CHAPTER 11 MOTION OF DISLOCATIONS The motion of dislocations by glide and climb is fundamental to many important kinetic processes in materials. Gliding dislocations are responsible for plastic defor- mation of crystalline materials at relatively low temperatures, where any dislocation climb is negligible. They also play important roles in the motion of glissile interfaces during twinning and diffusionless martensitic phase transformations. Both gliding and climbing dislocations cause much of the deformation that occurs at higher tem- peratures where self-diffusion rates become significant, and significant climb is then possible. Climbing dislocations act as sources and sinks for point defects. This chapter establishes some of the basic kinetic features of both dislocation glide and climb. 11.1 GLIDE AND CLIMB The general motion of a dislocation can always be broken down into two compo- nents: glide motion and climb motion. Glide is movement of the dislocation along its glide (slip) plane, which is defined as the plane that contains the dislocation line and its Burgers vector. Climb is motion normal to the glide plane. Glide motion is a conservative process in the sense that there is no need to deliver or remove atoms at the dislocation core during its motion. In contrast, the delivery or removal of atoms at the core is necessary for climb. This is illustrated for the simple case of the glide and climb of an edge dislocation in Fig. 11.1, The glide along IC in Fig. 11.1 a and b is accomplished by the local conservative shuffling of atoms at the disloca- Kinetics of Materials. By Robert W. Balluffi, Samuel M. Allen, and W. Craig Carter. 253 Copyright @ 2005 John Wiley & Sons, Inc. 254 CHAPTER 11: MOTION OF DISLOCATIONS .&. 0 0 @ (b) Y t Figure 11.1: Glide and climb of edge dislocation in primitive cubic crystal (g = [boo]. ( = [OOl]) [l]. (a) and (b) Glide from left to right. (c)-(f) Downward climb along -y. In (d), the lighter-shaded substitutional atom shown adjacent to the dislocation core in (c) has joined the extra half plane and created a vacancy. In (e), the vacancy has migrated away from the dislocation core by diffusion. In (f), the vacancy has been annihilated at the surface step. This overall process is equivalent to removing an atom from the surface and transporting it to the dislocation at its core. A new site was created at the dislocation, which acted as a vacancy source. This site was subsequently annihilated at the surface. which acted as an atom source. tion core as it moves. The climb along -y, however, requires that the extra plane associated with the edge dislocation be extended in the -y direction. This requires a diffusive flux of atoms to the dislocation core, and when self-diffusion occurs by a vacancy mechanism, the corresponding creation of an equivalent number of new lattice sites in the form of vacancies. In this case, the dislocation acts as a sink for atoms and, equivalently, as a source for vacancies. Glide can therefore occur at any temperature, whereas significant climb is possible only at elevated temperatures where the required diffusion can 0ccur.l Defects such as dislocations can be sources or sinks for atoms or for vacancies. Whether such point entities are created or destroyed depends on the type of defect, its orientation, and the stresses acting on it. It is convenient to adopt a single term source, which describes a defect’s capability for creation and destruction of crystal sites and vacancies in the crystal. “Source” will generically indicate creation of point entities (i.e., “positive” source action) as well as destruction of point entities (i.e., “negative” source action). Thus, a climbing edge dislocation that destroys vacancies will be, equivalently, both a (positive) source of atoms and a (negative) source of vacancies. If the sense of climb is reversed, the dislocation would be a (negative) source of atoms. lProvided that the Peierls force is not too large (see Section 11.3.1). 11 2. DRIVING FORCES ON DISLOCATIONS 255 11.2 DRIVING FORCES ON DISLOCATIONS Dislocations in crystals tend to move in response to forces exerted on them. In gen- eral, an effective driving force is exerted on a dislocation whenever a displacement of the dislocation causes a reduction in the energy of the system. Forces may arise in a variety of ways. 11.2.1 Mechanical Force In general, a segment of dislocation in a crystal in which there is a stress field is subjected to an effective force because the stress does an increment of work (per unit length), bW, when the dislocation is moved in a direction perpendicular to itself by the vector, 67. In this process, the material on one side of the area swept out by the dislocation during its motion is displaced relative to the material on the opposite side by the Burgers vector, b', of the dislocation. Work bW is generally done by the stress during this displacement. This results in a corresponding reduction in the potential energy of the system. The magnitude of the effective force on the dislocation (often termed the "mechanical" force) is then just f = bW/br. A detailed analysis of this force yields the Peach-Koehler equation: (11.1) where L is the mechanical force exerted on the dislocation (per unit length), u the stress tensor in the material at the dislocation, and ( the unit vector tangent to the dislocation along its positive direction [2]. Equation 11.1 is consistent with the convention that the Burgers vector of the dislocation is the closure failure (from start to finish) of a Burgers circuit taken in a crystal in a clockwise direction around the dislocation while looking along the dislocation in the positive direction.2 When written in full, Eq. 11.1 has the form where (11.2) (11.3) With this result, the mechanical force exerted on any straight dislocation by any stress field can be calculated. For example, if the edge dislocation (b' = [boo], ( = [OOl]) in Figure 11.1 is subjected to a shearing stress uxy, it experiences a force urging it to glide on its slip plane in the 2 direction. However, if the dislocation is subjected to the tensile stress, gxx, Eq. 11.1 shows that it will experience the force f,, = -jbaxx (i.e., a force urging it to climb in the -9 direction). In a more general stress field, the force (which is always perpendicular to the dislocation line) can have a component in the glide plane of the dislocation as well as a compor!ent normal to the glide plane. In such a case, the overall force will tend to produce both glide and climb. However, if the temperature is low enough that no significant diffusion is possible, only glide will occur. 2The Burgers circuit is constructed so that it will close if mapped step by step into a perfect reference crystal. See Hirth and Lothe [2]. .+ 256 CHAPTER 11 MOTION OF DISLOCATIONS 11.2.2 Osmotic Force A dislocation is generally subjected to another type of force if nonequilibrium point defects are present (see Fig. 11.2). If the point defects are supersaturated vacancies, they can diffuse to the dislocation and be destroyed there by dislocation climb. A diffusion flux of excess vacancies to the dislocation is equivalent to an opposite flux of atoms taken from the extra plane associated with the edge dislocation. This causes the extra plane to shrink, the dislocation to climb in the fy direction, and the dislocation to act as a vacancy sink. In this situation, an effective “osmotic” force is exerted on the dislocation in the fy direction, since the destruction of the excess vacancies which occurs when the dislocation climbs a distance by causes the free energy of the system to decrease by 66. The osmotic force is then given by By evaluating 66 and by when SNv vacancies are destroyed, an expression for f;. can be obtained. The quantity 66 is just -pvGNv, where the chemical potential of the vacancies, pv, is given by Eq. 3.66. If a climbing edge dislocation destroys SNv vacancies per unit length, the climb distance will be by = (R/b)bNv. The osmotic force is therefore f;. = -j 66lSy. (11.4) This result is easily generalized for mixed dislocations which are partly screw- type and partly edge-type, and also for cases having subsaturated vacancies. For a mixed dislocation, b must be replaced by the edge component of its Burgers vector Figure 11.2: Oblique view of edge dislocation climb due to destruction of excess vacancies. The extra plane associated with the edge dislocation is shaded. At A, a vacancy from the crystal is destroyed directly at a jog. At B, a vacancy from the crystal jumps into the core. At C, an attached vacancy is destroyed at a jog. At D, an attached vacancy diffuses along the core. 11 2 DRIVING FORCES ON DISLOCATIONS 257 and the result (see Exercise 11.1) is where - -kT R B = b - In ($) (11.5) (11.6) If the vacancies are subsaturated, the dislocation tends to produce vacancies and therefore acts as a vacancy source. In that case, Eq. 11.5 will still hold, but pv will be negative and the climb force and climb direction will be reversed. Equation 11.5 also holds for interstitial point defects, but the sign of 6 will be reversed. 11.2.3 Curvature Force Still another force will be present if a dislocation is curved. In such cases, the dislocation can reduce the energy of the system by moving to decrease its length. An effective force therefore tends to induce this type of motion. Consider, for example, the simple case of a circular prismatic dislocation loop of radius, R. The energy of such a loop is W=R- 2(1 pb2 - u) [In(:) - 11 (11.7) where R, is the usual cutoff radius (introduced to avoid any elastic singularity at the origin) [2]. The energy of such a loop can be reduced by reducing its radius and therefore its length. Thus, a climb force, fl;, exists which is radial and in the direction to shrink the loop. A calculation of the reduction in the loop energy achieved when its radius shrinks by 6R shows that (dW/dR) 6R = 27rR fK 6R. The force is therefore (11.8) This result may be generalized. Any segment of an arbitrarily curved dislocation line will be subjected to a curvature force of similar magnitude because the stress fields of other segments of the dislocation line at some distance from the segment under consideration exert only minimal forces on it. For most curved dislocation geometries, the magnitude of the right-hand side of Eq. 11.8 is approximately equal to pb2 (l/R). Therefore, for a general dislocation with radius of curvature, R, (11.9) The quantity pb2 has the dimensions of a force (or, equivalently, energy per unit length) and is known as the line tension of the dislocation. Equation 11.9 can also be obtained by taking the line tension to be a force acting along the dislocation in a manner tending to decrease its length.3 This approximation is supported by detailed calculations for other forms of curved dislocations [2]. 3This is explored further in Exercise 11.2. 258 CHAPTER 11: MOTION OF DISLOCATIONS The vector form of Eq. 11.9 is readily obtained. If r'is the position vector tracing out the dislocation line in space and ds is the increment of arc length traversed along the dislocation when r' increased by dr',4 (11.10) where at the point r' on the line, fi is the principal normal, which is a unit vector perpendicular to ( and directed toward the concave side of the curved line, K is the curvature, and R is the radius of curvature. Therefore, (11.11) 11.2.4 The total driving force on a dislocation, f: is the sum of the forces previously Total Driving Force on a Dislocation considered and, therefore, $=flr+$+.L = (IX t) + 11.3 DISLOCATION GLIDE df 4 x B + pb2 - = x (2 - 4 + pb2 - (11.12) (' '> ds ds Of central interest is the rate at which a dislocation is able to glide through a crystal under a given driving force. Many factors play potential roles in determining this rate. In perfect crystals, relativistic effects can come into play as dislocation velocities approach the speed of sound in the medium. At elevated temperatures, dissipative phonon effects can produce frictional drag forces opposing the motion. Also, the atom shuffling at the core, which is necessary for the motion, may be difficult in certain types of crystals and thus inhibit glide. In imperfect crystals, any point, line, and planar defects and inclusions can serve as additional obstacles hindering dislocation glide. We begin by discussing glide in a perfect single crystal, which for the present is taken to be a linear elastic continuum. 11.3.1 Relativistic Effects. Consider the relatively simple case of a screw dislocation mov- ing along 5 at the constant velocity v' (see Fig. 11.3). The elastic displacements, ul, u2, and 213, around such a dislocation may be determined by solving the Navier equations of isotropic linear elasticity [3].5 For this screw dislocation, the only non- zero displacements are along z, and for the moving dislocation the Navier equations Glide in Perfect Single Crystals therefore reduce to (11.13) where p is the density of the medium, p is the shear modulus, and on the left is the inertial term due to the acceleration of mass caused by the moving dislocation. 4See Appendix C for a brief survey of mathematical relations for curves and surfaces. 5See standard references on dislocation mechanics [2, 4, 51. 11 3 DISLOCATION GLIDE 259 Y Y‘ i A I Figure 11.3: = [OOl] moving in the +z direction at a constant velocity v’. The origin of the primed (d, y’, z’) coordinate system is fixed to the niovi ng dislocation. Screw dislocation with b’ = [OOb], Equation 11.13 is readily solved after making the changes of variable I - r-vt x yL E 7- Y’ = Y 2‘ = z I - t-vx (11.14) where c = is the velocity of a transverse shear sound wave in the elastic medium. The origin of the (XI, y‘, z’) coordinate system is fixed on the moving dislocation as in Fig. 11.3. These changes of variable transform Eq. 11.13 into d2u3 32213 -++,=o dXl2 dy (11.15) because ug is a not a function of tl in the moving coordinate system and du3/dt’ = 0. Equation 11.15 has the form of the Navier equation for a static screw dislocation and its solution6 has the form 2.n Transforming this solution back to x, y, t space, b 2i7 uz(x, y, t) = - tan- (1 1.16) (11.17) The shear stress of the dislocation in cylindrical coordinates, goz, may now be found by using the standard relations orz = p(au,/dx), uyz = p(du,/dy), and Ooz = oyz cose - ozz sine. The result is pb YL (x; + yg)’” goz = - 2.n x;+r;y; 6Further discussion of this can be found in Hirth and Lothe [2]. (11.18) 260 CHAPTER 11: MOTION OF DISLOCATIONS where the distances 20 and yo (measured from the moving dislocation) have been introduced. Equation 11.18 indicates that the stress field is progressively contracted along the 20 axis and extended along the yo axis as the velocity of the dislocation is increased. This distortion is analogous to the Lorentz contraction and expansion of the electric field around a moving electron, and the quantity y~ plays a role similar to the Lorentz-Einstein term (1 - w2/c2)lI2 in the relativistic theory of the electron, where c is the velocity of light rather than of a transverse shear wave. In the limit when w + c and y~ -+ 0, the stress around the dislocation vanishes everywhere except along the y'-axis, where it becomes infinite. Another quantity of interest is the velocity dependence of the energy of the dislocation. The energy density in the material around the dislocation, w, is the sum of the elastic strain-energy density and the kinetic-energy density, w = 2w,, 2 + 2pLEyz + zp (;;)2=;[(EE)2+(L!%)2+;(a,'l - (1 1.19) where the first two terms in each expression make up the elastic strain-energy density and the third term is the kinetic-energy density [3]. The total energy may then be found by integrating the energy density over the volume surrounding the (11.20) where W" is the elastic energy of the dislocation per unit length at rest [2, 4, 51, (1 1.21) Here, R, is again the usual cutoff radius at the core and R is the dimension of the crystal containing the dislocation. According to Eq. 11.20, the energy of the moving dislocation will approach infinity as its velocity approaches the speed of sound. Again, the relationship for the moving dislocation is similar to that for a relativistic particle as it approaches the speed of light. These results indicate that in the present linear elastic model, the limiting ve- locity for the screw dislocation will be the speed of sound as propagated by a shear wave. Even though the linear model will break down as the speed of sound is approached, it is customary to consider c as the limiting velocity and to take the relativistic behavior as a useful indication of the behavior of the dislocation as w + c. It is noted that according to Eq. 11.20, relativistic effects become important only when w approaches c rather closely. The behavior of an edge dislocation is more complicated since its displacement field produces both shear and normal stresses. The solution consists of the super- position of two terms, each of which behave relativistically with limiting velocities corresponding to the speed of transverse shear waves and longitudinal waves, re- spectively [a, 4, 51. The relative magnitudes of these terms depend upon w. Drag Effects. Dislocations gliding in real crystals encounter dissipative frictional forces which oppose their motion. These frictional forces generally limit the dislo- cation velocity to values well below the relativistic range. Such drag forces originate from a variety of sources and are difficult to analyze quantitatively. 11.3: DISLOCATION GLIDE 261 Drag by Emission of Sound Waves. When a straight dislocation segment glides in a crystal, its core structure varies periodically with the periodicity of the crystal along the glide direction. The potential energy of the system, a function of the core structure, will therefore vary with this same periodicity as the dislocation glides. Because of this position dependence, there is a spatially periodic Peierls force that must be overcome to move a dislocation. Therefore, the force required to displace a dislocation continuously must exceed the Peierls force, indicated by the positions where the derivative of potential energy in Fig. 11.4 is maximal [2].’As the dislocb tion traverses the potential-energy maxima and minima, it alternately decelerates and accelerates and changes its structure periodically in a “pulsing” manner. These structural changes radiate energy in the form of sound waves (phonons). The energy required to produce this radiation must come from the work done by the applied force driving the dislocation. The net effect is the conversion of work into heat, and a frictional drag force is therefore exerted on the dislocation. I I b X Position of dislocation Figure 11.4: Variation of potential energy of crystal plus dislocation w a function of dislocation position. Periodicity of potential energy corresponds to periodicity of crystal structure. In a crystal, sound waves of a given polarization and direction of propagation are dispersive-their velocity is a decreasing function of their wavenumber, which produces a further drag force on a dislocation. The dispersion relation is (11.22) where w is the angular frequency, d is the distance between successive atomic planes in the direction of propagation, k = 27r/X is the wavenumber, and X is the wave- length.8 In the long-wavelength limit (A >> d) corresponding to an elastic wave in a homogeneous continuum, the phase velocity is c (as expected). However, at the shortest wavelength that the crystal can transmit (A = 24, the phase veloc- ity is lower and, according to Eq. 11.22, is given by 2c/7r. The displacement field of the dislocation can now be broken down into Fourier components of different wavelengths. If the dislocation as a whole is forced to travel at a velocity lower than c but higher than 2c/7r, the short-wavelength components will be compelled to travel faster than their phase velocity and will behave as components of a su- ‘However, dislocations will still move by thermally activated processes below the Peierls force. *For more about the dispersion relation, see a reference on solid-state physics, such as Kittel [6]. 262 CHAPTER 11: MOTION OF DISLOCATIONS I<,,, IIIII IIIII IIIII IIIII IIIII IIIII IIIII IIIII IIIII IIIII IIIII IIIII IIIII IIIII IIIII IIIII IIIII IIIII IIIII IIIII personic dislocation. These components will radiate energy and therefore impose a viscous drag force on the dislocation (see Section 11.3.4). Drag by Scattering of Phonons and Electrons. A dislocation scatters phonons by two basic mechanisms. First, there are density changes in its displacement field which produce scattering. Second, the dislocation moves under the influence of an impinging sound wave and, as it oscillates, re-radiates a cylindrical wave. If the dislocation undergoes no net motion and is exposed to an isotropic flux of phonons it will experience no net force. However, if it is moving, the asymmetric phonon scattering will exert a net retarding force, since, in general, any entity that scatters plane waves experiences a force in the direction of propagation of the waves. If, in addition, free electrons are present, they will be scattered by an effective scattering potential produced by the displacement field of the dislocation. This produces a further retarding force on a moving dislocation. Peierls Force: Continuous vs. Discontinuous Motion. In some crystals (e.g., covalent crystals) the Peierls force may be so large that the driving force due to the applied stress will not be able to drive the dislocation forward. In such a case the dislocation will be rendered immobile. However, at elevated temperatures, the dislocation may be able to surmount the Peierls energy barrier by means of stress-aided thermal activation, as in Fig. 11.5. ,I,,, IIIII IIIII IIIII IIIII IIIII Ill11 IIIII 111Il IIIII IIIII IIIII IIIII Ill11 IIIII I/ I1 I1 /I I1 I1 IIIII ::: Figure 11.5: Movement of dislocation across a Peierls energy barrier by thermally activated generation of double kinks. Dashed lines represent positions of energy minima shown in Fig. 11.4. In Fig. 11.5a, the dislocation is forced up against the side of a Peierls “hill” by an applied stress as in Fig. 11.4. With the aid of thermal activation, it then generates a double kink in which a short length of the dislocation moves over the Peierls hill into the next valley (Fig. 11.5b).9 The two kinks then glide apart transversely under the influence of the driving force (Fig. 11.5~)~ and eventually, the entire dislocation advances one periodic spacing. By repeating this process, the dislocation will advance in a discontinuous manner with a waiting period between each advance, and the overall forward rate will be thermally activated. This is an 9A kink is an offset of the dislocation in its glide plane; it differs fundamentally from a jog, an offset normal to the glide plane. [...]... waves Mech Muter., 5(1):1328, 1986 14 B.L Holian Modeling shock-wave deformation via molecular-dynamics Phys Rev A , 37( 7):256 2-2 568, 1988 15 A S Nowick and B.S Berry Anelastic Relaxation in Crystalline Solids Academic Press, New York, 1 972 EXERCISES 275 16 J Lothe Theory of dislocation climb in metals J Appl Phys., 31(6):1 07 7- 1 0 87, 1960 17 R.M Thomson and R.W Balluffi Kinetic theory of dislocation... pages 85 2-8 74 , Springfield, VA, 1 975 National Technical Information Service, US.Department of Commerce 23 R.W Balluffi and D.N Seidman Diffusion-limited climb rate of a dislocation: Effect of climb motion on climb rate J Appl Phys., 36 (7) : 270 8-2 71 1, 1965 24 D.N Seidman and R.W Balluffi Sources of thermally generated vacancies in single crystal and polycrystalline gold Phys Rev., 139(6A):182 4-1 840, 1965... v ( r ) - c"v"m) = cy(dis1) - c y ( m ) ln(R/a) (11.66) The vacancy current diffusing away from the two dislocations is then I = -2 7rrDv- dcv 27rDv [cy(disl) - c y ( m ) ] = dr ln(R/a) (11. 67) Finally, making use of Eqs 8. 17 and 11.63, the velocity o f approach is 2, = -1 = s1 27r*D b [ efn/(kTb) - fbI n ( R / a ) ( 11.68) 11.12 Consider a segment of dislocation in the dislocation network of a crystal... dislocation velocities may be achieved at the start of even low-strain-rate deformation if the initial concentration of mobile dislocations is unusually low [ll] In such cases, a small number of dislocations must move very rapidly to accom- ii 3: DISLOCATION GLIDE 265 1 07 105 13 0 I0 1 0-1 1 0-3 1 0-5 1 0 -7 0.1 1 10 100 Applied shear stress (kg mm-2) Figure 11 .7: crystals Velocity vs resolved shear stress for... screw dislocations J Appl Phys., 33(3):80 3-8 17, 1962 18 R.W Balluffi Mechanisms of dislocation climb Phys Status Solidi, 31(2):44 3-4 63, 1969 19 R.W Balluffi and A V Granato Dislocations, vacancies and interstitials In F.R.N Nabarro, editor, Dislocations in Solids, volume 4, pages 1-1 33, Amsterdam, 1 979 North-Holland 20 A.P Sutton and R.W Balluffi Interfaces in Crystalline Materials Oxford University... L [c7(lOOp) - c ~ ( c o ) ] I = 4 r r 2 D v - [cv(r)]= 47rDv dr In(~ R/ R o ) L (11.39) in agreement with the results of the previous analysis Applications of Eq 11.35 and closely related equations to the observed annealing rate of loops have been described [19, 281 Bibliography 1 S.M Allen and E.L Thomas The Structure of Materials John Wiley & Sons, New York, 1999 2 J.P Hirth and J Lothe Theory of. .. Solid State Physics John Wiley & Sons, New York, 3rd edition, 19 67 7 F.R.N Nabarro, editor Dislocations in Solids (Series), volume 1-1 2 Elsevier NorthHolland, New York, 1 97 9-2 004 8 J Friedel Dislocations Pergamon Press, Oxford, 1964 9 U.F Kocks, A.S Argon, and M.F Ashby Thermodynamics and kinetics of slip Prog Muter Sci., 19:l-288, 1 975 10 W.G Johnston and J.J Gilman Dislocation velocities, dislocation... the lower-stacking-fault-energy metals The efficiencies 1 for the higher-stacking-fault-energy metals appear to fall off less rapidly with Jgs This may be understood on the basis of the tendency of the dislocations to contain more jogs as lgsl increases and the greater difficulty in forming jogs on dissociated dislocations than on undissociated dislocations because of the larger jog energies of the former... loops tend to shrink and be eliminated by means of climb during subsequent thermal annealing A number of measurements of loop shrinkage rates have been made, and analysis of this phenomenon is therefore of interest [2] In this section we calculate the isothermal annealing rate of such a loop located near the center of a thin film in a high-stacking-fault-energy material (such as Al) where the climb efficiency... Ham Stress assisted precipitation on dislocations J Appl Phys., 30(6):91 5-9 26, 1959 26 C.P Flynn Monodefect annealing kinetics Phys Rev., 133(2A):A5 87, 1964 27 H Buchholz Electrische und Magnetische Potentialfelder Springer-Verlag, Berlin, 19 57 28 D.N Seidman and R.W Balluffi On the annealing of dislocation loops by climb Phil Mag., 13:64 9-6 54, 1966 29 J Bardeen and C Herring Diffusion in alloys and . therefore possess stress fields that interact with dislocation stress fields, causing localized dislocation-solute-atom attraction or repulsion. If a dis- persion of solute atoms is present. the velocity of edge and screw dislocation seg- ments in LiF single crystals as a function of applied force (stress) [lo]. Stresses above a yield threshold stress were required for any motion temperatures where the required diffusion can 0ccur.l Defects such as dislocations can be sources or sinks for atoms or for vacancies. Whether such point entities are created or destroyed