Website Info

1
Website Info

• Go to http//mse.stanford.edu/faculty/nixres.html
• Click on Julia Rosolovsky Greer
• Click on Dislocations Course 2005
• Get the notes!

2
Questions from Last Lecture
gt Are there real knots where 4, 5, 6, or even
more dislocations meet? gt Are there really
dislocations with all kinds of translation
vectors, e.g. b alt100gt or b alt123gt? gt Is
the geometry of a network arbitrary, i.e. are the
angles between dislocations in a knot arbitrary?
gt Are real dislocations really arbitrarily
curved? gt Do dislocations repel or attract each
other? gt How do they interact with other
defects including point defects, other
dislocations, grain boundaries, precipitates and
so on?
Have to consider the elastic energy of a
dislocation!
3
Basics of Elasticity Theory
displacement field u(x,y,z) u is a vector that
defines the displacement of atoms or displacement
of any point P in a strained body from its
original (unstrained) position to the position P'
in the strained state.
SHEAR STRAINS
NORMAL STRAINS
4
Sometimes Polar Coordinates Work Best
5
Displacements of a Screw Dislocation
In the z-direction, the displacement varies
smoothly from 0 to b as the angle q goes from 0
to 2p
6
Stress and Strain Fields of Screw Dislocations
Pure shear, no tensile or compressive
components Buta serious problem with these
equations
stresses and strains are to 1/r and therefore
diverge to 8 as r ?0
CORE RADIUS, r0 small strains gt equate the core
region with the region were the strain is gt 10.
Reasonable value for the dislocation core radius
r0 lies in the range b to 4b r01nm
7
Stress Fields of EDGE Dislocations

• The stress field has both dilational and shear
  components
• The largest normal stress is sxx which acts
  parallel to the Burgers vector
• Since the slip plane can be defined as y 0,
  maximum compressive stress (- sxx) acts
  immediately above the slip plane and the maximum
  tensile stress ( sxx) acts immediately below
  the slip plane

8
Illustration of Stress Fields
9
Strain Energy of Dislocations
Strain energy Eel per unit volume  
The line energy of an edge dislocation is always
larger that that of a screw dislocation since (1
n) lt 1. If n 1/3, Escrew 2/3 Eedge.
10
Dislocation Energy Properties
1. The total energy U of a dislocation is
proportional to its length L.
2. A dislocation tends to be straight between its
two "end points," (usually dislocation knots).
3. The line energy of an edge dislocation is
always larger that that of a screw dislocation
gt a dislocation tends to have as large a screw
component as possible.
4. The elastic energy depends (logarithmically)
on the crystal size and the core radius, r0.
5. The energy is a weak function of the crystal
(or grain size) R.
For r0 0.1 nm and R 10nm 100 mm gt
ln(R/r0) 13.8 - 4.6
Good approximation r0 b
11
Dislocation Energy Properties, contd
6. Dislocations always tend to have the smallest
possible Burgers vector
7. If bi larger than the smallest translation
vector gt bi b1 b2 where b1,2 some shorter
vectors of the lattice. Splitting into smaller
Burgers vectors is always energetically
favorable!
8. The line energy 5 eV (per b), which makes
dislocations non-equilibrium defects. They will
not come into being out of nothing like point
defects for free enthalpy reasons.
9. The line energy ( energy per length) has the
same dimension as a a force, it expresses a line
tension, i.e. a force in the direction of the
line vector which tries to shorten the
dislocation.
12
Forces on Dislocations

• Only the shear stress on this plane will make
  the dislocation glide
• normal components of the the stress in the glide
  plane system act perpendicular to the glide plane
  and will only contribute to the dislocation climb
  (at higher temperatures)
• resolved shear stress, tresolved, points in the
  direction of the Burgers vector
• Peach-Koehler Formula

13
Gliding Dislocations
IMPORTANT The direction of the force component
acting on the dislocation is always perpendicular
to the line direction!
14
Work-Force Relationship
Total work W    A t b
dW the incremental work done on an incremental
area that consists of an incremental piece dl of
the dislocation moving an incremental distance ds
Define the force per unit length dl on a
dislocation to be F dW/dl ds
F (magnitude) tb
Lets do an example of a force calculation
15
Line Tension
In reality, dislocations can rarely move in
total because they are usually firmly anchored
somewhere
A mechanical equilibrium is established as soon
as the force pulling the dislocation back (its
own line tension) cancels the applied external
force.
16
Line Tension, contd
Energy increment dE T dl (G b2dq) dl
Work done dW F dl (tbRdq) dl
At equilibrium dW dE G b2 dq tbRdq
17
Line Tension Consequences

• gt The sum of the line tensions at a knot must be
  zero
• (otherwise the knot and the dislocations with it
  will move)
• Therefore, 3-knots will always show angles of
  120o.
• Knots with more than 3 dislocations will split
  into 3-knots, otherwise there can be no easy
  balance of line tensions.

18
Observation of Dislocations
When a dislocation line intersects the surface of
a metallic material, the associated strain field
locally increases the relative susceptibility of
the material to acidic etching and an etch pit of
regular geometrical format results. If the
material is strained (deformed) and repeatedly
re-etched, a series of etch pits can be produced
which effectively trace the movement of the
dislocation in question. Transmission electron
microscopy can be used to observe dislocations
within the microstructure of the material. Thin
foils of metallic samples are prepared to render
them transparent to the electron beam of the
microscope. The electron beam suffers diffraction
by the regular crystal lattice planes of the
metal atoms and the differing relative angles
between the beam and the lattice planes of each
grain in the metal's microstructure result in
image contrast (between grains of diffent
crystallographic orientation). The less regular
atomic structures of the grain boundaries and in
the strain fields around dislocation lines have
different diffractive                           
                                                  
                                                  
                                       Transmis
sion Electron Micrograph of Dislocations propertie
s than the regular lattice within the grains, and
therefore present different contrast effects in
the electron micrographs. (The dislocations are
seen as dark lines in the lighter, central region
of the micrographs on the right). Transmission
electron micrographs of dislocations typically
utilise magnifications of 50,000 to 300,000 times
(though the equipment itself offers a wider range
of magnifications than this). Some microscopes
also permit the in-situ heating and/or
deformation of samples, thereby permitting the
direct observaion of dislocation movement and
their interractions. Field ion microscopy and
atom probe techniques offer methods of producing
much higher magnifications (typically 3 million
times and above) and permit the observation of
dislocations at an atomic level. (By contrast,
traditional optical microscopy, which is not
appropriate for the observation of dislocations,
typically offers magnifications up to a maximum
of only around 2000 times).
19
Etch pits
Impurity decoration
20
Observing Dislocations TEM

• Dislocations are visible in TEM by diffracting
  electrons away from the image
• (fast electrons used in TEM have the wavelength
  in the X-ray region and therefore can diffract)

21
Electron micrograph of dislocation loops formed
by aggregation and collapse of vacancies in Al
5 Mg. The helical dislocations are formed by
vacancy condensation on a screw dislocation.
43,000 x. C. Kittel, Introduction to Solid State
Physics
A transmission electron micrograph of a titanium
alloy in which the dark lines are dislocations.
51,450 x. W.D. Callister, Materials Science and
Engineering An Introduction
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Scanning Tunneling Microscopy