tilt boundary that is only comprised of the

first set of dislocation. (the second set of dislocation

causes the deviation from symmetry)

Low angle twist boundaries require at least 2

sets of screw dislocations.

Mixed boundaries are composed of

dislocation networks of 3 Burgers vectors.


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3.4.2.2 High angle grain boundaries

For small angles of rotation, the specific grain boundaries energy (energy per unit area) increases

exactly as predicted by the dislocation model. (Fig. 3.28). For the real measurement, the angles of rotation > of

15O grain boundary energy does not change with increasing rotation angle. In contrast to dislocation model, the

grain boundary energy increases as increasing rotation angle.

For rotation angles > 15O the dislocation model fails, because the dislocation cores tend to overlap.

Therefore, grain boundaries with rotation angles > 15O are considered as high angle grain boundaries. The

structure of high angle grain boundaries has been derived from geometrical concepts, based on dislocation

models of low angle grain boundaries.A reason for the failure because the requirement of a strictly periodic

dislocation arrangement is to minimize grain boundary energy

The dislocation spacing, D, changes discretely at least 1 atomic distance. As a consequence, the

angle of rotation = b/D also changes in steps rather than continually.

For small angles b


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In a perfect crystal the atoms have a defined position which is determined by the minimum of the

free energy. Any deviation from this position the free energy will increase. In fact, the crystal will try to keep the

atoms in their ideal position. This is also the case in grain boundary.

There are atomic positions in the grain boundary which coincide with ideal positions of both

adjacent lattices. Such lattice points are called coincidence sites . Fig. 3.29.

For example, a rotation angle of 36.87O about a axis in a cubic lattice is equivalent to

53.13O because of the 90O crystal symmetry.

From fig. 3.29, if the grain boundary plane is (100) ( the rotation axis is along this plane and the

atoms on both grains are on this plane, grain boundary plane), atomic positions perpendicular to the rotation

axis (right part of fig. 3.29) then the occurrence of many coincidence sites can be seen. Since both crystallattices are periodic, the coincidence sites also must be periodic - the coincidence site lattice (CSL).

The elementary cell of the CSL is larger than the

elementary cell of the crystal lattice. The density of

coincidence sites of for the size of the elementary cell of

CSL can be calculated

For the rotation 36.87O (100) is = a(a5)2/a3 = 5, everyfifth lattice site is a coincidence site, fig. 3.29.

latticecrystalofcellelementaryvolume

CSLofcellelementaryvolume

_____

____=


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Fig. 3.29 is a 2-dimensional case but in reality the coincidence site lattice is a 3-dimensional lattice.

A crystal lattice, each lattice point carries 2 atoms one round () and one triangle (). Then rotate thetriangular atoms while the round atoms remain unchanged. After rotation there again lattice points where

triangular and round atoms coincide because of the periodicity of the crystal lattices.

To apply the crystallographic construct to grain boundaries, the spatial orientation of the grain

boundary plane has to be defined. To define this plane

- remove round atoms on one side of the plane

- remove triangular atoms on the other side.

- bicrystal with a boundary, and the structure of the boundary is given by the atoms located in theboundary.

- if good fit of the atoms (and coincidence sites represent atoms with ideal fit) is associated with a

low energy. The grain boundary strongly favors running through coincidence sites rather than non-

coincidence sites.

Grain boundaries between crystals which have an orientation relationship corresponding to a high densityof coincidence sites are called CSL boundaries or spatial boundaries.

A smaller value which is always an odd integer implies a more ordered grain boundary

= 1 : low angle grain boundaries, almost all lattice points except for the atoms of

the dislocation cores are coincidence sites. = 3 : twin orientation, and in the coherent twin boundary all lattice site arecoincidence sites. Atoms on the very third twin plane are in perfect coincidence.

Table 3.2, does not change monotonically with the rotation angle. changesdiscretely as well as rotation angle because of a periodic crystal dislocation

arrangement.


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Low angle grain boundary, dislocations at grain boundary

Form CSL on both side of the grains

But some atoms may deviate from their suitable positions

Crystals try to compensate a small misorientation by periodic arrangement of dislocations.

These dislocations must have Burgers vector that conserves the CSL

These displacement vectors which satisfy the above condition define the DSC-lattice

(Displacement shift complete) Fig.3.31

[If one of the 2 adjacent crystal lattices is shifted by a translation vector of the DSC-

lattice, the CSL will displace as a whole]

- All translation vectors or CSL and the crystal lattice are also the vectors of DSC lattice,

but vectors of DSC lattice are much smaller.- The dislocation energy increases with the square of the Burgers vector therefore, the

vector of DSC lattice qualify for Burgers vectors of the secondary grain boundary

dislocation (SGBDs).

- Dislocations with DSC Burgers vectors are referred to as SGBDs.


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At grain boundary, the primary grain boundary dislocations are crystal lattice dislocations.

The periodic arrangement of which generates the CSL.

SGBDs are confined to grain boundaries, since their Burgers vectors are not translation

vectors of the crystal lattice and their introduction into the crystal lattice could cause local disruption ofthe crystal structure.

The primary dislocations can compensate a misorientation in a perfect crystal by a low angle

grain boundary.

The second grain boundary dislocations (SGBD) can compensate an orientation difference

to a CSL relationship while conserving the CSL.

GB CSL DSC lattice (displacement vector causes) SGBDs

A special property of the SGBDs is that at the location of the dislocation core, the grain boundary

usually has a step (fig. 3.33)

This step is a consequence of the fact that the CSL is displaced when an SGBD is introduced

If SGBD moves along the grain boundary, the step moves along with the dislocation and thus the

grain boundary is displaced perpendicular to the its plane. The grain boundary will migrate by thedistance of the step height.

The motion of a grain boundary dislocation will cause a combination of grain boundary migration

and grain boundary sliding.


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In fact for the geometrical considerations, coincidence is almost always lost upon relaxation, but

the periodicity remains.

The analysis reveals that the arrangement of atoms at the grain boundary can be described by

polyhedra. Only 7 polyhedra are necessary to describe all possible arrangements of atoms in agrain boundary.


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3.5 Phase boundaries

3.5.1 Classif ication of Phase Boundaies

The adjacent crystals can have different crystal structures as well as different orientation.

1. A coherent phase boundary - the lattice parameters of both phases are slightly different and no

orientation difference. All lattice planes are continuous in both phase. Fig. 3.38

2. A partially coherent phase boundary the adjacent crystals have the same structure but

different lattice parameters. This mismatch is compensated by the formation of edge

dislocations which will reduce the coherency stresses. Not all lattice planes are continuous

through the interface. Fig. 3.393. Incoherent phase boundary both phases have different crystal structures. Fig. 3.40.

4. Interface in the composite not in equilibrium, large perturbations around the interface and

inhomogeneities. Fig. 3.41.

3.5.2 Phenomenological Characterization of Phase Boundaries

The interfacial tension, , (J/m2 = N/m) force per unit length. The surface tension or force per

unit length is the force needed to keep the 2 phases in contact.

For the internal interface the interfacial tension.

The interfacial tension determines the equilibrium shape of interfaces in the phase mixtures.


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Along the lines of contact of the phases the surface tensions act as forces in order to establish the

configuration with the smallest energy. Horizontal equilibrium of forces is obtained if

cos.lg+= slgs

If = 0O :the droplet will spread on the surface as a film complete wetting

= 180O :the droplet will have the shape of a sphere complete non-wetting

can be affected by chemical composition.

- Immiscible system: is usually large

- Chemical reaction: is usually very small

- for the composite with excellent adhesion: is small

contact angle

(3.27)


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Small prefer - water on the windshield

Large prefer - inclusions with low melting point in metal.

Bi in brass or FeS in steel low contact angle

Bi segregates to the grain boundary of brass. If the temperature is increased above the melting

temperature of Bi. Liquid Bi will spread along the grain boundaries because of it low surface tension

(low contact angle) which causes the hot shortness. By adding Pb the contact angle increases and

hot shortness of brass can be avoid. Fig. 3.44


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General case of the equilibrium of 3 phases, eq.

3.28a.

idepends on interfacial energy on theinterface orientation. Fig. 3.46.

If the energies of all three boundaries ar

independent of interfacial orientation, the equation

is

2sinsinsin

31

1

23

3

12

==

For many high angle grain boundaries, energydoesnt vary strongly with misorientation. In this

case ij = constant (k = 120O). In equilibrium,

microstructure of homogeneous phases the contact

angle of grain boundaries is 120O. Fig. 3.47.

Chapter 3 Crystal Defects

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