EXERCISES 299 12.5 A crystal growing from the vapor phase possesses a singular surface with two screw dislocations intersecting it. The dislocations are very close to one another (relative to the dimensions of the surface) and have opposite Burgers vectors. Describe the form of the step structure that is produced because of the presence of the two dislocations. Solution. Before any growth, the two dislocations are associated with steps that may be as indicated in Fig. 12.9~. During growth, each dislocation rotates about its point of intersection to produce a spiral step, as in Fig. 12.5. However, the spirals will rotate in opposite directions, and sections will annihilate one another when they meet as in (a) and (b). The process will then continue as in (c)-(f) generating a potentially unlimited series of concentric steps. I I Figure 12.9: two screw dislocations with opposite Burgers vectors. Step structure generated on growing crystal surface at the intersections of 12.6 Suppose that the Burgers vectors of the two screw dislocations in Exercise 12.5 are now the same. Describe the ledge structure that is produced. Solution. Start with the ledge structure in Fig. 12.10~. The growth spirals of the two dislocations now rotate in the same direction as indicated. This will generate an inter- leaved double growth spiral of potentially unlimited extent as illustrated in Fig. 12.10b-d. Figure 12.10: of two screw dislocations with the same Burgers vectors. Step structures generated on growing crystal surface at the intersections 12.7 Consider a ledge on a surface which, on average, lies parallel to a closely spaced row of atoms. The ledge contains N sites, with spacing a, where either a positive or a negative kink can exist. (When traveling along a ledge, positive and negative kinks displace the ledge in opposite directions.) Show that the total number of kinks present in thermal equilibrium, NEq, is (12.34) 300 CHAPTER 12 MOTION OF CRYSTALLINE SURFACES where GL is the kink free energy of formation (excluding any configurational entropy). Solution. The increment of free energy due to the presence of NZ positive kinks and Ni kinks is Ag = NZGL + NiGL - kTlnP (12.35) where the last term is the configurational entropy. P is the number of distinguishable ways of arranging the positive and negative kinks on the N sites and is given by N! P= (NZ)! (NF)! (N - Nk+ - NL)! (12.36) Using Eqs. 12.35 and 12.36 and the approximation lnz! = zlnz - z, d Ag = G{ dN$ + G[ dNi (12.37) N' ) dNF N - N,$ - NF NZ ) dNz + kTIn ( N - N? - ~i + kTln ( Numbers of positive and negative kinks are equal and therefore dN$ = dNi. The condition for equilibrium is then Therefore, because N >> NZ + N;, and N,'q N - - [ $1 eq + [ g] eq = 2 e-G{/(lcT) (12.38) (12.39) (12.40) 12.8 Exercise 12.7 showed that the total equilibrium concentration of positive and negative kinks on a ledge running, on average, along a relatively close-packed direction is given by Eq. 12.34. Find the concentrations of positive and neg- ative kinks on a ledge lying at an angle 0 with respect to the close-packed direction. Assume that the direction of 8 requires the building-in of positive kinks. Solution. The population of kinks consists of BaNld built-in positive kinks, where d is the kink height, along with equal numbers of thermally generated positive and negative kinks, represented by N$ and Ni, respectively. Following the same procedure as in Exercise 12.7, the derivative of the total kink free energy, AG, is d AG = GidN$ + G,fdNi (BaNld) + N,' N - (BaN/d) - NZ - Ni + kTln [ + kTln 1 N - (OaN/d) - NZ - Ni EXERCISES 301 The numbers of thermally generated positive and negative kinks are equal and therefore dN,f = dNL and the condition of equilibrium is dng -0=2GL+kTln } (12.41) dN,' [N - (eaN/d) - N,' - Nil2 Assuming that 0 is small enough so that N >> (OaN/d) + N,' + NF, (12.42) 12.9 Consider the energy of a vicinal surface at a low temperature that consists of an array of straight parallel ledges separated by patches of singular terraces as in Fig. B.l. Express the form of the energy cusp in which the surface lies (i.e., express the surface energy as a function of 0, the angle by which the inclination of the vicinal surface deviates from that of the singular terraces). Note that this model will break down at higher temperatures where the sys- tem's entropy increases and free energy decreases as the ledges wander and become nonparallel and other roughening processes occur. Solution. The structure of the vicinal surface is shown in Fig. 12.11. The energy of unit area of surface can be expressed as the sum of two parts: the first is that of the singular terraces, which is ys cose , where ys is the energy per unit area of the terraces; the second is that of the ledges, which is (gL sinB)/h, where h is the ledge height and gL is the energy per unit length of a ledge. Therefore, gLsine h = ys + - h e i b t t - I ' (12.43) Figure 12.11: Structure of vicinal surface. CHAPTER 13 MOTION OF CRYSTAL/CRYSTAL I N T E R FAC ES Crystal/crystal interfaces possess more degrees of freedom than vapor/crystal or liquid/crystal interfaces. They may also contain line defects in the form of inter- facial dislocations, dislocation-ledges, and pure ledges. Therefore, the structures and motions of crystal/crystal interfaces are potentially more complex than those of vapor/crystal and liquid/crystal interfaces. Crystal/crystal interfaces experience many different types of pressures and move by a wide variety of atomic mecha- nisms, ranging from rapid glissile motion to slower thermally activated motion. An overview of crystal/crystal interface structure is given in Appendix B. 13.1 THERMODYNAMICS OF CRYSTALLINE INTERFACE MOTION In Section 12.1, common sources of driving pressures for the motion of vapor/crystal and liquid/crystal interfaces were described. These and additional sources of pres- sure exist for crystal/crystal interfaces. For example, during recrystallization, the interfaces between the growing recrystallized grains and the deformed matrix are subjected to a pressure that is due to the bulk free-energy difference (per atom) AG between the free energy of the deformed matrix and that of the recrystallized grains.' Also, compatibility stresses are often generated in stressed polycrystals so Recrystallization occurs when a crystalline material is plastically deformed at a relatively low temperature and then heated [l]. The as-deformed material possesses excess bulk free energy resulting from a high density of dislocations and point-defect debris produced by the plastic Kinetics of Materials. By Robert W. Balluffi, Samuel M. Allen, and W. Craig Carter. 303 Copyright @ 2005 John Wiley & Sons, Inc. 304 CHAPTER 13. MOTION OF CRYSTALLINE INTERFACES that two crystals adjacent to a grain boundary possess markedly different stresses. Equation 12.1 then holds, with AF equal to the difference between the elastic strain energy per atom in the two adjoining crystals. Pressures on crystal/crystal interfaces can also arise when the motion of the interface causes a change in the shape of the body in which it moves. Figure 13.1 shows the motion of a small-angle symmetric tilt grain boundary under applied shear stress. The motion occurs by the forward glide motion of the edge dislocations which comprise the boundary along x, causing the specimen to shear in the y direction. When this occurs, the applied stresses perform work, and the potential energy of the system is reduced. A displacement of the interface by Sx allows the applied stress, crzy, to perform the work, oxyO 62. The pressure on the interface is oxyo 6s p=- = oxyo SX (13.1) There is a close similarity between this type of pressure and the mechanical force exerted on a dislocation by a stress (see Section 11.2.1 and Exercise 13.2). Estimated values of the magnitudes of the pressures commonly applied to crys- tal/crystal interfaces extend over a wide range of values, spanning about six orders of magnitude [2]. Generally, the pressures generated by phase transformations are the highest (107-109 Pa), whereas those generated by interface curvature are rela- tively small (i03-io5 Pa). Figure 13.1: Motion of small-angle symmetric tilt boundary by means of glissile motion of its edge dislocations. Dislocation spacing, d, is equal to b/B, where b is the magnitude of the Burgers vector and B is the misorientation between the two grains. 13.2 CONSERVATIVE AND NONCONSERVATIVE INTERFACE MOTION The motion of a crystal/crystal interface is either conservative or nonconservative. As in the case of conservative dislocation glide, conservative interface motion occurs in the absence of a diffusion flux of any component of the system to or from the deformation. Upon heating, nuclei of relatively perfect recrystallized material form and then grow into the deformed material, eliminating most of the crystal defects at the moving interfaces between the recrystallized and deformed material. 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. [...]... 23(10):171 8- 1 721, 1 981 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, 1 980 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, 1 980 24 Y Komem, P Petroff, and R.W Balluffi Direct observation of grain boundary dislocation climb in ion-irradiated... 26:23 9-2 52, 1972 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, 1 989 26 G.C Weatherly The structure of ledges at plate-shaped precipitates Acta Metall., 19(3): 18 1-1 92, 1971 27 F.S Ham Theory of diffusion limited precipitation J Phys Chem Solids, 6(4):335351, 19 58 EXERCISES... 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 from which the boundary must repeatedly escape (e) Figure 13. 18 indicates... mechanism of grain boundary migration Acta Metall., 17(5):56 5-5 73, 1969 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 , 1979 10 J.W Cahn The impurity-drag effect in grain boundary motion 10(9) : 78 9-7 98, 1962 Acta Metall., 11 E Nes, N Ryum, and 0 Hunderi On the Zener drag Acta Metall., 33:ll-22, 1 985 12 M.F Ashby The influence of particles... Trans AIME, 224:lll-115, 1962 15 E.M Fridman, C.V Kopezky, and L.S.Shvindlerman Effects of orientation and concentration factors on migration of individual grain-boundaries in aluminum Z Metallkd., 66(9):53 3-5 39, 1975 16 P.R Howell, J.O Nilsson, and G.L Dunlap The effect of creep deformation on the structure of twin boundaries Phil Mag A , 38( 1):3 9-4 7, 19 78 17 D.A Smith, C.M.F Rae, and C.R.M Grovenor Grain... Lothe Theory of Dislocations John Wiley & Sons, New York, 2nd edition, 1 982 5 C Herring Diffusional viscosity of a polycrystalline solid J Appl Phys., 21:43 7-4 45, 1950 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, 1951 McGraw-Hill 7 W.T Read Dislocations in Crystals McGraw-Hill, New York, 1953 8 H Gleiter... 23 3-2 34:40 5-4 12, 1997 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 3 18, 1990 21 D.A Molodov, C.V Kopetskii, and L.S Shvindlerman Detachment of a special ( C = 19, (111))tilt boundary from an impurity in iron-doped aluminum bicrystals Sow Phys Solid State, 23(10):171 8- 1 721,... c(R, t ) = ce"4p (13.15) =0 The separation -of- variables method (Section 5.2.4) then gives the series solution [27] c ( r , t ) = ce"9p where the eigenvalues, Xi, + OL? i=O 5 e-'TDBt r sin [&(r - R)] (13.16) are the roots of tan[Xi(R, - R)] = XiR, ( 13.17) and the coefficients, ai, are given by a = i + 2(c0 - c , * , ~ ) R ( x ~ R ~ 1) Xi[XiR?(R, - R) - R] (13. 18) The diffusion current into the particle... into the more convenient form - D RdR = -dt RL -R3 R Z (13.62) The solution of Eq 13.62, subject t o the initial condition R = 0 when t = 0, is ln(’ R2+RwR+R &-& tan-’( ) R,-R 2R+ R, & R, ) + G3R,5 T (13.63) rr = t A few calculations show t h a t R increases with time at an ever-decreasing rate and approaches R, asymptotically as t -+ m 13.9 Consider a small-angle tilt boundary of the type shown in Fig 13.1,... and C.R.M Grovenor Grain boundary migration In R.W Balluffi, editor, Grain Boundary Structure and Kinetics, pages 33 7-5 71, Metals Park, OH, 1 980 American Society for Metals 18 H Ichinose and Y Ishida In situ observation of grain boundary migration of silicon C3 boundary and its structural transformation at 1000 K J Phys Colloq (Paris), 5l(suppl no l):C1: 18 5-1 90, 1990 19 T Kizuka, M Iijima, and N Tanaka . energy of unit area of surface can be expressed as the sum of two parts: the first is that of the singular terraces, which is ys cose , where ys is the energy per unit area of the terraces;. additional sources of pres- sure exist for crystal/crystal interfaces. For example, during recrystallization, the interfaces between the growing recrystallized grains and the deformed matrix are subjected. EXERCISES 299 12.5 A crystal growing from the vapor phase possesses a singular surface with two screw dislocations intersecting it. The dislocations are very close to one another (relative