Defectsareimperfectionsinthestructure.Theymaybeone-dimensional pointdefects(Fig.2.6),linedefects(Fig.2.7),two-dimensionalplane defects(Fig.2.8),orthree-dimensionalvolumedefectssuchasinclusions orporosity,Fig.1.16(d).Thedifferenttypesofdefectsaredescribedbriefly in this section. 2.3.1 One-Dimensional Point Defects One-dimensional point defects [Fig. 2.6) may include vacancies [Fig. 2.6(a)], interstitials [Figs 2.6(a) and 2.6(b)], solid solution elements [Fig. 2.6(b)], and pairs or clusters of the foregoing, Fig. 2.6(c). Pairs of ions (Frenkel defects) or vacancies (Schottky defects) are often required to maintain charge neu- trality, Fig. 2.6(c). Point defects can diffuse through a lattice, especially at temperatures above approximately 0.3–0.5 of the absolute melting tempera- ture. If the movement of point defects produces a net state change, it causes thermally activated stress-induced deformation, such as creep. The diffusion of point defects such as vacancies may also lead to the growth of grains in a polycrystalline material. 2.3.2 Line Defects Line defects consist primarily of dislocations, typically at the edges of patches where part of a crystallographic plane has slipped by one lattice F IGURE 2.6 Examples of point defects: (a)] vacancy and interstitial elements; (b) substitutional element and interstitial impurity element; (c) pairs of ions and vacancies. [(a) and (c) are adapted from Shackleford, 1996—reprinted with permission from Prentice-Hall; (b) is adapted from Hull and Bacon, 1984. Reprinted with permission from Pergamon Press.] Copyright © 2003 Marcel Dekker, Inc. spacing (Fig. 2.7). The two pure types of dislocations are edge and screw, Figs 2.7(a) and 2.7(b). Edge dislocations have slip (Burgers) vectors perpen- dicular to the dislocation line [Fig. 2.7a)], while screw dislocations have translation vectors parallel to the dislocation line, Fig. 2.7(b). In general, however, most dislocations are mixed dislocations that consist of both edge and screw dislocation components, Fig. 2.7(c). Note that the line segments along the curved dislocation in Fig. 2.7(c) have both edge and screw com- ponents. However, the deflection segments are either pure edge or pure screw at either end of the curved dislocation, Fig. 2.7(c). F IGURE 2.7 Examples of line defects: (a) edge dislocations; (b) screw disloca- tions; (c) mixed dislocations. (Adapted from Hull and Bacon, 1980. Reprinted with permission from Pergamon Press.) Copyright © 2003 Marcel Dekker, Inc. 2.3.3 Surface Defects Surface defects are two-dimensional planar defects (Fig. 2.8). They may be grain boundaries, stacking faults, or twin boundaries. These are surface boundaries across which the perfect stacking of atoms within a crystalline lattice changes. High- or low-angle tilt or twist boundaries may involve changes in the crystallographic orientations of adjacent grains, Figs 2.8(a) and 2.8(b). The orientation change across the boundary may be described using the concept of coincident site lattices. For example, a Æ ¼ 5or F IGURE 2.8 Examples of surface defects: (a) low-angle tilt boundary; (b) high- angle tilt boundary; (c) S ¼ 5 boundary; (d) twin boundary; (e) intrinsic stack- ing fault; (f) extrinsic stacking fault. (Adapted from Shackleford, 1996. Reprinted with permission from Prentice-Hall.) Copyright © 2003 Marcel Dekker, Inc. Æ À1 ¼1=5boundaryisoneinwhich1in5ofthegrainboundaryatoms match,asshowninFig.2.8(c). Twinboundariesmayformwithincrystals.Suchboundarieslieacross deformationtwinplanes,asshowninFig.2.8(d).Notethattheatomson eithersideofthetwinplanesaremirrorimages.Stackingfaultsmayalsobe formedwhentheperfectstackinginthecrystallinestackingsequenceis disturbed,Figs2.8(e)and2.8(f).Thesemaybethoughtofastheabsence ofaplaneofatoms(intrinsicstackingfaults)ortheinsertionofrowsof atomsthatdisturbthearrangementofatoms(extrinsicstackingfaults). IntrinsicandextrinsicstackingfaultsareillustratedschematicallyinFigs 2.8(e)and2.8(f),respectively.NotehowtheperfectABCABCstackingof atomsisdisturbedbytheinsertionorabsenceofrowsofatoms. 2.3.4VolumeDefects Volumedefectsareimperfectionssuchasvoids,bubble/gasentrapments, porosity,inclusions,precipitates,andcracks.Theymaybeintroducedintoa solidduringprocessingorfabricationprocesses.Anexampleofvolume defectsispresentedinFig.2.9.ThisshowsMnSinclusionsinanA707 steel.AnotherexampleofavolumedefectispresentedinFig.1.16(d). This shows evidence of $1–2 vol % of porosity in a molybdenum disilicide composite. Such pores may concentrate stress during mechanical loading. Volume defects can grow or coalesce due to applied stresses or temperature fields. The growth of three-dimensional defects may lead ultimately to cat- astrophic failure in engineering components and structures. F IGURE 2.9 MnS inclusions in an A707 steel. (Courtesy of Jikou Zhou.) Copyright © 2003 Marcel Dekker, Inc. 2.4THERMALVIBRATIONSAND MICROSTRUCTURALEVOLUTION Asdiscussedearlier,atomsinacrystallinesolidarearrangedintounitsthat arecommonlyreferredtoasgrains.Thegrainsizemaybeaffectedbythe controlofprocessingandheattreatmentconditions.Grainsmayvaryinsize fromnanoscale($10–100nm)tomicroscale($1–100m),ormacroscale ($1–10mm).SomeexamplesofmicrostructuresarepresentedinFigs 1.13(a–d).Notethatthemicrostructuremayconsistofsinglephases[Fig. 1.13(a)]ormultiplephases[Figs1.13(b–d)].Microstructuresmayalso change due to diffusion processes that occur at temperatures above the so-called recrystallization temperature, i.e., above approximately 0.3–0.5 of the melting temperature in degrees Kelvin. Since the evolution of microstructure is often controlled by diffusion processes, a brief introduction to elementary aspects of diffusion theory is presented in this section. This will be followed by a simple description of phase nucleation and grain growth. The kinetics of phase nucleation and growth and growth in selected systems of engineering significance will be illustrated using transformation diagrams. Phase diagrams that show the equilibrium proportions of constituent phases will also be introduced along with some common transformation reactions. 2.4.1 Statistical Mechanics Background At temperatures above absolute zero (0 K), the atoms in a lattice vibrate about the equilibrium positions at the so-called Debye frequency, ,of $ 10 13 s À1 . Since the energy required for the lattice vibrations is supplied thermally, the amplitudes of the vibration increase with increasing tempera- ture. For each individual atom, the probability that the vibration energy is greater than q is given by statistical mechanics to be P ¼ e Àq=kT ð2:7Þ where k is the Boltzmann constant (1:38  10 À23 JÁatom À1 K À1 ) and T is the absolute temperature in degrees Kelvin. The vibrating lattice atoms can only be excited into particular quantum states, and the energy, q, is given simply by Planck’s law (q ¼ h). Also, at any given time, the vibrational energy varies statistically from atom to atom, and the atoms continuously exchange energy as they collide with each other due to atomic vibrations. Nevertheless, the average energy of the vibrating atoms in a solid is given by statistical mechanics to be 3kT at any given time. This may be sufficient to promote the diffusion of atoms within a lattice. Copyright © 2003 Marcel Dekker, Inc. 2.4.2 Diffusion Diffusion is the thermally- or stress-activated movement of atoms or vacan- cies from regions of high concentration to regions of low concentration (Shewmon, 1989). It may occur in solids, liquids, or gases. However, we will restrict our attention to considerations of diffusion in solids in the current text. Consider the interdiffusion of two atomic species A and B shown schematically in Fig. 2.10; the probability that n A atoms of A will have energy greater than or equal to the activation barrier, q, is given by n A e Àq=kT . Similarly, the probability that n B atoms of B will have energy greater than or equal to the activation barrier is given by n B e Àq=kT . Since the atoms may move in any of six possible directions, the actual frequency in any given direction is =6. The net number of diffusing atoms, n, that move from A to B is thus given by n d ¼  6 ðn A À n B Þe Àq=KT ð2:8Þ If the diffusion flux, J, is defined as the net number of diffusing atoms, n d , per unit area, i.e., J ¼ n d =ðl 1 l 2 Þ, and the concentration gradient, dC=dx, F IGURE 2.10 Schematic illustration of diffusion: activation energy required to cross a barrier. (Adapted from Ashby and Jones, 1994. Reprinted with per- mission from Pergamon Press). Copyright © 2003 Marcel Dekker, Inc. whichisgivensimplybyÀðC A ÀC B Þ=r 0 ,thediffusionflux,J,maythenbe expressedas J¼D 0 exp Àq kT  dC dx  ð2:9Þ IfwescalethequantityqbytheAvogadronumber,thentheenergyterm becomesQ¼N A qandR¼kN A .Equation(2.9)maythusbeexpressedas J¼ÀD 0 exp ÀQ RT  dC dx  ð2:10Þ IfwenowsubstituteD¼ÀD 0 exp ÀQ RT  intoEq.(2.10),weobtainthe usualexpressionforJ,i.e.,Jisgivenby J¼ÀD dC dx ð2:11Þ TheaboveexpressionisFick’sfirstlawofdiffusion.Itwasfirstpro- posedbyAdolfHicksin1855.Itisimportanttonoteherethatthediffusion coefficientforself-diffusion,D,canhaveastrongeffectonthecreepproper- ties,i.e.,thetime-dependentflowofmaterialsattemperaturesgreaterthan $0.3–0.5ofthemeltingtemperatureindegreesKelvin.Also,theactivation energy,Q,inEq.(2.10)isindicativeoftheactualmechanismofdiffusion, whichmayinvolvethemovementofinterstitialatoms[Fig.2.11(a)]and vacancies[Fig.2.11(b)]. Diffusionmayalsooccuralongfastdiffusionpathssuchasdislocation pipesalongdislocationcores[Fig.2.12(a)]orgrainboundaries[Fig.2.12(b)]. This is facilitated in materials with small grain sizes, d g , i.e., a large number of grain boundaries per unit volume. However, diffusion in most crystalline materials occurs typically by vacancy movement since the activation ener- gies required for vacancy diffusion ($1 eV) are generally lower than the activation energies required for interstitial diffusion ($2–4 eV). The activa- tion energies for self-diffusion will be shown later to be consistent with activation energies from creep experiments. 2.4.3 Phase Nucleation and Growth The random motion of atoms and vacancies in solids, liquids, and gases are associated with atomic collisions that may give rise to the formation of small embryos or clusters of atoms, as shown in Figs 2.13(a) and 2.13(b). Since the initial free-energy change associated with the initial formation and growth of such clusters is positive (Read-Hill and Abbaschian, 1992), the initial clusters of atoms are metastable. The clusters may, therefore, disintegrate due to the effects of atomic vibrations and atomic collisions. However, a Copyright © 2003 Marcel Dekker, Inc. F IGURE 2.11 Schematic illustration of diffusion mechanisms: (a) movement of interstitial atoms; (b) vacancy/solute diffusion. (Adapted from Shewmon, 1989. Reproduced with permission from the Minerals, Metals, and Materials Society.) F IGURE 2.12 Fast diffusion mechanisms: (a) dislocation pipe diffusion along dislocation core; (b) grain boundary diffusion. (Adapted from Ashby and Jones, 1980. Reprinted with permission from Pergamon Press.) Copyright © 2003 Marcel Dekker, Inc. statistical number of clusters or embryos may grow to a critical size, beyond which further growth results in a lowering of the free energy. Such clusters may be considered stable, although random atomic jumps may result in local transitions in cluster size to dimensions below the critical cluster dimension. Beyond the critical cluster size, the clusters of atoms may be consid- ered as nuclei from which new grains can grow primarily as a result of atomic diffusion processes, Figs 2.13(c) and 2.13(d). The nuclei grow until the emerging grains begin to impinge on each other, Fig. 2.13(e). The growth results ultimately in the formation of a polycrystalline structure, Fig. 2.13(f). Subsequent grain growth occurs by interdiffusion of atoms and vacan- cies across grain boundaries. However, grain growth is mitigated by inter- stitial and solute ‘‘atmospheres’’ that tend to exert a drag on moving grain boundaries. Grain growth is also associated with the disappearance of smal- ler grains and the enhanced growth of larger grains. Due to the combined effects of these factors, a limiting grain size is soon reached. The rate at which this limiting grain size is reached depends on the annealing duration and the amount of prior cold work introduced during deformation proces- sing via forging, rolling, swaging, and/or extrusion. F IGURE 2.13 Schematic illustration of nucleation and growth: (a, b) formation of embryos; (c,d) nuclei growth beyond critical cluster size; (e) impingement of growing grains; (f) polycrystalline structure. (Adapted from Altenpohl, 1998.) Copyright © 2003 Marcel Dekker, Inc. The simple picture of nucleation and growth presented above is gen- erally observed in most crystalline metallic materials. However, the rate of nucleation is generally enhanced by the presence of pre-existing nuclei such as impurities on the mold walls, grain boundaries, or other defects. Such defects make it much easier to nucleate new grains heterogeneously, in contrast to the difficult homogeneous nucleation schemes described earlier. In any case, the nuclei may grow by diffusion across grain boundaries to form single-phase or multi-phase microstructures, such as those shown in Fig. 1.13. A simple model of grain growth may be developed by using an analogy of growing soap bubbles. We assume that the growth of the soap bubbles (analogous to grains) is driven primarily by the surface energy of the bubble/ grain boundaries. We also assume that the rate of grain growth is inversely proportional to the curvature of the grain boundaries, and that the curva- ture itself is inversely proportional to the grain diameter. We may then write: dðDÞ=dt ¼ k =d ð2:12Þ where D is the average grain size, t is time elapsed, and k is a proportionality constant. Separating the variables and integrating Eq. (2.12) gives the following expression: D 2 ¼ kt þ c ð2:13Þ where c is a constant of integration. For an initial grain size of D 0 at time t ¼ 0, we may deduce that c ¼ D 2 0 . Hence, substituting the value of c into Eq. (2.13) gives D 2 À D 2 0 ¼ kt ð2:14Þ Equation (2.14) has been shown to fit experimental data obtained for the growth of soap bubbles under surface tension forces. Equation (2.14) has also been shown to fit the growth behavior of metallic materials when grain growth is controlled by surface energy and the diffusion of atoms across the grain boundaries. In such cases, the constant k in Eq. (2.14) exhibits an exponential dependence which is given by k ¼ k 0 expðÀQ=RT Þð2:15Þ where k 0 is an empirical constant, Q is the activation energy for the grain growth process, T is the absolute temperature, and R is the universal gas constant. By substituting Eq. (2.15) into Eq. (2.14), the grain growth law may be expressed as D 2 À D 2 0 ¼ tk 0 expðÀQ=RT Þð2:16Þ Copyright © 2003 Marcel Dekker, Inc. . disturbed,Figs2.8(e)and2.8(f).Thesemaybethoughtofastheabsence ofaplaneofatoms(intrinsicstackingfaults)ortheinsertionofrowsof atomsthatdisturbthearrangementofatoms(extrinsicstackingfaults) primarily of dislocations, typically at the edges of patches where part of a crystallographic plane has slipped by one lattice F IGURE 2.6 Examples of point