Lecture 23: Polycrystalline Materials

PHYS 430/603 materialLaszlo Takacs

UMBC Department of Physics

Continuity has to be maintained by• slip and twinning in individual grains,• geometrically necessary dislocations,• sliding along grain boundaries, grain rotation.

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Grain structure before & after plastic deformation

Differently oriented grains slip along different directions, try to deform and rotate differently. The grain structure changes and surface roughness develops - a problem in rolling metals into sheets.

From the Ph.D. Thesis of Eric Moore, 2006

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~100 m

=0.140 = 0

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Tensile Direction

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10o

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Sample Normal

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Sample annealed 425°C for 1 hGrain structure and the orientation change of

individual grains during deformation

From the Ph.D. Thesis of Eric Moore, 2006

Dislocation pile-upDislocations cannot cross grain boundaries (except for rare cases involving coherent boundaries.)There are several dislocations moving toward the grain boundary in a slip plane. They repel each other with a force

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F = Gb2

2π (1−ν )1r

A pile-up forms with increasing distance between dislocations with increasing distance from the grain boundary. The first dislocation is pushed toward the grain boundary by the effect of the external stress plus the force from all the other dislocations. This large stress is transmitted into the neighboring grain also.

Typically strength increases with decreasing grain size - an important benefit that drives the development of ultrafine grained materials. These are typical stress - strain curves for Cu to 22 nm grain size. How can this be explained by dislocation pile-up?

From Khan, Farrokh, Takacs, J. Mater. Sci. 43 (2008) 3305

The Hall-Petch relationThe number of dislocations that can fit into a grain of diameter D is limited to

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n = π (1−ν )Gb

D2

τ

At the tip of the pile-up, the external stress plus (n-1) dislocations act, each with a force b. The local stress at the grain boundary is n. This stress penetrates neighboring grain 2, where at x0

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2(x0) = m2σ + β (x0) π (1−ν )2Gb

Dτ 2

First term is from external stress with the Schmid factor of grain 2, second term is stress from grain 1, describes geometry. At the yield point 2 reaches 0, the ultimate shear stress. If nothing but the grain size changes and the first term is negligible

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Dτ 2 = const. = ′ k y2

For a single crystal the ultimate stress is 0, Adding the term from grain 2

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= 0 +′ k yD

Data available for Cu as a function of grain size. Notice that hardness begins to decline for very small grains. There is an optimum where ultimate shear stress, tensile stress, and hardness are the largest.

• For grain size larger than 100 nm - 1µm depending on the material, H-P applies.• Deviations become increasingly significant, until a maximum is reached at about 20 nm

(less than 500,000 atoms).• Softening takes place for smaller grain sizes. The deformation mechanism relies more

on grain boundary processes than dislocations. This is an area of current research.

How is it possible to make materials with such small grain size?• Inert gas condensation, solution processing.• Rapid solidification, crystallization of an amorphous initial state.• Sever plastic deformation, ball milling.

Grain size refinement is an important strengthening mechanism. Are there others?

Dislocations move relatively freely in defect-free single crystals. Any deviation - solute, other dislocation, grain boundary, etc. - acts as an obstacle, impeding dislocation motion and thus increasing strength.

Notice that the large increase of critical shear stress in Ag-Au is in spite of the chemical and structural similarity of Au and Ag.

Solid solution hardening• Parelastic interaction - lattice parameter effect

Solute - both larger and smaller than the matrix - can find an energetically favorable place in a dislocation. The energy difference of the solute between a general lattice site and a site in the core of a dislocation becomes a barrier to dislocation motion. Affects mostly edge dislocations, although screw can be affected if impurity causes non-isotropic distortion (such as C in bcc Fe.)

• Dielastic interaction - shear modulus effect.Dislocation energy is proportional to the shear modulus. If an impurity changes G, it changes the dislocation energy, providing a barrier to dislocation motion.

• Suzuki interaction - chemical effect.Impurity changes the stacking fault energy, thus the separation between partial dislocations. Dislocation motion requires restoring the original separation.

• The magnitude of solid solution hardening (from parelastic an dielastic effect) is proportional to the square root of the concentration:

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Δ c = G δ + β η( )3

2 c a /3

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where δ = d ln ad c a , η = d lnG

d c a , β = const. ≈1

Solid solution hardening of Cu by several solutes.

Notice that the classical bronze-forming elements Sn and As provide much hardening. The good properties of brass (Zn) are due to the formation of compound phases, not due to solid solution hardening.

The effect of a metal as alloying element is very different from its own mechanical properties.

Usually we assume that the dislocations move but the motion of the solutes is negligible.

If the solutes diffuse fast - e.g. C in iron - time-dependent phenomena result, such as the Lüders maximum.Once the dislocation breaks free from the impurities, it moves easier and the stress decreases. But if the test is stopped for some time, the impurities diffuse into the dislocations again and the dislocation must break free again.

Dispersion hardening

Finely dispersed particles are obstacles for dislocation motion.Orowan mechanism: If the dislocation meets a pair of particles, it can only proceed by bowing out, similar to the principle of the Frank-Reed source. The stress needed to move the dislocation across this barrier is

= Gb/(l-2r)Introducing finely dispersed particles into a metal is often a difficult task, as the particles must be wetted by the matrix but not dissolved by it. Searching for a way to disperse aluminum oxide particles in Ni-based superalloys lead to the development of mechanical alloying in the late 1960s.

Precipitation hardeninga form of dispersion hardening

Precipitates are often coherent or semi-coherent. A dislocation can pass through a coherent precipitate, but it requires extra stress as a stacking fault results and creating it requires energy. This mode dominates, if the material contains many small particles. If the same total volume is in fewer but large particles, the Orowan mechanism is preferred. The largest hardness is achieved when the two mechanisms require the same stress; this happens at around 20 nm.

Precipitates form when a solid solution becomes supersaturated during cooling and a second phase crystallizes in the matrix in the form of small crystallites. They are often coherent (like Ni3Al in Ni.)

Strain rate dependence

The time / strain rate dependence of the stress-strain curve is intuitively anticipated and clearly observed, but it is very difficult to explain quantitatively. Typically it is characterized by the strain rate sensitivity, m, defined as

Ball milled and consolidated Cu, average particle size 32 nm.

Babak Farrokh, UMBC

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m = d ln σd ln ˙ ε

Temperature dependence

• The strain rate sensitivity is low at low temperature (T < 0.5 Tm) but increases at higher temperature. This is understandable, as atomic motion is more vigorous at higher temperature, diffusion is faster.

• High strain rate sensitivity is usually associated with larger strain to failure. Consider a tensile experiment. If a random cross section decreases in diameter, the strain rate at that cross section increases, with enough strain rate sensitivity the section becomes harder and no further reduction leading to failure occurs.

• In fine grained (<10 µm) materials close to Tm very large strain rate sensitivity and strain to failure (up to 100-fold elongation) can be observed. This is called superplasticity. It depends on grain boundary sliding, rather than dislocation mechanisms.

• Nanocrystalline materials contain many grain boundaries, superplasticity should be more easily achieved.

Superplasticity of electrodeposited nc Ni andnc Al-1420 alloy and Ni3Al by severe plastic deformation

Notice that superplasticity was achieved at a temperature much below typical for conventional materials; 350°C for Ni corresponds to 0.36 Tm!

McFadden et al. (UC Davis, Ufa, Russia)Nature 398 (1999) 684-686

High RT ductility of a hcp Mg-5%Al-5%Nd alloy

Ball milling results in repeated fracturing and agglomeration of grains, resulting in a nanometer scale microstructure (mean grain size probably 25 nm). Grain rotation and sliding results in high ductility even at room temperature and 3x10-4 s-1 stress rate. (Recall that a hcp material is normally brittle.)

L. Lu and M.O. Lai, Singapore

Ceramic nanocomposite of 40 vol.% ZrO2, 30% Al2MgO4, and 30% Al2O3 shows superplastic behavior at 1650°C.

Kim et al. (Tsukuba, Japan)

Creep

Nabarro-Herring creep Coble creepmediated by

volume diffusion grain boundary diffusion

mass flux

Anelasticity and viscoelasticity

• Small time dependent effects can be observed also for elastic deformation - e.g. related to reversible diffusion of C in a steel under stress.

• If a sample is vibrated close to resonance, the deviation from perfect elasticity, i.e. the existence of dissipative processes, results in a change of the resonance curve.

• While technologically unimportant, this is the way one can gain information about diffusion and other time dependent phenomena at low temperature, where their rate is very low and the macroscopic effects are not detectable.