Interfacial free energy 111 Tper unit area is given by Crystal Interfaces and Microstructure Basically three different types of interface are important in metallic systems- 1. The free surfaces of a crystal (solid/vapour interface) 2. Grain boundaries (a/a interfaces) 3. Interphase interfaces (a/p interfaces). All crystals possess the first type of interface. The second type separates*! crystals with essentially the same composition and crystal structure, but ai different orientation in space. The third interface separates two different] phases that can have different crystal structures and/or compositions and I therefore also includes solid/liquid interfaces. j The great majority of phase transformations in metals occur by the growth of a new phase (|3) from a few nucleation sites within the parent phase (a)—a I nucleation and growth process. The a/p interface therefore plays an impor-j tant role in determining the kinetics of phase transformations and is the most important class of interface listed. It is, however, also the most complex and . least understood, and this chapter thus begins by first considering the simpler | interfaces, (1) and (2). I The solid/vapour interface is of course itself important in vaporization and condensation transformations, while grain boundaries are important in re-| crystallization, i.e. the transformation of a highly deformed grain structure into new undeformed grains. Although no new phase is involved in recrystal-lization it does have many features in common with phase transformations. The importance of interfaces is not restricted to what can be called the , primary transformation. Since interfaces are an almost essential feature of the | transformed microstructure, a second (slower) stage of most transformations is the microstructural coarsening that occurs with time1. This is precisely. analogous to the grain coarsening or grain growth that follows a recrystalhza-tion transformation. 3.1 Interfacial Free Energy It is common practice to talk of interfacial energy. In reality, however, whal is usually meant and measured by experiment is the interfacial free energy, 7. -The free energy of a system containing an interface of area A and free energy G = Go + Ay (3.1) where G0 is the free energy of the system assuming that all material in the system has the properties of the bulk—7 is therefore the excess free energy arising from the fact that some material lies in or close to the interface. It is ilso the work that must be done at constant T and P to create unit area of interface. Consider for simplicity a wire frame suspending a liquid film, Fig. 3.1. If one bar of the frame is movable it is found that a force Fper unit length must be applied to maintain the bar in position. If this force moves a small distance go'that the total area of the film is increased by dA the work done by the force is FdA. This work is used to increase the free energy of the system by dG. From Equation 3.1 dG = ydA + Ady Equating this with FdA gives ■ F=y + A^ (3.2) In the case of a liquid film the surface energy is independent of the area of the interface and dy/dA = 0. This leads to the well-known result F = y (3.3) i.e. a surface with a free energy y J m 2 exerts a surface tension of 7 N itT1. In the case of interfaces involving solids, however, it is not immediately obvious that 7 is independent of area. Since a liquid is unable to support shear stresses, the atoms within the liquid can rearrange during the stretching process and thereby maintain a constant surface structure. Solids, however, are much more viscous and the transfer of atoms from the bulk to the surface, which is necessary to maintain an unchanged surface structure and energy, will take much longer. If this time is long in comparison to the time of the experiment then d7/dd 0 and surface free energy and surface tension will not be identical. Nevertheless, at temperatures near the melting point the atomic mobility is usually high enough for Equation 3.3 to be applicable. Liquid film Fig. 3.1 A liquid film on a wire frame. t 112 Crystal interfaces and microstructure 3.2 Solid/Vapour Interfaces To a first approximation the structure of solid surfaces can be discussed m terms of a hard sphere model. If the surface is parallel to a low-index crystal | plane the atomic arrangement will be the same as in the bulk, apart from perhaps a small change in lattice parameter. (This assumes that the surface is uncontaminated: in real systems surfaces will reduce their free energies by the I adsorption of impurities.) Figure 3.2 for example shows the {111} {200} {220} I atom planes in the fee metals. Note how the density of atoms in these planes decreases as Qt2 + k2 + I2) increases. (The notation {200} and {220} has been used instead of {100} and {110} because the spacing of equivalent atom planes i is then given by a/J(h2 + k2 + I2) where a is the lattice parameter.) coooo ooooo ooooo 220 Fig. 3.2 Atomic configurations on the three closest-packed planes in fee crystals. (Ill), (200) and (220). The origin of the surface free energy is that atoms in the layers nearest the surface are without some of their neighbours. Considering only nearest neighbours it can be seen that the atoms on a {111} surface, for example, are deprived of three of their twelve neighbours. If the bond strength of the metal is e each bond can be considered as lowering the internal energy of each atom by e/2. Therefore every surface atom with three 'broken bonds' has an excess internal energy of 3e/2 over that of the atoms in the bulk. For a pure metal e can be estimated from the heat of sublimation Ls. (The latent heat of sublimation is equal to the sum of the latent heat of melting (or fusion) and the latent heat of vaporization.) If 1 mol of solid is vaporized 12 ATa broken bonds are formed. Therefore Ls = 12 jVa e/2. Consequently the energy of a {ill} surface should be given by 111 200 £sv = 0.25 LJNa J/surface atom (3.4) This result will only be approximate since second nearest neighbours have been ignored and it has also been assumed that the strengths of the remaining bonds in the surface are unchanged from the bulk values. From the definition of Gibbs free energy the surface free energy will be given by E + PV - TS (3-5) Thus even if the 'PV term is ignored surface entropy effects must be taken into account. It might be expected that the surface atoms will have more freedom of movement and therefore a higher thermal entropy compared to Solid!vapour interfaces 113 atoms in the bulk. Extra configurational entronv rthe surface by the formation of surface vacancies alS° be 'ntr"duced into a crystal should therefore be associated with a Dn • examPle- The surface of iwiil partly compensate for the high internal en e excess entropy which Experimental determination of -y5V js difficult^^ °f ^uation 3-4-' pure metals indicate that near the melting the measured values for ;energy averaged over many surface nlane* if .Cmpf aCure the surface free yaucb is given by 7SV = 0.15 Ls/Na J/surface atom As a result of entropy effects y< is sJJohti,, a RFrom Equation 1.10 ghtly indent on temperature. dT = -S (3.7) Measured values of 5 are positive and vary between 0 and 3 ml m'2 K"'. Some selected values of -y5V at the melting point are listed in Table 3.1. Note that metals with high melting temperatures have high values for and high surface energies. Table 3.1 Average Surface Free Energiesof Selected Metals Values selected from H. Jones The surface energy of solid metals', Metal Science Journal, 5:15 (1971). Experimental errors are generally about 10%. The values have been extrapolated to the melting temperature, 7m. Crystal m'_ 1084 1536 It can be seen from a w § -Q ■> ° xi CD C in V- cose/a broken bonds Fig. 3.3 The 'broken-bond' model for surface energy. A crystal plane at an angle 9 to the close-packed plane will contain broken bonds in excess of the close-packed plane due to the atoms at the steps. For unit length of interface in the plane of the diagram and unit length out of the paper (parallel to the steps) there will be (cos 6/a)(l/a) broken bonds out of the close-packed plane and (sin |9|/fl)(l/«) additional broken bonds from the atoms on the steps. Again attributing e/2 energy to each broken bond, then (cos 6 + sin |e|)e/2a2 (3.8) This is plotted as a function of 8 in Fig. 3.4. Note that the close-packed orientation (9 = 0) lies at a cusped minimum in the energy plot. Similar arguments can be applied to any crystal structure for rotations about any axis from any reasonably close-packed plane. All low-index planes should therefore be located at low-energy cusps. If 7 is plotted v. 9 similar cusps are found, but as a result of entropy effects o ♦ e Fig. 3.4 Variation of surface energy as a function of 9 in Fig. 3.3. Solid/vapour interfaces 115 they are less prominent than in the £-6 plot, and for the higher index planes they can even disappear. A convenient method for plotting the variation of 7 with surface orientation in three dimensions is to construct a surface about an origin such that the free energy of any plane is equal to the distance between the surface and the origin when measured along the normal to the plane in question. A section through such a surface is shown in Fig. 3.5a. This type of polar representation of 7 is known as a 7-plot and has the useful property of being able to predict the equilibrium shape of an isolated single crystal. For an isolated crystal bounded by several planes A1; A2, etc. with energies 71; 72, etc. the total surface energy will be given by the sum Ai7i + A272 + • • * The equilibrium shape has the property that SA^ is a minimum and the shape that satisfies this condition is given by the following, so-called Wulff construction3. For every point on the 7 surface, such as A in Fig. 3.5a, a plane is drawn through the point and normal to the radius vector A Wulff plane ✓ (001) I (111) (b) (110) Equilibrium shape (a) ~ _ ' ~ Fig. 3.5 (a) A possible (110) section through the 7-plot of an fee crystal. The length OA represents the free energy of a surface plane whose normal lies in the direction OA. Thus OB = 7<00i), OC = 7(in). etc. Wulff planes are those such as that which lies normal to the vector OA. In this case the Wulff planes at the cusps (B, C, etc.) give the inner envelope of all Wulff planes and thus the equilibrium shape, (b) The equilibrium shape in three dimensions showing {100} (square faces) and {111} (hexagonal faces). 116 Crystal interfaces and microstructure Boundaries in single-phase solids 111 OA. The equilibrium shape is then simply the inner envelope of all such planes. Therefore when the 7-plot contains sharp cusps the equilibrium shape is a polyhedron with the largest facets having the lowest interfaciai free energy. Equilibrium shapes can be determined experimentally by annealing small single crystals at high temperatures in an inert atmosphere, or by annealing small voids inside a crystal". Fee crystals for example usually assume a form showing {100} and {111} facets as shown in Fig. 3.5b. Of course when 7 is isotropic, as for liquid droplets, both the 7-plot and equilibrium shapes are spheres. When the equilibrium shape is known it is possible to use the Wulff theorem in reverse to give the relative interfaciai free energies of the observed facet planes. In Fig. 3.5 for example the widths of the crystal in the (111) and <100) directions will be in the ratio of 7(111): 7(100). {110} facets are usually missing from the equilibrium shape of fee metals, but do however appear for bec metals5. The aim of this section has been to show, using the simplest type of interface, the origin of interfaciai free energy, and to show some of the methods available for estimating this energy. Let us now consider the second type of interface, grain boundaries. 3.3 Boundaries in Single-Phase Solids The grains in a single-phase polycrystalline specimen are generally in many different orientations and many different types of grain boundary are therefore possible. The nature of any given boundary depends on the misorienta-tion of the two adjoining grains and the orientation of the boundary plane relative to them. The lattices of any two grains can be made to coincide by rotating one of them through a suitable angle about a single axis. In general the axis of rotation will not be simply oriented with respect to either grain or the grain-boundary plane, but there are two special types of boundary that are relatively simple. These are pure tilt boundaries and pure twist boundaries, as illustrated in Fig. 3.6. A tilt boundary occurs when the axis of rotation is parallel to the plane of the boundary, Fig. 3.6a, whereas a twist boundary is formed when the rotation axis is perpendicular to the boundary, Fig. 3.6b. 3.3.1 Low-Angle and High-Angle Boundaries It is simplest to first consider what happens when the misorientation between two grains is small. This type of boundary can be simply considered as an array of dislocations. Two idealized boundaries are illustrated in Fig. 3.7. These are symmetrical low-angle tilt and low-angle twist boundaries. The i Rotation axis 6 1 (a) Boundary ptane axis ^____ ^3 (b) Boundary-plane Fig. 3.6 The relative orientations of the crystals and the boundary forming (a) a tilt boundary (b) a twist boundary. low-angle tilt boundary is an array of parallel edge dislocations, whereas the twist boundary is a cross-grid of two sets of screw dislocations. In each case the atoms in the regions between the dislocations fit almost perfectly into both adjoining crystals whereas the dislocation cores are regions of poor fit in which the crystal structure is highly distorted. The tilt boundary need not be symmetrical with respect to the two adjoining crystals. However, if the boundary is unsymmetrical dislocations with different Burgers vectors are required to accommodate the misfit, as illustrated in Fig. 3.8. In general boundaries can be a mixture of the tilt and twist type in which case they must contain several sets of different edge and screw dislocations. The energy of a low-angle grain boundary is simply the total energy of the dislocations within unit area of boundary. (For brevity the distinction between internal energy and free energy will usually not be made from now on except where essential to understanding.) This depends on the spacing of the dislocations which, for the simple arrays in Fig. 3.7, is given by D = sin 9 b 6 (3.9) where b is the Burgers vector of the dislocations and e is the angular mis-orientation across the boundary. At very small values of 6 the dislocation 118 Crystal interfaces and microstructure spacing is very large and the grain boundary energy 7 is approximately proportional tb the density of dislocations in the boundary (1/Z>), i.e. 7 a e (3.10) However as 9 increases the strain fields of the dislocations progressively cancel out so that 7 increases at a decreasing rate as shown in Fig. 3.9. In general when 6 exceeds 10-15° the dislocation spacing is so small that the; dislocation cores overlap and it is then impossible to physically identify the I individual dislocations (see Fig. 3.10). At this stage the grain-boundary en- { ergy is almost independent of misorientation, Fig, 3.9. When 6 > 10-15° the boundary is known as a random high-angle grain I boundary. The difference in structure between low-angle and high-angle grain \ boundaries is lucidly illustrated by the bubble-raft model in Fig. 3.11. High- j angle boundaries contain large areas of poor fit and have a relatively open structure. The bonds between the atoms are broken or highly distorted and | consequently the boundary is associated with a relatively high energy. In j; (a) Fig. 3.7 (a) Low-angle tilt boundary, (b) low-angle twist boundary: O atoms in crystal below boundary, • atoms in crystal above boundary. (After W.T. Read Jr Dislocations in Crystals, McGraw-Hill, New York, 1953.) Boundaries in single-phase solids 119 low-angle boundaries, however, most of the atoms fit very well into both lattices so that there is very little free volume and the interatomic bonds are only slightly distorted. The regions of poor fit are restricted to the dislocation cores which are associated with a higher energy similar to that of the random high-angle boundary. Fig. 3.7 (b) Boundaries in single-phase solids 121 120 Crystal interfaces and microstructure Fi, ,8 An -»->;ÄftSÄ'äS vectors are present. {Atter wi. ~ New York, 1953.) Grain boundary Energy Y 10-15 /Low High /angle j angle Misorienlation 0 Fig. 3.9 Variation of grain boundary energy with misorientation (schematic). Grain boundary transition zone Rg. 3.10 Disordered grain boundary structure (schematic). Hg7n" Rafts of so^Z^o^^J^^^T^ Note that the boundary with the fallest misone^a™ smicture l0 whlch Scaüons, whereas the ^^^^^^^^ Transformations individual dislocations cannot be identified. A ^ } in Metals, McGraw-H.U, New York, WW, 122 Crystal interfaces and microstructure Measured high-angle grain boundary energies yb are often found to be roughly given by 7b (3.11) Some selected values for yb and 7b/7sv are listed in Table 3.2. As for surface energies yb is temperature dependent decreasing somewhat with increasing temperature. Table 3.2 Measured Grain Boundary Free Energies Values selected from compilation given in Interfacial Phenomena in Metals and Alloys, by L.E. Murr, Addison-Wesley, London, 1975. Crystal 7b/mJ m 2 T/°C 7b/7Sv Sn 164 223 0.24 Al 324 450 0.30 Ag 375 950 0.33 Au 378 1000 0.27 Cu 625 925 0.36 7-Fe 756 1350 0.40 S-Fe 468 1450 0.23 Pt 660 1300 0.29 W 1080 2000 0.41 3.3.2 Special High-Angle Grain Boundaries Not all high-angle boundaries have an open disordered structure. There are some special high-angle boundaries which have significantly lower energies than the random boundaries. These boundaries only occur at particular misorientations and boundary planes which allow the two adjoining lattices to fit together with relatively little distortion of the interatomic bonds. The simplest special high-angle grain boundary is the boundary between two twins. If the twin boundary is parallel to the twinning plane the atoms in the boundary fit perfectly into both grains. The result is a coherent twin boundary as illustrated in Fig. 3.12a. In fee metals this is a {111} close-packed plane. Because the atoms in the boundary are essentially in undistorted positions the energy of a coherent twin boundary is extremely low in comparison to the energy of a random high-angle boundary. If the twin boundary does not lie exactly parallel to the twinning plane, Fig. 3.12b, the atoms do not fit perfectly into each grain and the boundary energy is much higher. This is known as an incoherent twin boundary. The energy of a twin boundary is therefore very sensitive to the orientation of the Boundaries in single-phase solids 123 Twinning plane Rows of atoms (a) (b) Incoherent Coherent twin (c) Fig. 3.12 (a) A coherent twin boundary, (b) An incoherent twin boundary, (c) Twin-boundary energy as a function of the grain boundary orientation. boundary plane. If 7 is plotted as a function of the boundary orientation a sharp cusped minimum is obtained at the coherent boundary position as shown in Fig. 3.12c. Table 3.3 lists some experimentally measured values of coherent and incoherent twins along with high-angle grain boundary energies for comparison. Table 3.3 Measured Boundary Free Energies for Crystals in Twin Relationships (Units mj m-2) Values selected from compilation given in Interfacial Phenomena in Metals and Alloys, by L.E. Murr, Addison-Wesley, London, 1975. Crystal Coherent twin Incoherent twin Grain boundary energy boundary energy boundary energy Cu Ag Fe-Cr-Ni (stainless steel type 304) 21 8 19 498 126 209 623 377 835 124 Crystal interfaces and microstructure Twin orientations in fee metals correspond to a misorientation of 70.5° about a <110) axis. Therefore a twin boundary is a special high-angle grain boundary, and a coherent twin boundary is a symmetrical tilt boundary between the two twin-related crystals. Figure 3.13 shows measured grain-boundary energies for various symmetric tilt boundaries in aluminium. When the two grains are related by a rotation about a {100} axis, Fig. 3.13a, it can be seen that most high-angle boundaries have about the same energy and should therefore have a relatively disordered structure characteristic of random boundaries. However, when the two grains are related by a rotation about a (110) axis there are several large-angle orientations which have significantly lower energies than the random boundaries (Fig. 3.13b). 8 = 70.5° corresponds to the coherent twin boundary discussed above, but low-energy boundaries are also found for several other values of 9. The reasons for these other special grain boundaries are not well understood. However, it seems reasonable to suppose that the atomic structure of these boundaries is such that they contain extensive areas of good fit. A two-dimensional example is shown in Fig. 3.14. This is a symmetrical tilt boundary between grains with a misorientation of 38.2°. The boundary atoms fit rather well into both grains leaving relatively little free volume. Moreover, a small group of atoms (shaded) are repeated at regular intervals along the boundary. 3.3.3 Equilibrium in Polycrystalline Materials Let us now examine how the possibility of different grain-boundary energies affects the microstructure of a polycrystalline material. Figure 3.15 shows the microstructure of an annealed austenitic stainless steel (fee). The material contains high- and low-angle grain boundaries as well as coherent and incoherent twin boundaries. This microstructure is determined by how the different grain boundaries join together in space. When looking at two- 80 0 20 40 60 80 100 \20 (40 f60 180 Misorientation, B deg. Fig. 3.13 Measured grain boundary energies for symmetric tilt boundaries in Al (a) when the rotation axis is parallel to (100), (b) when the rotation axis is parallel to (110). (After G. Hasson and C. Goux, Scripta Metallurgica, 5 (1971) 889.) Boundaries in single-phase solids 125 Fig. 3.14 Special grain boundary. (After H. Gleiter, Physica Status Solidi (b) 45 (1971) 9.) dimensional microstructures like this it is important to remember that in reality the grains fill three dimensions, and only one section of the three-dimensional network of internal boundaries is apparent. Note that two grains meet in a plane (a grain boundary) three grains meet in a line (a grain edge) and four grains meet at a point (a grain corner). Let us now consider the factors that control the grain shapes in a recrystallized polycrystal. The first problem to be solved is why grain boundaries exist at all in annealed materials. The boundaries are all high-energy regions that increase the free energy of a polycrystal relative to a single crystal. Theiefore a polycrystalline material is never a true equilibrium structure. However the grain boundaries in a polycrystal can adjust themselves during annealing to produce a metastable equilibrium at the grain boundary intersections. The conditions for equilibrium at a grain-boundary junction can be obtained either by considering the total grain boundary energy associated with a particular configuration or, more simply, by considering the forces that each boundary exerts on the junction. Let us first consider a grain-boundary segment of unit width and length OP as shown in Fig. 3.16. If the boundary is mobile then forces Fx and Fy must act at O and P to maintain the boundary in Boundaries in stngle-phase solids 127 126 Crystal interfaces and microstructure equilibrium. From Equation 3.3, Fx = 7. Fy can be calculated as follows: if P is moved a small distance by while O remains stationary, the work done will be Fy?>y. This must balance the increase in boundary energy caused by the change in orientation 58, i.e. FySy = I- 66 Since 8y = /86 (3.12) This means that if the grain-boundary energy is dependent on the orientation of the boundary (Fig. 3.16b) a force d7/d0 must be applied to the ends of the boundary to prevent it rotating into a lower energy orientation. d7/d6 is therefore known as a torque term. Since the segment OP must be supported by forces Fx and Fy the boundary exerts equal but opposite forces -Fx and -Fy on the ends of the segment which can be junctions with other grain boundaries. 0 / P6* Fig. 3.16 (a) Equilibrium forces Fx and Fy supporting a length / of boundary OP. (b) The origin of Fy. If the boundary happens to be at the orientation of a cusp in the free energy, e.g. as shown in Fig. 3.12c, there will be no torque acting on the boundary since the energy is a minimum in that orientation. However, the boundary will be able to resist a pulling force Fy of up to (d7/d6)cusp without rotating. If the boundary energy is independent of orientation the torque term is zero and the grain boundary behaves like a soap film. Under these conditions the requirement for metastable equilibrium at a junction between three grains, Fig. 3.17, is that the boundary tensions yY, 72 and 73 must balance. In Fig. 3.17 The balance of grain boundary tensions for a grain boundary intersection in metastable equilibrium. 128 Crystal interfaces and microstructure mathematical terms 723 7l3 sin 6? 712 sin 61 sin 62 sin 63 (3-13) Equation 3.13 applies to any three boundaries so that grain 1 for example could be a different phase to grains 2 and 3. Alternatively grain 1 could be a vapour phase in which case 713 and -y12 would be the surface energies of the solid. This relationship is therefore useful for determining relative boundary energies. One method of measuring grain-boundary energy is to anneal a specimen at a high temperature and then measure the angle at the intersection of the surface with the boundary, see Fig. 3.18. If the solid-vapour energy (7SV) is the same for both grains, balancing the interfacial tensions gives 27sv cos- = 7b (3.14) Therefore if 7SV is known 7b can be calculated. When using Equation 3.14 it must be remembered that the presence of any torque terms has been neglected and such an approximation may introduce large errors. To illustrate the importance of such effects let us consider the junction between coherent and incoherent twin boundary segments, Fig. 3.19. As a result of the orientation dependence of twin boundary energy, Fig. 3.12c, it is energetically favourable for twin boundaries to align themselves parallel to the twinning plane. If, however, the boundary is constrained to follow a macroscopic plane that is near but not exactly parallel to the twinning plane the boundary will usually develop a stepped appearance with large coherent low-energy facets and small incoherent high-energy risers as shown in Fig. 3.19. Although this configuration does not minimize the total twin boundary area it does minimize the total free energy. It is clear that at the coherent/incoherent twin junction the incoherent twin boundary tension 7i must be balanced by a torque term. Since the maximum value of the resisting force is d7c/d8, the condition that the configuration shown in Fig. 3.19 is stable is (3-i5) de Vapour Solid Solid Fig. 3.18 The balance of surface and grain boundary tensions at the intersection of a grain boundary with a free surface. Boundaries in single-phase solids 129 Coherent j Torque forces -I incoherent (b) /_ P "---.J be l6y (c) Fig 3 19 (a) A twin boundary in a thin foil specimen as imaged in the transmasitm electron microscope. (After M.N. Thompson and C.W. Chen, Philips Electron Optics Eindhoven, 1979.) (b) and (c), the coherent and mcoherent segments of the twin boundary. Likewise the 'incoherent' facet must also be a special boundary showing rather good fit in order to provide a force resisting ye. That is 7c « de (3.16) However, since 7c is usually very small the incoherent interface need only lie in a rather shallow energy cusp. 130 Crystal interfaces and microstructure Boundaries in single-phase solids 131 The above can be obtained in another way from energy considerations. If (metastable) equilibrium exists at P in Fig. 3.19c, then a small displacement such as that shown should either produce no change, or an increase in the total free energy of the system, i.e. dG & 0 Considering unit depth a small displacement Sy at P will increase the total free energy by an amount 0 Since /56 = hy this leads to the same result as given by Equation 3.15 3.3.4 Thermally Activated Migration of Grain Boundaries In the previous section it was shown that metastable equilibrium at the grain boundary junctions requires certain conditions to be satisfied for the angles at which three boundaries intersect. For simplicity, if all grain boundaries in a polycrystal are assumed to have the same grain-boundary energy independent of boundary orientation, Equation 3.13 predicts that 01 = 02 = 83 = 120°. It can be similarly shown that the grain-boundary edges meeting at a corner formed by four grains will make an angle of 109° 28'. If these, or similar, angular conditions are satisfied then metastable equilibrium can be established at all grain boundary junctions. However, for a grain structure to be in complete metastable equilibrium the surface tensions must also balance over all the boundary faces between the junctions. If a boundary is curved in the shape of a cylinder, Fig. 3.20a, it is acted on by a force of magnitude y/r towards its centre of curvature. Therefore the only way the boundary tension forces can balance in three dimensions is if the boundary is planar (r = <») or if it is curved with equal radii in opposite directions, Fig. 3.20b and c. It is A (a) (b) (c) Fig. 3.20 (a) A cylindrical boundary with a radius of curvature r is acted on by a force 7/r. (b) A planar boundary with no net force, (c) A doubly curved boundary with no net force. theoretically possible to construct a three-dimensional polycrystal in which the boundary tension forces balance at all faces and junctions, but in a random polycrystalline aggregate, typical of real metallurgical specimens, there are always boundaries with a net curvature in one direction. Consequently a random grain structure is inherently unstable and, on annealing at tiigh temperatures, the unbalanced forces will cause the boundaries to migrate towards their centres of curvature. The effect of different boundary curvatures in two dimensions is shown in Fig. 3.21. Again for simplicity it has been assumed that equilibrium at each boundary junction results in angles of 120°. Therefore if a grain has six boundaries they will be planar and the structure metastable. However, if the total number of boundaries around a grain is less than six each boundary must be concave inwards, Fig. 3.21. These grains will therefore shrink and eventually disappear during annealing. Larger grains, on the other hand, will have Fig. 3.21 Two-dimensional grain boundary configurations. The arrows indicate the directions boundaries will migrate during grain growth. more than six boundaries and will grow. The overall result of such boundary migration is to reduce the number of grains, thereby increasing the mean grain size and reducing the total grain boundary energy. This phenomenon is known as grain growth or grain coarsening. It occurs in metals at temperatures above about 0.5 Tm where the boundaries have significant mobility. A soap froth serves as a convenient analogue to demonstrate grain growth as shown in Fig. 3.22. In the case of the cells in a soap froth the higher pressure on the concave side of the films induces the air molecules in the smaller cells to diffuse through the film into the larger cells, so that the small cells eventually disappear. A similar effect occurs in metal grains. In this case the atoms in the shrinking grain detach themselves from the lattice on the high pressure side of the boundary and relocate themselves on a lattice site of the growing grain. 132 Crystal interfaces and microstructure Boundaries in single-phase solids 133 Fig. 3.22 Two-dimensional cells of a soap solution illustrating the process of grain growth. Numbers are time in minutes. (After C.S. Smith, Metal Interfaces, American Society for Metals, 1952, p. 81.) For example in Fig. 3.23a if atom C jumps from grain 1 to grain 2 the boundary locally advances a small distance. The effect of the pressure difference caused by a curved boundary is to create a difference in free energy (AG) or chemical potential (Au.) that drives the atoms across the boundary, see Fig, 3.24. In a pure metal AG and Aw, are identical and are given by Equation 1.58 as AG 2yVtt (3.17) This free energy difference can be thought of as a force pulling the grain boundary towards the grain with the higher free energy. As shown in Fig. 3.25, if unit area of grain boundary advances a distance §x the number of moles of material that enter grain B is bx • 1/ Vm and the free energy released is given by AG ■ §x/Vm This can be equated to the work done by the pulling force Fbx. Thus the pulling force per unit area of boundary is given by AG -TT- N m " m (3.18) In other words the force on the boundary is simply the free energy difference per unit volume of material. In the case of grain growth AG arises from the boundary curvature, but Equation 3.18 applies equally to any boundary whose migration causes a Boundary motion (a) Grain 1 Grain 2 (b) Fig. 3.23 (a) The atomic mechanism of boundary migration. The boundary migrates to the left if the jump rate from grain 1 2 is greater than 2 -» 1. Note that the free volume within the boundary has been exaggerated for clarity, (b) Step-like structure where close-packed planes protrude into the boundary. decrease in free energy. During recrystallization, for example, the boundaries between the new strain-free grains and the original deformed grains are acted on by a force AG/Vm where, in this case, AG is due to the difference in dislocation strain energy between the two grains. Figure 3.26 shows a dislocation-free recrystallized grain expanding into the heavily deformed surroundings. In this case the total grain-boundary area is increasing, therefore the CD c 0) 0 Grain 1 Grain 2 Distance Fig. 3.24 The free energy of an atom during the process of jumping from one grain to the other. crystal interfaces and microstructure High-energy groin Low-energy grain "7> Boundaries in single-phase solids 135 Unit-area 6x Grain boundary Fig. 3.25 A boundary separating grains with different tree energies is subjected to a pulling force F. Fig. 3.26 Grain boundary migration in nickel pulled 10% and annealed 10 min at 425 °C. The region behind the advancing boundary is dislocation-free. (After J. Bailey and P. Hirsch, Proceedings of the Royal Society, London, A267 (1962) 11 ) driving force for recrystallization must be greater than the opposing boundary tension forces. Such forces are greatest when the new grain is smallest, and the effect is therefore important in the early stages of recrystallization. Let us now consider the effect of the driving force on the kinetics of boundary migration. In order for an atom to be able to break away from grain 1 it must acquire, by thermal activation, an activation energy AGa, Fig 3.24. If the atoms vibrate with a frequency vx the number of times per second that an atom has this energy is vx exp (-AG*/RT). If there are on average nx atoms per unit area in a favourable position to make a jump there will be mvi exp (-AGa/RT) jumps m-2 s_1 away from grain 1. It is possible that not all these atoms will find a suitable site and 'stick' to grain 2. If the probability of being accommodated in grain 2 is A2 the effective flux of atoms from grain 1 to 2 will be A2n1v1 exp (-AGa/RT) m~2 s-1 There will also be a similar flux in the reverse direction, but if the atoms in grain 2 have a lower free energy than the atoms in grain 1 by AG (moP1) the flux from 2 to 1 will be Aln2v2 exp - (AGa + AG)/RT m~z s"1 When AG = 0 the two grains are in equilibrium and there should therefore be no net boundary movement, i.e. the rates at which atoms cross the boundary in opposite directions must be equal. Equating the above expressions then gives Axn2v2 = A2ni0i For a high-angle grain boundary it seems reasonable to expect that there will not be great problems with accommodation so that Ax — A2 = 1. Assuming the above equality also holds for small non-zero driving forces, with AG > 0 there will be a net flux from grain 1 to 2 given by ynet = A2nlvi exp - AGa ~RT 1 - exp | -^p (3.19) If the boundary is moving with a velocity v the above flux must also be equal to v/(Vm/Na), where (Vm/Wa) is the atomic volume. Therefore expanding exp (-AG/RT) for the usual case of AG <§ RT gives A^VJ ( AGa\ AG v = ~i^^{--Rf)v: (3-20) In other words v should be proportional to the driving force AG/Vm (N m~2). Equation 3.20 can be written more simply as v = M ■ AG/Vm (3.21) where M is the mobility of the boundary, i.e. the velocity under unit driving force. Substituting for AGa gives 136 Crystal interfaces and microstructure Boundaries in single-phase solids 137 exp -AH' RT (3.22) 1 Note how this simple model predicts an exponential increase in mobility with temperature. This result should of course be intuitively obvious since the boundary migration is a thermally activated process like diffusion. Indeed boundary migration and boundary diffusion are closely related processes The only difference is that diffusion involves transport along the boundary whereas migration requires atomic movement across the boundary. The model used to derive Equations 3.20 and 3.22 is particularly simple and gross assumptions are involved. In real grain boundaries it is likely ihat not all atoms in the boundary are equivalent and some will jump more easily than others. For example atoms may jump preferably to and from atomic steps or ledges, like atoms A, B and C in Fig. 3.23a. In fee metals such ledges should exist where the close-packed {111} planes protrude into the boundary. Boundary migration could then be effected by the growth of the ledges in one grain combined with the shrinking of corresponding ledges in the other grain as shown in Fig. 3.23b. From our discussion of grain-boundary structure it might be argued that the relatively open structure of a random high-angle boundary should lead to a high mobility whereas the denser packing of the special boundaries should be associated with a low mobility. Indeed, the coherent twin boundary, in which the atoms fit perfectly into both grains, has been found to be almost entirely immobile6. However, experiments have shown that the other special boundaries are usually more mobile than random high-angle boundaries. The reason for this is associated with the presence of impurity or alloying elements in the metal. Figure 3.27 shows data for the migration of various boundaries in zone-refined lead alloyed with different concentrations of tin. For a given driving force the velocity of the random boundaries decreases rapidly with increasing alloy content. Note that only very low concentrations of impurity are required to change the boundary mobility by orders of magnitude. The special grain boundaries on the other hand are less sensitive to impurities. It is possible that if the metal were 'perfectly' pure the random boundaries would have the higher mobility. The reason for this type of behaviour arises from differences in the interactions of alloy elements or impurities with different boundaries. Generally the grain boundary energy of a pure metal changes on alloying. Often (though not always) it is reduced. Under these circumstances the concentration of alloying element in the boundary is higher than that in the matrix. In grain boundary segregation theory, grain boundary solute concentrations (Xb) are expressed as fractions of a monolayer. One monolayer (Xb — 1) means that the solute atoms in the boundary could be arranged to form a single close-packed layer of atoms. Approximately, for low mole fractions of solute in the matrix (Af0), tne boundary solute concentration Xb is given by Special boundaries 36°- 23' 26- 42° (ill) 4 (III) * - 28 (100) D 'Random' grain boundaries 0 0-002 0-004 0'006 Concentration of tin, wt% Fig. 3.27 Migration rates of special and randomJtojjjjarte «300 lead alloyed with tin under equal driving forces. (After K. Aust and J.w. Transactions of the Metallurgical Society of AWE, 215 (1959) 119.) AGb Xb = A^exp ^ (3.23) AGb is the free energy released per mole when a solute atom is moved from the matrix to the boundary. AGb is usually positive and roughly increases as the size misfit between the solute and matrix increases and as the solute-solute bond strength decreases. Equation 3.23 shows how grain boundary segregation decreases as temperature increases, i.e. the solute 'evaporates' into the matrix. For sufficiently low temperatures or high values of AGb, Xb increases towards unity and Equation 3.23 breaks down as Xb approaches a maximum saturation value. The variation of boundary mobility with alloy concentration varies markedly from one element to another. It is a general rule that AGb, which measures the tendency for segregation, increases as the matrix solubility decreases. This is illustrated by the experimental data in Fig. 3.28. ^10* J3 CO i c 10 CD E -C o i_ c CD TD c o c 10 - AISI3340-Sb eCu-Sb' CMn steel-S? ' * orFe Fe-Sn o«Fe-Si Si 1 icF2" 10 Atomic solid solubility *c0 Fig. 3.28 Increasing grain boundary enrichment with decreasing solid solubility in a range of systems. (After E.D. Hondras and M.P. Seah, International Metallurgical Reviews, December 1977, Review 222.) When the boundary moves the solute atoms migrate along with the boundary and exert a drag that reduces the boundary velocity. The magnitude of the drag will depend on the binding energy and the concentration in the boundary. The higher mobility of special boundaries can, therefore, possibly be attributed to a low solute drag on account of the relatively more close-packed structure of the special boundaries. The variation of boundary mobility with alloy concentration varies markedly from one element to another. It is a general rule that QB, which measures the tendency for segregation, increases as the matrix solubility decreases. This is illustrated by the experimental data in Fig. 3.28. It is possible that the higher mobility of special grain boundaries plays a role in the development of recrystallization textures. If a polycrystalline metal is heavily deformed, by say rolling to a 90% reduction, a deformation texture develops such that the rolled material resembles a deformed single crystal. On heating to a sufficiently high temperature new grains nucleate and begin to grow. However, not all grains will grow at the same rate: those grains which - are specially oriented with respect to the matrix should have higher mobility boundaries and should overgrow the boundaries of the randomly oriented grains- Consequently the recrystallized structure should have a special iirfentation with respect to the original 'single crystal'. Thus a new texture results which is called the recrystallization texture. Recrystallization is, however, incompletely understood and the above explanation of recrystalliza-noh texture may be an oversimplification. It is possible for example that the nuclei for recrystallization are themselves specially oriented with respect to i lie deformed matrix. A recrystallization texture is sometimes an advantage. For example the ■■roper texture in Fe-3wt% Si alloys makes them much better soft magnets rur use in transformers. Another application is in the production of textured sheet for the deep drawing of such materials as low-carbon steel. The only • ay to avoid a recrystallization texture is to give an intermediate anneal before a deformation texture has been produced. 3 3.5 The Kinetics of Grain Growth It was shown in the previous section that at sufficiently high temperatures the grain boundaries in a recrystallized specimen will migrate so as to reduce the total number of grains and thereby increase the mean grain_diameter. In a single-phase metal the rate at which the mean grain diameter D increases with time will depend on the grain boundary mobility and the driving force for boundary migration. If we assume that the mean radius of curvature of all the grain boundaries is proportional to the mean grain diameter D the mean driving force for grain growth will be proportional to 2y/D (Equation 3.17). Therefore fcc-<1120> hep. Fig. 3.33 The close-packed plane and directions in fcc and hep structures. .....unu rmcrostructure Fig. 3.32. This can only be achieved if, disregarding chemical species, ft interfacial plane has the same atomic configuration in both phases, and thi requires the two crystals to be oriented relative to each other in a special wa*j For example such an interface is formed between the hep silicon-rich k ph.^ and the fee copper-rich a-matrix in Cu~Si alloys. The lattice parameters c these two phases are such that the (lll)fcc plane is identical to the (OOOl) I plane. Both planes are hexagonally close-packed (Fig. 3.33) and in this pail ticular case the interatomic distances are also identical. Therefore when thj two crystals are joined along their close-packed planes with the close-pa^Ked directions parallel the resultant interface is completely coherent. The req hi 1 ment that the close-packed planes and directions are parallel produces a| orientation relationship between the two phases such that 1 (lll)a//(0001)K 1 [iiO]a//[H20L Note that the relative orientation of two crystals can always be specified bJ giving two parallel planes (hkl) and two parallel directions [uvw] that he irf those planes. % Within the bulk of each phase every atom has an optimum arrangement ■■1 nearest neighbours that produces a low energy. At the interface, however % there is usually a change in composition so that each atom is partly bonded to§ wrong neighbours across the interface. This increases the energy ol the| interfacial atoms and leads to a chemical contribution to the interfacial energy (ich)- F°r a coherent interface this is the only contribution, i.e. : ^(coherent) = -ych (3.29) $ In the case of the a-K interface in Cu-Si alloys the interfacial energy has been estimated to be as low as 1 mJ m-2. In general coherent interfacial energies range up to about 200 mJ m-2. Interphase interfaces in solids 145 a^oinfng *** ™™tCh,eadS t0 COheren<* »the tjje case of a hep/fee interface there is only one plane that can form a %herent interface: no other plane is identical in both crystal lattices. If, «f *" ver the two adjoining phases have the same crystal structure and lattice arJneter then, apart from differences in composition, all lattice planes are * When the distance between the atoms in the interface is not identical it is possible to maintain coherency by straining one or both of the two lattices %»*illustrated in Fig. 3.34. The resultant lattice distortions are known as 'coherency strains. Semicoherent Interfaces The strains associated with a coherent interface raise the total energy of the - system, and for sufficiently large atomic misfit, or interfacial area, it becomes energetically more favourable to replace the coherent interface with a semi-coherent interface in which the disregistry is periodically taken up by misfit dislocations, Fig. 3.35. If da and dp are the unstressed interplanar spacings of matching planes in the a and p phases respectively, the disregistry, or misfit between the two lattices (8) is defined by , b = d±zJa (3.30) It can be shown that in one dimension the lattice misfit can be completely accommodated without any long-range strain fields by a set of edge dislocations with a spacing D given by Fig. 3.35 A semicoherent interface. The misfit parallel to the interface is accommodated by a series of edge dislocations. 146 Crystal interfaces and microstructure D = or approximately, for small 8 (3.31) (3.32j Interphase interfaces in solids 147 where" b = (da + dp)/2 is the Burgers vector of the dislocations. The matching in the interface is now almost perfect except around the dislocation cores where the structure is highly distorted and the lattice planes are discontinuous. In practice misfit usually exists in two dimensions and in this case the coherency strain fields can be completely relieved if the interface contains two ^ non-parallel sets of dislocations with spacings Dx = &1/&1 and D2 = b2/^2, as shown in Fig. 3.36. If, for some reason, the dislocation spacing is greater than given by Equation 3.32, the coherency strains will have been only partially relieved by the misfit dislocations and residual long-range strain fields will still be present. The interfacial energy of a semicoherent interface can be approximately considered as the sum of two parts: (a) a chemical contribution, -ych, as for a fully coherent interface, and (b) a structural term y$t, which is the extra energy due to the structural distortions caused by the misfit dislocations, i.e. } -y (semicoherent) = 7ch + ys (3.33) Equation 3.32 shows that as the misfit S increases the dislocation spacing diminishes. For small values 0/8 the structural contribution to the interfacial energy is approximately proportional to the density of dislocations in the interface, i.e. 7s, « 8 (for small 8) (3.34) However 7st increases less rapidly as 8 becomes larger and it levels out when 8 - 0.25 in a similar way to the variation of grain-boundary energy with 6 D2=b2/62 misfit -> Fig. 3.36 Misfit in two directions (8X and B2) can be accommodated by a cross-grid of edge dislocations with spacings Y>x = bt/5i and D2 = b2/82 - shown in Fig. 3.9. The reason for such behaviour is that as the misfit dislocation spacing decreases the associated strain fields increasingly overlap and annul each other. The energies of semicoherent interfaces are generally in the range 200-5Q0 mJ m-2. When 8 > 0.25, i.e. one dislocation every four interplanar spacings, the regions of poor fit around the dislocation cores overlap and the interface cannot be considered as coherent, i.e. it is incoherent. Incoherent Interfaces When the interfacial plane has a very different atomic configuration in the two adjoining phases there is no possibility of good matching across the interface. The pattern of atoms may either be very different in the two phases or, if it is similar, the interatomic distances may differ by more than 25%. In both cases the interface is said to be incoherent. In general, incoherent interfaces result when two randomly oriented crystals are joined across any interfacial plane as shown in Fig. 3.37. They may, however, also exist between crystals with an orientation relationship if the interface has a different structure in the two crystals. Very little is known about the detailed atomic structure of incoherent interfaces, but they have many features in common with high-angle grain boundaries. For example they are characterized by a high energy (~500-1000 mJ m-2) which is relatively insensitive to the orientation of the interfa- An incoherent interface. 148 Crystal interfaces and microstructure rial plane. They probably have a disordered atomic structure in that the interface lacks the long-range periodicity of coherent and semicoherent interfaces; although, like high-angle grain boundaries, they may have a step-like structure caused by low-index planes protruding into the interface, as in Fig. 3.23b. Complex Semicoherent Interfaces The semicoherent interfaces considered above have been observed at boundaries formed by low-index planes whose atom patterns and spacings are clearly almost the same. However, semicoherent interfaces, i.e. interfaces containing misfit dislocations, can also form between phases when good lattice matching is not initially obvious. For example, fee and bec crystals" often appear with the closest-packed planes in each phase, (lll)fcc and; (110)bcc, almost parallel to each other. Two variants of this relationship are ■ found: the so-called Nishiyama-Wasserman (N-W) relationship: (110)bcc//(lll)fcc, [001]bcc//[ÍOl]fcc and the so-called Kurdjumov-Sachs (K-S) relationship: (110)bcc//(lll)fcc, [lll]bcc//[011]fcc Interphase interfaces in solids 149 ^t># o# o# o# o% o»o«o#o #o #o ##o#o#o*o«c*o« *#o#o#o#o «o «o *o #o c* aar*^ #o#o#o#o»o»o*o#o#o a a ^nr-~-o#qto#o«o«^*'"* * - v -•^•o«o«o^ • • > ^ Si ^ O#q,o#c*o«o«o*o#o* • * * ^ 0 fc<*<*o*o»c*o#o#J».* ^ o ■----' Fig. 3.38 Atomic matching across a (lll)fcc/(llO)bcc interface bearing the NW orientation relationship for lattice parameters closely corresponding to the case of fee and bec iron (M.G. Hall et al., Surface Science, 31 (1972) 257). FCC O BCC (The only difference between these two is a rotation in the closest-packed planes of 5.26°.) Figure 3.38 shows that the matching between a {111}fcc and {110}bcc plane bearing the N-W relationship is very poor. Good fit is restricted to small diamond-shaped areas that only contain —8% of the interfacial atoms. A similar situation can be shown to exist for the K-S orientation relationship. Thus it can be seen that a coherent or semicoherent interface between the two phases is impossible for large interfaces parallel to {lll}fCC and {110}bcc. Such interfaces would be incoherent. The degree of coherency can, however, be greatly increased if a macro-scopically irrational interface is formed, (i.e. the indices of the interfacial plane in either crystal structure are not small integers). The detailed structure of such interfaces is, however, uncertain due to their complex nature8,9. 3.4.2 Second-Phase Shape: Interfacial Energy Effects In a two-phase microstructure one of the phases is often dispersed within the other, for example p-precipitates in an a-matrix. Consider for simplicity a system containing one p-precipitate embedded in a single a crystal, and assume for the moment that both the precipitate and matrix are strain free. Such a system will have a minimum free, energy when the shape of the precipitate and its orientation relationship to the matrix are optimized to give the lowest total interfacial free energy (SAj-y;). Let us see how this can be achieved for different types of precipitate. Fully Coherent Precipitates If the precipitate (j3) has the same crystal structure and a similar lattice parameter to the parent a phase the two phases can form low-energy coherent interfaces on all sides—provided the two lattices are in a parallel orientation relationship—as shown in Fig. 3.39a. This situation arises during the early stages of many precipitation hardening heat treatments, and the (3 phase is then termed a fully coherent precipitate or a GP zone. (GP for Guinier and Preston who first discovered their existence. This discovery was made independently by Preston in the USA and Guinier in France, both employing X-ray diffraction techniques. Their work was later confirmed by transmission electron microscopy.) Since the two crystal structures match more or less perfectly across all interfacial planes the zone can be any shape and remain fully coherent. Thus a 7-plot of the a/3 interfacial energy would be largely spherical and, ignoring coherency strains, the equilibrium shape of a zone should be a sphere. Figure 3.39b shows an example of GP zones, ~10 nm in diameter, in an Al-4 atomic % Ag alloy. The zones are a silver-rich fcc region within the aluminium-rich fcc matrix. Since the atomic diameters of aluminium and silver differ by only 0.7% the coherency strains make a negligible contribution to the total free energy of the alloy. In other systems such as Al-Cu where the atomic size difference is much larger strain energy is Crystal interfaces and microstructure n rl > e M > d 7 \ ) ^ ;?—6 ) \y \ /\ f 1 c ) \ y ^ r \ n ( \ / -> ^ (• \ O f •) r \ < •*—t 1 ( "\ (~\ ■> \ 7~< M J \) Interphase interfaces in solids 151 (a) -** *J*if ?***tt Fig. 3.39 (a) A zone with no misfit (0 Al, • Ag, for example), (b) Electron micrograph ot Ag-nch zones in an Al-4 atomic % Ag alloy (x 300 000). (After R B 1959) 431 )°' Th°maS and J" Nuttin£' Journal of the Institute of Metals, 87 (1958- foand to be more important than interfacial energy in determining the equilibrium shape of the zone. This point will be discussed further in Section 3.4.3. Partially Coherent Precipitates From an interfacial energy standpoint it is favourable for a precipitate to be surrrounded by low-energy coherent interfaces. However, when the precipitate and matrix have different crystal structures it is usually difficult to find a lattice plane that is common to both phases. Nevertheless, for certain phase combinations there may be one plane that is more or less identical in each crystal, and by choosing the correct orientation relationship it is then possible for a low-energy coherent or semicoherent interface to be formed. There are, however, usually no other planes of good matching and the precipitate must consequently also be bounded by high-energy incoherent interfaces. A 7-plot of the interfacial energy in this case could look like that in Fig. 3.40, i.e. roughly a sphere with two deep cusps normal to the coherent interface. The Wulff theorem would then predict the equilibrium shape to be a disc with a thickness/diameter ratio of yjjj, where 7C and -y; are the energies of the (semi-) coherent and incoherent interfaces. Triangular, square, or hexagonal plate shapes would be predicted if the 7 plot also contained smaller cusps at symmetrically disposed positions in the plane of the plate. The precipitate shapes observed in practice may deviate from this shape for two main reasons. Firstly the above construction only predicts the equilibrium shape if misfit strain energy effects can be ignored. Secondly the precipitate may not be able to achieve an equilibrium shape due to constraints on how it can grow. For example disc-shaped precipitates may be much wider than the equilibrium shape if the incoherent edges grow faster than the broad faces. Plate-like precipitates occur in many systems. For example the hep 7'-phase in aged Al-4 atomic % Ag alloys forms as plates with semicoherent Equilibrium shape Fig. 3.40 A section through a 7-plot for a precipitate showing one coherent or semicoherent interface, together with the equilibrium shape (a disc). 152 Crystal interfaces and microstructure broad faces parallel to the {lll}a matrix planes and with the usual hcp/fCc orientation relationship. The tetragonal 8' phase in aged Al-4 wt% Cu alloys; are also plate-shaped, but in this case the broad faces of the plate (known as the habit plane) are parallel to {100}a matrix planes. Figure 3.41 shows that the {100}a planes are almost identical to the (001)r plane so that the orientation relationship between the 8' and the aluminium-rich matrix (a) is . (ooiv//(ooi)a [100]9.//[100]n Examples of the precipitate shapes that are formed in these two systems are shown in Figs. 3.42 and 3.43. Note that as a result of the cubic symmetry of the aluminium-rich matrix there are many possible orientations for the precipitate plates within any given grain. This leads to a very characteristic crystallographic microstructure known after its discoverer as a Widmanstatten morphology. Besides plate-like habits precipitates have also been observed to be lath-shaped (a plate elongated in one direction) and needle-like. For example the S phase in Al-Cu-Mg alloys forms as laths and the (3' phase in Al-Mg-Si alloys as needles10. In both cases the precipitates are also crystallographically related to the matrix and produce a Widmanstatten structure. Incoherent Precipitates When the two phases have completely different crystal structures, or when the two lattices are in a random orientation, it is unlikely that any coherent or semicoherent interfaces form and the precipitate is said to be incoherent. Since the interfacial energy should be high for all interfacial planes, the -/-plot and the equilibrium inclusion shape will be roughly spherical. It is possible Interphase interfaces in solids 153 ro O 1 o ----2 o 4-0 4 A 1 I*- 4-Ö4Ä -»( Fig. 3.41 (a) The unit cell of the 0' precipitate in Al-Cu alloys, (b) The unit cell of the matrix. (After J.M. Silcock, T.J. Heal and H.K. Hardy, Journal of the Institute of Metals, 82 (1953-1954) 239.) Fig. 3.42 Electron micrograph showing the Widmanstatten morphology of y' precipitates in an Al-4 atomic % Ag alloy. GP zones can be seen between the e.g. at H (x 7000). (R.B. Nicholson and J. Nutting, Acta Metallurgien, 9 (1961) 332.) that certain crystallographic planes of the inclusion lie at cusps in the 7-plot so that polyhedral shapes are also possible. Such faceting, however, need not imply the existence of coherent or semicoherent interfaces. The 8(CuAl2) precipitate in Al-Cu alloys is an example of an incoherent precipitate, Fig. 3.44. It is found that there is an orientation relationship between the 8 and aluminium matrix but this is probably because 6 forms from the 9' phase and does not imply that 0 is semicoherent with the matrix. Precipitates on Grain Boundaries Rather special situations arise when a second-phase particle is located on a grain boundary as it is necessary to consider the formation of interfaces with two differently oriented grains. Three possibilities now arise (Fig. 3.45): the precipitate can have (i) incoherent interfaces with both grains, (ii) a coherent or semicoherent interface with one grain and an incoherent interface with the other, or (iii) it can have a coherent or semicoherent interface with both grains. The first two cases are commonly encountered but the third possibility is unlikely since the very restrictive crystallographic conditions imposed by coherency with one grain are unlikely to yield a favourable orientation relationship towards the other grain. 154 Crystal interfaces and microstructure 4 TV «• Interphase interfaces in solids 155 0*2l* Fig. 3.43 Electron micrograph of a single coherent 6' plate in an Al-3.9 wt% Cu alloy aged 24 h at 200 °C. (x 80 000) (R. Sankaran and C. Laird, Acta Metallureica 22 (1974) 957.) The minimization of interfacial energy in these cases also leads to planar semicoherent (or coherent) interfaces and smoothly curved incoherent interfaces as before, but now the interfacial tensions and torques must also balance at the intersection between the precipitate and the boundary. (The shape that produces the minimum free energy can in fact be obtained by superimposing the 7 plots for both grains in a certain way11.) An example of a boundary precipitate is shown in Fig. 3.46. grain- 3.4.3 Second-Phase Shape: Misfit Strain Effects Fully Coherent Precipitates It was pointed out in the previous section that the equilibrium shape of a coherent precipitate or zone can only be predicted from the 7-plot when the misfit between the precipitate and matrix is small. When misfit is present the formation of coherent interfaces raises the free energy of the system on account of the elastic strain fields that arise. If this elastic strain energy is denoted by AGS the condition for equilibrium becomes SAi7i + AGS = minimum (3.35) Fig. 3-44 Electron micrograph showing incoherent particles of 6 in an Al-Cu alloy. (After G.A. Chadwick, Metallography of Phase Transformalions, Butterworths, London, 1972, from C. Laird.) The origin of the coherency strains for a misfitting precipitate is demonstrated in Fig. 3.47. If the volume of matrix encircled in Fig. 3.47a is cut out and the atoms are replaced by smaller atoms the cut-out volume will undergo a uniform negative dilatational strain to an inclusion with a smaller lattice parameter, Fig. 3.47b. In order to produce a fully coherent precipitate the matrix and inclusion must be strained by equal and opposite forces as shown in Fig. 3.47c12. If the lattice parameters of the unstrained precipitate and matrix are a$ and aa respectively the unconstrained misfit 5 is defined by Semicoherent (a) Incoherent (b) (c) Fig. 3.45 Possible morphologies for grain boundary precipitates. Incoherent interfaces smoothly curved. Coherent or semicoherent interfaces planar. ^rju^a una microstructure Interphase interfaces in solids 157 If the elastic moduli of the matrix and inclusion are equal and Poisson's ratio is 1/3, e and 8 are simply related by (3.38) >1 a (3.36) However, the stresses maintaining coherency at the interface, Hi 0) has a very low strain energy, and a needle shape (c/a = <») lies between the two. If elastic anisotropy is included14 it is found that the same general form for f(c/a) is preserved and only small changes in the exact values are required. Therefore the equilibrium shape of an incoherent inclusion will be an oblate spheroid with c/a value that balances the opposing effects of interfacial energy and strain 1 t /■^Sphere " 1 f (c/a) Needle c/a-co 0-5 / 0 / Disc _i-— 0 1 % Fig. 3.50 The variation of misfit strain energy with ellipsoid shape, f(c/a). (After F.R.N. Nabarro, Proceedings of the Royal Society A, 175 (1940) 519.) 160 Crystal interfaces and microstructure Interphase interfaces in solids 161 energy. When A is small interfacial energy effects should dominate and the inclusion should be roughly sphercial. Plate-Like Precipitates Consider a plate-like precipitate with coherent broad faces and incoherent or semicoherent edges, Fig. 3.51. (The criterion for whether these interfaces are coherent or semicoherent is discussed in the following section.) Misfit across the broad faces then results in large coherency strains parallel to the plate, but no coherency strains will exist across the edges. The in situ misfit across the broad faces increases with increasing plate thickness which leads to greater strains in the matrix and higher shear stresses at the corners of the plates'5 Eventually it becomes energetically favourable for the broad faces to become semicoherent. Thereafter the precipitate behaves as an incoherent inclusion with comparatively little misfit strain energy. An example of a precipitate that can be either coherent or semicoherent in this way is 9' in Al-Cu alloys (see Section 5.5.1). 3.4.4 Coherency Loss Precipitates with coherent interfaces have a low interfacial energy, but in the presence of misfit, they are associated with a coherency strain energy. On the other hand, if the same precipitate has non-coherent interfaces it will have a higher interfacial energy but the coherency strain energy will be absent. Let Us now consider which state produces the lowest total energy for a spherical precipitate with a misfit S and a radius r. The free energy of a crystal containing a fully coherent spherical precipitate has contributions from (i) the coherency strain energy given by Equation 3.39, and (ii) the chemical interfacial energy 7ch. The sum of these two terms is given by AG(coherent) = 4(jlS2 • -w3 + 4-rrr2 7ch (3.43) If the same precipitate has incoherent or semicoherent interfaces that completely relieve the unconstrained misfit there will be no misfit energy, but Fig. 3.51 Coherency strains caused by the coherent broad faces of 6' precipitates. there will be an extra structural contribution to the interfacial energy 7st. The total energy in this case is given by AG(non-coherent) = 0 + 4nr2(7ch + 7st) (3.44) For a given 8, AG (coherent) and AG (non-coherent) vary with r as shown in Fig. 3.52. When small, therefore, the coherent state gives the lowest total energy, while it is more favourable for large precipitates to be semicoherent or incoherent (depending on the magnitude of S). At the critical radius (rcrit) AG(coherent) = AG(non-coherent) giving 37st 4u.S2 (3.45) If we assume that 8 is small, a semicoherent interface will be formed with a structural energy 7st a 8- In which case 'en. « \ (3-46) If a coherent precipitate grows, during ageing for example, it should lose coherency when it exceeds rcrit. However, as shown in Fig. 3.53 loss of coherency requires the introduction of dislocation loops around the precipitate and in practice this can be rather difficult to achieve. Consequently coherent precipitates are often found with sizes much larger than rcrit. There are several ways in which coherency may be lost and some of them are illustrated in Fig. 3.54. The most straightforward way is for a dislocation loop to be punched out at the interface as shown in Fig. 3.54a. This requires the stresses at the interface to exceed the theoretical strength of the matrix. However, it can be shown that the punching stress ps is independent of the precipitate size and depends only on the constrained misfit e. If the shear modulus of the matrix is u, ps = 3u,e (3.47) Non-coherent rcrit Fig. 3.52 The total energy of matrix + precipitate v. precipitate radius for spherical coherent and non-coherent (semicoherent or incoherent) precipitates. 162 Crystal interfaces and microstructure Dislocation (a) (b) (o Fig 3.53 Coherency loss for a spherical precipitate. Ca) Coherent (h\ Cnh^nm strains replaced by dislocation loop, (c) In perspective U"lerent' (b) Cohe™cy Dislocation / Matrix dislocation wraps around Precipitate Punched-out loop Interfacial dislocation (a) (b) (c) New dislocation as plate lengthens > Misfit dislocation (d) l^C^Lr^?™?™^ coherency loss, (a) Dislocation punching from interface S^&dH1^ dlSl°CatT (c> N^deation at ^ of plate repeated as plte lengthens, (d) Loop expansion by vacancy condensation in the precipitate. interphase interfaces in souus ±u_, It has been estimated that the critical value of e that can cause the theoretical strength of the matrix to be exceeded is approximately given by = 0.05 (3.48) f Consequently precipitates with a smaller value of e cannot lose coherency by this mechanism, no matter how large. There are several alternative mechanisms but all require the precipitate to reach a larger size than rent16. For example, the precipitate can attract a matrix dislocation with a suitable Burgers vector, and cause it to wrap itself around !; the precipitate, Fig. 3.54b. This mechanism is difficult in annealed specimens but is assisted by mechanical deformation. In the case of plate-like precipitates the situation is different and it is now possible for the high stresses at the edges of the plates to nucleate dislocations «§ by exceeding the theoretical strength of the matrix. The process can be repeated as the plate lengthens so as to maintain a roughly constant interdis-v location spacing, Fig. 3.54c. Another mechanism that has been observed for plate-like precipitates is the nucleation of dislocation loops within the precipitate17. Vacancies can be attracted to coherent interfaces18 and 'condense' to form a prismatic dislocation loop which can expand across the precipitate, as shown in Fig. 3.54d. 3,4.5 Glissile Interfaces In the treatment of semicoherent interfaces that has been presented in the previous sections it has been assumed that the misfit dislocations have Burgers vectors parallel to the interfacial plane. This type of interface is referred to as epitaxial. Glide of the interfacial dislocations cannot cause the interface to advance and the interface is therefore non-glissile. It is however possible, under certain circumstances, to have glissile semicoherent interfaces which can advance by the coordinated glide of the interfacial dislocations. This is possible if the dislocations have a Burgers vector that can glide on matching planes in the adjacent lattices as illustrated in Fig. 3.55. The slip planes must be continuous across the interface, but not necessarily parallel. Any gliding dislocation shears the lattice above the slip plane relative to that below by the Burgers vector of the dislocation. In the same way the gliding of the dislocations in a glissile interface causes the receding lattice, a say, to be sheared into the p-structure. As an aid to understanding the nature of glissile boundaries consider two simple cases. The first is the low-angle symmetric tilt boundary, shown in Figs. 3.7a and 3.11. In this case the Burgers vectors are all pure edge in nature and as they glide one grain is rotated into the other grain. Strictly speaking this is not an interphase interface as there is no change in crystal structure, just a rotation of the lattice. A slightly more complex example of a glissile interface between two different lattices is that which can arise between 164 Crystal interfaces and microstructure Interfacial dislocations Interphase interfaces in solids 165 Macroscopic plane of interface responding planes Fig. 3.55 The nature of a glissile interface. the cubic and hexagonal close-packed lattices. To understand the structure of this interface requires a slight digression to consider the nature of Shockley partial dislocations. Both fee and hep lattices can be formed by stacking close-packed layers of atoms one above the other. If the centres of the atoms in the first layer are denoted as A-positions, the second layer of atoms can be formed either by filling the B-positions, or C-positions as shown in Fig. 3.56. Either position produces the same atomic configuration at this stage. Let us assume therefore that the atoms in the second layer occupy B-sites. There are now two non-equivalent ways of stacking the third layer. If the third layer is placed directly above the first layer the resulting stacking sequence is ABA and the addition of further layers in the same sequence ABABABABAB . . . has hexagonal symmetry and is known as a hexagonal close-packed arrangement. The unit cell and stacking sequence of this structure are shown in Fig. 3.57. The close-packed plane can therefore be indexed as (0001) and the close-packed directions are of the type (1120). If the atoms in the third layer are placed on the C-sites to form ABC and the same sequence is then repeated, the stacking sequence becomes ABCABCAB . . . which produces a cubic close-packed arrangement with a face-centred cubic unit cell as shown in Fig. 3.58. The close-packed atomic planes in this case become the {111} type and the close-packed directions the (110) type. In terms of the fee unit cell the distance between the B- and C-sites measured parallel to the close-packed planes corresponds to vectors of the type § (112). Therefore if a dislocation with a Burgers vector § [112] glides between two (111) layers of an fee lattice, say layers 4 and 5 in Fig. 3.59, all layers above the glide plane (5, 6, 7 . . .) will be shifted relative to those below the glide plane by a vector % [112]. Therefore all atoms above the glide plane in B-sites are moved to C-sites, atoms in C-sites move to A-sites, and atoms in A-sites move to B-sites, as shown in Fig. 3.59, This type of dislocation with b = f(112) is known as Shockley partial dislocation. They are called partial dislocations because vectors of the type |(112) do not connect lattice Fig. 3.56 The location of A, B and C sites in a close-packed layer.of atoms. See also Figs. 3.57 and 3.58. (After J.W. Martin and R.D. Dohertv, Stability of Microstnictwe in Metallic Systems, Cambridge University Press, Cambridge, 1976 ) B Fig. 3.57 A hexagonal close-packed unit cell . -I t ^ ***** 166 Crystal interfaces and microstructure Interphase interfaces in solids 167 Fig. 3.58 A cubic close-packed structure showing fee unit cell and stacking sequence. b=§-[112] /y~--~$Kp plane *S (III) 7 6 5 4 3 2 1 A C B A C B A B A C A C B A 'hef (a) (b) Fig. 3.59 (a) An edge dislocation with a Burgers vector b = f [112] on (111). (Shock-ley partial dislocation.) (b) The same dislocation locally changes the stacking sequence from fee to hep. points in the fee structure. The gliding of Shockley partial dislocations therefore disrupts the crystal lattice and causes a stacking fault over the area of glide plane swept by the dislocation. Figure 3.59 shows that the nature of this fault is such that four layers of material are converted into a hexagonal close-packed sequence CACA. Therefore in thermodynamically stable fee lattices the stacking fault is a region of high free energy. On the other hand if the fee lattice is only metastable with respect to the hep structure the stacking fault energy will be effectively negative and the gliding of Shockley partial dislocations will decrease the free energy of the system. Consider now the effect of passing another f[112] dislocation between layers 6 and 7 as shown in Fig. 3.60. It can be seen that the region of hep stacking is now extended by a further two layers. Therefore a sequence of Shockley partial dislocations between every other (111) plane will create a glissile interface separating fee and hep crystals, Fig. 3.61. The glide planes of the interfacial dislocations are continuous from the fee to the hep lattice and the Burgers vectors of the dislocations, which necessarily lie in the glide plane, are at an angle to the macroscopic interfacial ■ fee ' 10 A B C 9 C A B 8 B C A 7 A B C 6 C A A \ 5 B C C 4 A A A 3 c ; C C 2 B B B . 1 A A A hep Fig- 3.60 Two Shockley partial dislocations on alternate (111) planes create six layers of hep stacking. Fig. 3.61 An array of Shockley partial dislocations forming a glissile interface between fee and hep crystals. plane. If the dislocation network glides into the fee crystal it results in a transformation of fee —> hep, whereas a hep —» fee transformation can be brought about by the reverse motion. Macroscopically the interfacial plane lies at an angle to the (111) or (0001) planes and need not be parallel to any low-index plane, i.e. it can be irrational. Microscopically, however, the interface is stepped into planar coherent facets parallel to (lll)fcc and (0001)hcp with a step height the thickness of two closed-packed layers. An important characteristic of glissile dislocation interfaces is that they can produce a macroscopic shape change in the crystal. This is illustrated for the fee -»■ hep transformation in Fig. 3.62a. If a single fee crystal is transformed into an hep crystal by the passage of the same Shockley partial over every (111) plane then there is a macroscopic shape change, in this case a simple shear, as shown. There are, however, two other Shockley partials which can also be used to transform fee hep stacking, and if the transformation is achieved using all three partials in equal numbers there will be no overall shape change, Fig. 3.62b. : 168 Crystal interfaces and microstructure (a) fee hep (b) Fig, 3.62 Schematic representation of the different ways of shearing cubic close-packed planes into hexagonal close-packed (a) using only one Shockley partial, (b) using equal numbers of all three Shockley partials. The formation of martensite in steel and other alloy systems occurs by the motion of glissile-dislocation interfaces. These transformations are characterized by a macroscopic shape change and no change in composition. Usually, however, the interface must be more complex than the fec/hep case discussed above, although the same principles will still apply. Martensitic transformations are dealt with further in Chapter 6. 3.4.6 Solid I Liquid Interfaces19 Many of the ideas that were discussed with regard to solid/vapour interfaces can be carried over to solid/liquid interfaces, only now the low density vapour phase is replaced by a high density liquid, and this has important consequences for the structure and energy of the interface. There are basically two types of atomic structure for solid/liquid interfaces. One is essentially the same as the solid/vapour interfaces described in Section 3.1, i.e. an atomically flat close-packed interface, Fig. 3.63a. In this case the transition from liquid to solid occurs over a rather narrow transition zone approximately one atom layer thick. Such interfaces can also be described as smooth, faceted, or sharp. The other type is an atomically diffuse interface, Fig. 3.63b, in which the transition from liquid to solid occurs over several atom layers. Thus there is a gradual weakening of the interatomic bonds and an increasing disorder across the interface into the bulk liquid Interphase interfaces in solids Liquid 169 (a) Liquid (c) (b) Solid Fie 3 63 Solid/liquid interfaces: (a) atomically smooth, (b) and (c) atomically rough' or diffuse interfaces. (After M.C. Flemings, Solidification Processing, McGraw-Hill, New York, 1974.) phase; or in thermodynamic terms, enthalpy and entropy gradually change from bulk solid to bulk liquid values across the interface as shown in Fig 3 64 When the solid and liquid are in equilibrium (at Tm) the high enthalpy of the liquid is balanced by a high entropy so that both phases have the same free energy. In the interface, however, the balance is disturbed thereby giving rise to an excess free energy, -/$L. Diffuse interfaces are also known as rough or non-faceted. The dotted line Distance across interface Fig. 3.64 The variation of H, -TmS and G across the solid/liquid interface, at: the equilibrium melting temperature Tm, showing the origin of the sohd/hquid interface energy 7. 170 Crystal interfaces and microstructure Interface migration 111 in Fig. 3.63b is an attempt to show the rough nature of the interface bj jS I dividing the atoms into the 'solid' and 'liquid'. If this is done the schemalic representation of Fig. 3.63c can be used. The type of structure chosen by a particular system will be that which % \ minimizes the interfacial free energy. According to a simple theory developed ; by Jackson20 the optimum atomic arrangement depends mainly on the latent ^ ' heat of fusion (Lt) relative to the melting temperature (Tm). This theoij ^ predicts that there is a critical value of Lt/Tm — 4 R above which the interface should be flat and below which it should be diffuse. Most metals ha\e * * Lt/Tm = R and are therefore predicted to have rough interfaces. On the ^ other hand some intermetallic compounds and elements such as Si, Ge, Sb as • well as most non-metals have high values of Lt/Tm and generally have flat close-packed interfaces. If the model is applied to solid/vapour interfaces / (the heat of sublimation) should be used instead of Lf and then flat surfaa. are predicted even for metals, in agreement with observations. If the broken-bond model is applied to the calculation of the energy of a solid/liquid interface it can be argued that the atoms in the interface ai roughly half bonded to the solid and half to the liquid so that the interfacht enthalpy should be ~0.5 Lt/N& per atom. This appears to compare rather favourably with experimentally measured values of 7SL which are ~0.45 £f/Na per atom for most metals. However the agreement is probabh only fortuitous since entropy effects should also be taken into account. Fig. 3.64. Some experimentally determined values of 7SL are listed in Table 3.4. Table 3.4 Experimentally Determined Solid/Liquid Interfacial Free Energies Values selected from D. Turnbull, Journal of Applied Physics, Vol. 21: 1022(1950). Material TJK 7SL/mJ m 2 Sn 505.7 54.5 Pb 600.7 33.3 Al 931.7 93 Ag 1233.7 126 Au 1336 132 Cu 1356 177 Mn 1493 206 Ni 1725 255 Co 1763 234 Fe 1803 204 Pd 1828 209 Pt 2043 240 These values were determined by indirect means from homogeneous nuclea-tion experiments (see Chapter 4) and may contain systematic errors. Comparison of Tables 3.2 and 3.3 indicates 7SL — 0.307b (for a grain boundary). More direct experiments21 imply that 7SL ~ 0.45 7b (~ 0.15 7SV). Another useful empirical relationship is that Isv > 7sl + 7lv which means that for a solid metal close to Tm it is energetically favourable for the surface to melt and replace the solid/vapour interface with solid/liquid and liquid/vapour interfaces. It is found experimentally that the free energies of diffuse interfaces do not vary with crystallographic orientation, i.e. 7-plots are spherical22. Materials with atomically flat interfaces, however, show strong crystallographic effects and solidify with low-index close-packed facets, Fig. 3.65. 3.5 Interface Migration The great majority of phase transformations in metals and alloys occur by a process known as nucleation and growth, i.e. the new phase (ß) first appears Fig. 3.65 Examples of solid-liquid interface structure in metallic systems, (a) Non-faceted dendrites of silver in a copper-silver eutectic matrix (x 330); (b) faceted cuboids of ß'-SnSb compound in a matrix of Sn-rich material (x 110). (After G.A. Chadwick, Metallography of Phase Transformations, Butterworths, London, 1972.) 172 Crystal interfaces and microstructure at certain sites within the metastable parent (a) phase (nucleation) and this is subsequently followed by the growth of these nuclei into the surrounding matrix. In other words, an interface is created during the nucleation stage and then migrates into the surrounding parent phase during the growth stage. This type of transformation is therefore essentially heterogeneous, i.e. at any time during the transformation the system can be divided into parent and product phases. The nucleation stage is very important and determines many features of the transformation. However, most of the transformation product is formed during the growth stage by the transfer of atoms across the moving parent/product interface. There are basically two different types of interface: glissile and non-glissile. Glissile interfaces migrate by dislocation glide that results in the shearing of the parent lattice into the product. The motion of glissile interfaces is relatively insensitive to temperature and is therefore known as athermal migration. Most interfaces are non-glissile and migrate by the more or less random jumps of individual atoms across the interface in a similar way to the migration of a random high-angle grain boundary. The extra energy that the atom needs to break free of one phase and attach itself to the other is supplied by thermal activation. The migration of non-glissile interfaces is therefore extremely sensitive to temperature. A convenient way of classifying nucleation and growth transformations is to divide them according to the way in which the product grows. Therefore two. major groupings can be made by dividing the transformations according the whether growth involves glissile or non-glissile interfaces. Transformations produced by the migration of a glissile interface are referred to as military transformations. This emphasizes the analogy between the coordinated motion of atoms crossing the interface and that of soldiers moving in ranks on the parade ground. In contrast the uncoordinated transfer of atoms across a non-glissile interface results in what is known as a civilian transformation. During a military transformation the nearest neighbours of any atom are essentially unchanged. Therefore the parent and product phases must have the same composition and no diffusion is involved in the transformation. Martensitic transformations belong to this group. Glissile interfaces are also involved in the formation of mechanical twins and twinning therefore has much in common with martensitic transformations. During civilian transformations the parent and product may or may not have the same composition. If there is no change in composition, e.g. the a —> y transformation in pure iron, the new phase will be able to grow as fast as the atoms can cross the interface. Such transformations are said to be interface controlled. When the parent and product phases have different compositions, growth of the new phase will require long-range diffusion. For example, the growth of the B-rich |3 phase into the A-rich a phase shown in Fig, 3.66 can only occur if diffusion is able to transport A away from, and B towards the advancing interface. If the interfacial reaction is fast, i.e. the transfer of atoms across the interface is an easy process, the rate at which the Interface migration 173 #18 P= I O B • A Fig. 3.66 Composition changes in a substitutional alloy caused by interface migration when the two adjoining phases have different compositions. P phase can grow will be governed by the rate at which lattice diffusion can remove the excess atoms from ahead of the interface. This is therefore known as diffusion-controlled growth. However, if for some reason the interfacial reaction is slow, the growth rate will be governed by the interface kinetics. Under these circumstances growth is said to be interface controlled and a very small concentration gradient in the matrix is sufficient to provide the necessary flux of atoms to and from the interface. It is also possible that the interface reaction and diffusion process occur at similar rates in which case the interface is said to migrate under mixed control. The above discussion of interface migration and classification of nucleation and growth transformations (also known as heterogeneous transformations) is summarized in Table 3.5, together with some examples of each class. This classification is adapted from that first proposed by Christian23-24. Non-glissile interfaces can be considered to include solid/liquid and solid/vapour interfaces as well as solid/solid (coherent, semicoherent and incoherent) interfaces. Therefore solidification and melting can be included in the classification of civilian transformations under diffusion control (although the concept of diffusion may sometimes need to be extended to include the diffusion of 174 Crystal interfaces and microstructure > 0 4j & 00 b c u co *-* j= ^ cj ° o CJ 3 » « 3 0 C "5 .5* "5 O 2 ' u S j= O o z 2 « Si ■§ £ o E O ^ 8 o o d. £ o ° I c o. .2 -a o. c E £ • . o a o 2 c £ 3 - s o e u e O u t u ,a .9 1-1 s 3 u £ ^ '£ c "5 .2 -o - .£-•3 o -f 'a. 2-S Si £ c o 5 - o CO 'Zi 'M-o cj «1 £ 5 c =5 ^ o s o c 3 ■§ c o c *s o co 'S. -2 a u 5 a. Q m U W. Interface migration 175 c .2 .H S f c s 2 g.-s 4> c o « « c- £ t3 X? - Ct .> s o £ £ o o £ c c | E 3 H t/5 .£> e CO x beat). Condensation and evaporation at a free solid surface are also included although they will not be treated in any depth25. While many transformations can be easily classified into the above system, there are other transformations where difficulties arise. For example, the bainite transformation takes place by thermally activated growth, but it also produces a shape change similar to that produced by the motion of a glissile interface. At present the exact nature of such transformations is unresolved. There is a small class of transformations, known as homogeneous transformations that are not covered by Table 3.5. This is because they do not occur by the creation and migration of an interface, i.e. no nucleation stage is involved. Instead the transformation occurs homogeneously throughout the parent phase. Spinodal decomposition and certain ordering transformations are examples of this category and they will be discussed in Chapter 5. 3.5.1 Diffusion-Controlled and Interface-Controlled Growth26 Let us now look more closely at the migration of an interface separating two phases of different composition. Consider for simplicity a p precipitate of ■almost pure B growing behind a planar interface into A-rich a with an initial composition X0 as illustrated in Fig. 3.67. As the precipitate grows, the a /.adjacent to the interface becomes depleted of B so that the concentration of ; B in the a phase adjacent to the interface X[ decreases below the bulk concentration, Fig, 3,67a. Since growth of the precipitate requires a net flux |bf B atoms from the a to the p phase there must be a positive driving force across the interface AuB as shown in Fig. 3.67b. The origin of this chemical potential difference can be seen in Fig. 3.67c. Clearly for growth to occur the interface composition must be greater than the equilibrium concentration Xe. By analogy with the migration of a high-angle grain boundary ; (Section 3.3.4) the net flux of B across the interface will produce an interface velocity v given by (3.49) where M is the interface mobility and Vm is the molar volume of the (3 phase. The corresponding flux across the interface will be given by PB = -AfA|ij,/V* moles of B m-2 s_1 (3.50) (The negative sign indicates that the flux is in the negative direction along the jc-axis.) As a result of the concentration gradient in the a phase there will also be a flux of B atoms towards the interface given by dCt (3.51) j interface If a steady state exists at the interface these two fluxes must balance, i.e. JB — J] (3.52) Crystal interfaces and microstructure (C) | ■ I_ 0 Xe Xi x0 XB__ 1 acr ^^nterface migration with long-range diffusion, (a) Composition profiles interface, (b) The origin of the driving force for boundary migration into . « phase, (c) A schematic molar free energy diagram showing the relationship the ten Xi and x*- ^Note that the solubiUtyof A in the P Phase is so low that ue shape of the free energy curve cannot be drawn on this scale.) Interface migration 111 |* If the interface mobility is very high, e.g. an incoherent interface, Au,'B can be very small and X± — Xe. Under these circumstances there is effectively local equilibrium at the interface. The interface will then move as fast as y diffusion allows, and growth takes place under diffusion control. The growth rate can then be evaluated as a function of time, say, by solving the diffusion equation with the boundary conditions X{ = Xe and XB(<*>) = X0. Simple examples of this problem will be given in subsequent chapters in connection with solidification and diffusive transformations in solids. When the interface has a lower mobility a greater chemical potential difference (Au,B) is required to drive the interface reaction and there will be a departure from local equilibrium at the interface. The value of Z; that is chosen will be that which enables Equation 3.52 to be satisfied and the interface will then be migrating under mixed control. In the limit of a very low mobility it is possible that Xt ~ X0 and (dC/dx)iMeiface is almost zero. Under these conditions growth is said to be interface controlled and there is a maximum possible driving force Au^ across the interface. It can easily be shown that for a dilute or ideal solution, the driving force Au-'b is given by provided (X-, - Xs) « Xe (see exercise 3.20). Thus the rate at which the interface moves under interface control should be proportional to the deviation of the interface concentration from equilibrium (X[ — Xc). Let us now consider the question of why interface control should occur at all when the two phases have a different composition. At first sight it may appear that interface control should be very unlikely in practice. After all, the necessary long-range diffusion involves a great many atom jumps while the interface reaction essentially involves only one jump. Furthermore the activation energy for diffusion across the interface is not likely to be greater than for diffusion through the lattice—quite the contrary. On this basis, therefore, all interface reactions should be very rapid in comparison to lattice diffusion, i.e. all growth should be diffusion controlled. In many cases the above arguments are quite valid, but under certain conditions they are insufficient and may even be misleading. Consider again the expression that was derived for the mobility of a high-angle grain boundary, Equation 3.22. A similar expression can be derived for the case of an interphase interface with A(x'B replacing AG, (see exercise 3.19). It can be seen, therefore, that the above arguments neglect the effect of the accommodation factor (A), i.e. the probability that an atom crossing the boundary will be accommodated on arrival at the new phase. It is likely that incoherent interfaces and diffuse solid/liquid interfaces, as high-angle grain boundaries, will have values of A close to unity. These interfaces should therefore migrate under diffusion control. However, as will be demonstrated later, it is possible for certain types of coherent or semicoherent crystal interfaces and microstructure interfaces, as well as smooth solid/liquid interfaces to have such low values of A that some degree of interface control is easily possible. If two phases with different compositions, but the same crystal structure are separated by a coherent interface as shown in Fig. 3.32a, the interface can advance by the replacement of the a atoms in plane AA' with fi atoms by normal lattice diffusion involving vacancies. There is no need for a separate! interface reaction and the migration of this type of interface is therefore? diffusion controlled. This situation arises during the growth of GP zones for example. The same arguments will apply if the interface is semicoherent provided the misfit dislocations can climb by vacancy creation or annihilation. Quite a different situation arises when the two phases forming a coherent or semicoherent interface have different crystal structures. Consider for example the coherent close-packed interface between fee and hep crystals, Fig. 3.68a. If growth of the hep phase is to occur by individual atomic jumps (i.e. so-called continuous growth) then an atom on a C site in the fee phase must change into a B position as shown in Fig. 3.68b. It can be seen, however, that this results in a very high energy, unstable configuration with two atoms directly above each other on B sites. In addition a loop of Shockley partial dislocation is effectively created around the atom. An atom attempting such a jump will, therefore, be unstable and be forced back to its original position. The same situation will be encountered over the coherent regions of semi-coherent interfaces separating phases with different crystal structures. Solid/vapour as well as smooth solid/liquid interfaces should behave in al similar manner, though perhaps to a lesser extent. If a single atom attaches itself to a flat close-packed interface it will raise the interfacial free energy and will therefore tend to detach itself again. It can thus be seen that continuous growth at the above type of interfaces will be very difficult, i.e. very low accommodation factors and low mobility are expected. A A A A A A f.c.c. b b b b b b C C C C - m . C A A A A~ J L A ~A hep. b b b b b b A A A A A A (a) (b) Fig. 3.68 Problems associated with the continuous growth of coherent interfaces ST differem CrySta' StmCtUreS' b 179 D Fig. 3.69 The ledge mechanism. A way of avoiding the difficulties of continuous growth encountered in the above cases is provided by the 'ledge' mechanism shown in Fig. 3.69. If the interface contains a series of ledges BC, DE normal to the facets AB, CD, EF, atoms will be able to transfer more easily across the ledges than the immobile facets and interface migration is therefore effected by the transverse migration of the ledges as shown. Growth ledges have in fact been seen with the aid of the electron microscope on the surfaces of growing precipitates. For example Fig. 3.70 shows an electron micrograph and a schematic drawing of the growth ledges on an Mg2Si plate in an Al~Mg-Si alloy27. Note that growth ledges are usually hundreds of atom layers high. When existing ledges have grown across the interface there is a problem of generating new ones. In Fig. 3.70 the source of new ledges is thought to be heterogeneous nucleation at the point of contact with another precipitate. The same problem will not be encountered if the precipitate is dissolving, however, since the edges of the plate will provide a continual source of ledges28. It is thought that once nucleated, the rate at which ledges migrate across the planar facets should be diffusion controlled, i.e. controlled by how fast diffusion can occur to and from the ledges. However, the problem of nucleating new ledges may often lead to a degree of interface control on the overall rate at which the coherent or semicoherent interface can advance perpendicular to itself. Growth ledges are by no means restricted to solid/solid systems. The first evidence for the existence of growth ledges came from studies of solid/vapour interfaces. They are also found on faceted solid/liquid interfaces. The mechanism of interface migration can have important effects on the shape of second-phase inclusions. It was shown in Section 3.4.2 that in the absence of strain energy effects the equilibrium shape of a precipitate should be determined by the relative energies of the bounding interfaces. For example, a partially coherent precipitate should be disc or plate shaped with an aspect ratio of yi/yc where y{ is the energy of the incoherent edges and yc is the energy of the coherent or semicoherent broad faces. However, the precipitate shape observed in practice may be prevented from achieving this equilibrium shape by the relative rates at which the coherent and incoherent interfaces can migrate. For example if there are problems of ledge nucleation the easier growth of the incoherent plate edges may lead to a larger aspect ratio than the equilibrium. 180 Crystal interfaces and microstructure References 181 Fig. 3.70 (a) Growth ledges at an Mg2Si plate in Al-1.5 wt% Mg2Si, solution treated and aged 2 h at 350 °C. Dark field micrograph, (b) Schematic diagram of (a) showing ledges on Mg2Si plate, (After G.C. Weatherly, Acta Metatlurgica, 19 (1971) 181.) .. References 1. This subject is covered in detail in J.W. Martin and R.D. Doherty Stability of Microstructure in Metallic Systems, Cambridge University Press, 1976, and in M. McLean 'Miscrostructural instabilities in metallurgical systems—a review' Metal Science, March 1978, p. 113. 2. Methods are described in J.M. Blakely, Introduction to the properties of crystal surfaces, Pergamon, Oxford, 1973, p. 53. 3. For a more detailed discussion, see J.W. Christian, The Theory of Transformations in Metals and Alloys, Second Edition, Part I, Pergamon, Oxford, 1975, p. 153. 4. R.S. Nelson, D.J. Mazey and R.S. Barnes, Philosophical Magazine 11:91 (1965). 5. Sundquist, Acta Metallurgica, 12:67 (1964). 6. C.J. Simpson, K..T. Aust and W.C. Winegard, Scripta Met. 3-171 (1969). 7. See for example J.W, Martin and R.W. Doherty, Stability of Micro-structure in Metallic Systems, Cambridge University Press, 1976, p. 221. 8. J.W. Christian, Metallurgical Transactions, 21A: 799 (1990). 9. J.W. Christian, Materials Science and Engineering, A127: 215 (1990). 10. G.C. Weatherly and R.B. Nicholson, Phil. Mag. 17:801 (1968). .11. The interested reader is referred to M.I. Aaronson and J.K. Lee, 'Lectures on the theory of phase transformations', H.I. Aaronson (Ed.), Met. Soc. AIME, New York, 1975. 12. A quite general method for calculating the stress and strain fields in and around misfitting inclusions has been elucidated by J.D. Eshelby in Proc. Roy. Soc, London, A241 (1957) p. 376, and in Progress in Solid Mechanics, II, Chapter HI (1961). 13. F.R.N. Nabarro, Proc. Roy. Soc. A175: 519 (1940). 14. Jong K. Lee and William C. Johnson, Acta Metallurgica 26:541-545 (1978). 15. R. Sankaran and C. Laird, J. Mech. Phys. Solids, 24:251-262 (1976) and Phil. Mag., 29:179 (1974). 16. L.M. Brown and G.R. Woolhouse, Phil. Mag., 21:329 (1970). 17. R. Sankaran and C. Laird, Phil. Mag., 29:179 (1974). 18. K.E. Easterling and T. Johannesson, Phil. Mag., 24:981 (1971). 19. A detailed treatment of solid/liquid interfaces can be found in D.P, Woodruff The Solid-Liquid Interface, Cambridge University Press, 1973. 20. K.A. Jackson, Liquid Metals and Solidification, ASM, Cleveland, Ohio (1958), p. 174. 21. W.A. Miller and G.A Chadwick, Acta Metallurgica, 15:607 (1967). 22. W.A. Miller and G.A. Chadwick, Proc. Roy. Soc. A312:251 (1969). 23. J.W. Christian, The Theory of Transformations in Metals and Alloys, Edition 2, Part 1, Pergamon, 1975, p. 9. 24. J.W. Christian, 'Phase transformations in metals and alloys—an introduction', in Phase Transformations, 1, p. 1, Institute of Metallurgists, 1979. 25. The interested reader is referred to E. Rutner et al. (Eds.), Condensation and Evaporation of Solids, Gordon and Breach, New York, 1964. 26. For an advanced discussion of this subject see M. Hillert, 'Diffusion and interface control of reactions in alloys', Met. Trans. A., 6A:5 (1975). 27. G.C. Weatherly, Acta Metallurgica, 19:181 (1971). 182 Crystal interfaces and microstructure Exericses 183 28. J.W. Martin and R.D. Doherty, Stability of Microstructure in Metallic Systems, Cambridge University Press, 1976, p. 17. Further Reading J.M. Blakely, Introduction to the Properties of Crystal Surfaces, Pergamon Press, Oxford, 1973. G. A. Chadwick and D.A. Smith (Eds.), Grain Boundary Structure and Properties, Academic Press, London, 1976. H. Gleiter and B. Chalmers, 'High angle grain boundaries' in Progress in Materials Science, B. Chalmers, J.W. Christian and T.B. Massalski, (Eds.), Vol. 16, Pergamon, 1972. J.W. Martin and R.D. Doherty, Stability of Microstructure in Metallic Systems, Cambridge University Press, Cambridge, 1976. L.E. Murr, Interfacial Phenomena in Metals and Alloys, Addison-Wesley, London,1975. D.P. Woodruff, The Solid-Liquid Interface, Cambridge University Press Cambridge, 1973. Exercises 3.1 Use the method of Section 3.1 to estimate the surface energy of {111}, {200} and {220} surface planes in an fee crystal. Express your answer in J/surface atom and in J/m2. 3.2 Differentiate Equation 3.8 to obtain the slope of the ESv ~ 8 curve at 8 = 0. 3.3 If a two-dimensional rectangular crystal is bounded by sides of lengths l\ and l2 show by differentiation that the equilibrium shape is given by ll = "ll where 71 and y2 are tne energies of the sides ^ and l2 respectively. (The area of the crystal /t/2 is constant.) 3.4 (a) Measure 6 for the low-angle tilt boundary in Fig. 3.11. (b) Determine the Burgers vector of the interface dislocations by making a Burgers circuit around one of the dislocations. Does the mean spacing of the dislocations agree with that predicted by Equation 3.9? 3.5 Explain why grain boundaries move towards their centre of curvature during grain growth but away from their centre of curvature during recrystallization. 3.6 (a) Suppose a recrystallized, dislocation-free grain is growing into a deformed matrix containing a dislocation density of 1016 m" (i.e. 1016 m/m3). If the dislocations have an energy of ui>2/4 J m" calculate the pulling force acting on the recrystallized grain boundary. (Assume a shear modulus p, = 1010 N m-2 and a Burgers vector b = 0.28 nm.) (b) If the recrystallized grains grow from spherically shaped nuclei, what is the diameter of the smallest nucleus that can expand into the surrounding matrix? (Assume a grain boundary energy of 0.5 J m-2.) 3.7 Look up the equilibrium phase diagrams for the Al-Fe and Al-Mg systems. On the basis of these diagrams would you expect the grain boundary enrichment of Fe in dilute Al-Fe alloys to be greater or less than for Mg in dilute AI-Mg alloys at the same temperatures? 3.8 Derive Equation 3,31. 3.9 When a precipitate is surrounded by a spherical interface of radius r it is subjected to a pressure above that of the matrix by 27/r. Consider a faceted precipitate with an equilibrium shape that of a square plate with a thickness of 2xx and width 2x2. If the free energies of the broad faces and edges are respectively 71 and 72, show that the broad faces exert a pressure on the precipitate (AP) given by AP = (Hint: consider the total force acting on the periphery of the broad faces.) Show that the same result can be obtained by considering the pressure exerted by one of the edge faces of the plate. 3.10 Explain the structure and energies of coherent, semicoherent and incoherent interfaces, with particular reference to the role of orientation relationships and misfit. 3.11 Fe-rich GP zones can form in dilute Al-Fe alloys. Given that the atomic radii are 1.43 A for Al and 1.26 A Fe, would you expect the zones to be spherical or disc shaped? 3.12 Mg can dissolve in Al to form a substitutional solid solution. Mg atoms are, however, bigger than Al atoms and each Mg atom therefore distorts the surrounding Al lattice, i.e. a coherency strain field effectively exists around each Mg atom. Using Equation 3.39 estimate the misfit strain energy. Express the answer in kJ mol-1 and eV atom-1. (The shear modulus of Al = 25 GPa, the radius of an Al atom = 1.43 A, the radius of a Mg atom = 1.60 A.) What assumptions are implicit in this calculation? 3.13 Explain why fully coherent precipitates tend to lose coherency as they grow. 3.14 Show that the passage of a Shockley partial dislocation over every one of a given set of close-packed planes in fee crystals produces a twin of the original crystal. 3.15 If the ledges on the planar semicoherent interface in Fig, 3.69 move with a transverse velocity u what will be the overall velocity of the interface