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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
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.