Materials Science and Engineering: Imperfections of Solids PDF

E N G G 4 1 2 : 3 M A T E R I A L S S C I E N C E A N D E N G I N E E R I N G Chapter Imperfections of Solids...

E N G G 4 1 2 : 3 M A T E R I A L S S C I E N C E A N D E N G I N E E R I N G Chapter Imperfections of Solids Intended Learning Outcomes After studying this chapter, you should be able to do the following: 1. Describe both vacancy and self-interstitial crystalline defects. 2. Calculate the equilibrium number of vacancies in a material at some specified temperature, given the relevant constants. 3. Name the two types of solid solutions and provide a brief written definition and/or schematic sketch of each. 4. Given the masses and atomic weights of two or more elements in a metal alloy, calculate the weight percent and atom percent for each element. 5. For each of edge, screw, and mixed dislocations: (a) describe and make a drawing of the dislocation, (b) note the location of the dislocation line, and (c) indicate the direction along which the dislocation line extends. 6. Describe the atomic structure within the vicinity of (a) a grain boundary and (b) a twin boundary This chapter primarily presents the various types of defects, classified on the basis of their geometry, which is realistic as defects are disrupted region in a volume of a solid. The present discussion is devoted to describe both vacancy and self- interstitial defects. Within this framework, types of solid solutions were enumerated together with its illustration. For dislocations, this chapter will differentiate and elaborate edge, screw and mixed dislocations. POINT DEFECTS VACANCIES AND SELF-INTERSTITIALS The simplest of the point defects is a vacancy, or vacant lattice site, one normally occupied but from which an atom is missing (Figure 3.1). Module No. 3 – Imperfections of Solids 1 E N G G 4 1 2 : M A T E R I A L S S C I E N C E A N D E N G I N E E R I N G Figure 3.1. Two-dimensional representations of a vacancy and a self-interstitial. (Adapted from W. G. Moffatt, G. W. Pearsall, and J. Wulff, The Structure and Properties of Materials, Vol. I, Structure, p. 77. Copyright © 1964 by John Wiley & Sons, New York, NY. Reprinted by permission of John Wiley & Sons, Inc.) All crystalline solids contain vacancies, and, in fact, it is not possible to create such a material that is free of these defects. The necessity of the existence of vacancies is explained using principles of thermodynamics; in essence, the presence of vacancies increases the entropy (i.e., the randomness) of the crystal. The equilibrium number of vacancies Ny for a given quantity of material (usually per meter cubed) depends on and increases with temperature according to 𝑄𝑣 𝑁𝑣 = 𝑁 exp(− ) 𝑘𝑡 Where N = total number of atomic sites (m3) 𝑄𝑣 = energy required for the formation of a vacancy (J/mol or eV/atom) T = absolute temperature in Kelvin k = gas or Boltzmann’s constant. The value of k is 1.38 1023 J/atom K A self-interstitial is an atom from the crystal that is crowded into an interstitial site—a small void space that under ordinary circumstances is not occupied. This kind of defect is also represented in Figure 3.1. In metals, a self-interstitial introduces relatively large distortions in the surrounding lattice because the atom is substantially larger than the interstitial position in which it is situated. Consequently, the formation of this defect is not highly probable, and it exists in very small concentrations that are significantly lower than for vacancies. Module No. 3 – Imperfections of Solids 2 E N G G 4 1 2 : M A T E R I A L S S C I E N C E A N D E N G I N E E R I N G Example Problem 3.1 Number-of-Vacancies Computation at a Specified Temperature Calculate the equilibrium number of vacancies per cubic meter for copper at 1000°C. The energy for vacancy formation is 0.9 eV/atom; the atomic weight and density (at 1000°C) for copper are 63.5 g/mol and 8.4 g/cm 3, respectively. 𝑁𝐴 𝜌 𝑁= 𝐴𝐶𝑢 𝑔 6.022 𝑥 1023 𝑎𝑡𝑜𝑚𝑠/𝑚𝑜𝑙)(8.4 3 )(106 𝑐𝑚3 /𝑚3 ) = 𝑐𝑚 𝑔 63.5 𝑚𝑜𝑙 𝑎𝑡𝑜𝑚𝑠 = 8 𝑥 1028 𝑚3 Thus, the number of vacancies at 1000°C (1273 K) is equal to 𝑄𝑣 𝑁𝑣 = 𝑁 exp(− ) 𝑘𝑡 (0.9 𝑒𝑉) = 8 𝑥 1028 𝑎𝑡𝑜𝑚𝑠/𝑚3 ) 𝑒𝑥𝑝 ⌈ 10−5𝑒𝑉 ⌉ (8.62 𝑋 𝐾 )(1273 𝐾) 𝒗𝒂𝒄𝒂𝒏𝒄𝒊𝒆𝒔 = 2.2 x𝟏𝟎𝟐𝟓 𝒎𝟑 IMPURITIES IN SOLIDS The addition of impurity atoms to a metal results in the formation of a solid solution and/or a new second phase, depending on the kinds of impurity, their concentrations, and the temperature of the alloy. With regard to alloys, solute and solvent are terms that are commonly employed. Solvent is the element or compound that is present in the greatest amount; on occasion, solvent atoms are also called host atoms. Solute is used to denote an element or compound present in a minor concentration. Module No. 3 – Imperfections of Solids 3 E N G G 4 1 2 : M A T E R I A L S S C I E N C E A N D E N G I N E E R I N G Figure 3.2. Two-dimensional schematic representations of substitutional and interstitial impurity atoms. (Adapted from W. G. Moffatt, G. W. Pearsall, and J. Wulff, The Structure and Properties of Materials, Vol. I, Structure, p. 77. Copyright © 1964 by John Wiley & Sons, New York, NY. Reprinted by permission of John Wiley & Sons, Inc.) Solid Solutions A solid solution forms when, as the solute atoms are added to the host material, the crystal structure is maintained and no new structures are formed. If two liquids that are soluble in each other (such as water and alcohol) are combined, a liquid solution is produced as the molecules intermix, and its composition is homogeneous throughout. A solid solution is also compositionally homogeneous; the impurity atoms are randomly and uniformly dispersed within the solid. Impurity point defects are found in solid solutions, of which there are two types: substitutional and interstitial. For the substitutional type, solute or impurity atoms replace or substitute for the host atoms (Figure 4.2). Several features of the solute and solvent atoms determine the degree to which the former dissolves in the latter. These are expressed as four Hume–Rothery rules, as follows: 1. Atomic size factor. Appreciable quantities of a solute may be accommodated in this type of solid solution only when the difference in atomic radii between the two atom types is less than about >15%. Otherwise, the solute atoms create substantial lattice distortions and a new phase form. 2. Crystal structure. For appreciable solid solubility, the crystal structures for metals of both atom types must be the same. 3. Electronegativity factor. The more electropositive one element and the more electronegative the other, the greater the likelihood that they will form an intermetallic compound instead of a substitutional solid solution. 4. Valences. Other factors being equal, a metal has more of a tendency to dissolve another metal of higher valency than to dissolve one of a lower valency. Module No. 3 – Imperfections of Solids 4 E N G G 4 1 2 : M A T E R I A L S S C I E N C E A N D E N G I N E E R I N G For interstitial solid solutions, impurity atoms fill the voids or interstices among the host atoms (see Figure 3.2). For both FCC and BCC crystal structures, there are two types of interstitial sites—tetrahedral and octahedral—these are distinguished by the number of nearest neighbor host atoms—that is, the coordination number. Tetrahedral sites have a coordination number of 4; straight lines drawn from the centers of the surrounding host atoms form a four-sided tetrahedron. However, for octahedral sites the coordination number is 6; an octahedron is produced by joining these six sphere centers.3 For FCC, there are two types of octahedral sites with representative 1 1 1 1 point coordinates of 0 1 and. Representative coordinates for a single tetrahedral site type 2 2 2 2 131 are Locations of these sites within the FCC unit cell are noted in Figure 3.3a. One type of each 444 of octahedral and tetrahedral interstitial sites is found for BCC. Representative coordinates are 1 11 as follows: octahedral, 1 0 and tetrahedral, 1. Figure 3.3b shows the positions of these sites 2 24 within a BCC unit cell. Figure 3.3. Locations of tetrahedral and octahedral interstitial sites within (a) FCC and (b) BCC unit cells. Example Problem 3.2 Computation of Radius of BCC Interstitial Site Compute the radius r of an impurity atom that just fits into a BCC octahedral site in terms of the atomic radius R of the host atom (without introducing lattice strains). Solution As Figure 3.3b notes, for BCC, the octahedral interstitial site is situated at the center of a unit cell edge. In order for an interstitial atom to be positioned in this site without introducing lattice strains, the atom just touches the two adjacent host atoms, which are corner atoms of the unit cell. The drawing shows atoms on the (100) face of a BCC unit cell; the large circles represent the host atoms—the small circle represents an interstitial atom that is positioned in an octahedral site on the cube edge. Module No. 3 – Imperfections of Solids 5 E N G G 4 1 2 : M A T E R I A L S S C I E N C E A N D E N G I N E E R I N G On this drawing is noted the unit cell edge length—the distance between the centers of the corner atoms—which, from Equation 3.4, is equal to 4𝑅 𝑈𝑛𝑖𝑡 𝑐𝑒𝑙𝑙 𝑒𝑑𝑔𝑒 𝑙𝑒𝑛𝑔𝑡ℎ = √3 Also shown is that the unit cell edge length is equal to two times the sum of host atomic radius 2R plus twice the radius of the interstitial atom 2r; i.e., 𝑛𝑖𝑡 𝑐𝑒𝑙𝑙 𝑒𝑑𝑔𝑒 𝑙𝑒𝑛𝑔𝑡ℎ = 2𝑅 + 2𝑟 Now, equating these two-unit cell edge length expressions, we get 4𝑅 2R + 2r = √3 and solving for r in terms of R 4𝑅 2 2𝑟 = − 2𝑅 = ( − 1) (2𝑅) √3 √3 2 𝑟 = ( − 1) 𝑅 − 0.155𝑅 √3 SPECIFICATION OF COMPOSITION or It is often necessary to express the composition (or concentration) 5 of an alloy in terms of its constituent elements. The two most common ways to specify composition are weight (or mass) percent and atom percent. The basis for weight percent (wt%) is the weight of a particular element relative to the total alloy weight. For an alloy that contains two hypothetical atoms denoted by 1 and 2, the concentration of 1 in wt%, 𝐶1 , is defined as 𝑚1 𝐶1 = 𝑋 100 𝑚1+ 𝑚2 where 𝑚1 and 𝑚2 represent the weight (or mass) of elements 1 and 2, respectively. The basis for atom percent (at%) calculations is the number of moles of an element in relation to the total moles of the elements in the alloy. The number of moles in some specified mass of a hypothetical element 1, 𝑛𝑚1 , may be computed as follows: Module No. 3 – Imperfections of Solids 6 E N G G 4 1 2 : M A T E R I A L S S C I E N C E A N D E N G I N E E R I N G 𝑚′1 𝑛𝑚1 = 𝐴1 Here, 𝑚′1 and 𝐴1 denote the mass (in grams) and atomic weight, respectively, for element 1. Concentration in terms of atom percent of element 1 in an alloy containing element 1 and element 2 atoms, 𝐶′1 is defined by 𝑛𝑚1 𝐶′1 = 𝑥100 𝑛𝑚1 + 𝑛𝑚2 Composition Conversions It is necessary to convert from one composition scheme to another—for example, from weight percent to atom percent (i.e., weight percent denoted by C 1 and C2, atom percent by C’1 and C’2, and atomic weights as A1 and A2). 𝐶1 𝐴2 𝐶′1 = 𝑥 100 𝐶1 𝐴2 + 𝐶2 𝐴1 𝐶2 𝐴1 𝐶′2 = 𝑥 100 𝐶1 𝐴2 + 𝐶2 𝐴1 In addition, it sometimes becomes necessary to convert concentration from weight percent to mass of one component per unit volume of material (i.e., from units of wt% to kg/m 3). Concentrations in terms of this basis are denoted using a double prime (i.e., C 1’’ and C2’’), and the relevant equations are as follows: 𝐶1 𝐶1′′ = ( ) 𝑥 103 𝐶1 𝐶2 + 𝜌1 𝜌2 𝐶2 𝐶2′′ = ( ) 𝑥 103 𝐶1 𝐶2 + 𝜌1 𝜌2 Furthermore, to determine the density and atomic weight of a binary alloy, given the composition in terms of either weight percent or atom percent. If we represent alloy density and atomic weight by 𝜌𝑎𝑣𝑒 and 𝐴𝑎𝑣𝑒 , respectively, then 100 𝜌𝑎𝑣𝑒 = 𝐶1 𝐶2 + 𝜌1 𝜌2 𝐶′1 𝐴1 𝜌𝑎𝑣𝑒 = 𝐶′1 𝐴1 𝐶′2 𝐴2 + 𝜌1 𝜌2 Module No. 3 – Imperfections of Solids 7 E N G G 4 1 2 : M A T E R I A L S S C I E N C E A N D E N G I N E E R I N G 100 𝐴𝑎𝑣𝑒 = 𝐶1 𝐶 + 2 𝐴1 𝐴2 𝐶 ′ 1 𝐴1 + 𝐶 ′ 2 𝐴′ 2 𝐴𝑎𝑣𝑒 = 100 Example Problem 3.3 Composition Conversion—From Weight Percent to Atom Percent Determine the composition, in atom percent, of an alloy that consists of 97 wt% aluminum and 3 wt% copper. Solution If we denote the respective weight percent compositions as C Al = 97 and CCu = 3, 𝐶𝐴𝑙 𝐴𝐶𝑢 𝐶 ′𝐴𝑙 = 𝑥 100 𝐶𝐴𝑙 𝐴𝐶𝑢 + 𝐶𝐶𝑢 𝐴𝐴𝑙 𝑔 (97) (63.55 ) = 𝑚𝑜𝑙 𝑔 𝑔 𝑥100 (97) (63.55 ) + (3) (26.98 ) 𝑚𝑜𝑙 𝑚𝑜𝑙 = 98.7 at% DISLOCATIONS—LINEAR DEFECTS A dislocation is a linear or one-dimensional defect around which some of the atoms are misaligned. One type of dislocation is represented in Figure 3.4: an extra portion of a plane of atoms, or half-plane, the edge of which terminates within the crystal. This is termed an edge dislocation; it is a linear defect that centers on the line that is defined along the end of the extra half-plane of atoms. This is sometimes termed the dislocation line, which, for the edge dislocation in Figure 4.4, is perpendicular to the plane of the page. Within the region around the dislocation line there is some localized lattice distortion. The atoms above the dislocation line in Figure 4.4 are squeezed together, and those below are pulled apart; this is reflected in the slight curvature for the vertical planes of atoms as they bend around this extra half-plane. The magnitude of this distortion decreases with distance away from the dislocation line; at positions far removed, the crystal lattice is virtually perfect. Dislocations more commonly originate during plastic deformation, during solidification, and as a consequence of thermal stresses that result from rapid cooling. Edge dislocation arises when there is a slight mismatch in the orientation of adjacent parts of the growing crystal. A screw dislocation allows easy crystal growth because additional atoms can be added to the ‘step’ of the screw. Thus, the term screw is apt, because the step swings around the axis as growth proceeds. Unlike point defects, these are not thermodynamically stable. They can be removed by heating to high temperatures where they cancel each other or move out through the crystal to its surface. Module No. 3 – Imperfections of Solids 8 E N G G 4 1 2 : M A T E R I A L S S C I E N C E A N D E N G I N E E R I N G Virtually all crystalline materials contain some dislocations. The density of dislocations in a crystal is measures by counting the number of points at which they intersect a random cross-section of the crystal. These points, called etch-pits, can be seen under microscope. In an annealed crystal, 8 10 -2 the dislocation density is the range of 10 -10 m. Figure 3.4 The atom positions around an edge dislocation; extra half-plane of atoms shown in perspective. (Adapted from A. G. Guy, Essentials of Materials Science, McGraw-Hill Book Company, New York, NY, 1976, p. 153.) Another type of dislocation, called a screw dislocation, may be thought of as being formed by a shear stress that is applied to produce the distortion shown in Figure 3.5a: the upper front region of the crystal is shifted one atomic distance to the right relative to the bottom portion. The atomic distortion associated with a screw dislocation is also linear and along a dislocation line, line AB in Figure 3.5b. The screw dislocation derives its name from the spiral or helical path or ramp that is traced around the dislocation line by the atomic planes of atoms. Module No. 3 – Imperfections of Solids 9 E N G G 4 1 2 : M A T E R I A L S S C I E N C E A N D E N G I N E E R I N G Figure 3.5 (a) A screw dislocation within a crystal. (b) The screw dislocation in (a) as viewed from above. The dislocation line extends along line AB. Atom positions above the slip plane are designated by open circles, those below by solid circles. [Figure (b) from W. T. Read, Jr., Dislocations in Crystals, McGrawHill Book Company, New York, NY, 1953.] Most dislocations found in crystalline materials are probably neither pure edge nor pure screw but exhibit components of both types; these are termed mixed dislocations. All three dislocation types are represented schematically in Figure 3.6; the lattice distortion that is produced away from the two faces is mixed, having varying degrees of screw and edge character. Module No. 3 – Imperfections of Solids 10 E N G G 4 1 2 : M A T E R I A L S S C I E N C E A N D E N G I N E E R I N G Figure 3.6 (a) Schematic representation of a dislocation that has edge, screw, and mixed character. (b) Top view, where open circles denote atom positions above the slip plane, and solid circles, atom positions below. At point A, the dislocation is pure screw, while at point B, it is pure edge. For regions in between where there is curvature in the dislocation line, the character is mixed edge and screw. [Figure (b) from W. T. Read, Jr., Dislocations in Crystals, McGraw-Hill Book Company, New York, NY, 1953.] The magnitude and direction of the lattice distortion associated with a dislocation are expressed in terms of a Burgers vector, denoted by b. Burgers vectors are indicated in Figures 3.4 and 3.5 for edge and screw dislocations, respectively. Furthermore, the nature of a dislocation (i.e., edge, screw, or mixed) is defined by the relative orientations of dislocation line and Burgers vector. For an edge, they are perpendicular (Figure 3.4), whereas for a screw, they are parallel (Figure 3.5); they are neither perpendicular nor parallel for a mixed dislocation. Also, even though a dislocation changes direction and nature within a crystal (e.g., from edge to mixed to screw), the Burgers vector is the same at all points along its line. INTERFACIAL DEFECTS Interfacial defects are boundaries that have two dimensions and normally separate regions of the materials that have different crystal structures and/or crystallographic orientations. These imperfections include external surfaces, grain boundaries, phase boundaries, twin boundaries, and stacking faults. External Surfaces Surface atoms are not bonded to the maximum number of nearest neighbors and are therefore in a higher energy state than the atoms at interior positions. The bonds of these surface atoms that are not satisfied give rise to a surface energy, expressed in units of energy per unit area (J/m2 or erg/cm2). To reduce this energy, materials tend to minimize, if at all possible, the total surface area. Module No. 3 – Imperfections of Solids 11 E N G G 4 1 2 : M A T E R I A L S S C I E N C E A N D E N G I N E E R I N G Grain Boundaries A grain boundary is represented schematically from an atomic perspective in Figure 3.7. Within the boundary region, which is probably just several atom distances wide, there is some atomic mismatch in a transition from the crystalline orientation of one grain to that of an adjacent one. Various degrees of crystallographic misalignment between adjacent grains are possible (Figure 3.7). When this orientation mismatch is slight, on the order of a few degrees, then the term small- (or low-) angle grain boundary is used. These boundaries can be described in terms of dislocation arrays. One simple small-angle grain boundary is formed when edge dislocations are aligned in the manner of Figure 3.8. This type is called a tilt boundary; the angle of misorientation, 𝜃, is also indicated in the figure. When the angle of misorientation is parallel to the boundary, a twist boundary results, which can be described by an array of screw dislocations. Figure 3.7. Schematic diagram showing small and high-angle grain boundaries and the adjacent atom positions. Module No. 3 – Imperfections of Solids 12 E N G G 4 1 2 : M A T E R I A L S S C I E N C E A N D E N G I N E E R I N G Figure 3.8. Demonstration of how a tilt boundary having an angle of misorientation u results from an alignment of edge dislocations. Phase Boundaries Phase boundaries exist in multiphase materials, in which a different phase exists on each side of the boundary; furthermore, each of the constituent phases has its own distinctive physical and/or chemical characteristics. Phase boundaries play an important role in determining the mechanical characteristics of some multiphase metal alloys. Twin Boundaries A twin boundary is a special type of grain boundary across which there is a specific mirror lattice symmetry; that is, atoms on one side of the boundary are located in mirror image positions to those of the atoms on the other side (Figure 3.9). The region of material between these boundaries is appropriately termed a twin. Twins result from atomic displacements that are produced from applied mechanical shear forces (mechanical twins) and also during annealing heat treatments following deformation (annealing twins). Twinning occurs on a definite crystallographic plane and in a specific direction, both of which depend on the crystal structure. Module No. 3 – Imperfections of Solids 13 E N G G 4 1 2 : M A T E R I A L S S C I E N C E A N D E N G I N E E R I N G Figure 3.9. Schematic diagram showing a twin plane or boundary and the adjacent atom positions (colored circles). BULK OR VOLUME DEFECTS Volume defects as name suggests are defects in 3-dimensions. These include pores, cracks, foreign inclusions and other phases. These defects are normally introduced during processing and fabrication steps. All these defects are capable of acting as stress raisers, and thus deleterious to parent metal’s mechanical behavior. However, in some cases foreign particles are added purposefully to strengthen the parent material. The procedure is called dispersion hardening where foreign particles act as obstacles to movement of dislocations, which facilitates plastic deformation. The second-phase particles act in two distinct ways – particles are either may be cut by the dislocations or the particles resist cutting and dislocations are forced to bypass them. Strengthening due to ordered particles is responsible for the good high-temperature strength on many super-alloys. However, pores are detrimental because they reduce effective load bearing area and act as stress concentration sites. ATOMIC VIBRATIONS Every atom in a solid material is vibrating very rapidly about its lattice position within the crystal. In a sense, these atomic vibrations may be thought of as imperfections or defects. At any instant of time, not all atoms vibrate at the same frequency and amplitude or with the same energy. At a given temperature, there exists a distribution of energies for the constituent atoms about an average energy. Over time, the vibrational energy of any specific atom also varies in a random manner. With rising temperature, this average energy increases, and, in fact, the temperature of a solid is really just a measure of the average vibrational activity of atoms and molecules. At room temperature, a typical vibrational frequency is on the order of 10 13 vibrations per second, whereas the amplitude is a few thousandths of a nanometer. However, in most materials the constituent grains are of microscopic dimensions, having diameters that may be on the order of microns, and their details must be investigated using some type of microscope. Grain size and shape are only two features of what is termed the microstructure. MICROSCOPIC TECHNIQUES With optical microscopy, the light microscope is used to study the microstructure; optical and illumination systems are its basic elements. For materials that are opaque to visible light (all metals and many ceramics and polymers), only the surface is subject to observation, and the light Module No. 3 – Imperfections of Solids 14 E N G G 4 1 2 : M A T E R I A L S S C I E N C E A N D E N G I N E E R I N G microscope must be used in a reflecting mode. Contrasts in the image produced result from differences in reflectivity of the various regions of the microstructure. Investigations of this type are often termed metallographic because metals were first examined using this technique. Normally, careful and meticulous surface preparations are necessary to reveal the important details of the microstructure. The specimen surface must first be ground and polished to a smooth and mirror-like finish. This is accomplished by using successively finer abrasive papers and powders. The microstructure is revealed by a surface treatment using an appropriate chemical reagent in a procedure termed The chemical reactivity of the grains of some single- phase materials depends on crystallographic orientation. Consequently, in a polycrystalline specimen, etching characteristics vary from grain to grain. Figure 3.10b shows how normally incident light is reflected by three etched surface grains, each having a different orientation. Figure 3.14a depicts the surface structure as it might appear when viewed with the microscope; the luster or texture of each grain depends on its reflectance properties. A photomicrograph of a polycrystalline specimen exhibiting these characteristics. Figure 3.10. (a) Polished and etched grains as they might appear when viewed with an optical microscope. (b) Section taken through these grains showing how the etching characteristics and resulting surface texture vary from grain to grain because of differences in crystallographic orientation. (c) Photomicrograph of a polycrystalline brass specimen, 60 characteristics is shown in Figure 3.10c. Module No. 3 – Imperfections of Solids 15 E N G G 4 1 2 : M A T E R I A L S S C I E N C E A N D E N G I N E E R I N G Figure 3.11. (a) Section of a grain boundary and its surface groove produced by etching; the light reflection characteristics in the vicinity of the groove are also shown. (b) Photomicrograph of the surface of a polished and etched polycrystalline specimen of an iron–chromium alloy in which the grain boundaries appear dark, 100. [Photomicrograph courtesy of L. C. Smith and C. Brady, the National Bureau of Standards, Washington, DC (now the National Institute of Standards and Technology, Gaithersburg, MD.)] Also, small grooves form along grain boundaries as a consequence of etching. Because atoms along grain boundary regions are more chemically active, they dissolve at a greater rate than those within the grains. These grooves become discernible when viewed under a microscope because they reflect light at an angle different from that of the grains themselves; this effect is displayed in Figure 3.11a. Figure 3.11b is a photomicrograph of a polycrystalline specimen in which the grain boundary grooves are clearly visible as dark lines. ELECTRON MICROSCOPY The upper limit to the magnification possible with an optical microscope is approximately 2000. Consequently, some structural elements are too fine or small to permit observation using optical microscopy. Under such circumstances, the electron microscope, which is capable of much higher magnifications, may be employed. An image of the structure under investigation is formed using beams of electrons instead of light radiation. According to quantum mechanics, a high-velocity electron becomes wavelike, having a wavelength that is inversely proportional to its velocity. When accelerated across large voltages, electrons can be made to have wavelengths on the order of 0.003 nm (3 pm). High magnifications and resolving powers of these microscopes are consequences of the short wavelengths of electron beams. The electron beam is focused and the image formed with magnetic lenses; otherwise, the geometry of the microscope components is essentially the same as with optical systems. Both transmission and reflection beam modes of operation are possible for electron microscopes. Module No. 3 – Imperfections of Solids 16 E N G G 4 1 2 : M A T E R I A L S S C I E N C E A N D E N G I N E E R I N G Transmission Electron Microscopy The image seen with a transmission electron microscope (TEM) is formed by an electron beam that passes through the specimen. Details of internal microstructural features are accessible to observation; contrasts in the image are produced by differences in beam scattering or diffraction produced between various elements of the microstructure or defect. Because solid materials are highly absorptive to electron beams, a specimen to be examined must be prepared in the form of a very thin foil; this ensures transmission through the specimen of an appreciable fraction of the incident beam. The transmitted beam is projected onto a fluorescent screen or a photographic film so that the image may be viewed. Magnifications approaching 1,000,000 are possible with transmission electron microscopy, which is frequently used to study dislocations. Scanning Electron Microscopy A more recent and extremely useful investigative tool is the scanning electron microscope (SEM). The surface of a specimen to be examined is scanned with an electron beam, and the reflected (or back-scattered) beam of electrons is collected and then displayed at the same scanning rate on a cathode ray tube (CRT; similar to a CRT television screen). The image on the screen, which may be photographed, represents the surface features of the specimen. The surface may or may not be polished and etched, but it must be electrically conductive; a very thin metallic surface coating must be applied to nonconductive materials. Magnifications ranging from 10 to in excess of 50,000 are possible, as are also very great depths of field. Accessory equipment permits qualitative and semi-quantitative analysis of the elemental composition of very localized surface areas. Scanning Probe Microscopy The field of microscopy has experienced a revolution with the development of a new family of scanning probe microscopes. The scanning probe microscope (SPM), of which there are several varieties, differs from optical and electron microscopes in that neither light nor electrons are used to form an image. Rather, the microscope generates a topographical map, on an atomic scale, that is a representation of surface features and characteristics of the specimen being examined. Some of the features that differentiate the SPM from other microscopic techniques are as follows: Examination on the nanometer scale is possible inasmuch as magnifications as high as 109 are possible; much better resolutions are attainable than with other microscopic techniques. Three-dimensional magnified images are generated that provide topographical information about features of interest. Some SPMs may be operated in a variety of environments (e.g., vacuum, air, liquid); thus, a particular specimen may be examined in its most suitable environment. Module No. 3 – Imperfections of Solids 17 E N G G 4 1 2 : M A T E R I A L S S C I E N C E A N D E N G I N E E R I N G Figure 3.12.(a) Bar chart showing size ranges for several structural features found in materials. (b) Bar chart showing the useful resolution ranges for four microscopic techniques discussed in this chapter, in addition to the naked eye. (Courtesy of Prof. Sidnei Paciornik, DCMM PUC-Rio, Rio de Janeiro, Brazil, and Prof. Carlos Pérez Bergmann, Federal University of Rio Grande do Sul, Porto Alegre, Brazil.) Module No. 3 – Imperfections of Solids 18 E N G G 4 1 2 : M A T E R I A L S S C I E N C E A N D E N G I N E E R I N G GRAIN-SIZE DETERMINATION The grain size is often determined when the properties of polycrystalline and singlephase materials are under consideration. In this regard, it is important to realize that for each material, the constituent grains have a variety of shapes and a distribution of sizes. Grain size may be specified in terms of average or mean grain diameter, and a number of techniques have been developed to measure this parameter. We now briefly describe two common grain-size determination techniques: (1) linear intercept—counting numbers of grain boundary intersections by straight test lines; and (2) comparison—comparing grain structures with standardized charts, which are based upon grain areas (i.e., number of grains per unit area). For the linear intercept method, lines are drawn randomly through several photomicrographs that show the grain structure (all taken at the same magnification). Grain boundaries intersected by all the line segments are counted. Example Problem 3.4 Grain Size Determination Using the intercept method, determine the average grain size, in millimeters, of the specimen whose microstructure is shown in the figure. Use at least seven straight-line segments. Solution In order to determine the average grain diameter, it is necessary to count the number of grains intersected by each of these line segments. These data are tabulated below. Module No. 3 – Imperfections of Solids 19 E N G G 4 1 2 : M A T E R I A L S S C I E N C E A N D E N G I N E E R I N G Line Number No. of Grains Intersected 1 11 2 10 3 9 4 8.5 5 7 6 10 7 8 The average number of grain boundary intersections for these lines was 9.1. Therefore, the average line length intersected is just 60 𝑚𝑚 = 6.59 𝑚𝑚 9.1 Hence, the average grain diameter, d, is 𝒂𝒗𝒆𝒓𝒂𝒈𝒆 𝒍𝒊𝒏𝒆 𝒊𝒏𝒕𝒆𝒓𝒔𝒆𝒄𝒕𝒆𝒅 𝟔. 𝟓𝟗 𝒎𝒎 𝒅= = = 𝟔. 𝟓𝟗𝒙 𝟏𝟎−𝟐 𝒎𝒂𝒈𝒏𝒊𝒇𝒊𝒄𝒂𝒕𝒊𝒐𝒏 𝟏𝟎𝟎 Reference: Materials Science and Engineering: An Introduction, 10th Edition by William D. Callister Jr and David G. Rethwisch Module No. 3 – Imperfections of Solids 20