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## Microstructure-Properties: I Lecture 6: The Effect of Grain Size?

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**Microstructure-Properties: ILecture 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 • 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**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.**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!).**Examples of Varistor Circuit Components**Electroceramics**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**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**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**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.**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**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.**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, ~ 106m. • 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.**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). p-type n-type**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. Electron Energy Electron Energy Ee Ee Conduction band BandGap BandGap valence band p-type n-type**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. Junction of p- & n-types Ee Electron Energy BandGap p-type n-type**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) ~0.8V Ee p-type n-type**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**Conductivity in a semiconductor**• Typical values for n-type doped silicon (subscript “n” denotes quantity in n-type):majority carrier concentration, nn, = 1022electrons.m-3 mobility, µn, = 0.35 m2V-1s-1 and charge,e, = -1.6.10-19C.minority carrier concentration, pn, = 2.3.1010holes.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**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]**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. p-type n-type Forward bias - + Forward bias Reverse bias**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).**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]**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**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**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.**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, a, is a measure of varistor quality. Electroceramics**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.**Typical Varistor Application**Electroceramics**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:sy = s0 + 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.**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]**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.**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]**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**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).**Nanocrystalline materials**• Limitations of nanocrystalline materials?**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,0is the resistance to dislocation motion in each grain, andappliedis 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 * √rd-1/2 = 0 + kd-1/2.**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.**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.**Diffusivity, Activation Energy**Porter & Easterling:diffusivity at the meltingpoint is constant for a given class ofmaterial. Similarly, the activation energynormalized by RTmis constant for a given class ofmaterial.**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.**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]**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.**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.**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.**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-{sW/kT}. Compression decreases the concentration (slightly) and tension raises it. • Given small stresses (1-50 MPa),W~10-29m3, kT~1.4.10-20J,sW/kT~0.02 «1. This permits us to linearize the driving force.**N-H Creep: derivation: 2**• Given small stresses (1-50 MPa),W~10-29m3, kT~1.4.10-20J,sW/kT~0.02 «1. This permits us to linearize the driving force.∆NV ~sW/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 (sW/kT)(1/d) exp-{Qm/kT}.**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}.dV/dt = JV d2dV/dt = d2D0Vexp-{Qm+QVacancy/kT} (sW/kT)(1/d).**N-H Creep: derivation: 4**• Rate of change in length = volume rate (dV/dt) / area; area ~ d2.dd/dt = D0Vexp-{Qm+QVacancy/kT} (sW/kT)(1/d). • Strain = Change in length / length =dd/d.dd/dt = D0Vexp-{Qm+QVacancy/kT} (sW/kT)(1/d2). • Collecting terms: Note the grain size dependence!**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.**Deformation Mechanism Maps**Deformation maps provide a convenient graphical view of the different regimes of deformation behavior as a function of temperature and stress.