Kinetics of Materials - R. Balluff S. Allen W. Carter (Wiley 2005) WW Part 9 ppt

13.3: CONSERVATIVE MOTION 305 interface. On the other hand, nonconservative motion occurs when the motion of the interface is coupled to long-range diffusional fluxes of one or more of the components of the system. Conservative motion can be achieved under steady-state conditions only when the atomic fraction of each component is the same in the adjoining crystals (see Exercise 13.1). For sharp interfaces, atoms are simply transferred locally across the interface from one adjoining crystal to the other and there is no need for the long-range diffusion of any species to the boundary. This local transfer can occur by the simple shuffling of atoms across the interface and/or by the creation of crystal defects (vacancies or interstitials) in one grain which then diffuse across the boundary and are destroyed in the adjoining grain, thus transferring atoms across the interface.* Examples of conservative motion are the glissile motion of martensitic interfaces (see Chapter 24) and the thermally activated motion of grain boundaries during grain growth in a polycrystalline material. During nonconservative interface motion, the boundary must act as a source for the fluxes. To accomplish this for sharp interfaces, atoms must be added to, or removed from, one or both of the the crystals adjoining the interface. This generally causes crystal growth or shrinkage of one or both of the adjoining crystals and hence interface motion with respect to one or both of the crystals. This can occur by the creation at the interface of the point defects necessary to support the long-range diffusional fluxes of substitutional atoms or by atom shuffling to accommodate the addition or removal of interstitial atoms. Nonconservative interface motion and the role of interfaces as sources or sinks for diffusional fluxes are of central importance in a wide range of phenomena in materials. For example, during diffusional creep and sintering of polycrystalline materials (Chapter 16), and the thermal equilibration of point defects, atoms diffuse to grain boundaries acting as point-defect sources. In these cases, the fluxes require the creation or destruction of lattice sites at the boundaries. In multicomponent-multiphase materials, the growth or shrinkage of the phases adjoining heterophase interfaces often occurs via the long-range diffusion of components in the system. In such cases, heterophase interfaces again act as sources for the diffusing components. Further aspects of the conservative and nonconservative motion of sharp inter- faces are presented below. The mechanism for the motion of a diffuse interface is discussed in Section 13.3.4. 13.3 CONSERVATIVE MOTION 13.3.1 Sharp boundaries of several different types can move conservatively by the glide of interfacial dislocations. In many cases, this type of motion occurs over wide ranges of temperature, including low temperatures where little thermal activation is available. Glissile Motion of Sharp Interfaces by Interfacial Dislocation Glide Small-Angle Grain Boundaries. As described in Appendix B, these semicoherent boundaries are composed of arrays of discretely spaced lattice dislocations. For 2Shuffles are small displacements of atoms (usually smaller than an atomic spacing) in a local region, such as the displacements that occur in the core of a gliding dislocation. 306 CHAPTER 13: MOTION OF CRYSTALLINE INTERFACES certain small-angle boundaries, these dislocations can glide forward simultaneously, allowing the boundary to move without changing its structure. The simplest ex- ample is the motion of a symmetric tilt boundary by the simultaneous glide of its edge dislocations as in Fig. 13.1. An important aspect of this type of motion is the change in the macroscopic shape of the bicrystal specimen which occurs because the transfer of atoms across the boundary from grain 2 to grain 1 by shuffling is a highly correlated process. Each atom in the shrinking grain is moved to a prede- termined position in the growing grain as it is overrun by the displacement field of the moving dislocation array and shuffled across the boundary. The positions of all the atoms in the bicrystal are therefore correlated with the position of the interface and there is a change in the corresponding macroscopic shape of the specimen as the boundary moves. This type of interface motion has been termed military to distinguish it from the disorganized civilian type of interface motion that occurs when an incoherent general interface moves as described in Section 13.3.3 [3]. In the latter case, there is no change in specimen shape. Numerous experimental observations of the glissile motion of small-angle bound- aries have been made [2]. Most general small-angle boundaries possess more than one family of dislocations having different Burgers vectors. Glissile motion of such boundaries without change of structure is possible only when the glide planes of all the dislocation segments in the array lie on a common zone with its axis out of the boundary plane. When this is not the case, the boundary can move conser- vatively only by the combined glide and climb of the dislocations as described in Section 13.3.2. Large-Angle Grain Boundaries. Semicoherent large-angle grain boundaries contain- ing localized line defects with both dislocation and ledge character can often move forward by means of the lateral glissile motion of their line defects. A classic ex- ample is the motion of the interface bounding a (111) mechanical twin in the f.c.c. structure illustrated in Fig. 13.2. This boundary can be regarded alternatively as a large-angle grain boundary having a misorientation corresponding to a 60" rota- tion around a [lll] axis. The twin plane is parallel to the (111) matrix plane, and the twin (i.e., island grain) adopts a lenticular shape in order to reduce its elastic energy (discussed in Section 19.1.3). The macroscopically curved upper and lower sections of the interface contain arrays of line defects that have both dislocation and ledge character, as seen in the enlarged view in Fig. 13.2b. Note that the interface is semicoherent with respect to a reference structure (see Section B.6) taken to be a bicrystal containing a flat twin boundary parallel to (111). The line defects are glissile in the (111) plane and their lateral glissile motion across the interface in the directions of the arrows causes the upper and lower sections of the interface to move normal to themselves in directions that expand the thickness of the lenticu- lar twin. In essence, the gliding line defects provide special sites where atoms can be transferred locally across the interface relatively easily by a military shuffling process, making the entire boundary glissile. This type of glissile interface motion produces a macroscopic shape change of the specimen for the same geometric rea- sons that led to the shape changes illustrated in Fig. 13.1. When a line defect with Burgers vector b' passes a point on the interface, the material is sheared parallel to the interface by the amount b. At the same time, the interface advances by h, the height of the ledge associated with the line defect. These effects, in combination, produce the shape change. A pressure urging the interface sections to move to 13.3 CONSERVATIVE MOTION 307 fY IX Matrix Twin / Matrix / Figure 13.2: (a) A lenticular twin in an f.c.c. structure bounded by glissile interfaces containing dislocations possessing ledge character viewed along [TlO]. (b) An enlarged view of the dislocation-step region. The interface is semicoherent with respect to a reference structure. corresponding to the bicrystal formed by a 60” rotation around [lll]. The Burgers vector of the dislocation is a translation vector of the DSC-lattice of the reference bicrystal. which is the fine grid shown in the figure (see Section B.6). (c) The same atomic structure as in (b). The interface now is considered to be coherent with respect to a reference structure. corresponding to the f.c.c. matrix crystal. In this framework. the dislocation is regarded as a coherency dislocation (see Section B.6). (d) The shape change produced by formation of a twin across the entire specimen cross section. expand the twin and produce this shape change can be generated by applying the shear stress, oxy, shown in Fig. 13.2~. The magnitude of this pressure is readily found through use of Eq. 12.1. The force (per unit length) tending to glide the line defects laterally is given by Eq. 11.1, f = baxy. The work done by the applied force in moving a unit area of the boundary a distance 6s is then (bxlh) boxy, and the pressure is therefore (13.2) This type of glissile boundary motion occurs during mechanical twinning when twins form in matrix grains under the influence of applied shear stresses [4]. The glissile lateral motion of the line defects can be very rapid, approaching the speed of sound (see Section 11.3.1), and the large number of line defects that must be generated on successive (111) planes can be obtained in a number of ways, including a dislocation “pole” mechanism. Glissile motion of other types of large-angle grain boundaries by the same basic mechanism have been observed [2]. Heterophase Interfaces. In certain cases, sharp heterophase interfaces are able to move in military fashion by the glissile motion of line defects possessing dislocation character. Interfaces of this type occur in martensitic displacive transformations, which are described in Chapter 24. The interface between the parent phase and the newly formed martensitic phase is a semicoherent interface that has no long- range stress field. The array of interfacial dislocations can move in glissile fashion and shuffle atoms across the interface. This advancing interface will transform 308 CHAPTER 13: MOTION OF CRYSTALLINE INTERFACES the parent phase to the martensite phase in military fashion and so produce a macroscopic shape change. 13.3.2 Thermally Activated Motion of Sharp Interfaces by Glide and Climb of Interfacial Dislocations The motion of many interfaces requires the combined glide and climb of interfacial dislocations. However, this can take place only at elevated temperatures where sufficient thermal activation for climb is available. Small-Angle Grain Boundaries. As mentioned, a small-angle grain boundary can move in purely glissile fashion if the glide planes of all the segments in its dislocation structure lie on a zone that has its axis out of the boundary plane. However, this will not usually be the case, and the boundary motion then requires both dislocation glide and climb. Figure 13.3 illustrates such an interface, consisting of an array of two types of edge dislocations with their Burgers vectors lying at 45" to the boundary plane, subjected to the shear stress oZy. Equation 11.1 shows that the shear stress exerts a pure climb force f = bnZy on each dislocation, which therefore tends to climb in response to this force. However, mutual forces between the dislocations in the array will tend to keep them at the regular spacing corresponding to the boundary structure of minimum energy. All dislocations will then move steadily along +z by means of combined glide and climb. The boundary as a whole will therefore move without changing its structure, and its motion will produce a specimen shape change, the same as that produced by the glissile motion of the boundary in Fig. 13.1. Successive dislocations in the array must execute alternating positive and negative climb, which can be accomplished by establishing the diffusion currents of atoms between them as shown in Fig. 13.3. Each current may be regarded as crossing the boundary from the shrinking crystal to the growing crystal. An approximate model for the rate of boundary motion can be developed if it is assumed that the rate of dislocation climb is diffusion limited [2]. Neglecting any effects of the dislocation motion and the local stress fields of the dislocations on Figure 13.3: Thermally activated conservative motion of a small-angle symmetric tilt boundary containing two arrays of edge dislocations with orthogonal Burgers vectors. f is the force exerted on each dislocation, by the applied stress. Arrows indicate atom fluxes between dislocations. 13.3. CONSERVATIVE MOTION 309 the diffusion] a flux equation for the atoms can be obtained by combining Eqs. 3.71 (13.3) Under diffusion-limited conditions, the vacancies can be assumed to be maintained at equilibrium at the dislocations. The dislocations act as ideal sources (Sec- tion 11.4.1) and, therefore] at the dislocations pv = 0. When an atom is inserted at a dislocation of type 2 acting as a sink (Fig. 13.3), the dislocation will move forward along x by the distance fi R/b. The force on it acting in that direction is axyb/fi, and the work performed by the stress is therefore (fi R/b)(oxyb/fi) = oxy R. The boundary value for the diffusion potential @A at the cores of these dislocations is, therefore, +:(sink) = pi - oxy R (13.4) where pi is the chemical potential of atoms in stress-free material. Similarly, at dislocations of type 1, acting as sources, @(source) = pi + gzy R. The average potential gradient in the region between adjacent dislocations is then (V@A) = 2R ozy/d, where d is the dislocation spacing. The approximate area per unit length, A, through which the diffusion flux passes is of order A E d. Using these quantities and Eq. 13.3, and assuming that the variations in *D due to local variations in the vacancy concentration are small enough to be neglected, the total atom current per unit length entering a dislocation of type 2 is given by (13.5) where *D is the self-diffusivity as measured under equilibrium conditions. The volume of atoms causing climb (per unit length per unit time) is then IAR, and the corresponding climb rate is therefore vc = IAR/b. Each dislocation moves along z by combined climb and glide at a rate that exceeds its climb rate by fi, and the boundary velocity is then v = five, or 4fi R*D bfkT gxy V= (13.6) Since d = b/(Ofi) [7] and the pressure on the boundary is P = tbzyl Eq. 13.6 may be expressed (13.7) Equation 13.7 shows that the velocity is proportional to the pressure through a boundary mobility, MBl itself proportional to the self-diffusivity, *D. The activa- tion energy for boundary motion will therefore be that for crystal self-diffusion as expected for a crystal diffusion-limited process. Large-Angle Grain Boundaries. Semicoherent large-angle boundaries may move con- servatively through the lateral motion of their dislocations (which also generally possess ledge character) by means of combined glide and climb. In these bound- aries, the coherent patches of the boundary between the dislocations are relatively 3Equation 13.3 was first obtained by Herring and is useful in modeling the kinetics of diffusional creep [5] and sintering [6] in pure metals. 310 CHAPTER 13: MOTION OF CRYSTALLINE INTERFACES stable and therefore resistant to any type of motion. The dislocations, however, are special places in the boundary that support the transfer of atoms across the inter- face from the shrinking to the growing crystal relatively easily as the dislocations glide and climb. The example in Fig. 13.4 is an extension of the model for the motion of a small- angle boundary by the glide and climb of interfacial dislocations (Fig. 13.3). Fig- ure 13.4 presents an expanded view of the internal “surfaces” of the two crystals that face each other across a large-angle grain boundary. Crystal dislocations have I V Figure 13.4: Expanded view of the “internal surfaces” of two crystals facing each other across a grain boundary. Lattice dislocations AB and DE have impinged upon the boundary, creating line defects with both ledge and dislocation character which may glide and climb in the boundary in the directions of the arrows creating growth or dissolution spirals. impinged upon the boundary from crystals 1 and 2, causing the formation of extrin- sic dislocation segments in the boundary along CB and EF, re~pectively.~ These extrinsic segments have Burgers vector components perpendicular to the boundary plane and possess ledge character. Crystal 1 can grow and crystal 2 can shrink if the segments CB and EF climb and glide in the directions of the arrows under the influence of the pressure driving the boundary. This can be achieved by the diffu- sion of atoms across the boundary from segment EF to segment CB, thus allowing the boundary to move conservatively. The continued motion of the segments in these directions will cause them to wrap themselves up into spirals around their pole dislocations in the grains (i.e., AB and ED). The dislocations will therefore form crystal growth or dissolution spirals in the boundary similar to the growth spi- rals that form on crystal free surfaces at points where lattice dislocations impinge on the surface (see Fig. 12.5). There is therefore a close similarity between this mode of dislocation-induced boundary motion and the motion of free surfaces due to the action of growth or dissolution ledge spirals as discussed in Section 12.2.2. Probable observed examples of such dislocation growth or dissolution spirals on grain boundaries are shown in Fig. 13.5. The rates of boundary motion will depend strongly upon the available densities of boundary dislocations with ledge character. The formation of such dislocations by 4See Section B.7 for a discussion of extrinsic vs. intrinsic interfacial dislocations. 13 3 CONSERVATIVE MOTION 311 Figure 13.5: grain boundaries. From Gleiter 181 and Dingley and Pond 191. Observed examples of apparent dislocation growth or dissolution spirals on the homogeneous nucleation of dislocation loops in the boundary is highly unlikely at the pressures that are usually exerted on boundaries [2]. An important source may then be impinged lattice dislocations, as described above. However, under many conditions, the rate of this type of boundary motion may be very slow. 13.3.3 Shuffling at Pure Ledges. Interfaces capable of supporting pure ledges (see Sec- tion B.7) may migrate by the transverse motion of the ledges across their faces much like the motion of free surfaces described in Section 12.2.2. However, the ledges in interfaces can move conservatively by the shuffling of single atoms or small groups of atoms from the shrinking crystal to the growing crystal at kinks in the interface ledges. This type of motion does not produce a specimen shape change. Its conservative nature is in contrast to the nonconservative nature of free-surface motion via surface-ledge migration. The shuffles will be thermally activated, and a simple analysis shows that the interface velocity can then be written Thermally Activated Motion of Sharp Interfaces by Atom Shuffling (13.8) where NI, is the number of kink sites per unit interface area, N, is the average number of atoms transferred per shuffle, vo is a frequency. and Ss and ES are the activation entropy and energy for the shuffling. As in Eq. 13.7, the velocity is proportional to the driving pressure, P, through a boundary mobility, MB. This mobility is critically dependent upon the density of kink sites, which may vary widely for different interfaces. Ledges will be present initially in vicinal interfaces, 312 CHAPTER 13: MOTION OF CRYSTALLINE INTERFACES but these will tend to be grown off during the interface motion and can therefore support only a limited amount of motion. Ledges cannot be nucleated homoge- neously in the form of small pillboxes at significant rates at the driving pressures usually encountered. However, heterogeneous nucleation could be of assistance in certain cases. In general, widely different boundary mobilities may be expected under different circumstances [2]. Uncorrelated Shuffling at General Interfaces. Interfaces that are general with respect to all degrees of freedom possess irregular structures and cannot support localized line defects of any significant strength. However, in many places along an irregular general interface, the structure can be perturbed relatively easily to allow atoms to be shuffled from the shrinking crystal to the growing crystal by means of thermal activation. In this case, a simple analysis of the interface velocity leads again to a relationship of the form of Eq. 13.8 [2]. However, the quantity Nk appearing in the mobility MB is now the density of sites in the interface at which successful shuffles can occur. Under most circumstances, the intrinsic density of these sites will be considerably larger than the density of kink sites on vicinal stepped boundaries, and the mobility of general interfaces will be correspondingly larger. 13.3.4 Diffuse interfaces of certain types can move by means of self-diffusion. One example is the motion of diffuse antiphase boundaries which separate two ordered regions arranged on different sublattices (see Fig. 18.7). Self-diffusion in ordered alloys allows the different types of atoms in the system to jump from one sublattice to the other in order to change the degree of local order as the interface advances. This mechanism is presented in Chapter 18. Thermally Activated Motion of Diffuse Interfaces by Self-Diffusion 13.3.5 The conservative motion of interfaces can be severely impeded by a variety of mech- anisms, including solute-atom drag, pinning by embedded particles, and pinning at grooves that form at the intersections of the interfaces with free surfaces. We take up the first two of these mechanisms below and defer discussion of surface grooving and pinning at surface grooves to Section 14.1.2 and Exercise 14.3. Impediments to Conservative Interface Motion Solute-Atom Drag. Solute atoms, which are present either by design or as un- wanted impurities, often segregate to interfaces where they build up “atmospheres” or segregates. This effect is similar in many respects to the buildup of solute-atom atmospheres at dislocations (discussed in Section 3.5.2). For the interface to move, it must either drag the solute atmosphere along with it or tear itself away. The dragging process requires that the solute atoms diffuse along with the moving in- terface under the influence of the attractive interaction forces exerted on them by the interface. In many cases, the forced diffusive motion of the solute atmosphere will be slow compared to the rate at which the interface would move in the absence of the solute atoms. The solute atoms then exert a solute-atom drag force on the moving interface and impede its motion. In cases where the applied pressure mov- ing the interface is sufficiently large, the interface will be torn away from the solute atmosphere. A number of models for solute-atom drag, involving various simpli- 13 3 CONSERVATIVE MOTION 313 fications, have been developed [2]. Figure 13.6 shows some of the main behavior predicted by Cahn's model [lo]. When the driving pressure, P, is zero, the steady-state interface velocity, w, is also zero and the distribution of solute atoms around the interface, shown in Fig. 13.6a, is symmetric. No net drag force is therefore exerted on the interface by the solute atoms in the atmosphere. However, as w increases, the atmosphere becomes increasingly asymmetric and increasing numbers of atoms cannot keep up the pace and are lost from the atmosphere. Figure 13.6b shows the steady-state velocity as a function of P. For the pure material (cxL = 0), the velocity is simply proportional to the pressure. This is known as intrinsic behavior. When solute atoms are added to the system, the velocity is reduced by the drag effect and the system now exhibits extrinsic behavior. At low pressures, the extrinsic velocity increases monotonically with increasing pressure, but at high pressures the interface eventually leaves behind its atmosphere and the velocity approaches the intrinsic velocity. When the solute concentration is sufficiently high, a region of instability appears in which the interface suddenly breaks free of its atmosphere as the pressure is increased. Figure 13.6~ shows that essentially intrinsic behavior is obtained at elevated temperatures at all solute concentrations because of thermal desorption of the atmospheres. However, extrinsic behavior appears at the lawer temperatures in a manner that is stronger the higher the solute concentration. Finally, Fig. 13.6d shows that essentially intrinsic behavior can be obtained over a range of solute concentrations as long as the driving pressure is sufficiently high. To summarize, the drag effect becomes more important as the solute concentration increases and the driving pressure and temperature decrease. x=o x c Constant T P , Constant T , Constant P IIT + log (.XI. - Figure 13.6: Grain-boundary solute-drag phenoinena predicted by Cahn's model. (a) Segregated solute concentration profile c(z) across boundary as a function of increasing boundary velocity v (the z axis is perpendicular to the boundary). cxL is the solute concentration in the adjoining crystals. (b) Bouiidary velocity vs. pressure, P, on boundary as u, function of increasing cxL. (c) In v VS. 1/T as a function of increasing cxL. (d) In v vs. hi cayL as a function of increasing P. From Cahn [lo] 314 CHAPTER 13 MOTION OF CRYSTALLINE INTERFACES Pinning Due to Embedded Second- Phase Particles. A single embedded second-phase particle can pin a patch of interface as illustrated in Fig. 13.7. Here, an interface between matrix grains 1 and 2, in contact with a spherical particle, is subjected to a driving pressure tending to move the interface forward along y past the particle. Interfacial energy considerations cause the interface to be held up at the particle. as analyzed below, and therefore to bulge around it. Inspection of the figure shows that static equilibrium of the tangential capillary forces exerted by the particle/grain 1 interface, the particle/grain 2 interface, and the grain l/grain 2 interface requires that the angle Q satisfy the relation (13.9) The net restraining force along y exerted on the interface by the particle (i.e., the negative of the force exerted by the interface on the particle) is F = ~TRCOS Q 7'' COS(Q - 4) ( 13.10) The maximum force, F,,,, occurs when dF/dQ = 0, corresponding to Q = a/2. Applying this condition to Eq. 13.10, the maximum force is F,,, = .irRy12(1 + COSQ) 0 and Q (13.11) For the simple case where ypl E yp2, COSQ 7r/4 and thus F,,, E rRy 12 (13.12) and F,,, depends only on R and Consider now the pressure-driven movement of an interface through a dispersion of randomly distributed particles. At any instant, the interface will be in contact with a certain number of these particles (per unit area). each acting as a pinning point and restraining the interface motion as in Fig. 13.7. Additionally, the particles themselves may be mobile due to diffusional transport of matter from the particle's leading edge to its trailing edge [a, 121. Each particle's mobility depends upon its size and the relevant diffusion rates. A wide range of behavior is then possible depending upon temperature, particle sizes, and other factors. If the particles are Matrix grain 1 f Y12 \ Interface ~ Figure 13.7: Spherical particle inning an interface between grains 1 and 2. The interface is subjected to a driving pressure tRat tends to move it in the y direction. From Nes et a1 [Ill. [...]... misorientations of singular boundaries Data for A1 of 99 .99 995 %, 99 .99 92%, and 99 .98 % purity From Fridman et al [15] 5Detailed analyses of these processes are given by Sutton and Balluffi [2] 61f motion is unaffected by drag effects due to impurity atoms, it is called intrinsic 316 CHAPTER 13: MOTION OF CRYSTALLINE INTERFACES The degree of solute segregation and drag is a function of the intrinsic... Komem, P Petroff, and R.W Balluffi Direct observation of grain boundary dislocation climb in ion-irradiated gold bicrystals Phil Mag., 26:23 9- 2 52, 197 2 25 K.E Rajab and R.D Doherty Kinetics of growth and coarsening of faceted hexagonal precipitates in an fcc matrix 1 Experimental-observations Acta Metall Muter., 37( 10):270 9- 2 722, 198 9 26 G.C Weatherly The structure of ledges at plate-shaped precipitates... York, 195 3 8 H Gleiter The mechanism of grain boundary migration Acta Metall., 17(5):56 5-5 73, 196 9 9 D.J Dingley and R.C Pond On the interaction of crystal dislocations with grain boundaries Acta Metal l., 27(4) :66 7-6 82, 197 9 10 J.W Cahn The impurity-drag effect in grain boundary motion 10 (9) :78 9- 7 98 , 196 2 Acta Metall., 11 E Nes, N Ryum, and 0 Hunderi On the Zener drag Acta Metall., 33:ll-22, 198 5 12... = 19, (111))tilt boundary from an impurity in iron-doped aluminum bicrystals Sow Phys Solid State, 23(10):171 8-1 721, 198 1 22 R.W Siegel, S.M Chang, and R.W Balluffi Vacancy loss at grain-boundaries in quenched polycrystalline gold Acta Metall Muter., 28(3):24 9- 2 57, 198 0 23 A.H King and D.A Smith On the mechanisms of point-defect absorption by grain and twin boundaries Phil Mag A , 42(4): 49 5-5 12, 198 0... l):C1:18 5-1 90 , 199 0 19 T Kizuka, M Iijima, and N Tanaka Grain boundary migration at atomic scale in MgO Muter Sci Forum, 23 3-2 34:40 5-4 12, 199 7 20 R.J Jahn and P.D Bristowe A molecular dynamic study of grain boundary migration without the participation of secondary grain boundary dislocations Scripta Metall., 24(7):131 3-1 318, 199 0 21 D.A Molodov, C.V Kopetskii, and L.S Shvindlerman Detachment of a special... Oxford, 197 5 4 J.P Hirth and J Lothe Theory of Dislocations John Wiley & Sons, New York, 2nd edition, 198 2 5 C Herring Diffusional viscosity of a polycrystalline solid J Appl Phys., 21:43 7-4 45, 195 0 6 C Herring Surface tension as a motivation for sintering In W.E Kingston, editor, The Physics of Powder Metallurgy, pages 14 3-1 79, New York, 195 1 McGraw-Hill 7 W.T Read Dislocations in Crystals McGraw-Hill,... relatively large activation energies for the motion of general tilt boundaries in the 99 .99 92% material (having misorientations between those of the singular boundaries indicated by the arrows) most probably arose from strong drag effects associated with relatively strong impurity segregation at these boundaries This effect disappears in the higher-purity, 99 .99 995 %, material At even higher purity, the situation... boundaries in 99 .99 92% pure Al Such results show that grain boundary mobilities are extremely sensitive to solute-atom drag effects, and can be strongly affected by them even at exceedingly small solute-atom concentrations Purity 99 .99 92 4 1 0 " " " " " ' 0 10 20 I 30 40 50 Misorientation 8 (degrees) Figure 13.8: Activation energy, E B , for the motion of (100) tilt boundaries in A1 as a function of tilt... curve y - h ( z )= constant, as illustrated in Fig 14.1 The rate of particle accumulation at a particular surface site is the integral of the flux over the surface enclosing each surface site Representing each surface site as a box with a square base of area l/cs (where cs is the number of surface sites per area) and height 6, the particle accumulation rate is ii/s =- J box-walls - dA - JS =- J box-volume... = 1273 K, the corresponding mobilities are proportional t o the slopes of the curves in Fig 13.16: M B ( T 1 ) 0: = 0.082 and 4 M B ( T ~ ) -= 0.23 0: 19 - 2 (13.37) From the Arrhenius law, (13.38) and hence - 1.054 x 1.38 x lo-'' - 1.46 x 1 0-4 K-I J K-' (13. 39) = 1.0 x 1 0 - l ~ J (d) The most likely explanation for jerky motion of the boundary is localized pinning by precipitates or small inclusions . arrows) most probably arose from strong drag effects associated with relatively strong impurity segregation at these boundaries. This effect disappears in the higher-purity, 99 .99 995 %, material embedded particles, and pinning at grooves that form at the intersections of the interfaces with free surfaces. We take up the first two of these mechanisms below and defer discussion of surface grooving. of (100) tilt boundaries in A1 as a function of tilt angle. The arrows at the top indicate misorientations of singular boundaries. Data for A1 of 99 .99 995 %, 99 .99 92%, and 99 .98 %