Upload
linus-rasmussen
View
35
Download
4
Embed Size (px)
DESCRIPTION
Microstructure-Properties: I Lecture 6: The Effect of Grain Size?. 27-301 18th September, 2002 Prof. A. D. Rollett. Bibliography. Electroceramics , A.J. Moulson & J.M. Herbert, Chapman & Hall, ISBN 0-412-29490-7, 621.381/M92e - PowerPoint PPT Presentation
Citation preview
1
Objective
Grain Size
Varistors
Hall-Petch
Creep
Microstructure-Properties: ILecture 6:
The Effect of Grain Size?
27-301
18th September, 2002
Prof. A. D. Rollett
2
Objective
Grain Size
Varistors
Hall-Petch
Creep
Bibliography
• Electroceramics, A.J. Moulson & J.M. Herbert, Chapman & Hall, ISBN 0-412-29490-7, 621.381/M92e
• Physical Ceramics (1997), Y.-T. Chiang, D.P. Birnie III, W.D. Kingery, Wiley, New York, 0-471-59873-9.
• Mechanical Behavior of Materials (1966), F. McClintock and A. S. Argon, Addison Wesley.
• Electronic Materials (1990), edited N. Braithwaite & G. Weaver, (The Open University) Butterworths.
• Mechanical Behavior of Materials, T.H. Courtney, McGraw-Hill, ISBN 0-07-013265-8, 620.11292,C86M
• Microstructure and Properties of Materials, J.C.M. Li, editor, World Scientific, ISBN 981-02-2403-6
3
Objective
Grain Size
Varistors
Hall-Petch
Creep
Objective
• This lecture is concerned with the effects of grain size on properties.
• Two examples will be given:
• (1) The effect of grain size on resistance in ceramics used for varistors (e.g. in surge protectors).
• (2) The effect of grain size on mechanical properties (Hall-Petch effect, Nabarro-Herring creep).
• If time permits, the discussion will be extended to magnetic hardness also.
4
Objective
Grain Size
Varistors
Hall-Petch
Creep
Varistors
• Varistor = variable resistor, i.e. a circuit element whose resistance varies with the voltage applied.
• As typically fabricated, they have highly non-linear response and are useful as voltage limiters.
• They operate by retaining high resistance to some voltage, above which their resistance drops rapidly.
• For short times they can pass large currents thereby preventing the voltage from rising much above the breakdown voltage.
• Varistors can therefore function as self-reseting circuit breakers (actually shunts, not breakers!).
5
Objective
Grain Size
Varistors
Hall-Petch
Creep
Examples of Varistor Circuit
Components
Electroceramics
6
Objective
Grain Size
Varistors
Hall-Petch
Creep
Macrostructure of a Surge Arrester
• The size and structure of the device depends on the application, e.g. at what voltage it is designed to limit to, and how much current it must be able to pass in a given surge.
Electroceramics
7
Objective
Grain Size
Varistors
Hall-Petch
Creep
Current-voltage characteristic
• At low voltages, the response is ohmic, i.e. the current is proportional to the voltage. At higher voltages the response is power-law, with a large exponent (compare this to the power-law relationship for plastic flow!). The better the device, the larger the exponent. The typical breakdown voltage ranges from tens to hundreds of volts.
Electroceramics
8
Objective
Grain Size
Varistors
Hall-Petch
Creep
Varistor application
• A varistor (“VDR” in the figure) is typically included in parallel with the load so that the latter never sees anything above some maximum voltage.
Electroceramics
9
Objective
Grain Size
Varistors
Hall-Petch
Creep
Material, microstructure
• Varistors can be made from a range of semiconducting ceramics: SiC, ZnO, TiO2 and SrTiO3.
• ZnO with Bi dopant and other oxides (Co, Sb, Fe) is standard material.
• Critical feature is the segregation of the dopant to the grain boundaries.
10
Objective
Grain Size
Varistors
Hall-Petch
Creep
Varistor microstructure
• The real microstructure contains a range of grain sizes and shapes (left). For the purposes of understanding varistor behavior, one can idealize the microstructure as a “brick” structure, i.e. a regular lattice of cubical grains.
Electroceramics
11
Objective
Grain Size
Varistors
Hall-Petch
Creep
ZnO
• ZnO has a 3.2eV band gap and so the presence of electron donor additions such as Co, Sb, Fe to make it an n-type extrinsic semiconductor are vital. The presence of the donor sites makes the grain interiors conductive.
• The Bi segregates strongly to grain boundaries (and other interfaces) where it provides acceptor states. The presence of the acceptor states locally depresses the Fermi level in the grain boundary.
12
Objective
Grain Size
Varistors
Hall-Petch
Creep
ZnO, contd.
• A typical ZnO compact has grain size 10-50µm, with an intergranular phase of thickness 1-1000nm.
• The high Bi-content intergranular phase has high resistance, ~ 106 m.
• Heating to high temperatures (typical = 1250°C) drives off oxygen, leaving vacancies on the oxygen sub-lattice (wurtzite structure). Thermal activation can ionize these vacancies, thereby releasing electrons into the conduction band (giving n-type conduction).
• Typical compositions include ~1mol% dopants: 96.5ZnO-0.5Bi2O3-1.0CoO-0.5MnO-1.0Sb2O3-0.5Cr2O3.
13
Objective
Grain Size
Varistors
Hall-Petch
Creep
p-n diode junctions (silicon)
• It is useful to go back to basics and consider how to form a p-n diode in terms of doped semiconductors.
• Consider a block of Si with two (adjacent) regions of doping - one p-type and one n-type.
• p-type means that conduction is hole-dominated (acceptor dopant atoms). n-type means electron-dominated conduction (donor dopant atoms).
n-typep-type
14
Objective
Grain Size
Varistors
Hall-Petch
Creep
Fermi levels
• For acceptor dopants (e.g. boron), the Fermi level is low in the gap. For donor dopants (e.g. phosphorus, arsenic) the Fermi level is high in the gap.
BandGap
BandGap
Ee Ee
Electron Energy
Electron Energy
p-type n-type
Conduction band
valence band
15
Objective
Grain Size
Varistors
Hall-Petch
Creep
Electron energies at junction
• When we join the p-type to the n-type, the rule is that the Fermi level is constant throughout the material (otherwise there would be a net flow of electrons in the material). The result is a bending of the energy levels in the junction region.
BandGap
Ee
Electron Energy
Junction of p- & n-types
p-type n-type
16
Objective
Grain Size
Varistors
Hall-Petch
Creep
Potential vs. electron energy
• Electric potential (voltage) is the opposite of electron energy (from the change in sign).
• Holes move down gradients in electric potential: electrons move down gradients in electron energy.
• By equilibrating Fermi levels, no net electron (or hole) flow will occur between the p- and n-type regions.
Potential (V)+
-
p-type n-type
~0.8V
Ee
17
Objective
Grain Size
Varistors
Hall-Petch
Creep
Junction Region
• In addition to the gradients in electron energy and potential, there is some flow of electrons from the n-type into the p-type region with recombination of the carriers.
• This depletes the concentrations of holes and electrons on either side of the junction.
• Carrier depletion obviously decreases conductivity.• Conduction: the conductivity depends (linearly) on
the carrier concentration, n, mobility, µ, and charge,e; = n e µ
18
Objective
Grain Size
Varistors
Hall-Petch
Creep
Conductivity in a semiconductor
• Typical values for n-type doped silicon (subscript “n” denotes quantity in n-type): majority carrier concentration, nn, = 1022 electrons.m-3
mobility, µn, = 0.35 m2V-1s-1
and charge,e, = -1.6.10-19C.
minority carrier concentration, pn, = 2.3.1010 holes.m-3
mobility, µh, = 0.044 m2V-1s-1
and charge,e, = +1.6.10-19C.• Remember: electric field = -1*gradient of potential;
E = -dV/dx
19
Objective
Grain Size
Varistors
Hall-Petch
Creep
Junction region
• The local electric field repels electrons on the n-type side, and repels holes on the p-type side.
• Only minority carriers on either side of the junction are available to carry current.
p-type n-type
[Electronic Materials]
20
Objective
Grain Size
Varistors
Hall-Petch
Creep
Biasing a p-n junction
Now we consider what happens when we apply an external voltage (electric potential) to the system and require a current to flow through the junction.
n-typep-type
+ -Forward bias
Forward bias
Reverse bias
21
Objective
Grain Size
Varistors
Hall-Petch
Creep
Biasing, contd.
• Forward bias = lowers the potential (voltage) on the n-type side, and raises it on the p-type side. This tends to diminish the depletion zone (from both sides).
• Reverse bias = as expected, this raises the potential (voltage) on the n-type side, and lowers it on the p-type side. This tends to widen the depletion zone (from both sides).
22
Objective
Grain Size
Varistors
Hall-Petch
Creep
Biasing: minority carrier conc.
• Bias voltage changes the density of minority carriers at the edge of the depletion zone and thus the current that can be carried across the zone.
• Increasing forward bias increases the number of majority carriers (holes) in the p-type side which flow into the n-type side, raise the (minority carrier) level on that side and increase current capacity. The density is proportional to the exponential of the voltage across the junction.
[Electronic Materials]
23
Objective
Grain Size
Varistors
Hall-Petch
Creep
Grain Boundary electric double layer• The electronic structure at a grain boundary in a ceramic
is understood as having acceptor states (not well understood!) that cause a local increase in the electron energy. This constitutes a barrier to electron motion through the material.
Electroceramics
24
Objective
Grain Size
Varistors
Hall-Petch
Creep
Band Structure at a Grain Boundary• Equilibration of the chemical
potential of electrons throughout the solid equalizes the Fermi levels inside and outside the boundaries. Charge redistribution occurs. Conduction electrons are depleted from the boundary vicinity (and go into the acceptor states in the boundary).
• A potential energy barrier at the boundary is created.
• Applying a voltage across the material tilts the energy levels until breakdown occurs.
Electroceramics
25
Objective
Grain Size
Varistors
Hall-Petch
Creep
Grain Boundary control
• As a consequence,the electrical properties depend on (a) the doping of the grain boundaries and (b) the microstructure through the number and arrangement of the boundaries.
• Chiang gives an example of estimating the breakdown voltage based on a 3V breakdown for an individual boundary. For a 1mm thick device with a 10µm grain size, one expects about 100 boundaries through the thickness, which predicts a breakdown voltage of ~300V.
26
Objective
Grain Size
Varistors
Hall-Petch
Creep
Voltage-Current Characteristic
• This is the characteristic that one can observe across a polycrystal, i.e. a breakdown voltage of about 300V. The inverse slope, , is a measure of varistor quality.
Electroceramics
27
Objective
Grain Size
Varistors
Hall-Petch
Creep
Relation to Diodes
• Each boundary can be regarded as a pair of back-to-back Schottky barriers, i.e. metal-semiconductor junctions.
• Chemistry of the boundaries is not well understood. Bi3+ is an electron donor solute, so it is not clear how it functions as an acceptor in the boundary!
• The oxidation state is important: quenched samples of ZnO exhibit little or no breakdown. Apparently, oxidation of the grain boundaries during post-sintering cool-down is important for development of the critical properties.
28
Objective
Grain Size
Varistors
Hall-Petch
Creep
Typical Varistor Application
Electroceramics
29
Objective
Grain Size
Varistors
Hall-Petch
Creep
Hall-Petch Effect
• The Hall-Petch effect is remarkably simple to express but still difficult to explain in fundamental terms.
• At ambient conditions (no creep), yield strength rises as the grain size decreases.
• The variation in strength can be described by a power-law relationship:
y = 0 + kd-1/2
• The Hall-Petch effect is named for E.O. Hall and N.J. Petch from their papers of the early 1950’s, e.g. “The Cleavage Strength of Crystals” N.J. Petch, J. Iron & Steel Inst., 174, 25-28.
30
Objective
Grain Size
Varistors
Hall-Petch
Creep
Dislocation Pile-ups
• The central idea is that dislocations are forced to pile up at grain boundaries, either because there is a barrier to crossing over into the next grain, or because a source must be activated in the next grain.
[Courtney]
31
Objective
Grain Size
Varistors
Hall-Petch
Creep
Pile-up at a Boundary
• The classical explanation for the Hall-Petch effect is that some stress concentration in a given grain is required to initiate slip in its neighboring grain. That stress concentration is most plausibly obtained through a dislocation pile-up, see figure 5.5. The essence of the argument is that stress is higher as the number of dislocations increases. Thus the larger the grain size, the more quickly (in terms of macroscopic strain) is the critical stress reached at which slip is initiated in the neighboring grain. The form of the equation describing the pile-up stress contains a term in √(d/r) where d is the grain diameter and r is the (average) distance to the source in the neighboring grain from the boundary.
32
Objective
Grain Size
Varistors
Hall-Petch
Creep
Material Dependence
• The Hall-Petch constant, k in the equation, varies considerably amongst materials. This in itself raises some questions about the mechanism(s) underlying the effect. The explanation given is purely geometrical and although the material dependence could be explained through the ratio d/r, it is not clear why this should be so!
• Solutes tend to enhance themagnitude of the Hall-Petch effect.
[Courtney]
33
Objective
Grain Size
Varistors
Hall-Petch
Creep
Grain Size and Fracture
• Grain size also has a marked effect on fracture, which was, in fact, part of Petch’s original contribution.
McClintock & Argon
34
Objective
Grain Size
Varistors
Hall-Petch
Creep
Nanocrystalline materials
• All this suggests that remarkably strong materials can be generated if very small grain sizes can be achieved. This, of course, is one aim of nanocrystalline materials in which grain sizes are obtained that are well less than one micron. The processing (in metals) relies on either compaction of fine powders (which requires second phase particles in order to maintain the small grain sizes at sintering temperatures) or heavy deformations allied with recrystallization. This is an exciting area and is a lively area of research and development.
• How to make nanocrystalline material? Powders, ball milling, equal-angle channel extrusion, thin film deposition (chemical vapor deposition, physical vapor deposition, laser ablation).
35
Objective
Grain Size
Varistors
Hall-Petch
Creep
Nanocrystalline materials
• Limitations of nanocrystalline materials?
36
Objective
Grain Size
Varistors
Hall-Petch
Creep
Stress concentration• The square root term is akin to the stress concentration at the tip
of a penny-shaped crack (in fracture mechanics). Thus,
(applied - 0) √(d/4r) = *,
where * is the critical stress for dislocation source activation, 0 is
the resistance to dislocation motion in each grain, and applied is the applied shear stress. Again, the larger the diameter, d, the more dislocations in the pile-up for a given applied stress (minus the resistance). Rearranging, we get {Courtney - Eq. 5.8}.
applied = 0 + 2 * √r d-1/2 = 0 + kd-1/2.
37
Objective
Grain Size
Varistors
Hall-Petch
Creep
Creep
• An important property of materials is their resistance to creep.
• Creep is irreversible (plastic) flow at low rates under low stresses.
• We will return to this issue in later lectures because of its importance.
• Creep is highly sensitive to temperature because thermal activation makes the largest contribution to plastic flow when the stress is too small to overcome mechanical barriers to dislocation motion.
38
Objective
Grain Size
Varistors
Hall-Petch
Creep
Diffusion
• Creep is therefore a phenomenon associated with high temperatures.
• High temperature is a relative term: one contribution that thermal activation makes is by increasing diffusion rates.
• Diffusion coefficients (D = D0exp-{Q/RT}) are strongly (exponentially!) dependent on temperature.
• The activation energy (enthalpy, strictly speaking) is approximately proportional to the melting point of the material.
• At the same temperature, a higher melting point material will exhibit slower diffusion than a lower melting point material.
39
Objective
Grain Size
Varistors
Hall-Petch
Creep
Diffusivity, Activation
Energy
Porter & Easterling:diffusivity at the meltingpoint is constant for a given class ofmaterial.
Similarly, the activation energynormalized by RTm
is constant for a given class ofmaterial.
40
Objective
Grain Size
Varistors
Hall-Petch
Creep
Homologous Temperature
• Therefore it is common to use homologous temperature as a measure of relative temperature:
T’ = T/Tmelt
• Therefore we expect materials tested at the same homologous temperature to show similar behavior.
• Materials will tend to creep at high homologous temperatures because diffusion allows changes in shape.
41
Objective
Grain Size
Varistors
Hall-Petch
Creep
Creep Mechanisms: diffusion
• For the purposes of this lecture, we will consider just one creep mechanism: self-diffusion between grain boundaries.
• Assumption: grain boundaries are perfect sources and sinks of vacancies.
• Therefore a tensile stress (for example) on a polycrystalline body sets up a driving force for vacancy motion.
[Courtney 7.5]
42
Objective
Grain Size
Varistors
Hall-Petch
Creep
Nabarro-Herring Creep
• The creep mechanism involving diffusion to/from grain boundaries through the bulk lattice is known as Nabarro-Herring creep for the scientists who identified it.
• The reason for the grain size dependence is simple: the diffusion path length is proportional to the grain size:- since the vacancy concentration at the boundaries is fixed by the stress and the path length is proportional to the grain size, the concentration gradient is inversely proportional to the grain size.- the creep rate (i.e. the strain rate) is proportional to the vacancy flux and is thus inversely proportional to the grain size.
43
Objective
Grain Size
Varistors
Hall-Petch
Creep
Other Creep Mechanisms• Dislocation Glide. This is self-explanatory: dislocations move (conservatively) in response to shear
stresses. • Nabarro-Herring Creep. Creep can occur by mass transport, i.e. diffusion of atoms from regions of
lower (algebraically) stress to regions of higher (more tensile) stress. This is equally effective in amorphous materials as in crystalline.
• Coble Creep. Mass transport can occur either in the bulk (leading to N-H Creep) or along interfaces such as grain boundaries. In the latter case it is known as Coble creep. Both of these mechanisms result in a significant grain size dependence.
• Solute Drag Creep. For dislocations gliding at high T, not only do the solute atoms interact with the dislocations but they can also move sufficiently rapidly for the drag effect to be significant.
• Dislocation Climb-Glide Creep. In between the (low) temperatures at which only dislocation glide is important, and the (high) temperatures at which diffusion dominates (at low stresses), a combination of glide and climb controls creep. That is to say, dislocation motion carries most of the strain but the dislocations circumvent obstacles by climb.
• Grain Boundary Sliding accommodated by diffusional flow. In superplasticity especially, sliding of one grain relative to another is very important.
• Grain Boundary Sliding accommodated by Dislocation Flow. This is the same mechanism of g.b. sliding but the accommodation is achieved by dislocation glide. Clearly one expects this to dominate over diffusion at lower temperatures.
44
Objective
Grain Size
Varistors
Hall-Petch
Creep
Nabarro-Herring Creep: grain size dependence
• The creep rate in Nabarro-Herring creep is inversely proportional to the square of the grain size.
• The quadratic dependence of creep rate on grain size arises from distributing the vacancy flux over the (average) area of a grain facet.
• Bottom line: small grain size lowers creep resistance, and large grain size increases creep resistance.
• Ideal microstructure (w.r.t. grain structure) is a single crystal.
45
Objective
Grain Size
Varistors
Hall-Petch
Creep
N-H Creep: derivation: 1
• Difference in vacancy concentration, Nv, (which provides the driving force). The activation energies are modified from the unstressed values by exp{+/- stress*atomic volume}=exp-{/kT}. Compression decreases the concentration (slightly) and tension raises it.
• Given small stresses (1-50 MPa), ~10-29m3, kT~1.4.10-
20J, /kT~0.02 «1. This permits us to linearize the driving force.
tensionNvacancy≈exp−Qvacancy
kT
⎧ ⎨ ⎩
⎫ ⎬ ⎭ exp
σΩkT
⎛ ⎝ ⎜
⎞ ⎠ ⎟
compressionNvacancy≈exp−Qvacancy
kT
⎧ ⎨ ⎩
⎫ ⎬ ⎭ exp−
σΩkT
⎛ ⎝ ⎜
⎞ ⎠ ⎟
46
Objective
Grain Size
Varistors
Hall-Petch
Creep
N-H Creep: derivation: 2
• Given small stresses (1-50 MPa), ~10-29m3, kT~1.4.10-20J, /kT~0.02 «1. This permits us to linearize the driving force.
∆NV ~ /kT exp-{Qvacancy/kT}.
• The vacancy flux is given by Fick’s first law:JV = DV(dNV/dx)
• The distance over which the diffusion occurs is approximated by the grain diameter, d:
JV = DV (/kT)(1/d) exp-{Qm/kT}.
47
Objective
Grain Size
Varistors
Hall-Petch
Creep
N-H Creep: derivation: 3
• If we multiply the flux by the area over which diffusion takes place, which we approximate by the area of a grain boundary facet, d2, we obtain the rate of change of volume. We can also include the diffusion coefficient in the expression, where Qm is the activation energy for vacancy motion; DV=D0Vexp-{Qm/kT}.
V/t = JV d2
V/t = d2D0Vexp-{Qm+QVacancy/kT} (/kT)(1/d).
48
Objective
Grain Size
Varistors
Hall-Petch
Creep
N-H Creep: derivation: 4
• Rate of change in length = volume rate (V/t) / area; area ~ d2.
d/t = D0Vexp-{Qm+QVacancy/kT} (/kT)(1/d).
• Strain = Change in length / length = d/d.
d/t = D0Vexp-{Qm+QVacancy/kT} (/kT)(1/d2).
• Collecting terms: ˙ ε =ANH
Dbulk
d2
⎛ ⎝ ⎜
⎞ ⎠ ⎟
σΩkT
⎛ ⎝ ⎜
⎞ ⎠ ⎟
Note the grain size dependence!
49
Objective
Grain Size
Varistors
Hall-Petch
Creep
Creep: general characteristics
• Low temperature deformation is characterized by work hardening: high temperature by a short transient hardening, followed by steady-state flow.
• Similarly, constant load leads to steady-state flow at high T, but cessation of flow at low T.
50
Objective
Grain Size
Varistors
Hall-Petch
Creep
Deformation Mechanism Maps
Deformation maps provide a convenient graphical view of the different regimes of deformation behavior as a function of temperature and stress.
51
Objective
Grain Size
Varistors
Hall-Petch
Creep
Grain Size and NH Creep
• Smaller grain sizes expand the range over which Nabarro-Herring (and Coble) creep are observed.
• Alternatively, the map can be drawn in the space of stress (normalized by modulus) and grain size (normalized by Burgers vector). In this map, the temperature is fixed.
52
Objective
Grain Size
Varistors
Hall-Petch
Creep
Creep Resistance: Superalloys
• Nickel-based “superalloys” originated with the Ni-Cr alloys used for heating elements in furnaces. In these, and the subsequent superalloys, their oxidation resistance is critical. Very few materials possess ductility, oxidation resistance and strength at high temperatures.
• The term superalloy refers - loosely - to the use of this alloy class at unusually high homologous temperatures.
• They are based on the Ni-Cr-Al ternary system but have many other alloy additions.
• The key to their success is the presence of a second phase, close to Ni3Al (“gamma-prime”) that is coherent with the matrix (more on this in 302!) and whose strength increases with temperature.
53
Objective
Grain Size
Varistors
Hall-Petch
Creep
Refractory Materials (Engines)
• Carbon 3800 oxidizesTungsten 3650 oxidizesMgO 3100 brittle (room T)SiC 3000 brittle (room T)Mo 2880 oxidizesNb 2740 oxidizesAl2O3 2290 brittle (room T)Cr 2160 brittleZr 2125 expensive
(nuclear fuel elements)Pt 2042 expensive (crucibles)
Fe 1808 oxidizesNi 1726 √
Tmelt
54
Objective
Grain Size
Varistors
Hall-Petch
Creep
Superalloys: Temperature of UseMicrostructure and Properties of Materials: Stoloff
Why is temperature important? The efficiency of the engine is very sensitive to hot zone temperature. Therefore even small increases in the temperature to which turbine blades can be exposed produces significant gains in engine efficiency
55
Objective
Grain Size
Varistors
Hall-Petch
Creep
Grain Size effect on creep rate
[Microstructure and Properties of Materials: Stoloff]
56
Objective
Grain Size
Varistors
Hall-Petch
Creep
Ni-Al phase diagram
Note the use of high Al contents in order to obtain very high volume fractions >50%) of the Ni3Al phase (gamma-prime) that exhibits INCREASING hardness/strength with increasing temperature (alloy d, right). The ordered crystal structure of the ’provides some of the enhanced strength.
Microstructure and Properties of Materials: Stoloff
57
Objective
Grain Size
Varistors
Hall-Petch
Creep
Superalloy strengthening
• The coherent interface between the ’ and the matrix is important because it means that the precipitate does not coarsen (again, more on this in 302).
• The increasing strength with temperature (of the ’is critical: most materials soften with temperature. This effect is not completely understood and is likely the result of dislocation-dislocation interactions.
• The phase relationships allow a very large volume fraction of cuboidal ’ to be precipitated in the matrix: this is a very effective barrier to dislocation creep.
Microstructure and Properties of Materials: Stoloff
58
Objective
Grain Size
Varistors
Hall-Petch
Creep
Contemporary Issues• An example of a contemporary issue is the effort being made
by the manufacturers of gas turbine engines to transfer the technology for single crystal turbine blades from aircraft engines over to land-based gas turbines (for power generation).
• What’s the issue? Hot stage turbine blades for an aircraft engine are comparatively small - a few centimeters long. The equivalent component for a land-based turbine is an order of magnitude large - 30-70 cm long! Making these depends on control of directional solidification: faults give rise to new orientations and grain boundaries which weaken the material.
• Solution? Either much improved furnaces for directional solidification (very expensive and perhaps not feasible) or fabricate the single crystal from smaller pieces.
59
Objective
Grain Size
Varistors
Hall-Petch
Creep
Bonded “Single” Crystals
• The approach adopted by one manufacturer is to grow thin single crystal pieces (slabs) of the nickel superalloy using standard directional solidification. Control of the crystal soldification direction is still feasible because of the thinness. Then two pieces, chosen for similar orientation, are bonded together.
• The bonding process relies on a boron-rich bonding alloy (braze) that produces transient melting.
Xtal A + bond + Xtal B Bonded Xtal
60
Objective
Grain Size
Varistors
Hall-Petch
Creep
Summary• Grain size is a critically important aspect of polycrystalline materials.
• In the case of varistors, a special electronic structure in the grain boundary layer produces a back-to-back diode that has a well-defined breakdown voltage. The electrical characteristics of the device are directly related to the electrical properties of the boundary and the grain size.
• In the case of the Hall-Petch effect, in most materials, both the strength and the toughness increase as the grain size is reduced. This effect can be explained by the resistance of the boundaries to plastic flow (in the case of strength) and/or the decreased microcrack size in the case of fracture.
• Grain size can play a major role in controlling creep resistance. Larger grain size increases creep resistance - hence the use of single crystals where feasible, especially for superalloys.
61
Objective
Grain Size
Varistors
Hall-Petch
Creep
Example Problem for Varistors
• How much Bismuth oxide must I add to ZnO (proportions by weight) in order to dope the grain boundaries to the desired level?
• Can estimate a minimum amount by assuming that we need, say, 2 layers of Bi atoms along every boundary in order to accomplish the required doping.
• From here on, it is college chemistry to make the estimate.
• Suppose that the grain size in the ZnO is 12µm (as in the exam question). That is, 3V grain boundary breakdown voltage, 250V device breakdown, 1mm thick.
62
Objective
Grain Size
Varistors
Hall-Petch
Creep
Bi doping levels in ZnO
• Step 1: grain boundary area per unit volume, AV, = 1/d = 8.3 104 m2/m3.
• Step 2: atomic area, Aatom ~ 0.32 10-18 m2
• Step 3: no. of atoms per unit volume = AV/Aatom = 8.3/9 104/10-20 ~ 1024 atoms/m3.
• Step 4: moles(Bi)/m3 = 1024/Nav = 1.7 moles/m3.
• Step 5: molecular weight of Bi2O3=464 gms
• Step 6: weight(Bi2O3) per m3 = 770 gms
• Step 7: density of ZnO = 5.6 Mg/m3.
• Step 8: weight proportions are 1:7200, Bi2O3:ZnO.
63
Objective
Grain Size
Varistors
Hall-Petch
Creep
Topology of Networks
• Consider the body as a polycrystal in which only the grain boundaries are of interest.
• Each grain is a polyhedron with facets, edges (triple lines) and vertices (corners).
• Typical structure has three facets meeting at an edge (triple line/junction or TJ); Why 3-fold junctions? Because higher order junctions are unstable [to be proved].
• Four edges (TJs) meet at a vertex (corner).
64
Objective
Grain Size
Varistors
Hall-Petch
Creep
Definitions
• G B = Grain = polyhedral object = polyhedron = body
• F = Facet = face = grain boundary• E = Edge = triple line = triple junction = TJ• C V = Corner = Vertex = points• n = number of edges around a facet• overbar or <angle brackets> indicates average
quantity
65
Objective
Grain Size
Varistors
Hall-Petch
Creep
Euler’s equations
• 3D: simple polyhedra (no re-entrant shapes)V + F = E + 2 [G = 1]
• 3D: connected polyhedra (grain networks)V + F = E + G +1
• 2D: connected polygons V + F = E + 1
• Proof: see What is Mathematics? by Courant & Robbins
(1956) O.U.P., pp 235-240.
66
Objective
Grain Size
Varistors
Hall-Petch
Creep
Grain Networks
• A consequence of the characteristic that three grain boundaries meet at each edge to form a triple junction is this:
3V = 2E
E
EE
E
EE
EE
E
EEEV
V
V
V
V
V
67
Objective
Grain Size
Varistors
Hall-Petch
Creep
2D sections• In a network of 2D grains, each grain boundary has
two vertices at each end, each of which is shared with two other grain boundaries (edges):
2/3 E = V, or, 2E = 3V E = 1.5V
• Each grain has an average of 6 boundaries and each boundary is shared:
n = 6: E = n/2 G = 3G, or, V = 2/3 E = 2/3 3G = 2G
_ _
68
Objective
Grain Size
Varistors
Hall-Petch
Creep
2D sections
6-sided grain= unit cell;each vertex has 1/3 in eachunit cell;each boundaryhas 1/2 in each cell
split each edge
divide each vertex by 3
69
Objective
Grain Size
Varistors
Hall-Petch
Creep
3D Topology: polyhedra
70
Objective
Grain Size
Varistors
Hall-Petch
Creep
2D Topology: polygons
71
Objective
Grain Size
Varistors
Hall-Petch
Creep
Typical section
• Correction terms (Eb, C1’,C2’) allow finite sections to be interpreted.
C1’:=no. incomplete corners against 1 polygon; C2’:= same for 2 polygons
[Underwood]
72
Objective
Grain Size
Varistors
Hall-Petch
Creep
Grain size measurement: area based
• Grain count method: <A>=1/NA
• Number of whole grains= 20Number of edge grains= 21Effective total=Nwhole+Nedge/2
= 30.5Total area= 0.5 mm2
Thus, NA= 61 mm-2; <A>=16.4 µm2
• Assume spherical (?!) grains, <A> mean intercept area= 2/3πr2
d = 2√(3<A>/2π)= 5.6 µm.
[Underwood]
73
Objective
Grain Size
Varistors
Hall-Petch
Creep
SV and 2nd phase particles• Convex particles:= any two points on particle
surface can be connected by a wholly internal line.
• Sometimes it is easier to count the number of particles intercepted along a line, NL; then the number of surface points is double the particle number. Also applies to non-convex particles if interceptions counted.
Sv = 4NL (2.32)
74
Objective
Grain Size
Varistors
Hall-Petch
Creep
S:V and Mean Intercept Length
• Mean intercept length from intercepts of particles of alpha phase:
<L3> = 1/N i (L3)i (2.33)
• Can also be obtained as:<L3> = LL/NL (2.34)
• Substituting: <L3> = 4VV/SV,
(2.35)where fractions refer to alpha phase only.
75
Objective
Grain Size
Varistors
Hall-Petch
Creep
S:V example: sphere
• For a sphere, the volume:surface ratio is D[iameter]/6.
• Thus <L3>sphere = 2D/3.
• In general we can invert the relationship to obtain the surface:volume ratio, if we know (measure) the mean intercept:
<S/V>alpha = 4/<L3> (2.38)
76
Objective
Grain Size
Varistors
Hall-Petch
Creep
Table 2.2
<L3>:= mean intercept length, 3D objects
<V>:= mean volume
l := length (constant) of test lines superimposed on structure
p:= number of (end) points of l-lines in phase of interest
LT:= test line length [Underwood]
77
Objective
Grain Size
Varistors
Hall-Petch
Creep
Grain size measurement: intercepts
• From Table 2.2 [Underwood], column (a), illustrates how to make a measurement of the mean intercept length, based on the number of grains per unit length of test line.
<L3> = 1/NL
• Important: use many test lines that are randomly oriented w.r.t. the structure.
• Assuming spherical (?!) grains, <L3> = 4r/3, [Underwood, table 4.1], if LT= 25µm, LTNL= 5 d = 6<L3>/4 = 6/NL4 = 6*5/4 = 7.5µm.
78
Objective
Grain Size
Varistors
Hall-Petch
Creep
Particles and Grains• “Where the rubber meets the road”, in
stereology, that is!• Mean free distance, := uninterrupted
interparticle distance through the matrix averaged over all pairs of particles (in contrast to interparticle distance for nearest neighbors only).
λ =1−VV
(α )
NL
(4.7)
Number of interceptions with particles is same asnumber of interceptions with the matrix. Thus linealfraction of occupied by matrix is NL, equal to thevolume fraction, 1-VV-alpha.
79
Objective
Grain Size
Varistors
Hall-Petch
Creep
Mean Random Spacing
• The number of interceptions with particles per unit test length = NL = PL/2. Reciprocal of this quantity is the mean random spacing, which is the mean uninterrupted center-to-center length between all possible pairs of particles. Thus,the particle mean intercept length, <L3>:
<L3> = [mm] (4.8)
80
Objective
Grain Size
Varistors
Hall-Petch
Creep
Particle Relationships• Application: particle
coarsening in a 2-phase material; strengthening of solid against dislocation flow.
• Eqs. 4.9-4.11, with LA=πPL/2=πNL= πSV/4
• dimension: lengthunits (e.g.): mm
• Use to calculate critical resolved shear stresses.
• Alternate notation for 3rd eq.: = d (1-f) / f
L3 =4VV
(α )
SV(α )
λ =41−VV
(α)
SV(α )
λ = L31−VV
(α )
VV(α )
λ =π1−VV
(α)
LA(α)
81
Objective
Grain Size
Varistors
Hall-Petch
Creep
Nearest-Neighbor Distances
• Also useful are distances between nearest neighbors: S. Chandrasekhar, “Stochastic problems in physics and astronomy”, Rev. Mod. Physics, 15, 83 (1943).
• 2D: ∆2 = 0.5 / √PA
(4.18a)
• 3D: ∆3 = 0.554 (PV)-1/3 (4.18)
• Based on ~1/NL, ∆3 0.554 (πr2 )1/3
for small VV, ∆2 0.500 (π/2 r)1/2
82
Objective
Grain Size
Varistors
Hall-Petch
Creep
Application of ∆2
• Percolation of dislocation lines through arrays of 2D point obstacles.
• Caution! “Spacing” has many interpretations: select the correct one!
Hull & Bacon;fig 10.17
83
Objective
Grain Size
Varistors
Hall-Petch
Creep
Measurement of Regularly Shaped Particles
• Purpose: how can we relate measurements in plane sections to what we know of the geometry of regularly shaped objects with a distribution of sizes?
• In general, the mean intercept length is not equal to the grain diameter, for example! Also, the proportionality factors depend on the (assumed) shape.
84
Objective
Grain Size
Varistors
Hall-Petch
Creep
Sections through dispersions of
spherical objectsEven mono-disperse spheresexhibit a variety of diametersin cross section.Only if you know that the second phase is monodispersemay you measure diameterfrom maximum cross-section!