Physical metallurgy principles

Chap. 6 Elements of Grain Boundaries

Three related quantities to define a interface.

Surface tension: Work required to form (or create) a unit area of new surface.

Surface free energy: The change in the free energy for the system per unit area of interface generated.

Surface stress: Work required deform (or stretch) a surface.

For pure metal, under some conditions these three quantities are equal.

Grain Boundary

• Surface tension:

* creating new surface by adding

additional atoms to the surface; the

work required to deform the solid

surface is a measure of the surface

stress.

* liquids to form the low energy state of

minimum surface area.

* On the surface each atom is only partly

surrounded by other atoms. On

bringing an atom from the interior to

the surface, bonds must be broken or

distorted and consequently there is an

increase in energy.

* defined as this increase in free energy

per unit area of new surface formed.

The difference in surroundings of

a surface atom and interior atom

Surface tension in a two-component

system

Effects of Impurities

There is a strong tendency for the distribution of materials to be such that the minimum surface energy results.

If a small amount of a low-surface-tension component is added, it tends to concentrate in the surface layer so that the surface energy is sharply decreased with but small additions.

If a high-surface-tension component is added to one of lower surface energy, it tends to be less concentrated in the surface layer than in the bulk and only a slight influence on the surface tension.

Ex., oxygen and sulfur can decrease the surface tension of liquid iron from 1835 dynes/cm to 1200dynes/cm with additions as small as 0.05%.

The excess free energy of a high-surface-area material is sufficient to provide the driving force for several processes. It is the driving force for the sintering of a powdered compact into a dense product.

Pressure due to curved surfaces :

• Many of the important effects of

surfaces and interfaces arise from

the fact that surface energy causes

a pressure difference across a

curved surface.

• The work of expanding a spherical

surface DPdv must equal the

increase in surface energy gdA. For

a sphere, dv and dA are given by

rdrdA

drrdv

dAPdv

g

8

4 2

D

the pressure exerted by the spherical

surface is

D

rdrr

rdr

dv

dAP

2

4

82

g

gg

In general when the surface is not spherical,

similar analysis gives

D

21

11

rrP g

Where r1 and r2 are the principal radii of curvature

*This fine particle size produces surface-energy forces which cause densification during

the firing process.

Tm

Grain Boundaries• The width of a grain boundary is

very small. By etched in an acid

solution in order to reveal their

presence.

• The grain boundaries play an

important part in determining the

properties of a metal.

• At low temperature the grain

boundaries are quite strong and do

not weaken metals. In facts, heavily

strained pure metals, and most

alloys, fail at low temperatures by

cracks that pass through the

crystals and not the boundaries.

• At high temperatures and slow

strain rates, the grain boundaries

lose their strength more rapidly

than do the crystals, with the result

that fractures no longer traverse the

crystals but run along the grain

boundaries.

A polycrystalline zirconium specimen photographed

with polarized light. In this photograph, individual

crystals can be distinguished by a difference in shading

, as well as by the thin dark lines representing grain

boundaries.

Small-angle grain boundary

• Greater the angular rotation greater the inclination of the planes closer the spacing of dislocations in the vertical boundary.

• If the orientation mismatch between two crystals is quite small the boundary between the crystals is called a small-angle boundary.

d:is the spacing between the dislocations

b:is the Burgers vector of a dislocation in the

boundary

q: is the degree of orientation mismatch

If the angle of rotation of the crystal structure

across the boundary is assumed to be small,

then sinq/2 may be replaced by q/2.

d

b

22sin

q

dbq

D: dislocation spacing

1. The small angle formula

gives the proper form of EB

versus curve all the way up

to 15~20o.

2. The energy of large-angle

GBs remain constant

around 500~600ergs/cm2.

3. In polycrystalline metals

over 90% of all the GBs are

high-angle boundaries.

A low-angle boundary in a copper-13.2 atomic percent

aluminum specimen deformed 0.7 percent.

1. The orientation of one lattice w.r.t. the other, q.

2. The orientation of the boundary w.r.t. a lattice, .

2 degree of

freedom

boundary.

Five degrees of freedom of a grain boundary

The general grain boundary has five degrees of freedom; three degrees specify the orientation of one grain relative to the other and two degrees specify the orientation of the boundary relative to one of the grains.

Stress field of a grain boundary

• The shear stress on the slip plane is :

))/((sinh)1(2

))((

))((

)1(2

extent. infinitean

have oboundary t theassume and boundary,tilt

thefrom x distance aat stressshear totalThe

)4(

)4(

)1(2

2 and if ex.

ratio sPoisson':

modulusshear the:

plane slip on the stressshear the:

)(

)(

)1(2

ndislocatioboundary grain single afor

22

222

22

222

22

222

22

dxd

xb

ndx

ndxxb

dx

dxxb

n-ndy

yx

yxxb

xy

n

n

xy

xy

xy

xy

Ex. Iron metal with =86GPa, v=0.3 and

b=0.248nm.

•The shear stress due to the boundary falls very

rapidly with increasing x.

•The boundary stress approaches that of the single

dislocation as x becomes very small.

•While the shear stress due to a single dislocation

exceeds the critical resolved shear stress(CRSS)

at all values of x in this diagram, note that the

boundary stress equals the CRSS at a distance of

only about 25b.

•Further note that the boundary stress is negligible

when x is greater than approximately 50b.

* A compressive stress exists above each

dislocation and a tensile stress below each

dislocation. Alternating tensile and compressive

regions exist along the boundary and these stresses

will tend to cancel each other.

* Consequently, except at distances very close to

the boundaries, the interiors of grains or subgrains

are free of long-range stresses due to their

boundaries, that is the boundaries do not possess

long-range stress fields.

Grain-boundary energy

• A strain energy is associated with a

dislocation when it is in either its screw

or edge orientation because the atoms of

the crystal around a dislocation are

displaced from their normal equilibrium

positions.

• Now suppose that there is a positive and

negative pair of edge dislocations on the

same slip plane. If a tilt angle smaller

than a few degrees, the energy per unit

area of the boundary gb:

)1ln2/(ln)1(4

qq

g

bb

gb: the energy per unit area of the boundary

: the shear modulus

b: the Burgers vector

q: the tilt angle of the boundary

: a factor accounting for the dislocation

core energy

v: Poisson’s ratio

1.The small-angle formula give the proper

form of the gb versus q curve all the way

up to 15-20o.

2.The energies of large-angle grain boundaries

are approximately constant at around 500-600

ergs/cm2.

3.In polycrystalline metals over 90% of all the

grain boundaries are high-angle boundaries

because the probability that all three

orientation are low is very small. The grain

boundary energy in polycrystalline metals as

constant around 500-600 ergs/cm2.

Low-energy dislocation structures LEDS

(ordinal number of dislocation) stress contribution of dislocation ( )

(ordinal number of dislocation) stress contribution of dislocation (-)

: the energy of a boundary dislocation/the energy gb

e

n

n

w

w

per unit length of

a random dislocation

Taylor LEDS lattice

* Due to the mutual screening action of the stress

fields of the various dislocations, there is a

significant decrease in the total strain energy

associated with the dislocations.

* Plastic deformation tendency to form cells

with a low internal dislocation density and

boundaries between the cells composed of

dislocation tangles.

* Driving force for the formation of this structure

is the strain energy decrease associated with the

formation of the tangles.

* Increased plastic deformation increased

dislocation density cell size decrease cells

number increases.

* Empirical relationship between the cell size

and the dislocation density:

densityn dislocatio:

constant:

diameter cell average the:

/

9 percent strain

26 percent strain

High temperature recovery: the movement of the dislocations resulting from plastic deformation into subgrain or grain boundaries.

Dynamic recovery: the movement of the dislocationscan actually start during plastic deformation. The applied stress causing the deformation is added to the stresses acting between the dislocations.

Dynamic recovery tends to lower the work hardening

rate.

May observed at very low temperatures, and at these temperatures the applied stresses can be very large.

Dynamic recovery occurs most strongly in metals of high stacking fault energy.

Static recovery: the movement of the dislocations into the cell walls occurs as a result of the interaction stresses.

Dynamic recovery

Alloying normally reduces the stacking-fault energy of a metal.

(A) Pure nickel strained 3.1% at 293K

(B) Nickel – 5.5wt% Aluminum alloy strained 2.7% at 293K

(A) (B)

Surface tension of the grain boundary

Unit of surface energy: ergs/cm2 or J/m2

• The grain-boundary surface tension is an increasing function of the angle of mismatch between grains to an angle of approximately 20o, and then it is essentially constant for all larger angles.

• If these three force vectors are in static equilibrium, the relationship as below:

cba

cba

sinsinsin

ggg

a,b,c are the dihedral angles between boundaries.

Crystal boundaries are regions of misfit or disorder

between crystals, it is to be expected that atom

movements across and along boundaries should

occur quite easily. The boundary is caused to move

by the simple process whereby atoms leave one

crystal and join another crystal on the other side of

a boundary.

The speed with which crystal boundaries move

depends on a number of factors:

1.Temperature:atom move to another site from

thermal vibrations.

2.reducing its grain-boundary area: A.boundary may

move to straighten out sharply curved region. B.

grain growth– some crystals disappear, while

others grow in size.

*If a metal is heated at a sufficiently high temperature

for a long enough time, the equilibrium relationship

between the surface tensions and the dihedral angles

can actually be observed.

*A well-annealed pure metal one that has been heated

for a long time at a high temperature, the grain-

boundary intersections form angles very close to 120o.

Energy change for atomic jump

Grain growth

Torque terms

If isotropic ⇒ torque terms = 0

⇒ surface tension balance

⇒

Neglect the torque terms Surface tension balance

Boundaries between crystals of

different phases

• In alloys of two phases, two types of

boundaries are possible:boundaries

separating crystals of the same phase, and

boundaries separating crystals of the two

phases. If the surface tensions in the

boundaries are in static equilibrium, then

2cos2 1211

qgg

2cos2

1

11

12

qg

g

As the surface tension of the boundary

between two phases approaches half of that

of the single phase, the dihedral angle falls

rapidly to zero.

dihedral angle=1o dihedral angle=10o

Ex. The surface tension of a bismuth-copper interface

is so low that the dihedral angle is zero, bismuth forms

a continuous film around copper crystals the copper

loses its ductility, even though the total amount of the

bismuth impurity is less than 0.05 percent.

Ex. Iron containing small quantities of sulfur as an

impurity. FeS is liquid below the freezing point of iron.

By incorporating in the steel a small amount of manganese

to combine with the sulfur in a steel to form globules,

which are solid at the rolling temperature of steel.

The shape of a second phase for three different

dihedral angles at a grain boundary and a grain

edge.

-

Grain size

• Linear intercept method:

l :mean grain intercept

lN :the average number of grain boundaries

intercepted per centimeter

lNl

1

The reciprocal of mean grain intercept is

directed related to the amount of grain-

boundary surface area in a unit volume.

lv NS 2

vS :the surface area of the grain boundaries

per unit volume

The effect of grain boundaries

on mechanical properties

• The smaller the grain size the greater

the hardness or flow-stress.

H: is the hardness

d: is the average grain diameter

kH: is the slope of the straight line drawn

through the data

Ho: is the intercept of the line with the

ordinate axis

• In the fine-grained materials a much

larger applied stress is needed to cause

slip to pass through the boundary than

is the case with coarse-grained

materials.

21 dkHH Ho

ns)dislocatio slip toobstacles asact boundary grain (

stress flow

density n dislocatiostrain

Coincidence site boundaries

• Coincidence sites:rotations could produce

surfaces, separating the new crystal from the

old, that contained a number of positions

where the atoms in both crystals were in

coincidence.

• A:fcc crystal rotation a <111> axis

• B: a secondarily recrystallized crystal are

rotated A by 22o about the [111] pole.

• The density of coincidence sites:

The fraction of atoms in coincidence, at a

boundary of this type. It is the reciprocal of

the density. Ex. S7

• Four basic factors:

• 1.the rotation axis [hkl]

• 2.rotation angle q

• 3.the coordinates of a coincidence site in the

coincident site net on (hkl)

• 4.S is the reciprocal of the density of

coincidence sites

222

22

211 )))(/(tan2(

lkhN

Nyx

Nxy

S

q

(100) plane

N= 1

q= 53.1o

S= 5

S can only take odd values. If S is even it should be divided by multiples

of 2 until an odd number is attained.

Twist boundaries

(100) plane

N= 1

q= 36.9o

S= 10

{111} plane ; x axis:[011]; y axis:[211]

N = 3

S = 84/12= 7

q = 21.8o

Tilt boundaries

p:structural periodicity

As suggested by Aust, the overlapping of the

atoms at the boundary could be relieved by a

relative translation of the lattices above and

below the boundary.

Relaxed coincidence boundary:

such a boundary is still considered

to have a structural periodicity

equivalent to that of the boundary

where the atoms of the two halves

are shared at the coincidence sites.

It should also have a lower grain

boundary energy.