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4-1

C H A P T E R 4

C R Y S T A L S T R U C T U R E S

4.1 ATOMIC ARRANGEMENTS

4.2 LATTICES AND UNIT CELLS

4.2.1 Unit Cel l s in Space

4.2.2 Atomic Packing in Crysta ls

4.3 METALLIC CRYSTAL STRUCTURES

4.3.1 Face-Centred Cubic (FCC)

Crysta l St ructure

4.3.2 Body-Centred Cubic (BCC) Crysta lSt ructure

4.3.3 Hexagonal Close-Packed (HCP)Structure

4.3.4 A l lot ropy/Polymorphism

4.4 CLOSE-PACKED CRYSTAL STRUCTURES

4.5 INTERSTITIAL POSITIONS AND SIZES

4. 6 CRYSTALLOGRAPHIC DIRECTIONS AND PLANES

4.7 CRYSTALLINE MATERIALS



4-2

4.1 ATOMIC ARRANGEMENTS

• The properties of a material depend not only on atomic

bonding and forces, but also, equally important, on how

atoms pack together. There are 3 levels in which atoms

may be arranged:

• No order: there is no special positional

relationship or interaction between the

atoms; e.g. inert gases (Fig. 4.1-1a).

• Short-range order: the specific

arrangement of atoms extends only toan atom's nearest neighbours. Materials

exhibiting short-range order are

amorphous (glassy) (Fig. 4.1-1b).

• Long-range order: there is a special

arrangement of atoms that is repeated

throughout the entire material.

Crystalline materials exhibit both short-

range and long-range order. The

repetitive pattern formed by atoms in a

crystalline solid is known as a lattice (Fig.

4.1-1c).

(a) No order.

(b) Short-rangeorder.

(c) Long-rangeorder.

Fig. 4.1-1 The levelsof atomic

arrangement.

4-3

4.2 LATTICES AND UNIT CELLS

• A lattice is a collection of points (positions in space)

arranged in a periodic pattern so that surroundings of each

point in the lattice are identical. The points that make up

the lattice are called lattice points (Figs. 4.2-1, 4.2-2). An

everyday example in 2-dimensions is wall-paper.

Fig. 4.2-1 Lattices and unit cells in 2D.

Fig. 4.2-2 Lattices and unit cells in 3D.

• A unit cell is the smallest subdivision of a lattice that

contains all the characteristics of the entire lattice. A

complete crystal can be formed by translating its unit cell

along each of its edges.



4-4

• The properties of a unit cell are therefore the same as those

of the whole crystal.

• Each unit cell is described by lattice parameters, which

are the lengths of the cell edges and the angles between

the axes. (Figs. 4.2-1, 4.2-2)

4.2.1 Unit Cells in Space

• By geometry, there are only 7 unique unit cell shapes that

may be stacked together in space. And there are only a

total of 14 possible ways in which atoms may be arranged

inside these unit cells. These 14 different unit cells are

known as Bravais lattices and they fall into one of 7 crystal

systems (Fig. 4.2-3).

• One or more atoms may be associated with each lattice

point. The "group of atoms" located at each lattice point is

the basis. The actual crystal structure itself is defined by a

combination of crystal lattice and crystal basis (Fig. 4.2-4).

Crystal lattice

+

+

crystalbasis

=

= crystal structure

4-5

Fig. 4.2-3 14 Bravaislattices grouped into 7

crystal systems.



4-6

4.2.2 Atomic Packing in Crystals

• When describing crystal structures, atoms are assumed to

be solid spheres that touch one another. This is known as

the hard sphere model. The centres of the solid spheres

coincide with the lattice points in unit cells (Fig. 4.2-5).

Fig. 4.2-5 Hardsphere model of

a unit cell.

• The coordination number is the number of nearest

neighbours (touching atoms) to any atom. For materials

with non-directional bonding (i.e. metals and ionic solids),

the lowest energy (most stable) configuration is obtained

when atoms pack as closely as possible, separated only by

their equilibrium bond lengths. In such crystals, the

number of nearest neighbours (i.e. coordination number)

would be as high as possible.

• The atomic packing factor (APF) is the fraction of space

occupied by atoms in a unit cell.

APF = Volume of atoms in unit cell

Volume of unit cell

=(Number of atoms in cell)(Volume of one atom)

Volume of unit cell

4-7

• The properties of an entire crystal, such as the theoretical

density, may be calculated from just one of its unit cells.

Density = Mass of unit cell

Volume of unit cell

=(Number of atoms in cell)(Mass of one atom)

Volume of unit cell

= (Number of atoms in cell)(Atomic mass)(Volume of unit cell)(Avogadro's number)

4.3 METALLIC CRYSTAL STRUCTURES

In pure metals, only one metal ion occupies each lattice

point. Many common metals may be defined by one of 3

crystal structures: face-centred cubic (FCC), body-centred

cubic (BCC) and hexagonal close-packed (HCP) crystal

structures (Table 4.3-1).



4-8

4. 3. 1 Face-Centred Cubic (FCC) Crystal Structure

• Cubic geometry with one atom at each corner and one at

the centre of each face. (Fig. 4.3-1)

Fig. 4.3-1 FCC structure.

• Corner atoms touch face-centred atoms (along the facediagonal) but corner atoms do not touch one another.

Face-centred atoms also touch adjacent (but not opposite)

face-centred atoms in the midplanes of the cube.

• Unit cell length: a = 2R 2 (where R is the atomic radius)

4-9

• Coordination number: consider a corner atom, which

touches 4 face-centred atoms on each of the 3 mutually

perpendicular planes passing through the corner atom

itself (Fig. 4.3-2), giving CN = 12.

Fig. 4.3-2 Finding thecoordination number in

FCC structures.

• Atomic packing factor: each atom at the centres of the

cube faces is shared by 2 cells; each corner atom is shared

by 8 cells (Fig. 4.3-3), such that:

• Atoms per cell = 4; !

APF = 0.74

Fig 4.3-3 Sharing of face and corner atoms in FCC structures.

• Examples of FCC metals: Al, Au, Ag, Cu, Ni, Pb, Pt.



4-10

4. 3. 2 Body-Centred Cubic (BCC) Crystal Structure

• Cubic geometry with one atom at each corner and one at

the centre of the cube. (Fig. 4.3-4)

Fig. 4.3-4 BCC structure.

• Atoms touch along the body diagonal.

• a = 4R

3

• CN = 8.

• Atoms per cell = 2; APF = 0.68

• Examples of BCC metals: Cr, Fe, W, Mo, V.

4-11

4.3. 3 Hexagonal Close-Packed (HCP) Structure

• Hexagonal faces at the top and bottom are linked by 6

rectangular side faces. There is one atom at each corner of

the top and bottom hexagons surrounding one atom at

the centre of each hexagon. 3 other atoms are located on

a plane midway between the hexagons (Fig. 4.3-5).

Fig. 4.3-5 HCP structure and its smaller primitive unit cell.

• Corner atoms at the top and bottom hexagons are shared

between 6 cells; the central atom in each hexagon is

shared between 2 cells; the 3 atoms in the midplane

belong to only one cell.

• Lattice parameters: a = 2R; c = 4

6

a

• By considering the central atom in basal plane, CN =12.

• Atoms per cell = 6 (or 2 per primitive unit cell); APF = 0.74

• Examples of HCP metals: Co, Mg, Ti, Zn.



4-12

4.3.4 Allotropy/Polymorphism

• Allotropy or polymorphism is the ability of an element or

compound to assume more than one crystal structure in

the solid state, depending on external conditions such as

temperature, pressure, magnetic and electric fields (Table 4.3-2)

• The change in crystal structure is usually accompanied by

changes in properties.

• Such property changes can be very useful; e.g. hardening/

softening of steel through controlled heating/cooling (Chps

9&10), piezoelectric transducers (Figs. 4.3-6 & 7).

• Or detrimental; e.g. distortion and cracking due to sudden

changes in volume, especially in brittle ceramics, but also

in metals (Fig. 4.3-8).

4-13

Fig. 4.3-6 PZT (lead zirconate titanate)ceramic changes its structure from

(a) cubic, to (b) tetragonal,in response to an electric field.

Fig. 4.3-7 Use of piezoelectric effectof PZT crystals in inkjet printer head.

Fig. 4.3-8 Tin changes from tetragonal to diamond structure below 13.2°C.The volume expansion accompanying this transformation from

soft, ductile white tin to hard, brittle grey tin causes it to disintegrate.



4-14

4.4 CLOSE-PACKED CRYSTAL STRUCTURES

• Atoms in metals pack as closely as

possible, separated only by their

equilibrium bond lengths, r 0, as this

gives the lowest energy (most

stable) configuration (Fig 4.4-1).

• Close-packed planes or directions

refer to the planes or directions in a

crystal, where the atoms are in

direct contact, assuming a hard

sphere model of atomic packing.

• Plastic deformation in metals

occurs most readily on close-

packed planes along close-packed directions in those

planes (Sec. 6.1). The number and relative positions of these

planes and directions influence properties such as ductility.

• Packing same-sized atoms in FCC or HCP structures gives

the smallest volume (i.e. highest density). FCC and HCPare known as close-packed structures.

• FCC and HCP differ only in the arrangements of their

close-packed planes (Fig. 4.4-2, 4.4-3); this difference affects

plastic deformation and ductility (Sec. 6.1).

Fig. 4.4-1 Bonding energyis higher whenatoms are away from equilibriumseparation r 0.

4-15

Fig. 4.4-2 Illustration of close-packed stacking sequence.

HCP FCC

Fig. 4.4-3 Close-packed stacking sequence and close-packed planes for HCP and FCC.



4-16

4.5 INTERSTITIAL POSITIONS AND S IZES

• Lattices not completely filled with atoms. Interstices are

the ‘holes’ between lattice atoms.

• Interstitial sites are classified by geometry, based on the

shape of the polyhedron formed by existing lattice atoms

surrounding a particular site. (Fig. 4.5-1 & Table 4.5-1)

• The size of an interstitial site is defined by the radius ratio,

r

R, where r is the radius of the largest sphere that can

completely fill the site without straining the adjacent lattice

atoms, and R is the radius of the lattice atoms.

Tetrahedral interstitial Octahedral interstitial Cubic interstitialr

R = 0.225

r

R = 0.414

r

R = 0.732

Fig. 4.5-1 Interstitial sites and sizes in close-packed crystal structures.

• Atoms (alloy or impurity) occupying interstitial sites (Sec.

6.3.2) must be larger than the size of the holes; smaller

atoms are not allowed to “rattle” around loose in the sites.

4-17

Table 4.5-1 The size and number of interstitial sites in FCC, BCC and HCP.

Crystalstructure

Size of interstitial sites, r

R No. of sites per unit cell

Octahedral Tetrahedral Octahedral Tetrahedral

FCC 0.414 0.225 4 8

BCC 0.155 0.291 6 12

HCP 0.414 0.225 6 12

Fig. 4.5-2 Locations of the interstitial sites in FCC, BCC and HCP structures.



4-18

4.6 CRYSTALLOGRAPHIC DIRECTIONS AND PLANES

• Some physical and mechanical material properties vary

with the direction or plane within a crystal in which they

are measured. Miller Indices provide a convenient

notation system for directions and planes in crystals.

• In cubic systems, 3 integers uniquely identify a direction

(Fig. 4.6-1) or plane (Fig. 4.6-2), when enclosed by a pair of

square brackets [hkl ], or parentheses (hlk ), respectively.

• In hexagonal crystals, a redundant axis is added to the

basal plane to reflect its symmetry, such that 4-integerMiller-Bravais indices of the form [uvtw ] and (uvtw ) are

often used instead (Fig. 4.6-3).

Fig. 4.6-1 Some common directions in a cubic unit cell.

4-19

Fig. 4.6-2 Some planes in a cubic unit cell.

Fig. 4.6-3 4-integer Miller-Bravais indices for directions and planes.



4-20

4.7 CRYSTALLINE MATERIALS

• In a single crystal, the same orientation and alignment of

unit cells is maintained throughout the entire crystal.

• Most materials are polycrystalline: their structures are

composed of many small crystals (grains) with identical

structures but different orientations (Fig. 4.7-1).

Fig. 4.7-1 The solidification of a polycrystalline material.

• Grain boundaries are the

interfaces where grains of

different orientations meet

(Fig. 4.7-2). Single crystals do

not contain grain boundaries.

4-21

• The absence of grain boundaries in single crystals impart

unique properties but such crystals are extremely difficult

to grow, requiring carefully controlled conditions, which

are expensive. Single crystals are essential to some

applications; e.g. semiconductors and jet turbine blades,piezoelectric transducers.

• Besides the absence of grain boundaries, single crystals

exhibit directionality in properties, such as magnetism,

electrical conductivity and elastic modulus and creep

resistance, which depend on the crystallographic direction

of measurement. This directionality is called anisotropy.

• The random orientation of individual grains in apolycrystalline material means that measured properties

are independent of crystallographic direction. Such

materials are said to be isotropic.

• Grain boundaries are disordered regions of atomic

mismatch where atoms are displaced from their

equilibrium positions (Fig. 4.4-1) and there are improper

coordination numbers (Sec. 4.2.2) across the boundaries. Atoms at grain boundaries hence possess higher energy.

• This interfacial energy makes grain boundaries

preferential sites for chemical reactions and other chemical

changes.



4-22

• Therefore, grain boundaries are attacked more aggressively

by chemical etchants. Under a microscope, the more

deeply etched grain boundaries scatter more light, and

appear darker (Figs. 4.7-3/4), thus revealing the microstructure.

This is the principle behind metallography.

Fig. 4.7-3 Observation of grains and grain boundaries in stainless steel sample.Note that different orientations of the grains result in differences in reflection.

Fig. 4.7-4 Observed microstructure in 2-D and the underlying 3-D structure.



5-1

C H A P T E R 5

CRYSTAL DEFECTS

AND DIFFUSION

5.1 CRYSTAL DEFECTS

5.1.1 Thermal V ibrat ion of Atoms

5.1.2 Vacanc ies

5.2 DIFFUSION

5.2.1 Vacancy Di f fus ion

5.2.2 Inters t i t ia l D i f fus ion

5.2.3 Concentrat ion and F lux

5.3 FACTORS AFFECTING RATE OF DIFFUSION

5.3.1 Temperature

5.3.2 Di f fus ion Mechanism

5.3.3 Atomic Bonding

5.3.4 Crysta l St ructure

5.3 .5 Crystal Defects (Short Circuit Diffusion)

5-2

5.1 CRYSTAL DEFECTS

• In a perfect crystal, atoms would exist only on lattice sites

and every lattice site would be occupied by an atom.

• A crystal defect or imperfection is an irregularity in the

crystal lattice, a departure from the perfect crystal. Defects

affect material properties, but not necessarily adversely.

• Point defects are zero dimensional imperfections that

involve only a few atoms at most. These include: vacancies

(Sec. 5.1.2) and impurities (Sec. 6.3.2).

• Linear defects are one-dimensional imperfections wherelocal faults in the atomic arrangement lie along a straight

line, curve or loop through the crystal. Linear defects are

collectively known as dislocations (Sec. 6.2).

• Planar defects are two-dimensional imperfections that

serve as boundaries between regions having different

crystal structures and/or crystallographic orientations.

Grain boundaries (Sec. 4.7) are one class of planar defects.

• Volume defects are macroscopic (large-scale) defects that

represent inhomogeneities in a solid. Among these are:

inclusions (unwanted foreign particles), precipitates (Sec.

6.3.3), cracks (Chp. 7) and voids.



5-3

5.1.1 Thermal Vibration of Atoms

• At temperatures above absolute zero (0 Kelvin), atoms

possess kinetic energy and vibrate about their equilibrium

lattice positions (see also Sec. 3.7).

• At any given temperature, the average kinetic energy of a

material is fixed; however, the kinetic energy of an

individual atom at any time is not constant but randomly

distributed over a wide range (Fig. 5.1-1).

Fig. 5.1-1 Distribution ofkinetic energy of atomschanging with temperature.N(E) represents the number

of atoms that possess Eamount of kinetic energy.

• Sometimes an atom in a solid may possess high enough

kinetic energy to enable it to break its bonds with its

neighbours and jump away from its original lattice position

to another site.

• The amount of energy required for breaking atomic bonds

and making a jump is known as the activation energy, Q

(Fig. 5.1-2).

5-4

Fig. 5.1-2 Illustrationshowing how an atom must

overcome an activationenergy, Q, to move fromone stable position to a

similar adjacent position.

• The number of atoms that possess enough kinetic energy

to overcome the activation energy barrier (i.e. E ! Q) and

make successful jumps may be found from the kinetic

energy distribution of atoms.

• The rate at which atoms make successful jumps must be

proportional to number of such atoms (with enough E ).

• This rate is described by the Arrhenius equation:

Rate = C exp

" QRT

#

$ %

&

' ( where C = material constant

Q = activation energy

R = gas constantT = absolute temperature

• The Arrhenius equation implies that a low activation

energy and/or a high temperature will result in faster rates.



5-5

5.1.2 Vacancies

• A vacancy is a vacant lattice site from which an atom is

missing (Fig. 5.1-3). Vacancies are defects that occur naturally

as a result of atomic vibrations; they may also beintroduced during processing, or arise from radiation

damage.

• There is an equilibrium number of vacancies, N v, at any

temperature, T :

N v = N exp

" QRT

#

$

%

&

'

( where

N = total no. of lattice sites

Qv = energy required to form avacancy (J/mol)

R = gas constant (J/mol.K)

T = absolute temperature (K)

Fig. 5.1-3 A vacancy.

• A crystal will always contain vacancies at any temperature

above absolute zero (0 K). The equilibrium number of

vacancies, N v, increases with temperature.

• Vacancies are the basis of an important mechanism for the

movement of atoms (diffusion) within metals and

ceramics.

5-6

5.2 D IFFUSION

• Diffusion is the spontaneous movement of atoms within a

material as a result of atomic vibrations. The manufacture

of many materials (e.g. surface hardening, microchips) andmost changes in microstructure (and therefore, properties)

are accomplished through diffusion.

• Self-diffusion is the constant, random, movement of

atoms of the same type within pure materials.

• It occurs in the absence of a concentration gradient; there

is no net flow of atoms.

• The effects on material properties are insignificant.

• Interdiffusion is the movement of atoms from one

material into another unlike material.

• It occurs in the presence of a concentration gradient; there

is a net flow of atoms from high to low concentration.

• Interdiffusion affects material properties.

• This type of diffusion is also known as impurity

diffusion

or heterogeneous

diffusion.



5-7

5.2.1 Vacancy Diffusion

• In vacancy diffusion, the diffusing atom moves from its

normal lattice position to an adjacent vacancy . (Fig. 5.2-1)

Fig. 5.2-1 Schematic representation of vacancy diffusion.

• Vacancy diffusion is the mechanism for self-diffusion, as

well as interdiffusion, when the diffusing atoms and host

atoms are of comparable size.

• Since vacancies must first be

available for atoms to move

into (Fig. 5.2-2), the rate of

diffusion is limited by the

number of vacancies.

Fig. 5.2-2 A mechanical analogy ofvacancy diffusion using a puzzle in a

frame, showing how a vacancy must firstexist before an atom can move into it.

5-8

5.2.2 Interstitial Diffusion

• In interstitial diffusion, the diffusing atom moves from

one interstitial site to an adjacent empty interstitial site. (Fig.

5.2-3)

• The interstitial diffusion mechanism is limited to small

solute atoms such as H, C, N, O, which are small enough

to squeeze into the interstitial sites between lattice atoms.

Fig. 5.2-3 Schematic representation of interstitial diffusion.

• No vacancies are required for interstitial diffusion to occur.

• At lower temperatures, interstitial diffusion is generally faster than vacancy diffusion, since interstitial sites are far

more abundant than vacancies. Furthermore, interstitial

atoms are smaller and more mobile.



5-9

5.2.3 Concentration and Flux

• When there is a difference in concentration (i.e.

composition), random atomic jumps will result in a net

flow of atoms from high to low concentration, until thediffusing atoms are uniformly distributed and the

concentration gradient is zero (Fig. 5.2-4).

Concentration profile Concentration profileof copper of nickel

Fig. 5.2-4 Interdiffusion of copper atoms into nickel and vice versa.

5-10

• The plot of concentration, C, against distance (along x, y

or z axis), is called the concentration

profile. The slope,

" C

" x , at a particular point on the concentration profile is the

concentration

gradient. These can change with time.

• Diffusion problems in solids often involve finding out how

fast diffusion occurs. This rate of mass transfer is measured

by the diffusion

flux, J , defined as the mass (or

equivalently, the number of atoms) passing through a unit

cross-sectional area of the solid per unit time (Fig. 5.2-5).

• The flux at any position along a diffusion path is related to

the instantaneous concentration gradient at that point:

J = -D

" C

" x

where D is the diffusion coefficient or diffusivity (m2/s)

Fig. 5.2-5 Illustration of diffusion flux. The units of J are kg/m2.s or atoms/m2.s



5-11

5.3 FACTORS AFFECTING RATE OF DIFFUSION

• The flux of atoms is proportional to the concentration

gradient, with the constant of proportionality being the

diffusion coefficient, D. The magnitude of D is indicative ofthe rate at which atoms diffuse.

• D is directly related to the frequency at which atoms jump

from site to site within a solid. Jumps can only occur when

atoms possess high enough kinetic energy to overcome

the activation energy barrier. D is thus described by the

Arrhenius equation:

D = Doexp

" QRT

#

$

%

&

'

(

where Do = material constant (m2/s)

Q = activation energy (J/mol)

R = gas constant (J/mol.K)

T = absolute temperature (K)

5.3.1 Temperature

With higher temperatures, the number of atoms withkinetic energy high enough to overcome the activation

energy barrier increases, so the jump frequency of the

atoms rises. Therefore, the higher T is, the larger D

becomes, and the faster the rate of diffusion (Fig.5.3-1).

5-12

Fig. 5.3-1 Arrhenius plot of logarithm of diffusion coefficientD as a function ofthe reciprocal of absolute temperature of some metals and ceramics.

A rise in T (i.e. lower 1

T

) increases D.



5-13

5.3. 2 Diffusion Mechanism

• The activation energy, Q, depends on the diffusion

mechanism (vacancy or interstitial). When Q is high, few

atoms possess the kinetic energy required to overcome Q

and make a jump, so D is low and diffusion is slow.

• In vacancy diffusion, a vacancy must first be created before

an atom can jump from an adjacent lattice site. Q then

consists of the energy required for vacancy formation plus

the energy required for an atom to jump.

• In interstitial diffusion, interstitial spaces are always

available, so Q is simply the energy an atom requires tojump into an adjacent interstitial site.

• In general, Qvacancy > Qinterstitial. (Fig. 5.3-2 & Table 5.3-1)

Fig. 5.3-2

Qvacancy > Qinterstitial.

5-14



5-15

5.3.3 Atomic Bonding

The activation energy, Q, also depends on the atomic

bond strength. Q is high for materials with strong atomic

bonds (reflected in the high melting points)(Table 5.3-2),

because an atom in such a material requires high kinetic

energy in order to break its bonds with its neighbours and

make a jump.

Table 5.3-2 Typical Activation Energies for Self-Diffusion, Q

5-16

5.3.4 Crystal Structure

The influence of atomic packing within the crystal structure

is reflected in Q and Do. It is more difficult for atoms to

squeeze through regions that are densely packed; in

general, Q is higher and/or Do is smaller in close-packed

structures (e.g. FCC), resulting in lower D and slower

diffusion. (Fig. 5.3-3 & Table 5.3-3)

Fig. 5.3-3 Arrhenius plots

showing approximate rangesof D for vacancy diffusion in

BCC and FCC alloys. T M is themelting point of the alloy.



5-17

5.3.5 Crystal Defects (Short Circuit Diffusion)

• Crystal defects such as dislocations (Sec. 6.2), grain boundaries

(Sec. 4.7) and surfaces provide open, disordered regions

through which atoms can move easily. Q for diffusion alongsuch short circuits is much lower than through the bulk of

the crystal (Fig. 5.3-4), so D is higher and diffusion faster.

• Since the cross-sectional areas of these easy diffusion paths

are usually small, volume diffusion is still dominant under

most temperatures. Only at low temperatures does short

circuit diffusion become significant.

Fig. 5.3-4 Arrhenius plots showing D for diffusion along various different paths. 5-18

• In nanocrystalline materials , which have a large proportion

of grain boundaries, grain boundary diffusion can

dominate. Similarly, in fine powders with large surface area,

surface diffusion can dominate; e.g. sintering of powders in

powder metallurgy or manufacture of ceramics (Fig. 5.3-5).

Fig. 5.3-5 Surface diffusion is significant in the sintering of powders.

• Short circuit diffusion in microelectronic devices is a

reliability issue due to the comparable dimensions of the

electronic circuits and crystal defects.



6-1

C H A P T E R 6

D I S L O C A T I O N S ,D E F O R M A T I O N A N D

S T R E N G T H E N I N G I NM E T A L S

6.1 PLASTIC DEFORMATION BY SLIP

6.2 DISLOCATIONS AND SLIP

6.3 STRENGTHENING MECHANISMS

6.3.1 Dis locat ion Stress F ie lds andStra in Energies

6.3 .2 Stra in Hardening

6.3.3 Gra in S ize Strengthening

6.3.4 Sol id Solut ion Strengthening

6.3.5 Dispers ion Hardening

6.3.6 Combined StrengtheningMechanisms

6.4 ANNEALING

6.4.1 Recovery

6.4 .2 Recrysta l l i zat ion

6.4.3 Gra in Growth



6-2

6.1 PLASTIC DEFORMATION BY SL IP

• Plastic deformation in a crystal mostly involves the sliding

of one plane of atoms over another under the action of a

shear stress (Fig. 6.1-1); this process is known as slip.

Fig. 6.1-1 Plastic deformation by slip in an ideal crystal occurs when one plane

of atoms slides over another, producing a step of one atomic spacing.

• The plane and direction in which slip occurs are the slip

plane and slip direction. The slip direction always lies

within the slip plane. The combination of a slip plane and a

slip direction forms a slip system.

• Slip does not take place in any arbitrary plane or direction.

The preferred slip planes and directions are those in which

the atoms are most densely packed. This is because slip

occurs in steps of one atomic spacing, so moving atoms

from one stable site to the next would involve the least

energy when the atoms are closest together (Figs. 6.1-2 & 3).

6-3

" max

#ba

Fig. 6.1-2 The maximum shear stress, ! max, required to move one plane of atomsover another by one atomic spacing is a function of the interatomic distances,

such that smaller stresses are necessary for closely-spaced atoms.

Fig. 6.1-3 Slip requires less energy on (a) a close-packed plane in a close-packed direction than on (b) a less closely-packed plane and direction.

• Since most engineering alloys are polycrystalline, the

change in orientation from grain to grain means that each

grain is strained differently by an applied stress (Fig. 6.1-4).



6-4

Fig. 6.1-4 Resolving a uniaxial tensile stress ! into shear stress " =F sin#

A/cos# = $ sin# cos#

• Slip will begin on a slip system when the resolved shear

stress acting on the slip plane in the slip direction reaches a

critical value. If two or more slip systems have the required

shear stress acting on them, they all slip together (Fig. 6.1-5).

Fig. 6.1-5 Slip lines on the surface of polycrystalline copper that has beendeformed. Slip lines are actually a series of fine steps on the surface.

Note that the slip lines change direction at grain boundaries.Note also intersecting sets of lines within the same grain,indicating the operation of more than one slip system.

6-5

• In a polycrystalline solid, the deformation in each grain

must be compatible with its neighbours to maintain

mechanical integrity and coherency along the grain

boundaries. This requires the grains of various orientations

to slip on 5 independent systems simultaneously.

• Metals with FCC and BCC structures are ductile because

they possess a relatively large number of slip systems (12 in

FCC; up to 48 in BCC) (Table 6.1-1).

• The slip systems in FCC and BCC are also well-distributed

in space, such that at least one slip system would be

favourably oriented for slip at low applied stresses.

Table 6.1-1 Primary slip systems in the common metal structures. BCC and HCP containsecondary slip systems, which are not shown.



6-6

• Furthermore, slip systems in FCC and BCC intersect, so if

one system be constrained, slip can continue on a different

intersecting slip system; this is known as cross-slip.

• However, unlike the FCC structure, the BCC structure does

not contain close-packed planes, so slipping atoms must

move greater distances from one equilibrium lattice

position to another. Higher shear stresses are thus

necessary for slip in BCC than in FCC metals (Fig. 6.1-2 & Table

6.1-2), which translates into higher strengths for BCC metals.

[Note: critical shear stress is the shear stress required to move a dislocation in its slip system.

• Although the HCP structure contains both close-packed

planes and directions, its geometry gives rise to fewer slip

systems. Furthermore, the slip systems, being parallel, do

not intersect, so cross-slip is not possible (Fig. 6.1-6). Most

polycrystalline HCP metals are relatively brittle.

6-7

Fig. 6.1-6 Comparison of the slip systems in(a) an FCC structure, and (b) an HCP s tructure.



6-8 6-9

6.2 D ISLOCATIONS AND SLIP

• In a perfect crystal, slip would involve a whole plane of

atoms sliding over another in a single movement, which

would require the simultaneous stretching, breaking, and

remaking of all atomic bonds in the slip plane. The

theoretical shear strengths of metals have been roughly

estimated to be in the order of 1010 N/m2 (10 GPa).

• However, the actual measured yield strengths of bulk

metals are at least 1,000–10,000 times lower than this

value (Table 6.2-1). This is because slip in real metal crystals

occurs via the movement of dislocations , during which only

a small fraction of atomic bonds are broken at any one

time, with minimal disruption to the crystal lattice.

Table 6.2-1 Comparison of theoretical and experimental yield strengths of some metals.

• Dislocations are linear or one-dimensional crystal defects

where local faults in the atomic arrangement lie along a

straight line, curve or loop through the crystal.



6-10

• Dislocations can be introduced into a crystal in a number

of ways: during solidification, during plastic deformation,

or as a result of thermal stresses arising from rapid cooling.

All bulk crystalline materials (metals and ceramics) contain

dislocations.

• There are two fundamental types of dislocations: edge and

screw. An edge dislocation may be thought of as an extra

half-plane of atoms inserted into the crystal (Fig. 6.2-1). The

bottom edge of half-plane that ends within crystal is the

edge dislocation line . [Note: the extra half-plane of atoms itself is not the

dislocation.]

Fig. 6.2-1 An edge dislocation, showing the extra half planes of atoms.Note the regions of compression and tension around the dislocation line.

6-11

• A screw dislocation may be thought of as making a cut

half-way through the crystal, and then skewing the two

halves by one atomic spacing (Fig.6.2-2).

Fig. 6.2-2 A screw dislocation, showing the spiral screw-likearrangement of atoms above and below the plane of the cut.

• Most dislocations, however, are mixed dislocations, whichcontain both edge and screw dislocation components with

a transition region in between (Fig. 6.2-3).

Fig. 6.2-3 A mixed dislocation. Fig. 6.2-4 Transmission electronmicrograph of a titanium alloy

in which dark lines are dislocations.



6-12

• When a shear stress is applied to the edge dislocation

shown in Fig. 6.2-5, the extra half-plane of atoms, plane A,

will be forced to the right; this in turn pushes the top

halves of planes B, C, D, etc., to the right.

• If the shear stress is high enough, plane A eventually

becomes closer to the bottom half of plane B than the top

half of plane B itself. It is then more favourable

energetically for the atomic bonds across the two halves of

plane B to be severed and for plane A to bond with the

bottom half of plane B.

• The extra half-plane moves by discrete steps through thecrystal and ultimately emerges from the surface, forming a

slip step that is one atomic distance wide (~10-10m).

Macroscopic plastic deformation is the cumulative effect of

the motion of large numbers of dislocations.

Fig. 6.2-5 The step-by-step movement of an edge dislocationunder a low shear stress produces a unit step of slip.

6-13

• Before and after the movement of a dislocation through a

region of the crystal, the atomic arrangement is perfect

and ordered; it is only during the passage of the dislocation

that the lattice structure is disrupted. Only a relatively small

shear stress is required to operate in the immediate vicinity

of the dislocation in order to produce a step-by-step shear.

Fig. 6.2-6 Heimlichthe caterpillar illustrating

(a) the difficulty ofmoving without (b) a

dislocation mechanism.

• Although the edge, screw and mixed dislocation move in

different directions, the result is the same shear (Fig. 6.2-7).

Partially sheared Totally sheared

Fig. 6.2-7 Shear produced by motion of (a) edge, (b) screw and (c) mixed dislocations.The dark arrows indicate the direction in which the dislocations move.



6-14

• While bulk ceramics and other crystalline compounds

contain dislocations, the shear stress required for

dislocation motion is at least 2-4 times that in metals. Not

only are covalent and ionic bonds stronger, but ions in the

more complex ceramic structures must also move greater

distances between equilibrium lattice positions.

• In addition, ceramics in which the bonding is

predominantly ionic contain very few slip systems. If slip

were to occur in some directions, ions of the same charge

would be brought close together (see also Sec. 3.8.4), generating

strong electrostatic repulsion that would resist slip (Fig. 6.2-8).

Fig. 6.2-8 (a) Before slip; (b) like charges repel in this slip direction; (c) slip possible.

• For ceramics with highly covalent bonding, the directional

nature of the bonds makes the displacement of atoms

from their lattice sites extremely difficult.

• The shear stress that must be applied to activate slip in

bulk ceramics is higher than that required to cause fracture

(Chp. 7). Ceramics are therefore hard and brittle, and do not

generally undergo plastic deformation by slip, except at

high temperatures (~ 0.5-0.7 T M [Note: T M is the melting temperature] ).

6-15

6.3 STRENGTHENING MECHANISMS IN METALS

• Because plastic deformation in metals corresponds to the

movement of large numbers of dislocations, the capacity

of a metal for plastic deformation depends on the ability ofdislocations to move.

• Since the hardness and strength of a metallic alloy are

related to the stress at which plastic deformation can be

made to occur (and thus, the stress at which dislocations

are able to move), there are two possible methods of

hardening or strengthening a metal:

! Eliminating all crystal defects, including dislocations –

this has only been achieved in “whiskers” (very thin

single crystals only a few µm in diameter), in which

strengths approaching theoretical values are possible.

! Creating so many crystal defects that they restrict or

hinder the passage of dislocations – this is the method

used to strengthen bulk metals, but yield strengths are

still much lower than theoretical levels.

• In the second approach, strengthening is achieved through

interactions of the stress fields of moving dislocations with

those created by other crystal defects.



6-16

6. 3. 1 Dislocation Stress Fields and Strain Energies

• Atoms surrounding a dislocation are displaced from their

equilibrium lattice positions. Such elastic strain produces

an elastic stress field around the dislocation.

• In an edge dislocation, the presence of the extra half plane

of atoms above the dislocation line means that atoms in its

vicinity are squeezed together, resulting in compressive

stresses. Conversely, the atoms below the dislocation line

experience tensile stresses due to an increase in

interatomic separation in this region (Fig. 6.3-1).

Fig. 6.3-1 (a) Regions of compression and tension located around an edge dislocation.(b) Detailed stress state of an edge dislocation showing

compressive, tensile and shear stresses.

• In a screw dislocation, the lattice spirals around the centre

of the dislocation. The stress field is one of pure shear and

is symmetrical about the dislocation line (Fig. 6.3-2).

6-17

Fig. 6.3-2 Shear stress and strain associated with a screw dislocation.

• The distortion of atomic bonds around any dislocation

increases potential energy because of non-equilibrium

interatomic separations (see also Sec. 3.7). This energy is known

as strain energy, since it is associated with the strain or

distortion of the crystal lattice.

• When a dislocation is in close proximity to another, the

stress fields surrounding each dislocation will interact. For

example, if the compressive and tensile stress fields of two

edge dislocations lie on the same sides of the slip plane (Fig.

6.3-3a), the overall strain energy will be raised if the two

fields overlap; this gives rise to mutual repulsion as the

dislocations approach each other.

• Conversely, if the compressive and tensile stress fields were

on opposite sides (Fig. 6.3-3b), the dislocations would

annihilate each other when they meet, with a lowering of

the overall strain energy; the dislocations would thus be

attracted to each other.



6-18

Fig. 6.3-3 (a) The interaction of two edge dislocations of the same sign causesrepulsion, (b) while that of different signs cause s attraction and annihilation.

C and T denote compressive and tensile regions, respectively.

• Two dislocations can attract and annihilate each other only

if they meet exactly on the same slip plane, and the

components of their stress fields match exactly and are of

opposite signs (Fig. 6.3-3b); i.e. tension cancels compression,

but tension/compression does not interact with shear.

• Since most dislocations are randomly curved mixed

dislocations, there is a only a very low probability that all

the conditions for dislocation annihilation will be fulfilled

simultaneously. Thus, dislocation interactions with one

another tend to be mutually repulsive.

6-19

• These repulsive interactions obstruct the motion of those

interacting segments of different dislocations, while non-

interacting segments continue to move, creating many

dislocation tangles (Figs. 6.3-4&5) during plastic deformation.

• Dislocations are therefore obstacles to the movement of

other dislocations.

Fig. 6.3-4 An edge dislocation (wavy black line) moving through a “forest” of otherdislocations (red verticle lines). Intersecting segments that are mutually obstructive

tangle with one another, distorting and lengthening the original dislocation.

Fig. 6.3-5 Tangling dislocations marked with a ‘b’.



6-20

6.3.2 Strain Hardening

• Strain hardening, or work hardening is the phenomenon

whereby a ductile metal becomes harder and stronger as it

is plastically deformed. It is also known as cold working

because the temperature at which deformation takes place

is “cold” relative to the melting temperature of the metal.

• During plastic deformation, dislocations move under the

action of a shear stress and encounter other dislocations.

Since their interactions are generally repulsive (Sec. 6.3.1), a

higher applied stress is necessary to overcome this mutual

repulsion such that dislocation movement can continue;

i.e. the metal has become stronger/harder.

• Furthermore, many new dislocations are continuously

created during plastic deformation (Fig. 6.3-6), significantly

increasing the dislocation density. The average distance

between dislocations decreases, and the mutual resistance

to motion becomes more pronounced, requiring an

increasingly higher applied stress for continued plastic

deformation; thus, the metal strengthens until fracture.

• Crystals that have intersecting slip systems, e.g. FCC and

BCC, often strain-harden rapidly because slip tends to

occur in more than one slip system, causing dislocations

on different systems to intersect, impeding mutual motion.

6-21

Fig. 6.3-6 The sequence of events for the multiplication of a dislocation from a Frank-

Read source. A segment of dislocation pinned at two points bows out into a loop.Continued stress will cause the loop to expand and the residual segment to bow out

again into another loop. This process repeats over and over, sending out a set ofconcentric loops away from the source, creating many new dislocations. This is

analogous to the ripples generated when a pebble is dropped into a quiet pond.

• The yield strength and tensile strength of a metal increases

with increasing cold work, but ductility decreases (Figs. 6.3-7 &

6.3-8). Physical properties such as thermal and electrical

conductivity are also reduced due to the scattering of

electrons and phonons by dislocations.

• Strain hardening in metals explains why the true stress-

strain curve obtained during a tensile test shows a rising

stress from the start of yielding to fracture (Sec. 2.2.5).



6-22

Fig. 6.3-7 Stress-strain diagram showing the effects of strain hardening.(a)

Initially, yielding beings at A; (b) upon unloading and re-loading, yielding now occurs at the higher stress B.

Fig. 6.3-8 The effects of cold work on the mechanical properties of copper.

• Metals may be shaped and strengthened at the same time

by cold working (Fig. 6.3-9).

6-23

Fig. 6.3-9 Common metalworking processes: (a) rolling, (b) forging(open and closed die), (c) extrusion (direct and indirect), (d) wire drawing, (e) stamping.



6-24

6.3.3 Grain Size Strengthening

• In a polycrystal, each grain is has a different orientation to

its neighbours. Since slip occurs only on specific planes,

dislocations cannot move from one grain into another in a

straight line (Fig. 6.3-10). Furthermore, the atomic disorder at

grain boundaries interrupts the continuity of the slip

planes, and acts as a barrier to dislocation motion.

Fig. 6.3-10 Slip planes arediscontinuous and changedirections across the grain

boundary. Dislocationscannot move through the

grain boundary.

• A dislocation can move only within the grain in which it

was created. Dislocations pile up at the grain boundary,

causing strain energy to increase locally, creating a back

stress that repels other dislocations approaching the pile-

up (Fig. 6.3-11). A higher applied stress is needed to overcome

this repulsion for continued dislocation movement.

Fig. 6.3-11 Dislocation pile-upat a grain boundary.

6-25

• The more grain boundaries there are (i.e. the smaller the

grain size), the more obstacles there are to dislocation

motion, and the higher the stress needed to cause plastic

deformation; i.e. the metal becomes stronger/harder.

• The relationship between yield strength and grain size is

expressed by the Hall-Petch equation:

! y = ! 0 +k y

d where ! y = yield strength

d = average grain diameter

! 0, k y = material constants

• Generally, polycrystals are stronger than single crystals (Fig.

6.3-12); fine-grained metals are stronger than coarse-grained

(Fig. 6.3-13). Effective strengthening can be realized only when

the grain size is of the order of 5 µm or less.

.

Fig. 6.3-12 Stress-strain curves for singlecrystal and polycrystalline copper.

Fig. 6.3-13 Hall-Petch plot for brass, showingthe effects of grain size on yield strength.



6-26

• Grain size strengthening is one of the major reasons for the

interest in nanocrystalline materials , in which grain sizes are

less than 100 nm. However, as grain size is reduced below

~ 20 nm and becomes comparable to the width of grain

boundaries, a reverse Hall-Petch effect is observed, where

decreasing grain size causes softening (Fig. 6.3-14).

Fig. 6.3-14 Hardness of a metal as a function of grain size.

• Grain size may be refined by cooling quickly from the

molten state, inoculation of the melt (i.e. adding numerous

impurity particles to the liquid to encourage solidification

on the particles), or by extensive plastic deformation followed by rapid annealing (Sec. 6.4). Other special

techniques are required to obtain grain sizes in the

nanometre range.

6-27

6.3.4 Solid Solution Strengthening

• All materials contain small amounts of foreign atoms

(element or compound). These impurities may arise

unintentionally from raw materials and processing, or may

be added intentionally to obtain specific properties.

• Impurities added intentionally are also known as alloying

elements in metals, additives in polymers and ceramics,

and dopants in semiconductors.

• Within a crystal, impurities (solute) may occupy interstitial

sites or substitute for atoms of the host material (solvent),

depending on the relative sizes of solute and solventatoms. The incorporation of solute atoms without altering

the crystal structure of the host results in a solid solution.

• Solute atoms distort the surrounding lattice and increase

the strain energy of the crystal (Fig. 6.3-15).

Fig. 6.3-15 Compressive strain imposed on host atoms by(a)

an interstitial solute atom, and (b) a large substitutional atom.(c) Tensile strain imposed by a small substitutional atom.



6-28

• The solute stress field could interact with that of an

approa-ching dislocation such that repulsion arises (Fig. 6.3-

16), similar to repulsion between dislocations of like signs

(Sec 6.3-1). A higher applied stress is needed to overcome this

repulsion.

Fig. 6.3-16 Repulsion betweencompressive stress fields of solute anddislocation.

• On the other hand, if an interstitial or large substitutional

solute atom with a compressive stress field were to be

located in the tensile region around a dislocation, lattice

strain is reduced (Fig. 6.3-17). A similar reduction in strain is

seen for a small substitutional atom with tensile strain field

located in the compressive region of a dislocation (Fig. 6.3-18).

• Once such a configuration of low strain are established

between a solute atom and dislocation, further movement

of the dislocation (i.e. away from the solute) would again

raise strain energy (Fig. 6.3-19). This increase in energy is met

by applying a higher stress; i.e. the metal strengthens.

6-29

Fig. 6.3-17 (a) Compressive strains imposed by a large substitutional solute atom.

(b)

Possible locations of large solute atoms relative to an edge dislocation,leading to a reduction in overall lattice strain.

Fig. 6.3-18 (a) Tensile strains imposed by a small substitutional solute atom.(b) Possible locations of small solute atoms relative to an edge dislocation, leading to a

reduction in overall lattice strain.

Fig. 6.3-19 Interaction with a suitable solute lowers dislocation strain energy.Continued plastic deformation requires the movement of dislocation away from the

solute, which returns the dislocation and solute to their states before interaction. A higher stress is needed to restore the original strain energies of dislocation and solute.



6-30

• Higher stresses are thus required for dislocation movement

in the presence of solute atoms, which act as obstacles.

The more solute added (without exceeding the solubility

limit), the greater the strengthening (Fig. 6.3-20). Metals are

seldom used pure, but are usually alloyed for strength.

• The degree of solid solution strengthening depends on the

relative sizes of the solute and solvent atoms. The larger

the size difference, the greater the distortion of the

surrounding lattice, and the stronger the strengthening

effect (Fig. 6.3-20). Too large a size difference, however, would

lower solute solubility in the host lattice (Sec. 9.3.1).

• The strengthening effect further depends whether the

solute is substitutional or interstitial, and the crystal

structure of the solvent. The stress field of a substitutional

solute atom in close-packed FCC or HCP crystals is

spherically symmetric, without any shear component, and

as such, does not interact with the shear stress fields of

screw dislocations. Conversely, interstitial solute atoms in

non-close-packed crystals such as BCC cause a non-

symmetric tetragonal distortion, generating a stress field

that can interact with both edge and screw (and thus,

mixed) dislocations.

6-31

• Only the very small non-metallic atoms, such as H, B, C, N

and O, tend to dissolve interstitially in metals (Sec. 9.3.1).

Fig. 6.3-20 The effects of several substitutional alloying elementson the yield strength of copper.



6-32

6.3.5 Dispersion Hardening

• Small, hard particles of a second phase dispersed in a

softer, ductile matrix are effective obstacles to dislocation

motion, and lead to significant strengthening. Such

particles may be introduced intentionally, or arise naturally

from precipitation reactions in an alloy, the latter

producing a precipitation or age hardening effect.

• The interaction between dislocation and dispersed particle

depends on the nature of the particle-matrix boundary. For

particles that are intentionally incorporated, and for many

precipitates, the particle-matrix interface is non-coherent

and disordered (Fig. 6.3-21a); i.e. there is no atomic matching

between the crystal lattice of the particle and matrix. Such

particles do not distort the surrounding lattice.

Fig. 6.3-21 (a) A particle that has no relationship with the crystal structure of thesurrounding matrix forms a non-coherent interface with the matrix.

(b) When there is a definite relationship between the crystal structures of the precipatateand matrix, a coherent or semi-coherent interface exists.

6-33

• At a non-coherent interface between particle and matrix

(Fig. 6.3-21a), there is a discontinuity of slip planes, much like

that at grain boundaries (Sec. 6.3.3). A dislocation would be

unable to move through such a particle.

• The dislocation may be forced to keep on moving by

extruding or bowing between the particles (Fig. 6.3-22). Since

a curve between two points is longer than a straight line,

the bowed dislocation introduces greater lattice distortion

and higher strain energy than the original, straight

dislocation. A larger shear stress must now be applied to

cause such bowing and continued plastic deformation.

Fig. 6.3-22 A view looking down on a slip plane showing the bowing of a dislocationpast particles having a non-coherent interface with the matrix.

[Note that the circles represent particles, not single atoms.]

• After a dislocation has bowed past, dislocation loops are

left around the particles (Fig. 6.3-22). The stress fields of these

loops would interact with subsequent dislocations, and

add resistance to their motion, leading to further

strengthening.



6-34

• Fine particles that are precipitated from an alloy often have

planes of atoms in their crystal structures that are related

to, or even continuous with, planes in the matrix lattice;

such precipitate-matrix interfaces are said to be coherent

(Fig. 6.3-21b).

• Since a coherent precipitate does not usually share the

same lattice parameters as the matrix, this results in lattice

strain. The stress field thus generated would interact with

passing dislocations in a manner analogous to that of solid

solution strengthening (Sec. 6.3.4).

• Because the stress field generated by a coherent precipitate

is relatively wide, interactions with dislocations would

occur wherever the stress fields impinge upon one

another. The precipitate does not need to be on the slip

plane of a dislocation to have a strengthening effect.

• When a coherent precipitate lies directly in the path of a

dislocation, the coherency at the interface would allow the

slip plane of the dislocation to pass from the matrix into

the precipitate and shear the precipitate (Fig. 6.3-23).

However, the creation of new matrix-precipitate surface

area after shearing raises interfacial energy. For such

shearing to occur, a higher stress must be applied to fund

the increase in energy, thus strengthening the alloy.

6-35

Fig. 6.3-23 The formation of new precipitate-matrix interfaces when adislocation cuts through a coherent precipitate.

• The degree of strengthening depends on the number and

distribution of dispersed particles and precipitates: these

should be as numerous as possible and uniformly

distributed, so that they are closely spaced.

• Dispersion hardening is the principle behind metal matrix

composites (MMCs), in which an alloy is strengthening by a

dispersion of fine, hard particles, usually of a ceramic

material. Ceramics retain their shape, distribution and

superior hardness when heated, and are more effective at

strengthening an alloy at high temperatures than

precipitates, which tend to agglomerate (and hence,

reduce in number) and re-dissolve into the matrix.



6-36

6.3.6 Combined Strengthening Mechanisms

Two or more strengthening mechanisms may operate

simultaneously to improve strength and hardness in metals

(Tables 6.3-1&2 & Fig. 6.3-24).

Table 6.3-1 The effectiveness of the various strengthening mechanisms on copper.

6-37

Fig. 6.3-24 Strengthening mechanisms in copper alloys and the variation in properties.

Table 6.3-2 Metal alloys with typical applications, and the strengthening mechanisms used.

Alloy Typical uses Solutionhardening

Precipitationhardening

Workhardening

Pure Al Kitchen foil !!!

Pure Cu Wire !!!

Cast Al, Mg Automotive parts !!! !

Bronze (Cu-Sn), Brass (Cu-Zn) Marine components !!! ! !!

Non-heat-treatable wrought Al Ships, cans, structures !!! !!!

Heat-treatable wrought Al Aircraft, structure s ! !!! !

Low-carbon steels Car bodies, structures,ships, cans

!!! !!!

Low alloy steels Automotive parts, tools ! !!! !

Stainless steels Pressure vessels !!! ! !!!

Cast Ni alloys Jet engine turbines !!! !!!

Symbols: !!!= Routinely used. != Sometimes used.



6-38 6-39

6.4 ANNEALING

• There is a limit to which metals may be plastically

deformed, beyond which fracture occurs.

• During forming operations, it is sometimes necessary torestore the ductility of work-hardened metals to their state

prior to deformation, in order to carry out further plastic

deformation.

• Work hardening also lowers the thermal and electrical

conductivity of metals, which might require restoring; e.g.

copper electrical wires.

• The effects of work hardening can be removed by heating

the metal to a sufficiently high temperature in a process

called annealing. Annealing replaces the highly distorted

work-hardened grains with new, equiaxed grains

containing few dislocations.

• The driving force for annealing is the reduction of strain

energy associated with the high density of dislocations in a

severely work-hardened metal.

• In annealing, there are three temperature ranges (from low

to high) in which different phenomena occur: recovery,

recrystallization and grain growth.



6-40

6.4.1 Recovery

• When heated sufficiently, dislocations in a strain hardened

metal rearrange themselves into configurations with lower

strain energy, forming the cell boundaries of a subgrain

structure within the old grains (Fig. 6.4-1c), in a process called

polygonization.

• Dislocation density is lowered slightly through mutual

annihilation, but because the reduction is not significant,

hardness, strength and ductility are almost unchanged (Fig.

6.4-2). However, thermal and electrical conductivity are

restored close to their pre-cold-worked states.

6.4.2 Recrystall ization

• After recovery is complete, the strain energy of the crystal

is still relatively high due to the large number of

dislocations remaining. If the temperature were raised

further, recrystallization will follow.

• New, small, dislocation-free grains nucleate at the high-

energy cell boundaries of the polygonized subgrain

structure (Fig. 6.4-1d), eliminating most of the dislocations as

they grow and replace the strain hardened grains (Fig. 6.4-1e).

6-41

Fig. 6.4-1 Microstructural changes in cold working and annealing:(a) original state with few dislocations; (b) high density of dislocations after

cold working; (c) recovery; (d) recrystallization; (e) fully recrystallizedstructure of new relatively strain-free grains with few dislocations.

• Since recrystallized grains are relatively free of dislocations,

the hardness, strength and ductility of the metal are

restored to their pre-cold-worked values; i.e. hardness and

strength decrease while ductility increases (Fig. 6.4-2).

• The low strength and high ductility of a recrystallized

metal are exploited in hot working, in which plasticdeformation of a metallic alloy is carried out at

temperatures above its recrystallization temparature

(usually above 0.6 T M ). The continually recrystallizing metal

allows extensive deformation without strain hardening.



6-42

• The main disadvantage of hot working is the poor surface

finish as a result of oxidation of the metal surface at high

temperatures. Dimensional accuracy is also an issue due to

the elastic recovery (springback) and thermal contraction

that occur when the component is cooled subsequently.

Fig. 6.4-2 The effects of annealing temperatureon mechanical properties and grain size.

6-43

6.4.3 Grain Growth

• If heating were to continue after complete recrystallization

has occurred, the new grains will grow in size. [Note that grain

growth occurs in all polycrystalline materials at sufficiently high temperatures; it is not related

to cold-working. Only in cold-worked metals do recovery and recrystallization take place

before grain growth.]

• The driving force for grain growth is the reduction of the

interfacial energy associated with the atomic disorder at

grain boundaries (Sec. 4.7). Grain growth results in fewer

grains, thereby decreasing the total area of grain

boundaries and lowering the interfacial energy.

• Grain growth involves the diffusion of atoms across thegrain boundary from one grain to another, such that some

grains grow at the expense of others (Fig. 6.4-3).

Fig. 6.4-3 Grain growth occurs as atoms diffuse across the brain boundary from one grain to another.

• Grain growth reduces the strength and hardness of

metallic alloys (Sec. 6.3.2), because the number of grain

boundaries, which are barriers to dislocation motion, are

now fewer.

material's properties under microscopic

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