physical metallurgy 11 th lecture ms&e 410 d.ast [email protected] 255 4140
TRANSCRIPT
Review
• New concepts learned . => Screw dislocations need to constrict to cross-slip . => Dislocations intersecting each other generate “kink’ and . “jogs” the latter frequently are sessile, “pinning” the . . . dislocation
As you have find out via your homework, the stacking fault energy has a large influence on fatigue resistance of metals. Self test: Why ?
Review
• In metals, b has preferred directions, the line direction less so
• In d.c. Si, Ge, GaAs etc the preferred direction of both Burgers vector and line direction is <110> thus you have edge, screw and 600
Dislocations from a higher point of view
Dislocations violate the incompatibility equation of continuum mechanics
Electrodynamics Theory Continuum Dislocation Theory
A very elegant theory, if we would know how to take the divergence of a tensor field. The operator to do so, is the “inc” operator, for “incompatibility”
The view that dislocations are vortex lines in tensor strain fields,
The theory, developed by E. Kroener, was the starting point of modern mechanics of materials on the material manifold.
Figure 1 : High resolution image of a quasi-periodical grain boundary in gold. The plane of the boundary is parallel to (110) in the upper crystal and to (100) in the lower one (x axis) (JEOL 3010).
Grain boundaries are periodic. Above is a low angle twist boundary in Au (diffusion bonded, Balluffi and Schober)
A five fold grain boundary arrangement in a small Pb particle in silica (amorphous SiO2).
The boundaries are first order twin boundaries with a very low energy (otherwise you could not have so many boundaries in such a small particle)
A =19 Grain Boundary in Al
The grain boundary is periodic.
In this case, if you would extend the left crystal, in 3 dimensions (!), into the right one, one out of 19 atoms would exactly wind up on the same place.
Small angle grain boundaries ( < ~ 16o )
• tilt
• twist
• mixed
Calculating the grain boundary energy, by adding up the energy of the dislocation works well to about 16o
Self test:
What is this material good for ?
Twist
(Made by putting to single crystal foils on top of each other and rotating top relative to bottom)
Low angle twist boundary in Si.
Network of pure screw dislocations
(The funny lines are due to the fact that the boundary plane is not perfectly plane but contains atomic steps)
(
(Foell and Ast)
Note
Five (5) degrees of freedom
Left object
Unit direction (e.g. hexagonal axis)
Right object
Vector = 3 coordinates
3 degrees of freedom
Constraint must be unit vector, x2+y2+z2 = 1, -1 DOF
Rotation around vector = + 1
Boundary plane =
Pseudo vector = 3 coordinates
Constraint must be unit vector, x2+y2+z2 = 1, -1 DOF
Boundary plane
Total Degrees of Freedom, DOF = 5
Left Object Right Object
Stress cause by a single edge dislo in the GB
Without any math, you can see that the tensile field below the first dislocation overlaps with the compressive field of the second. Thus the stresses will cancel on a macroscopic scale
Of course, if you are a single edge dislocation of the same sign, trying to move from the grain interior to the grain boundary, you will eventually see a high force.
This is the famous Shockley Read model of a grain boundary
Shockley went on to transistor fame, then became infamous for doing I.Q. research…… a long story
Will discuss soon!
In 1963 Shockley left the electronics industry and accepted an appointment at Stanford. There he became interested in the origins of human intelligence. Although he had no formal training in genetics or psychology, he began to formulate a theory of what he called dysgenics. Using data from the U.S. Army's crude pre-induction IQ tests, he concluded that African Americans were inherently less intelligent than Caucasians — an analysis that stirred wide controversy among laymen and experts in the field alike. Nonetheless, Shockley pursued his inflammatory ideas in a series of articles and speeches. Regularly interrupted by boos and catcalls, he argued that remedial educational programs were a waste of time. He suggested that individuals with IQs below 100 be paid to undergo voluntary sterilization.
William Shockley
The Shockely-Read model of the small angle tilt boundary.
From the smooth decline of energy with angle, Aust and Chalmer concluded, long before HRTEM that the boundary could not be amorphous but had to have s structure
A very famous paper on Gb in Pb
Chalmers, born in London, moved from Great Britain, to Toronto (where he wrote this paper) and then to Harvard and became one of the most famous metallurgists of all time
Bruce Chalmers
Calculated grain boundary energies for varies boundary planes
• Low angel regime follows Shockley Read model
• High angle regime “flat” except for coherent twin ( =3) and =11
Typical values are ~ 200 (Al) to ~ 700 (Ni) ergs/cm2
0.2 J/m2 0.7 J/m2
It is interesting to compare GB to Surface energies of metal
Typical values are 3 to 5 times higher than grain boundary
In a bond picture, you break, in a grain boundary about 1 bond of 8 (bcc) or 1..2 out of 12 (fcc) “bonds”. At a surface, you break ~ 3..4 (bcc) or 4…6 (fcc) - varies with orientation
The simple bond breaking picture works very well for fcc metals. In bcc metals there is contribution from the second nearest neighbor
3 Grain boundaries, meeting in a triple point, will given the change (!) , establish an equilibrium angle. If you know the energy of 2 (say twins) you can derive the energy of the third
HW 11-1
A grain boundary is perpendicular to the surface. The grain boundary energy is 500 ergs/cm2 , the surface energy of the metal is 1500 ergs/cm2.
Calculate the angel alpha below
Measurement of grain boundary energy
(Thermal) Grain Boundary Grooving
Establish equilibrium and measure the angle a grain boundary makes with the surface
To demonstrate this, I will use the overhead !
The theory was developed by Bollman. It is strictly geometry. It does not give any information although it is frequently assumed that low boundaries have lower energies. This is only the case for the symmetric twin boundary. A twin boundary with an asymmetric boundary plane has a much higher energy. The boundary plane orientation, in addition to matters
The symetric tilt grain boundary (STGB)
•The Coincident Site Lattice is not the fraction of “common” atoms in the boundary plane
•The boundary plane has nothing to do with the value !
• The is the inverse of the number of common sites if one 3-D (!!) lattice would penetrate into the other
Blue atoms are on coincident sites
The General Theory is in Bollmann’s Book
The CSL lattice for a Sigma=5 boundary in a cubic lattice.
Note that this is an ‘unrelaxed’ boundary!
We can view this structure as being caused by screw dislocation (red lines) with the red burgers vectors.
This view leads to the displacement shift complete, or DSC dislocations.
They are important in G.B. theory
Extension
.
You may look upon CSL theory as an extension from twins and partial dislocations to more sophisticated structures than twins
DSC dislocations for fcc
Other ways to make Sigma 5’s : tilt
More tilt...
The Bollman theory only deals with the topological geometry. It does not give information on the core of a grain boundary.
Nor does it give information on the GB energy
Grain Boundary - Dislocation interactions
Stress will drive dislocations towards the boundary.
HW 11-2
Consider the square grain shown. The dislocation density is 107 per cm2 . The only dislocations are edge dislocation, with equal percentage having their extra half planes up and having the extra half plane down.
A stress, sufficient to drive all dislocations into the GB is applied.
A) will the grain boundary A, B, C, D tilt or not ?
B) if it should tilt, by how many degrees ? (use the sign convention: acw = negative, cw = positive)
is a shear stress working to the right on plane A, and to the left on plane C.
In an elastic deformation the cube would list to the right.
Contribution of dislocations bowing to elastic behavior
HW 10-3 Consider a dislocation which is periodic pinned
The material has a shear modulus G. The dislocation density is cmhe burgers vector is b = 2 Angstrom. The pinning distance is 5000 Angstrom.
The pseudo plastic strain produced by dislocation bowing is given by = (1/2) b d, where d is the average distance the dislocation moves forward.
By what % will the true elastic shear modulus increase due to this effect ?
b
Review of basic dislocation theory (MS&E 261)
• If the shear stress is spatially constant, the dislocation bow out as semicircles. The radius is given by :
Note that the relation is analogous to the pressure p inside a cylinder of surface tension
p = /r or rewritten r = p
Both pressure and shear stress have units of force per area.
is energy per area. A dislocation has an energy per unit length of Gb2/2. The derivative relative to the length, that is the line tension, is Gb. Think of GB as the surface tension of the dislocation
The pressure inside a bubble is given by p=2 /r
The problem is relevant to fatigue.
During fatigue, mobile dislocation bow “in and out” of cell walls. The dark line below are cell walls (dense tangle of dislocations with no dislocation movement). The white areas are where dislocation bow in and out. There is no applied stress here, thus you can not see them
Fatigued Cu
W-11-4
You have square dislocation wall cells, size 0.5 x 05 um.
Iron has a Young's modulus of 211 GPa and a shear modulus, 82 Gpa.
You load the specimen to 1/3 of its yield stress* of 0.211 Gpa.
a) what is the elastic shear strain (engineering shear strain). Answer in percent
b) If 10 dislocation with burgers vectors b = 2 Angstrom would slip back and forth in each cell, what would the plastic strain be ? Answer in %
1/3 of y is a typical value for the fatigue limit of steels. It is also the microstrain “yield” limit which can be measured via AC resistivity. The microstrain itself is too small to be measured
Grain boundaries impurities interactions
• Impurity segregations (e.g. Bi in Cu, dopants in Silicon)
Good : Can make alloys “machinable”
Good : Can “lock” GB in Al against electromigration
Bad: Corrosion, brittle failure
• New phases tend to “precipitate out” first at grain boundaries
Good; ’ in superalloys
Bad: sensitization of grain boundaries
Small amounts of Bi make grain boundaries in Cu much more brittle. Which is boon if you have to machine Cu !!!!!! Machining, is not “cutting” but “driving cracks”
At “service temperatures” below about 800 C, carbon in austenitic stainless steels precipitates out as chromium precipitates at grain boundaries. This sensitizes the steel to stress corrosion.
Happens during “wrong” welding
An excursion into practical metallurgy
T,t phase space to avoid welding of austenitic stainless steels.
Typicall, SS contains 0.08%C unless it is an L series SS
The End
Cementite is brittle. Austenite grains with cementite “skins” will fail brittle.
“Wrong” welding can result in this structure.
Cementite precipitating out at austenite grain boundaries